ssrn-id1711002

832019 SSRN-id1711002

httpslidepdfcomreaderfullssrn-id1711002 170Electronic copy available at httpssrncomabstract=1711002

A Theory of Asset Pricing and Performance Evaluation

for Minority Banks with Implications for Bank Failure

Prediction Compensating Risk and CAMELS Rating

Godfrey Cadogan lowastWorking Paper

First draft November 18 2010

This draft January 27 2011

lowastCorresponding address Information Technology in Finance Institute for Innovation and Technology

Management Ted Rogers School of Management Reyerson University 575 Bay Toronto ON M5G 2C5

e-mail gocadoggmailcom Research support from the Institute for Innovation and Technology Manage-

ment is gratefully acknowledged This paper builds on Cole (2010b) It benefited from several conversationswith John A Cole and I am thankful to him for introducing me to the minority bank topic for seeking clar-

ification of the relationship between cash flows and CAMELS rating and for suggesting that this writerlsquos

incipient technical appendix be expanded into what is now the instant paper I thank Samuel Myers Jr

for his encouragement and Walter E Williams for drawing my attention to early work he did on minority

bank portfolios The instant revision (1) corrects typographical errors (2) formalizes heretofore footnoted

heuristic statements (3) provides details on computing the optimal strike price for the put option on minority

bankslsquo assets (4) added an appendix of proofs and (5) includes some more references Any errors which

may remain are my own Electronic copy available at httpssrncomabstract=1711002

832019 SSRN-id1711002


Abstract

This paper introduces a comprehensive theory of performance evaluation and asset pricing for

minority (type-m) banks incuding but not limited to CAMELS rating bank failure prediction

compensating risks and risk adjusted internal rates of returns in a complete market Our the-ory predicts and explains several empirical regularities of minority banks First we provide a

novel mechanism for generating a synthetic cash flow stream in Hilbert space by comparing NPV

projects selected by minority and nonminority (type-n) peer banks Thus laying a foundation for

security design for sound type-m banks Second even when internal rates of return and managerial

ability are equal for type-m and type-n banks the expected cash flows for type-n banks uniformly

dominate that for type-m banks by a factor greater than 2 Third a behavioral mean-variance

analysis of liquidity preference explains type-m bankslsquo seeming zero covariance frontier portfolio

and asset pricing anomalies like concavity of their internal rates of return in project risk Such

anomalies affect computation of unlevered betas imply undercapitalization and inefficiency that

crowds out positive NPV projects and suggest that type-m banks overinvest in residual [second

best] projects eschewed by type-n banks See eg Ruback (2002) Jensen and Meckling (1976)

Myers (1977) and Williams-Stanton (1998) Even so the returns on minority bankslsquo portfolios are

viable provided that the risk free rate is less than that for the minimum variance [frontier] portfo-

lio Fourth we introduce a compensating risk factor for type-m banksndashpriced by a two-factor asset

pricing model induced by Vasicek (2002) for loan portfolio value ie return on assets (ROA) Our

theory predicts that that compensating transfer scheme is tantamount to a put option on type-m

bank assets in accord with Merton (1977) And we use Deelstra et al (2010) model to identify the

optimal strike price for that option According to Merton (1992) and Glazer and Kondo (2010)

to mitigate moral hazard in such schemes either (1) type-m banks must bear the associated costs

or (2) the costs must not be bourne by other banks Arguably our put option prices type-m banks

under FIRREA sect 308 (1989) Fifth on the fixed income side Vasicek (1977) interest rate modelpredicts that the term structure of bonds issued by type-m banks is a floor for the term structure of

bonds issued by type-n peer banks Moreover if risky cash flows follow a Markov process then

under Rothschild and Stiglitz (1970) type mean preserving spread analysis type-m bank bonds are

riskier provide lower yield and need to be issued over shorter durationndashaccording to Barclay and

Smith (1995) empirical findings on asymmetric information and the ldquocontracting-cost hypothesisrdquo

Sixth we extend Vasicek (2002) to an independently important conditional probit representation

for bank failure predictionndashmore granular than Cole and Gunther (1998) ad hoc probit model Sev-

enth we establish a nexus between random utility analysis the conditional probit and a simple

bivariate CAMELS rating function that (1) predict bank examiners attitudes towards type-m banks

and (2) identifies root causes of their bias towards high CAMELS scores for type-m banks In par-

ticular we show how bank examinerslsquo CAMELS scores for return on assets risk is predicted bybank specific variables

Keywords minority banks synthetic cash flows internal rates of return behavioral asset pric-

ing CAMELS rating random utility options pricing bank regulation return on assets

JEL Classification Code D03 D81 G11 G12 G21 G28 G31 G32 M48

832019 SSRN-id1711002

httpslidepdfcomreaderfullssrn-id1711002 370

Contents

1 Introduction 2

2 The Model 7

21 Diagnosics for IRR from NPV Projects of Minority and Nonmi-

nority Banks 8

211 Fundamental equation of project comparison 10

22 Yield spread for nonminority banks relative to minority peer banks 10

3 Term structure of bond portfolios for minority and nonminority banks 19

31 Minority bank zero covariance frontier portfolios 21

4 Probability of bank failure with Vasicek (2002) Two-factor model 23

5 Minority and Nonminority Banks ROA Factor Mimicking Portfolios 25

51 Computing unlevered beta for minority banks 26

52 On Pricing Risk Factors for Minority and Nonminority Peer Banks 29

521 Asset pricing models for minority and non-minority banks 29

53 A put option strategy for minority bank compensating risk 37

531 Optimal strike price for put option on minority banks 38

532 Regulation costs and moral hazard of assistance to minor-

ity banks 3954 Minority banks exposure to systemic risk 40

6 Minority Banks CAMELS rating 41

61 Bank examinerslsquo random utility and CAMELS rating 42

62 Bank examinerslsquo bivariate CAMELS rating function 44

7 Conclusion 49

A Appendix of Proofs 50A1 PROOFS FOR SUBSECTION 2 1 DIAGNOSICS FOR IRR FROM

NPV PROJECTS OF MINORITY AND NONMINORITY BANKS 50

A2 PROOFS FOR SUBSECTION 3 1 MINORITY BANK ZERO COVARI-

ANCE FRONTIER PORTFOLIOS 53

A3 PROOFS FOR SECTION 4 PROBABILITY OF BANK FAILURE WITH

VASICEK (2002) TWO -FACTOR MODEL 56

1

832019 SSRN-id1711002


A4 PROOFS FOR SECTION 5 MINORITY AND NONMINORITY BANKS

ROA FACTOR MIMICKING PORTFOLIOS 56

References 59

2

832019 SSRN-id1711002


1 Introduction

According to US GAO Report No 07-06 (2006) as of 2005 there were 195 mi-

nority banks 37 of them were Asian and 24 of them were African-American1

So the majority (61) are Asian and African-American owned Moreover 44of minority banks had asset size less than $100M Fifty-six (56) of all minor-

ity banks were regulated by the FDIC and the average loss reserve for African-

American banks was about 40 of assets id at page 11 Starting with Brim-

mer (1971) the literature on minority banks2 is based mostly on empirical studies

that surmise their viability and compare their performance to that of nonminority

banks See eg Hasan and Hunter (1996) for an earlier review and Hays and

De Lurgio (2003) for a more recent review of empirical regularities of minority

banks Thus performance evaluation of minority banks is relative to nonminority

peer banks In fact bank examination is based on a matched pair process in whichminority banks are compared to nonminority banks with a ldquosimilarrdquo profile 3 In

the overwhelming number of cases minority banks do not match up well 4

To the extent that minority banks are small banks they are subject to the

vagaries of the small bank market which includes but is not limited to the per-

formance of local economies in their domicile5 In fact the ldquoblack bank paradoxrdquo

is described as ldquoBlack-owned banks face a serious dilemma founded primarily to

help fill the gap between the demand for and supply of credit to the black commu-

nity the more they try to respond positively the greater is the probability that theywill failrdquo6 Therefore comparative analysis of minority banks performance is by

definition comparison of a substratum of small banks

To close the chasm between minority and nonminority small bank per-

1For the latest figures see the Federal Reserve Board website httpwwwfederalreservegovreleasesmob 2In this paper ldquominority banksrdquo and rdquoblack banksrdquo are used interchangeably However there is some variation

within minority banks as Asian banks tend to have a different profile for reasons outside the scope of this paper See

(US GAO Report No 07-06 2006 Appendix II) for heterogeniety in definition of ldquominority bankrdquo3 For instance meaningful comparison of return on assets (ROA) require that banks in the same peer group and

have similar asset size Thus a minority bank is compared with a nonminority bank in the same class See eg

memo dated May 27 2004 on changes made to Uniform Bank Performance Report by Federal Financial Institutions

Examination Counsel available at httpwwwffiecgovubpr˙memo˙200405htm See also (US GAO Report No07-06 2006 pg 11) for description of ldquopeerrdquo bank

4See eg Meinster and Elyasini (1996) (minority banks warrant concern in comparative analysis between foreign

minority and holding companies banks) Lawrence (1997) (poor performance of minority banks not due to operat-

ing environment) Iqbal et al (1999) (with a given set of inputs minority-owned banks produce less outputs than a

comparable group of nonminority-owned banks)5See eg Nakamura (1994)6See Brimmer (1992) Evidently this paradox is induced by minority bankslsquo attempt to alleviate credit rationing

problems in underserved communities See Stiglitz and Weiss (1981)

3

832019 SSRN-id1711002


formance Cole (2010a) presented qualitative arguments in favor of a designated

regulator of minority banks along the lines of recent Federal Reserve intervention

to correct market failure in credit markets The thesis of his argument appears to

be that given the peculiarities of minority banks that try to correct for credit market

discrimination in their domicile use of the Uniform Financial Institutions RatingSystem would inevitably produce lower CAMELS7 ratings which consequently

put these banks on ldquobank watchrdquo Implicit in that argument is that minority banks

are undercapitalized for the seemingly altruistic mission they undertake and that

perhaps some kind of regulatory subsidization of capital is needed8

To the best of our knowledge the literature is silent on theory geared

specifically towards explaining and or predicting empirical regularities of minority

bankslsquo performancendashat least in the context of microfoundations of financial eco-

nomics To be sure Henderson (1999) introduced a model of loan loss provision

based on changes in net interest income (∆ NII ) that identified proximate causes

of managerial inefficiency by minority banks There he found statistically signif-

icant constants for loan loss provisions (1) positive for African-American owned

banks and (2) negative for white banks operating in the same market Among

other things he also found that black banks had 5 times as much net charge-offs

than white banks during the period sampled As to managerial inefficiency he

found that managers at black banks appear to ldquoplace greater weight on other ex-

ogenous factors than on financial and environmental factors in their decisions to

allocate provisions forn loan lossrdquo9

By contrast our theory explains why even if managerial efficiency for minority and nonminority banks is the same minority

banks do not perform as well as nonminority banks

(Henderson 2002 pg 318) introduced a model that addressed the role of com-

munity banks in combating discrimination He presented an ldquoad hoc model as a

first step in better understanding [the Brimmer (1992)] paradoxrdquo However his

7The acronym stands for Capital adequacy Asset quality Management Earnings Liquidity Sensitivity to market

risk Details on computation of each component are described in UNIFORM FINANCIAL INSTITUTIONS RATING

SYSTEM (UFIRS) (eff 1996) available at httpwwwfdicgovregulationslawsrules5000-900html Last visited on

111220108It should be noted in passing that the UFIRS policy statement indicates ldquo Evaluations of [a banklsquos CAMELS]

components take into consideration the institutionrsquos size and sophistication the nature and complexity of its activities

and its risk profilerdquo So arguably regulatory transfers may fall under the so far nonimplemented Financial Institutions

Reform Recovery and Enforcement Act of 1989 (FIRREA) Section 308 See Cole (2010a) for further details on this

issue Additionally it is known that such transfer schemes would induce moral hazard in minority banks Nonetheless

that can be mitigated if regulation costs are not bourne by other banks See eg Glazer and Kondo (2010) and (Merton

1992 Chapter 20)9In private communications Prof Walter Williams pointed out that because of the business climate in black neigh-

borhoods most black banks had most of their asset portfolio outside of the neighborhood See Williams (1974)

4

832019 SSRN-id1711002


model is based on asymmetric response to loan demand and supply shocks in a

quadratic loss function for optimal loan loss provision His thesis is that greater

loan loss provisions and the inability of community banks and borrowers to accu-

rately predict the impact of shocks on loan performance reduces the profitability of

community banks It is not geared towards minority banks per se and it does notexplain empirical regularities of minority banks In our model we show how com-

pensating shocks to minority banks can alleviate some of the problems described

in Henderson (2002)

On a different note Cole (2010b) introduced a theory of minority banks

which is focused on ldquocapacity buildingrdquo However that catch phrase typically im-

plies provision of assistance with a view towards eventual self sustenance10 While

Cole (2010b) addresses several issues presented in here this paper is distinguished

by its presentation of inter alia (1) a theoretical estimate of the regulation cost of

providing assistance to minority banks (2) allocation of the burden of that cost to

preclude moral hazard by minority banks (3) behavioral framework for CAMELS

rating of minority banks (4) term structure of bond portfolios held by minority

banks and (5) the role of minority bank capital structure in determining its price

of debt and equities In a nutshell this paper attempts to fill a void in the literature

by providing a comprehensive theory of performance evaluation11 asset pricing

bank failure prediction and CAMELS rating for minority banks It also identifies

the amount of risk-adjusted capital transfers that may be required to address the

erstwhile minority bank paradox And it shows how those transfers are pricedWe choose the internal rate of return to motivate our theory because that

variable acts as a risk adjusted discount rate and it captures the efficiency quality

and earnings rate or yield of the projects undertaken by minority and nonminority

banks12 Arguably it also reflects a bankrsquos risk based capital (RBC) since capital

10See eg Robinson (2010) who advocates the creation of more minority banks as a vehicle for economic develop-

ment and self sufficiency11See eg U S GAO Report No 08-233T (2007) recommendations for establishing performance measures for

minority banks12According to capital budgeting theory See eg (Ross et al 2008 pg 241)

5

832019 SSRN-id1711002


adequacy is a function of RBC13 Using a simple project comparison argument in

infinite dimensional Hilbert space we generate a synthetic cash flow stream which

predicts that the cash flows of nonminority banks uniformly dominate that of mi-

nority peer banks in every period almost surely This implies that minority banks

overinvest in [second best] positive NPV projects that nonminority banks rejectby virtue of their [nonminority banks] underinvestment in the sense of Jensen and

Meckling (1976) Myers (1977) We show how the synthetic cash flow is tanta-

mount to a credit derivative of minority bank assets and provide some remarks on

its implications for security design for minority banks Further if the duration of

projects undertaken is finite then one can construct a theoretical term structure of

IRR for minority banks14 In fact Lemma 32 below plainly shows how Vasicek

(1977) model can be used to explain minority bank asset pricing anomalies like

negative price of risk

We derive an independently important bank failure prediction model by

slight modification of Vasicek (2002) two factor model There we show how the

model is reduced to a conditional probit on granular bank specific variables We

extend that paradigm to a two factor behavioral asset pricing model for minority

banks by establishing a nexus between it and factor pricing for minority banks

Cadogan (2010a) behavioral mean-variance asset pricing model is used to explain

minority banks asset pricing anomalies like ROA being concave in risk For in-

stance Cadogan (2010a) embedded loss aversion index acts like a shift parame-

ter in beta pricing for a portfolio of risky assets on Markowitz (1952a) frontierWhereupon minority banks risk seeking behavior over losses causes them to hold

inefficient zero covariance frontier portfolios Nonetheless minority bankslsquo port-

folios are viable if the risk free rate is less than that for the minimum variance

13A bankrsquos risk based capital is determined by a formula set by Basel II as follows The risk weights assigned to

balance sheet assets are summarized as follows

bull 0 risk weight cash gold bullion loans guaranteed by the US government balances due from Federal

Reserve Banks

bull 20 risk weight demand deposits checks in the process of collection risk participations in bankersrsquo accep-

tances and letters of credit and other short-term claims maturing in one year or less

bull 50 risk weight 1-4 family residential mortgages whether owner occupied or rented privately issued mort-

gage backed securities and municipal revenue bonds

bull 100 risk weight cross-border loans to non-US borrowers commercial loans consumer loans derivative

mortgage backed securities industrial development bonds stripped mortgage backed securities joint ventures

and intangibles such as interest rate contracts currency swaps and other derivative financial instruments

14See (Vasicek 1977 pg 178)

6

832019 SSRN-id1711002


portfolio on the portfolio frontier For there is a window of opportunity in which

the returns on minority bankslsquo portfolios is bounded from below by the risk free

rate and above by the returns from the minimum variance portfolio

Assuming that risky cash flows follows a Markov process we use a sim-

ple dynamical system introduced by Cadogan (2009) to show how the embeddedrisk in minority bank projects stochastically dominate that for nonminority peer

banks Accordingly to correct these asset pricing anomalies we show how risk

compensating regulatory shocks to minority banks transforms their negative price

of risk to a positive price15 And that isomorphic decomposition of such shocks

include a put option on minority bank assets Further we show how the amount

of compensating risk is priced in a linear asset pricing framework and use Merton

(1977) formulation to price the put option in a regulatory setting The strike price

for such an option is derived from considerations in Deelstra et al (2010) The

significance of our asset pricing approach is underscored by a recent GAO report

which states that even though FIRREA Section 308 requires that every effort be

made to preserve the minority character of failed minority banks upon acquisition

the ldquoFDIC is required to accept the bids that pose the lowest accepted cost to the

Deposit Insurance Fundrdquo16

Finally we provide a microfoundational model of the CAMELS rating

process which identify bank examinerlsquos subjective probability distributions and

sources of bias in their assignment of CAMELS scores In particular we establish

a nexus between random utility analysis and a simple bivariate CAMELS ratingfunction And use comparative statics from that setup to explain and predict bank

examiner attitudes towards minority banks

The rest of the paper proceeds as follows Section 2 introduces project

comparison for minority and nonminority peer banks on the basis of the IRR of

projects they undertake There we generate synthetic cash flows and the main

results are summarized in Lemma 211 and Lemma 217 Section 3 provides a

deterministic closed form solution for the term structure of bonds issued by mi-

nority banks which we extend to a bond pricing theory for minority banks In

15 As a practical matter ldquoregulatory shocksrdquo shocks can be implemented by legislation that encourage credit

agreements between minority banks and credit worthy borrowers in the private sector See ldquoExelon Credit Agree-

ments Expand Relationships with Minority and Community Banksrdquo October 25 2010 Business Wire c Avail-

able at httpseekingalphacomnews-article4181-exelon-credit-agreements-expand-relationships-with-minority-and-

community-banks16See (US GAO Report No 07-06 2006 pg 26) It should be noted in passing that empirical research by Dahl

(1996) found that when banks change hands between minority and noonminority ownership loan growth is slower

when banks are owned by minorities compared to when they are owned by non-minorities

7

832019 SSRN-id1711002


section 4 we introduce our bank failure prediction model In section 5 we intro-

duce asset pricing models for minority banks The main results there are in The-

orems 510 and 513 which explain asset pricing anomalies for minority banks

and proposal(s) to correct them In subsection 51 we show how standard formu-

lae for weighted average cost of capital (WACC) should be modified to computeunlevered beta for minority banks Section 6 provides our CAMELS rating model

which is summarized in Proposition 64 Bank regulator CAMELS rating bias is

summarized in Lemma 63 Finally we conclude in section 7 with conjectures for

further research Select proofs are included in the Appendix

2 The Model

In the sequel all variables are assumed to be deterministic unless otherwise statedLet C jt be the CAMELS rating for bank j at time t where

C jt = f (Capital adequacy jt Asset quality jt Management jt Earnings jt

Liquidity jt Sensitivity to market risk jt )

(21)

for some monotone [decreasing] function f

Let E [K m jt ] and E [K n jt ] be the t -th period expected cash flow for projects17 under

taken by the j-th matched pair of minority (m) and non-minority (n) peer banksrespectively and E [C m jt ] E [C n jt ] be their expected CAMELS rating over a finite

time horizon t = 01 T minus 1 Because the internal rate of return ( IRR) repre-

sents the risk adjusted discount rate for a project to break even18 ie

Net present value (NPV) of the project = Present value of assetminusCost = 0 (22)

it is perhaps an instructive variable for comparison of minority and non-minority

owned banks for break even projects We make the following

Assumption 21 The conditional distribution of CAMELS rating is known

17These include loan and mortgage payments received by the bank(s) every period Furthermore in accord with

(Modigliani and Miller 1958 pg 265) the expectations of each type of bank is with respect to its subjective probability

distribution These expectations may not necessarily coincide with the subjective components of bank regulators

CAMELS ratings18 See (Damodaran 2010 Chapter 5) and (Brealey and Myers 2003 pg 96-97)

8

832019 SSRN-id1711002


Proposition 22 (Cash flow CAMELS rating relationship) The conditional distri-

bution of CAMELS rating is inversely proportional to the distribution of a banklsquos

cash flows

Proof Let P(

middot) be the distribution of CAMELS rating and c

isin 12345

be a

CAMELS rating So that by definition of conditional probability

P(C jt = c| K jt ) = P(C jt = cK jt = k )P(K jt = k ) (23)

Since P is ldquoknownrdquo for given k the LHS is inversely proportional to P(K jt =k )

Corollary 23 (Cash flow sufficiency) K jt is a sufficient statistic for CAMELS

rating

Proof The proof follows from the definition of sufficient statistic in (DeGroot1970 pp 155-156)

Remark 21 An empirical study by Lopez (2004) supports an inverse relationship

between average asset correlation with a common risk factor and probability of

default Thus if average asset correlation is a proxy for cash flows and CAMELS

rating is a proxy for probability of default our proposition is upheld

Thus the cash flows for projects under taken by banks are a rdquoreasonablerdquo proxy

for CAMELS variables So that

C jt = f (K jt )prop (K jt )minus1 (24)

In other words banks with higher cash flows for NPV projects should have lower

CAMELS rating In the analysis that follows we drop the j subscript without loss

of generality

21 Diagnosics for IRR from NPV Projects of Minority and Nonminority

Banks

We start with the simple no arbitrage premise that NPV for projects undertaken by

minority and nonminority banks are zero at break even point(s) ie

NPV m = NPV n = 0 (25)

For if the NPV of a project is greater that zero then the project manager could

invest more in that project to generate more cash flow Once the NPV is zero

9

832019 SSRN-id1711002


further investment in that project should end otherwise NPV would be negative

In any event let Ω be the set of states of nature that affect cash flows F be a

σ -field of Borel measureable subsets of Ω F t t ge0 be a filtration over F and P

be a probability measure defined on Ω So that (Ω F F t t ge0 P) is a filtered

probability space19

So that for t fixed we have the cash flow mapping

Definition 21

K r t ΩrarrRminusinfin infin r = m n and (26)

E [K r t (ω )] =

E

K r t (ω )dP(ω ) E isinF t (27)

Remark 22

Implicit in the definition is that future cash flows follow a stochasticprocess see Lemma 511 infra and that for fixed t there exists an ldquoobjectiverdquo

probability measure P on Ω for the random variable K r t (ω ) In practice type r

banks may have subjective probabilities Pr about elementary events ω isinΩ which

affect their computation of expected cash flows Arguably the objective probabil-

ities P(ω ) are from the point of view of an ldquoobserverrdquo of type r bankslsquo behavior

In the sequel we suppress ω inside the expectation operator E [

middot]

Assumption 24 E [K mt ] gt 0 and E [K nt ] gt 0 for t = 0

Assumption 25 Minority and nonminority banks belong to the same peer group

based on asset size or otherwise

Assumption 26 The success rate for independent projects undertaken by minor-

ity and nonminority banks is the same

Assumption 27 The time value of money is the same for each type of bank so the

extended internal rate of return XIRR is superfluous

Assumption 28 The weighted average cost of capital (WACC) for each type of

bank is different

Assumption 29 Markets are complete

19See (Karatzas and Shreve 1991 pg 3) for details of these concepts

10

832019 SSRN-id1711002


211 Fundamental equation of project comparison

The no arbitrage condition gives rise to the fundamental equation of project com-

parisonT

minus1

sumt =0 E [K mt ](1 + IRRm)minust =

T

minus1

sumt =0 E [K nt ](1 + IRRn)minust = 0 (28)

Assume that K m0 lt 0 and K n0 lt 0 are the present value of the costs of the respective

projects So that the index t = 12 corresponds to the benefits or expected cash

flows which under 24 guarantees a single IRR20

22 Yield spread for nonminority banks relative to minority peer banks

Suppose that minority and nonminority banks in the same class competed for

the same projects and that nonminority banks are publicly traded while minor-ity banks are not The market discipline imposed on nonminority banks implies

that they must compensate shareholders for risk taking Thus they would seek

those projects that have the highest internal rate of return21 Non publicly traded

minority banks are under no such pressure and so they are free to choose any

project with positive returns However the transparency of nonminority banks

implies that they could issue debt [and securities] to increase their assets in or-

der to be more competitive than minority banks Consequently they tend to have

a higher return on equity (ROE)-even if return on assets (ROA) are comparablefor both types of banks-because risk averse shareholders must be compensated for

20See (Ross et al 2008 pg 246)21According to Hays and De Lurgio (2003) ldquoCompetition for the low-to-moderate income markets traditionally

served by minority banks intensified in the 1990s after passage of the FIRREA in 1989 which raised the bar for

compliance with CRA Suddenly banks in general had incentives to ldquoserve the underservedrdquo This resulted in instances

of ldquoskimming the creamrdquondashtaking away some of the best customers of minority banks leaving those banks with reduced

asset qualityrdquo See also Benston George J ldquoItrsquos Time to Repeal the Community Reinvestment Actrdquo September

28 1999 httpwwwcatoorgpub˙displayphppub˙id=4976 (accessed September 24 2010) (ldquosuburban banks often

make subsidized or unprofitable loans in central cities or to minorities in order to fulfill CRA obligations This

ldquocream-skimmingrdquo practice of lending to the most financially sound customers draws business (and complaints) from

minority-owned local banks that normally specialize in service to that clientelerdquo)

11

832019 SSRN-id1711002


additional risk 22 This argument23 implies that

IRRn = IRRm +δ (29)

where δ is the premium or yield spread on IRR generated by nonminority banks

relative to minority banks in order to compensate their shareholders So that theright hand side of Equation 28 can be expanded as

T minus1

sumt =0

E [K mt ](1 + IRRm)minust =T minus1

sumt =0

E [K nt ](1 + IRRm +δ )minust (210)

=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q

(1 + IRRm)qδ minust minusq (211)

=

T

minus1

sumt =0

E [K nt ]

infin

sumu=t (minus1)uminust

uminus

1

uminus t

(1 + IRRm)uminust δ minusu χ t gt0 (212)

For the indicator function χ Subtraction of the RHS ie Equation 212 from the

LHS of Equation 210 produces the synthetic NPV project cash flow

T minus1

sumt =0

E [K mt ]minus E [K nt ]infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u)(1 + IRRm)minust χ t gt0 = 0

(213)

These results are summarized in the following

Lemma 210 (Synthetic discount factor) There exist δ such that

infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u)(1 + IRRm)minust χ t gt0 lt 1 (214)

22According to (US GAO Report No 07-06 2006 pg 4) ldquoProfitability is commonly measured by return on assets

(ROA) or the ratio of profits to assets and ROAs are typically compared across peer groups to assess performancerdquo

Additionally Barclay and Smith (1995) provide empirical evidence in support of Myers (1977) ldquocontracting-cost

hypothesisrdquo where they found that firms with higher information asymmetries issue more short term debt In our case

this implies that relative asymmetric information about nontraded minority banks cause them to issue more short term

debt (relative to nonminority banks) which is reflected in their lower IRR and ROA It should be noted that Reuben

(1981) unpublished dissertation has the same title as Barclay and Smith (1995) paper However this writer was unable

to procure a copy of that dissertation or its abstract So its not clear whether that research may be brought to bear here

as well23Implicit in the comparison of minority and nonminority peer banks is Modigliani-Millerrsquos Proposition I that capital

structure is irrelevant to the average cost of capital

12

832019 SSRN-id1711002


Proof See Appendix subsection A1

Lemma 211 (Synthetic cash flows for minority nonminority bank comparison)

Let I RRn and IRRm be the internal rates of return for projects undertaken by non-

minority and minority banks respectively Let I RRn = IRRm + δ where δ is a

project premium and E [K mt ] and E [K nt ] be the t-th period expected cash flow for

projects undertaken by minority and nonminority banks respectively Then the

synthetic cash flow stream generated by comparing the projects undertaken by

each type of bank is given by

E [K m0 ]minus E [K n0 ]

E [K mt ]minus E [K nt ]

infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0

Corollary 212 (Synthetic call option on minority bank cash flow) Let Ω be the set

of possible states of nature affecting banks projects P be a probability measure

defined on Ω F be the σ -field of Borel measureable subsets of Ω and F s subeF t 0 le s le t lt infin be a filtration over F So that all cash flows take place over

the filtered probability space (ΩF F t P) For ω isinΩ let K = K t F t t ge 0be a cash flow process E [K mt (ω )] be the expected spot price of the cash flow for

minority bank projects σ K m be the corresponding constant cash flow volatility r

be a constant discount rate and E [K nt (ω )]suminfinu=t (

minus1)uminust (uminus1

u

minust )(1+ IRRm

δ )u

be the

comparable risk-premium adjusted cash flow for nonminority banks Let K nT bethe ldquostrike pricerdquo for nonminority bank cash flow at ldquoexpiry daterdquo t = T Then

there exist a synthetic European style call option with premium

C (K mt σ K m| K nT T IRRm δ ) =

E

max

E [K mt (ω )]minusK nT

T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)

on minority bank cash flows

Proof Apply Lemma 210 and European style call option pricing formula in

(Varian 1987 Lemma 1 pg 63)

13

832019 SSRN-id1711002


Remark 23 In standard option pricing formulae our present value factor

T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

is replaced by eminusr (T minust ) where r is a risk free discount rate

Remark 24 Even though the corollary is formulated as a bet on minority bank

cash flows it provides a basis for construction of an asset securitization and or

collateralized loan obligation (CLO) scheme which could take some loans off of

the books of minority banksndashprovided that they are packaged in a proper tranche

and rated accordingly See (Tavakoli 2003 pg 28) For example if investor A

has information about a minority banklsquos cash flow process K m [s]he could pay a

premium C (K mt σ K m

|K nT T IRRm δ ) to investor B who may want to swap for

nonminority bank cash flow process K n with expiry date T -periods ahead Sucharrangements may fall under rubric of option pricing credit default swaps and

security design outside the scope of this paper See eg Wu and Carr (2006)

(Duffie and Rahi 1995 pp 8-9) and De Marzo and Duffie (1999) Additionally

this synthetic cash flow process provides a mechanism for extending our analysis

to real options valuation which is outside the scope of this paper See eg Dixit

and Pindyck (1994) for vagaries of real options alternative to NPV analysis

Remark 25 The assumption of constant cash flow volatility σ K m is consistent

with option pricing solutions adapted to the filtration F t t ge0 See eg Cadogan(2010b)

Assumption 213 The expected cost of NPV projects undertaken by minority ad

nonminority banks are the same

Under the assumption that markets are complete there exists a cash flow stream

for every possible state of the world24 This assumption gives rise to the notion

of a complete cash flow stream which we define as follows

Definition 22 Let ( X ρ) be a metric space for which cash flows are definedA sequence K t infint =0 of cash flows in ( X ρ) is said to be a Cauchy sequence if

for each ε gt 0 there exist an index number t 0 such that ρ( xt xs) lt ε whenever

s t gt t 0 The space ( X ρ) is said to be complete if every Cauchy sequence in that

space converges to a point in X

24See Flood (1991) for an excellent review of complete market concepts

14

832019 SSRN-id1711002


Remark 26 This definition is adapted from (Hewitt and Stromberg 1965 pg 67)

Here the rdquometricrdquo ρ is a measure of cash flows and X is the space of realnumbers R

Assumption 214 The sequence



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X

That is every cash flow can be represented as the sum25

of a sub-sequence of cash-flows In a complete cash-flow sequence the synthetic NPV of zero implies

that

E [K m0 ]minus E [K n0 ] + E [K mt ]minus E [K nt ]infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)

for all t Because the cash flow stream is complete in the span of the discount fac-

tors (1 + IRRm)minust t = 01 we can devise the following scheme In particular

let

d t m = (1 + IRRm)minust (218)

be the discount factor for the IRR for minority banks Then the sequence

d t minfint =0 (219)

can be treated as coordinates in an infinite dimensional Hilbert space L2d m

( X ρ)of square integrable functions defined on the metric space ( X ρ) with respect

to some distribution function F (d m) In which case orthogonalization of the se-quence by a Gram-Schmidt process produces a sequence of orthogonal polynomi-

als

Pk (d m)infink =0 (220)

in d m that span the space ( X ρ) of cash flows

25By ldquosumrdquo we mean a linear combination of some sort

15

832019 SSRN-id1711002


Lemma 215 There exists orthogonal discount factors Pk (d m)infink =0 where Pk (d m)is a polynomial in d m = (1 + IRRm)minus1

Proof See (Akhiezer and Glazman 1961 pg 28) for theoretical motivation onconstructing orthogonal sequence of polynomials Pk (d m) in Hilbert space And

(LeRoy and Werner 2000 Cor 1742 pg 169)] for applications to finance

Under that setup we have the following equivalence relationship

infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u(1 + IRRm)minust χ t gt0 =

infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)

which gives rise to the following

Lemma 216 (Complete cash flow) Let

K k = E [K mk ]minus E [K nk ]infin

sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if

K k = 0 forallk (224)


Under Assumption 213 and by virtue of the complete cash flow Lemma 216

this implies that

E [K mk ]minus E [K nk ]infin

sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002


However according to Lemma 210 the oscillating series in the summand con-

verges if and only if

1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)

and substitution of the relation for δ in Equation 29 gives

IRRn gt 1 + 2 IRRm (228)

That is the internal rate of return for nonminority banks is more than twice that for

minority banks in a world of complete cash flow sequences26 The convergence

criterion for Equation 225 and Lemma 210 imply that the summand is a discount

factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)

that is less than 1 ie DF t lt 1 This means that under the complete cash flow

hypothesis for given t we have

E [K mt ]minus E [K nt ] DF t = 0 (230)

Whereupon

E [K mt ] le E [K nt ] (231)

So the expected cash flow of minority banks are uniformly dominated by that of

26Kuehner-Herbert (2007) report that ldquoReturn on assets at minority institutions averaged 009 at June 30 com-

pared with 078 at the end of 2005 For all FDIC-insured institutions the ROA was 121 compared with 13 atthe end of 2005rdquo Thus as of June 30 2007 the ROA for nonminority banks was approximately 1345 ie (121009)

times that for minority banks So our convergence prerequisite is well definedndasheven though it underestimates the ROA

multiple Not to be forgotten is the Alexis (1971) critique of econometric models that fail to account for residual ef-

fects of past discrimination which may also be reflected in the negative sign Compare ( Schwenkenberg 2009 pg 2)

who provides intergenerational income elasticity estimates which show that black sons have a lower probability of

upward income mobility with respect to parents position in income distribution compared to white sons Furthermore

Schwenkenberg (2009) reports that it would take anywhere from 3 to 6 generations for income which starts at half the

national average to catch-up to national average

17

832019 SSRN-id1711002


non-minority banks One implication of this result is that minority banks invest in

second best projects relative to nonminority banks27 Thus we have proven the

following

Lemma 217 (Uniformly Dominated Cash Flows) Assume that all sequences of

cash flows are complete Let E [K mt ] and E [K nt ] be the expected cash flow for the

t-th period for minority and non-minority banks respectively Further let IRRm

and IRRn be the [risk adjusted] internal rate of return for projects under taken by

the subject banks Suppose that IRRn = IRRm +δ δ gt 0 where δ is the project

premium nonminority banks require to compensate their shareholders relative to

minority banks and that the t -th period discount factor

DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1

Then the expected cash flows of minority banks are uniformly dominated by the

expected cash flows of nonminority banks ie

E [K mt ]le E [K nt ]

Remark 27 This result is derived from the orthogonality of Pk (d m) and telescop-

ing series


sumu=t

(minus1)uminust uminus1uminust

(1+ IRRm

δ )u = 0 (232)

which according to Lemma 210 converges by construction

Remark 28 The assumption of higher internal rates of return for nonminority

banks implies that on average they should have higher cash flows However the

lemma produces a stronger uniform dominance result under fairly weak assump-

tions That is the lemma says that the expected cash flows of minority banks ie

expected cash flows from loan portfolios are dominated by that of nonminority

banks in every period

27Williams-Stanton (1998) eschewed the NPV approach and used an information and or contract theory based

model in a multi-period setting to derive underinvestment results

18

832019 SSRN-id1711002


As a practical matter the lemma can be weakened to accommodate instances

when a minority bank cash flow may be greater than that of a nonminority peer

bank We need the following

Definition 23 (Almost sure convergence) (Gikhman and Skorokhod 1969 pg 58)

A certain property of a set is said to hold almost surely P if the P-measure of the

set of points on which the property does not hold has measure zero

Thus we have the result

Lemma 218 (Weak cash flow dominance) The cash flows of minority banks are

almost surely uniformly dominated by the cash flows of nonminority banks That

is for some 0 lt η 1 we have

Pr K mt le K nt gt 1minusη


The foregoing lemmas lead to the following

Corollary 219 (Minority bankslsquo second best projects) Minority banks invest in

second best projects relative to nonminority banks


Corollary 220 The expected CAMELS rating for minority banks is weakly dom-

inated by that of nonminority banks for any monotone decreasing function f ie

E [ f (K mt )] le E [ f (K nt )]

Proof By Lemma 217 we have E [K m

t ] le E [K n

t ] and f (middot) is monotone [decreas-ing] Thus f ( E [K mt ])ge f ( E [K nt ]) So that for any regular probability distribution

for state contingent cash flows the expected CAMELS rating preserves the in-

equality

19

832019 SSRN-id1711002


3 Term structure of bond portfolios for minority and nonmi-

nority banks

In (Vasicek 1977 pg 178) single factor model of the term structure or yield to

maturity R(t T ) is the [risk adjusted] internal rate of return (IRR) on a bond attime time t with maturity date s = t + T In that case if T m T n are the times to ma-

turity for time t bonds held by minority and nonminority banks then δ (t |T mT n) = R(t T n)minus R(t T m) is the yield spread or underinvestment effect A minority bank

could monitor its fixed income portfolio by tracking that spread relative to the LI-

BOR rate To obtain a closed form solution (Vasicek 1977 pg 185) modeled

short term interest rates r (t ω ) as an Ornstein-Uhlenbeck process

dr (t ω ) = α (r

minusr (t ω ))dt +σ dB(t ω ) (31)

where r is the long term mean interest rate α is the speed of reverting to the mean

σ is the volatility (assumed constant here) and B(t ω ) is the background driving

Brownian motion over the states of nature ω isinΩ Whereupon for a given state the

term structure has the solution

R(t T ) = R(infin) + (r (t )minus R(infin))φ (T α )

α + cT φ 2(T α ) (32)

where R(infin) is the asymptotic limit of the term structure φ (T α ) = 1T (1

minuseminusα T )

and c = σ 24α 3 By eliminating r (t ) (see (Vasicek 1977 pg 187) it can be shown

that an admissible representation of the term structure of yield to maturity for a

bond that pays $1 at maturity s = t + T m issued by minority banks relative to

a bond that pays $1 at maturity s = t + T n issued by their nonminority peers is

deterministic

R(t T m) = R(infin) + c1minusθ (T nT m)

[T mφ 2(T m)minusθ (T nT m)T nφ

2(T n)] (33)

where

θ (T nT m) =φ (T m)

φ (T n)(34)

with the proviso

R(t T n) ge R(t T m) (35)

20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)

We summarize the forgoing as aProposition 31 (Term structure of minority bank bond portfolio) Assume that

short term interest rates follow an Ornstein-Uhelenbeck process

dr (t ω ) = α (r minus r (t ω ))dt +σ dB(t ω )

And that the term structure of yield to maturity R(t T ) is described by Vasicek

(1977 ) single factor model

R(t T ) = R(infin) + (r (t )minus R(infin))

φ (T α )

α + cT φ 2

(T α )

where R(α ) is the asymptotic limit of the term structure φ (T α ) = 1T (1minus eminusα )

θ (T nT m) =φ (T m)φ (T n)

and c = σ 2

4α 3 Let s = t +T m be the time to maturity for a bond that

pays $ 1 held by minority banks and s = t + T n be the time to maturity for a bond

that pays $ 1 held by nonminority peer bank Then an admissible representation

of the term structure of yield to maturity for bonds issued by minority banks is

deterministic

R(t T m) = R(infin) +c

1minusθ (T nT m)[T mφ 2

(T mα )minusθ (T nT m)T nφ 2

(T nα )]

Sketch of proof Eliminate r (t ) from the term structure equations for minority and

nonminority banks and obtain an expression for R(t T m) in terms of R(t T n) Then

set R(t T n) ge R(t T m) The inequality induces a floor for R(t T n) which serves as

an admissible term structure for minority banks

Remark 31 In the equation for R(t T ) the quantity cT r = σ 2

α 3T r where r = m n

implies that the numerator σ 2T r is consistent with the volatility or risk for a type-rbank bond That is we have σ r = σ

radicT r So that cT r is a type-r bank risk factor

21

832019 SSRN-id1711002


31 Minority bank zero covariance frontier portfolios

In the context of a linear asset pricing equation in risk factor cT m the representa-

tion IRRm equiv R(t T m) implies that minority banks risk exposure β m =φ 2(T mα )

1minusθ (T nT m)

Let φ (T m) be the risk profile of bonds issued by minority banks If φ (T m) gt φ (T n)ie bonds issued by minority banks are more risky then 1 minus θ (T nT m) lt 0 and

β m lt 0 Under (Rothschild and Stiglitz 1970 pp 227-228) and (Diamond and

Stiglitz 1974 pg 338) mean preserving spread analysis this implies that mi-

nority banks bonds are riskier with lower yields Thus we have an asset pricing

anomaly for minority banks because their internal rate of return is concave in risk

This phenomenon is more fully explained in the sequel However we formalize

this result in

Lemma 32 (Risk pricing for minority bank bonds) Assume the existence of amarket with two types of banks minority banks (m) and nonminority banks (n)

Suppose that the time to maturity T n for nonminority banks bonds is given Assume

that minority bank bonds are priced by Vasicek (1977 ) single factor model Let

IRRm equiv R(t T m) be the internal rate of return for minority banks Let φ (middot) be the

risk profile of bonds issued by minority banks and


φ (T n)(37)

ϑ (T m) = θ (T nT m| T m) = a0φ (T m) where (38)

a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002


is now a relative constant of proportionality and

φ (T mα ) gt1

a0(310)

minusβ m = φ

2

(T mα )1minusθ (T nT m)

(311)

=φ 2(T mα )

1minusa0ϑ (T m)(312)

σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)

where ε m is an idiosyncracy of minority banks Then

E [ IRRm] = α mminusβ mσ m + γ mσ n (317)

Remark 32 In our ldquoidealizedrdquo two-bank world α m is a measure of minority bank

managerial ability σ n is ldquomarket wide riskrdquo by virtue of the assumption that time

to maturity T n is rdquoknownrdquo while T m is not γ m is exposure to the market wide risk

factor σ n and the condition θ (T m) gt 1a0

is a minority bank risk profile constraint

for internal consistency θ (T nT m) gt 1 and negative beta in Equation 312 For

instance the profile implies that bonds issued by minority banks are riskier than

other bonds in the market28 Moreover in that model IRRm is concave in risk σ m

More on point let ( E [ IRRm]σ m) be minority bank and ( E [ IRRn]σ n) be non-

minority bank frontier portfolios29

Then according to (Huang and Litzenberger1988 pp 70-71) we have the following

Proposition 33 (Minority banks zero covariance portfolio) Minority bank fron-

tier portfolio is a zero covariance portfolio

28For instance according to (U S GAO Report No 08-233T 2007 pg 11) African-American banks had a loan loss

reserve of almost 40 of assets See Walter (1991) for overview of loan loss reserve on bank performance evaluation29See (Huang and Litzenberger 1988 pg 63)

23

832019 SSRN-id1711002



In other words we can construct a an efficient frontier portfolio by short selling

minority bank assets and using the proceeds to buy nonminority bank assets

4 Probability of bank failure with Vasicek (2002) Two-factor

model

Consistent with Vasicek (2002) and Martinez-Miera and Repullo (2008) assume

that the return on assets (ROA) ie loan portfolio value for a minority bank is

described by the stochastic process30

dA(t ω )

A(t ω )= micro dt +σ dW (t ω ) (41)

where micro and σ are constant mean and volatility growth rates Starting at time t 0

the value of the asset at time T is lognormally distributed with

log A(T ) = log A(t 0) +micro T minus 1

2σ 2T +σ

radicT X (42)

where X is a normal random variable decomposed as follows

X = Z Sradicρ + Z

1minusρ (43)

Z S and Z are independent and normally distributed Z S is a common factor Z is

bank specific and ρ is the pairwise constant correlation for the distribution of X lsquos

Vasicek (2002) computed the probability of default ie bank failure in our case

if assets fall below a threshold liability B as follows

Pr A(T ) lt B= Pr X lt c = Φ(c) = p (44)

30Implicit in the equation is the existence of a sample space Ω for states of nature a probability measure P on that

space a σ -field of Borel measureable subsets F and a filtration F t t ge0 over F where F infin =F The P-negligiblesets are assumed to be in F 0 So that we have the filtered probability measure space (ΩF t t ge0F P) over which

solutions to the stochastic differential equation lie We assume that the solution to such an equation is adapted to the

filtration See eg Oslashksendal (2003) Karatzas and Shreve (1991) for technical details about solution concepts

24

832019 SSRN-id1711002


where Φ(c) is the cumulative normal distribution and

c =log Bminus log Aminusmicro T + 1

2 σ 2T

σ radic

T (45)

Furthermore (Vasicek 2002 pg Eq(3)) computed the conditional probability of

bank failure given the common factor Z S as follows Let

L =

1 if bank fails

0 if bank does not fail(46)

E [ L| Z S] = p( Z S) = Φ

Φminus1( p)minus Z S

radicρradic

1minusρ

(47)

By slight modification replace Z S with X minus Z radic1minusρradicρ ) to get the conditional probabil-

ity of bank failure for bank specific factor Z given X In which case we have

p( Z | X ) = Φ

Φminus1( p)minus X minus Z

radic1minusρradic

1minusρ

(48)

From which we get the unconditional probability of bank failure for bank specific

variable

p( Z ) = infinminusinfin

p( Z | X )φ ( X )dX (49)

where φ (middot) is the probability density function for the normal random variable X

Lemma 41 (Probit model for bank specific factors) p( Z ) is in the class of probit

models


Our theory based result is distinguished from (Cole and Gunther 1998 pg 104)

heuristic probit model ie monotone function with explanatory variables moti-vated by CAMELS rating used to predict bank failure In fact those authors

opined

Reflecting the scope of on-site exams CAMEL ratings incorporate a

bankrsquos financial condition its compliance with laws and regulatory poli-

cies and the quality of its management and systems of internal control

25

832019 SSRN-id1711002


Regulators do not expect all poorly rated banks to fail but rather fo-

cus attention on early intervention and take action designed to return

troubled banks to financial health Given the multiple dimensions of

CAMEL ratings their primary purpose is not to predict bank failures

In particular (Cole and Gunther 1998 pg 106) used proxies to CAMELS vari-

ables in their probit model by calculating them rdquoas a percentage of gross assets

(net assets plus reserves)rdquo Here we provide theoretical motivation which predict

a probit type model for bank failure probabilities However the theory is more

rdquogranularrdquo in that it permits computation of bank failure for each variable sepa-

rately or in combination as the case may be31

5 Minority and Nonminority Banks ROA Factor Mimicking

Portfolios

In a world of risk-return tradeoffs one would intuitively expect that nonminority

banks take more risks because their cash flow streams and internal rates of returns

are higher However that concordance may not hold for minority banks which

tend to hold lower quality assets and relatively more cash Consequently their

capital structure is more biased towards surving the challenges of bank runs32 So

the issue of risk management is paramount We make the following assumptions

Assumption 51 The rate of debt (r debt ) for each type of bank is the same

Assumption 52 Minority bank are not publicly traded

Assumption 53 Nonminority banks are publicly traded

31See Demyanyk and Hasan (2009) for a taxonomy of bank failure prediction models Rajan et al (2010) also

provide empirical support for the hypothesis that the inclusion of only hard variables reported to investors in bank

failure prediction models at the expense of rdquosoftrdquo but nonetheless informative variables subject those models to a

Lucas (1983) critique That is bank failure prediction models fail to incorporate behavioral changes in lenders over

time and consequently underestimate bank failure In fact Kupiec (2009) rejected Vasicek (2002) model for corporate

bond default in a sample of Moodyrsquos data for 1920-2008 However Lopez (2004) conducted a study of credit portfoliosfor US Japanese and European firms for year-end 2000 and reached a different conclusion Thus our design matrix

should include at least dummy variables that capture possible regime shifts in bank policy32See eg Diamond and Rajan (2000)

26

832019 SSRN-id1711002


51 Computing unlevered beta for minority banks

Because projects are selected when IRR geWACC we assume33 that

IRR = WACC +ϕ (51)

where ϕ ge 0 However

V = D + E (52)

WACC = DV

r debt +E V

r equity (53)

By assumption minority banks are not publicly traded Besides Bates and Brad-

ford (1980) report that the extreme deposit volatility in Black-owned banks leads

them to hold more liquid assets than nonminority owned banks in order to meet

withdrawal demand This artifact of Black-owned banks was reaffirmed by Haysand De Lurgio (2003) According to empirical research by (Dewald and Dreese

1970 pg 878) extreme deposit variability imposes constraints on a bankrsquos portfo-

lio mix by forcing it to borrow more from the Federal Reserve and hold more short

term duration assets These structural impediments to minority banks profitability

imply that a CAPM type model is inadmissible to price their cost of capital (or

return on equities) (r p) To see this we modify Ruback (2002) as follows Let

WACC m = R f +β U m ˜ R p (54)

β U m =

DV β Dm + E

V β E m

(55)

where R f is a risk free rate β Dm is the beta computed from debt pricing β E m is

the beta computed from equity pricing ˜ R p is a risk premium andβ U m is unlevered

beta ie beta for ROA with Instead of the CAPM our debt beta is computed by

Equation 317 in Lemma 32 and return on asset beta ie unlevered beta (β U m)

is computed by Equation 519 in Lemma 59 Since minority banks betas tend to

be negative it means that for IRRmgt 0 in Equation 51 we must have

φ geminus

DV β Dm + E

V β E m

˜ R pminus R f (56)

33See (Ross et al 2008 pg 241)

27

832019 SSRN-id1711002


In order for minority bank debt to be properly priced in an equilibrium in which

sgn(β E m) lt 0 and φ = 0 (57)

we need the constraint

|β E m|gt V

E

R f

˜ R p

(58)

By the same token in order for minority bank equity to be properly priced in an

equilibrium in which

sgn(β E m) lt 0 and φ = 0 we need (59)

|β Dm|gtV

D

R f

˜ R p (510)

Evidently for a given capital structure the inequalities are inversely proportional

to the risk premium on the portfolio of minority bank debt and equities Roughly

speaking up to a first order approximation we get the following

Proposition 54 (WACC and minority bank capital structure) For a given debt to

equity ratio D E

for minority banks the price of debt to equity is directly proportional

to the debt to equity ratio ie

|β

E m||β Dm| propD

E (511)

up to a first order approximation

Sketch of Proof For constants b E and b D let

|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)

Divide |β E m| by |β Dm| and use a first order approximation of the quotient

Remark 51 This result plainly shows that Modigliani and Miller (1958) capital

structure irrelevance hypothesis does not apply to pricing debt and equities of

minority banks

28

832019 SSRN-id1711002


In corporate finance if τ is the corporate tax rate then according to (Brealey

and Myers 2003 pg 535) we have

β E m = β U m[1 + (1minus τ ) D E ]minusβ Dm(1minus τ )

D E (514)

From Proposition 54 let c0 be a constant of proportionality so that

|β E m|= c0|β Dm| D E

(515)

Since classic formulae assumes that betas are positive upon substitution we get

β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)

However once we remove the absolute operation | middot | for minority banks the for-

mula must be modified so that

sgn(β U m) = sgn(β Dm) (517)

But β U m is the beta for return on assetsndashwhich tends to be negative for minority

banks This is transmitted to debt beta as indicated Thus we have

Proposition 55 For minority banks the sign of beta for return on assets is the

same as that for the beta for debt

In order to explain the root cause(s) of the characteristics of minority bank be-

tas we appeal to a behavioral asset pricing model In particular let λ be a loss

aversion index popularized by Khaneman and Tversky (1979) and x be fluctua-

tions in income ie liquidity and σ x be liquidity risk Then we introduce the

following

Proposition 56 Let x be change in income and v be a value function over gains

and losses with loss aversion index λ Furthermore let micro x and σ x be the average

change in liquidity ie mean and σ 2 x be the corresponding liquidity risk ie vari-

ance Then the behavioral mean-variance tradeoff between liquidity and liquidity

risk is given by

micro x = β ( xλ )σ x (518)

29

832019 SSRN-id1711002


where β ( xλ ) is the behavioral liquidity risk exposure

Corollary 57 There exists a critical loss aversion value λ c such that β ( xλ ) lt 0

whenever λ gt λ c

Proof Because the proofs of the proposition and corollary are fairly lengthy and

outside the scope of this paper we direct the reader to (Cadogan 2010a Sub-

section 21 pg 11) for details of the model and proofs

Proposition 56 and Corollary 57 are pertinent to this paper because they use

prospect theory and or behavioral economics to explain why asset pricing models

for minority banks tend to be concave in risk while standard asset pricing the-ory holds that they should be convex34 Moreover they imply that on Markowitz

(1952a) type mean-variance efficient frontier ie portfolio comprised of only

risky assets minority banks tend to be in the lower edge of the frontier (hyper-

bola) This is in accord with Proposition 33 which posits that minority banks

hold zero-covariance frontier portfolios Here the loss aversion index λ behaves

like a shift parameter such that beyond the critical value λ c minority banks risk

seeking behavior causes them to hold inefficient zero covariance portfolios

52 On Pricing Risk Factors for Minority and Nonminority Peer Banks

Our focus up to this point has been in relation to projectsrsquo internal rates of return

In this section we examine how these rates relate to relative risk for comparable

minority and non-minority banks in the context of an asset pricing model

521 Asset pricing models for minority and non-minority banks

Let σ mσ n be the specific project risks faced by minority and nonminority banks

and σ S be systemic ie market wide risks faced by all banks35 where as we34See Friedman and Savage (1948) Markowitz (1952b) and Tversky and Khaneman (1992) for explanations of

concave and convex portions of utility functions for subjectslsquo behavior towards fluctuations in wealth35 This has been describes thus

The sensitivity to market risk component reflects the degree to which changes in interest rates foreign

exchange rates commodity prices or equity prices can adversely affect a financial institutionrsquos earnings

or economic capital When evaluating this component consideration should be given to managementrsquos

ability to identify measure monitor and control market risk the institutionrsquos size the nature and

30

832019 SSRN-id1711002


demonstrate below σ m σ n Let K mt (σ m|σ S) and K nt (σ n|σ S) be the cash flow

streams as a function of project risk for given systemic risks By definition the

NPV relation in the fundamental equation(s) for project comparison Equation 28

plainly show that IRR(middot) is a function of K (middot) In which case IRR(middot)(σ (middot)|σ S) is

also a function of project risk σ (middot) for given systemic risk According to Brimmerthe minority bank paradox ldquoarises from the exceptional risks these lenders must

assume when they extend credit to individuals who suffer from above-average

instability of employment and income or to black-owned businesses which have

high rates of bankruptcyrdquo This observation is consistent with (Khaneman and

Tversky 1979 pg 268) prospect theory where it is shown that subjects are risk

seeking over losses and risk averse over gains36 In which case we would expect

returns of minority banks to be negatively correlated with project risk 37 Despite

their negative betas minority banks may be viable if we make the following

Assumption 58 ( Huang and Litzenberger 1988 pp 76-77)

The risk free rate is less than that attainable from a minimum variance portfolio

Thus even though their portfolios may be inefficient there is a window of op-

portunity for providing a return greater than the risk free rate Besides Lemma

32 provides a bond pricing scenario in which that is the case By contrast non-

minority banks select projects from a more rdquotraditionalrdquo investment opportunity

complexity of its activities and the adequacy of its capital and earnings in relation to its level of market

risk exposure

See eg Uniform Financial Institutions Ratings System page 9 available at

httpwwwfdicgovregulationslawsrules5000-900html as of 1112201036This type of behavior is not unique to minority banks Johnson (1994) conducted a study of 142 commercial

banks using Fishburn (1977) mean-risk method and found that bank managers take more risk when faced with below

target returns Anecdotal evidence from (US GAO Report No 07-06 2006 pg 20) found that 65 of minority

banks surveyed were optimistic about their future prospects despite higher operating expenses and lower profitability

than their nonminority peer banks Moreover despite the availability of technical assistance to ldquocorrect problems that

limit their financial and operational performancerdquo id at 35 most minority banks failed to avail themselves of the

assistance37See also (Tobin 1958 pg 76) who developed a theory of risk-return tradeoff in which ldquoa risk loverrsquos indifference

curve is concave downwardsrdquo in (returnrisk) space as opposed to the positive slope which characterizes Black (1972)

zero-beta CAPM More recently Kuehner-Herbert (2007) reported negative ROA for African-American banks andpositive ROA for all others Also (Henderson 1999 pg 375) articulates how inadequate assessment of risk leads to

loan losses (and hence lower IRR) for African-American owned banks Additionally (Hylton 1999 pg 2) argues that

bank regulation makes it difficult for inner city banks to compete For instance Hylton noted that regulators oftentimes

force banks to diversify their loan portfolios in ways that induce conservative lending and lower returns According to

empirical research by (Dewald and Dreese 1970 pg 878) extreme deposit variability imposes constraints on a bankrsquos

portfolio mix by forcing it to borrow more from the Federal Reserve and hold more short term duration assets These

structural impediments to minority banks profitability imply that a CAPM type model is inadmissible to price their

cost of capital (or return on equities) Intuitively these negative returns on risk imply that whereas the CAPM may

hold for nonminority banks it does not hold for African-American banks

31

832019 SSRN-id1711002


set in which the return-risk tradeoff is positively correlated These artifacts of as-

set pricing can be summarized by the following two-factor models of risk-return

tradeoffs38

IRRm = α m0minusβ mσ m + γ mσ S + ε m

(519)

IRRn = α n0 +β nσ n + γ nσ S + ε n (520)

where β mβ n gt 0 and ε (middot) is an idiosyncratic an error term The γ (middot) variable

is a measure of bank sensitivity to market risks ie exposure to systemic risks

and α (middot) is a proxy for managerial ability In particular Equation 519 and Equa-

tion 520 are examples of Vasicek (2002) two-factor model represented by Equa-

tion 42 with the correspondence summarized in

Lemma 59 (Factor prices) Let r = m n β r be the price of risk and γ r be exposure

to systemic risk for minority and nonminority banks is given by

sgn(β r )β r σ r = σ

T (1minusρr ) Z r (521)

γ r σ S = σ

T ρr Z S (522)

where as before Z S is a common factor Z (middot) is bank specific σ is volatility growth

rate for return on bank assets and ρr is the constant pairwise correlation coeffi-

cient for the distribution of background driving bank asset factors

Remark 52 ldquosgnrdquo represents the sign of β r

It should be noted in passing that implicit in Equation 521 and Equation 522

is that our variables are time subscripted Thus our parametrization employs Va-

sicek (2002) factor pricing theory and incorporates Brimmer (1992) black banksrsquo

paradox Let

ε lowastm = γ mσ S +ε m (523)

ε lowastn = γ nσ S + ε n (524)

38See (Elton et al 2009 pg 629) See also Barnes and Lopez (2006) King (2009) for use of CAPM in estimating

cost of equity capital (COE) in banking However Damodaran (2010) points out that estimates of CAPM betas are

backwards looking while the risk adjusted discount factor is forward looking To get around that critique one could

assume that beta follows a Markov process

32

832019 SSRN-id1711002


So the equations are rewritten as

IRRm = α m0minusβ mσ m +ε lowastm (525)

IRRn = α n0 +β nσ n + ε lowastn (526)

Assume arguendo that the IRR and managerial ability for minority and non-minority

banks are equal So that

α m0minusβ mσ m + ε lowastm = α n0 +β nσ n + ε lowastn (527)

where

α m0 = α n0 (528)

Thus of necessity

ε lowastm ε lowastn rArr variance(ε lowastm) variance(ε lowastn ) (529)

and consequently

σ ε lowastm σ ε lowastn (530)

We have just proven the following

Theorem 510 (Minority Bank Risk Profile) Assuming that Vasicek (2002) two

factor model holds let

IRRm = α m0minusβ mσ m + ε lowastm IRRn = α n0 +β nσ n + ε lowastn

be two-factor models used to price internal rates of returns for minority (m) and

nonminority (n) banks and

ε lowastm = γ mσ S + ε m

ε lowastn = γ nσ S +ε n

be the corresponding market adjusted idiosyncratic risk profile where ε m is id-

iosyncratic risk for minority banks ε n idiosyncratic risk for nonminority banks

σ S be a market wide systemic risk factor and γ (middot) be exposure to systemic risk

33

832019 SSRN-id1711002


If the internal rate of return and managerial ability is the same for each type of

bank then the market adjusted risk for projects undertaken by minority banks are

a lot higher than that undertaken by minority banks ie

σ

2

ε lowastm σ

2

ε lowastn

This theorem can also be derived from consideration of the risky cash flow

process K = K r (t ω )F t 0le t ltinfin r = m n For instance suppressing the

r superscript (Cadogan 2009 pg 9) introduced a risky cash flow process given

by

dK (t ω ) = r (K (t ω ))dt +σ K dB(t ω ) (531)

where the drift term r (K (t ω )) is itself a stochastic Arrow-Pratt risk process and

σ K is constant cash flow volatility He used a simple dynamical system to show

that

r (K (t ω )) =σ K

cosh(σ K ct )

t

0sinh(σ K cs)eminus RsK (sω )ds (532)

where c is a constant B(t ω ) is Brownian motion and R is a constant discount

rate So that the Arrow-Pratt risk process and stochastic cash flows for a project are

controlled by a nonlinear relationship in cash flow volatility and the discount ratefor cash flows In our case R is the IRR So other things equal since IRRn gt IRRm

in Equation 29 application of Equation 532 plainly shows that

r (K m(t ω )) gt r (K n(t ω )) (533)

This leads to the following

Lemma 511 (Arrow-Pratt risk process for minority and nonminority banks cash

flows) Let K =

K r (t ω )F t 0

let ltinfin

r = m n be a risky cash flow process

for minority (m) and nonminority (n) banks Then the Arrow-Pratt risk process

r (K m) = r (K m(t ω ))F t for minority banks stochastically dominates than that

for nonminority banks r (K n) = r (K n(t ω ))F t That is for r (K ) isin C [01]

and dyadic partition t (n)k = k 2minusn k = 012 2n of the unit interval [01]

the joint distribution of r (K m(t (n)1 ω )) r (K m(t

(n)q ω )) is greater than that for

r (K n(t (n)1 ω )) K n(t

(n)q ω )) for 0 le q le 2n under Diamond and Stiglitz (1974)

34

832019 SSRN-id1711002


mean preserving spread


Thus Theorem 510 and Lemma 511 predict Brimmerlsquos black bankslsquo para-

dox39ndashother things equal the idiosyncratic risks for projects undertaken by minor-

ity banks is a lot greater than that under taken by nonminority banks To alleviate

this problem40 quantified in Theorem 510 induced by structural negative price

of risk (presumably brought on by minority banks altruistic projects) we introduce

a compensating risk factor 41

∆gt 0 (534)

Let the risk relationship be defined by

σ 2ε lowastm minus∆ = σ 2ε lowastn (535)

Since each type of bank face the same systemic risk σ S we need another factor

call it ε ∆ such that

ε lowastm = ε lowastmminus ε ∆ = γ mσ S +ε mminus ε ∆ = ε lowastn (536)

where ε lowastm is compensating risk adjusted shock to minority banks that makes theiridiosyncratic risks functionally equivalent to that for nonminority banks after con-

trolling for market wide systemic risk ie

σ 2ε lowastm = σ 2ε lowastn (537)

39Compare (Henderson 2002 pg 318) introduced what he described as an ldquoad hocrdquo model that provides ldquoa first

step in better understanding [Brimmerlsquos] economic paradoxrdquo40According to (U S GAO Report No 08-233T 2007 pg 8) ldquomany minority banks are located in urban areas

and seek to serve distressed communities and populations that financial institutions have traditionally underservedrdquo

Arguably minority banks should be compensated for their altruistic mission Cf Lawrence (1997)41Arguably an example of such a compensating risk factor may be government purchase of preferred stock issued

by minority banks Kwan (2009) outlined how the federal reserve banks instituted a Capital Purchase Program (CPP)

which used a preferred stock approach to capitalizing banks in the Great Recession This type of hybrid subordinated

debt instrument falls between common stock and bonds It falls under rubric of so called Pecking Order Theory in

Corporate Finance It is a non-voting share which has priority over common stock in the payment of dividends and

the event of bankruptcy (Kwan 2009 pg 3) explained that a convertible preferred stock is akin to a call option the

banksrsquos common stock Furthermore many small banks that did not have access to capital markets chose this route

35

832019 SSRN-id1711002


In that case

σ 2ε lowastm = σ 2ε lowastm +σ 2ε ∆minus2cov(ε lowastmε ∆) (538)

This quantity on the right hand side is less than σ 2

ε lowastm if and only if the followingcompensating risk factor relationship holds

σ 2ε ∆minus2cov(ε lowastmε ∆) le 0 (539)

That is

∆ = 2cov(ε lowastmε ∆)minusσ 2ε ∆ gt 0 (540)

This compensating equation has several implications First it provides a measure

of the amount of compensated risk required to put minority banks on par with

nonminority banks Second it shows that the compensating shock ε ∆ to minority

banks must be positively correlated with the market adjusted idiosyncratic risks ε lowastmof minority banks That is the more idiosyncratic the risk the more the compen-

sation Third the amount of compensating shocks given to minority banks must

be modulated by σ 2ε ∆ before it gets too large We summarize this result with the

following

Theorem 512 (Minority bank risk compensation evaluation) Suppose that the

internal rates of return (IRR) of minority (m) and nonminority (n) banks are priced

by the following two factor models

IRRm = α m0minusβ mσ m + ε m

IRRn = α n0 +β nσ n +ε n

Let α (middot) be the factor for managerial ability at each type of bank Further let the

market adjusted idiosyncratic risk for each type of bank be


ε lowastn = γ nσ S + ε n

36

832019 SSRN-id1711002


Assume that managerial ability α (middot) and IRR are the same for each type of bank

Let ∆gt 0 be a compensating risk factor such that

σ 2ε lowastm minus∆ = σ 2ε lowastn

Let ε ∆ be a risk compensating shock to minority banks Then the size of the com-

pensating risk factor is given by

∆ = 2cov(ε lowastmε ∆)minusσ 2ε ∆ gt 0

where σ 2ε ∆ is the variance or risk of compensating shock

The covariance relation in Equation 540 suggests the existence of a linear pric-

ing relationship in which ε ∆ is a dependent variable and ε lowastm and σ 2ε ∆ are explanatory

variables In fact such a pricing relationship subsumes a mean-variance analysis

for the pair (ε ∆σ 2ε ∆) augmented by the factor ε lowastm Formally this gives rise to the

following

Theorem 513 (Minority bank risk compensation shock) Minority bank compen-

sating shock is determined by the linear asset pricing relationship

ε ∆ = β m∆ε lowastm

minusψσ 2ε ∆ +η∆ (541)

where ψ is exposure to compensating shock risk and η∆ is an idiosyncratic error

term

Remark 53 This pricing relationship resembles Treynor and Mazuy (1966) oft

cited market timing modelndashexcept that here the timing factor is convex down-

wards ie concave in minority bank compensation risk

Corollary 514 ( Ex ante compensation price of risk) The ex-ante compensation

price of risk for minority banks is positive ie sgn(β m∆) gt 0

37

832019 SSRN-id1711002


Remark 54 This corollary implies that the compensating shock ε ∆ caused the

price of risk for minority banks to go from minusβ m to +β m∆ That is the price of

risk is transformed from negative to positive From an econometric perspective

OLS estimation of β m∆ in Equation 541 suffers from classic ldquoerrors-in-variablesrdquo

problem which can be corrected by standard procedures See (Kmenta 1986pp 348-349) and (Greene 2003 pp 84-86)

53 A put option strategy for minority bank compensating risk

The asset pricing relationship in Equation 541 implies that the compensating

shock to minority banks can be priced as a put option on the idiosyncratic adjusted-

risk in minority banks projects To wit the asset pricing relation is concave incompensating shock risk So isomorphically a put strategy can be employed to

hedge against such risks See eg Merton (1981) Henriksson and Merton (1981)

In fact we propose the following pricing strategy motivated by Merton (1977)

Lemma 515 (Value of regulatory guarantee for compensating risk for minority

bank) Let Bε ∆ be the value of the bonds held by loan guarantors or regulators

of minority banks and V ε lowastm be the value of the assets or equity in minority banks

Let Φ(middot) be the cumulative normal distribution and σ V ε lowastmbe the variance rate per

unit time for log changes in value of assets Suppose that bank assets follow thedynamics

dV ε lowastm(t ω )

V ε lowastm= micro ε lowastmdt +σ V ε lowastm

dW (t ω )

where micro ε lowastm and σ V ε lowastm are constants and W (t ω ) is Brownian motion in the state ω isinΩ Let G∆(T ) be the value of a European style put option written on minority bank

compensating risk with exercise date T The Black and Scholes (1973) formula for

the put option is given by

G∆(T ) = Bε ∆eminusrT Φ( x1)minusV ε lowastmΦ( x2) (542)

38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm

minus (r +σ 2V ε lowastm

2 )T

σ V ε lowastm

radicT

(543)

x2 = x1 +σ V ε lowastmradic

T (544)

Remark 55 It should be noted in passing that the put option dynamics are func-

tionally equivalent to Vasicek (2002) loan portfolio model for return on assets

(ROA) Except that (Vasicek 2002 pg 2) decomposed the loan portfolio value

into 2-factors (1) a common factor and (2) company specific factor Thus we es-

tablish a nexus between a put option on the assets of minority banks and concavityin the asset pricing model for compensating shock in Equation 541

531 Optimal strike price for put option on minority banks

Recently (Deelstra et al 2010 Thm 42) introduced a simple pricing model for

the optimal strike price for put options of the kind discussed here Among other

things they considered a portfolio comprised of exposure to a risky asset X and a

put option P They let P(0T K ) be the time 0 price of a put option with strike priceK exercised at time T for exposure to risky asset X (T ) Additionally h 0le h lt 1

is the allocation to the put option ρ(minus X (T )) is a coherent measure of risk 42

and C is the budget allocated for ldquohedging expenditurerdquo for the portfolio H = X hP They assumed that the impact of short term interest rate was negligible

and formulated an approximate solution to the optimal strike price problem

minhK

[ X (0) +C minus ((1minush) X (T ) + hK )] (545)

st C = hP(0T K ) h isin (01) (546)42Given a probability space (ΩF P) and a set of random variables Γ defined on Ω the mapping ρ Γ rarr R is a

risk measure The coherent measure of risk concept which must satisfy certain axioms was popularized by Artzner

et al (1999)

39

832019 SSRN-id1711002


Whereupon the optimal strike price K lowast is determined by solving the put option

relation

P(0T K )minus (K +ρ(minus X (T )))part P(0T K )

part K = 0 (547)

Their put option P(0T K ) is our G∆(T ) their asset X (T ) is our V ε lowastm(T ) and their

strike price K is our Bε ∆ Thus a bank regulator can estimate Bε ∆ accordingly in

order to determine the put option premium on minority bankslsquo assets for insurance

purposes

532 Regulation costs and moral hazard of assistance to minority banks

Using a mechanism design paradigm Cadogan (1994) introduced a theory

of public-private sector partnerships with altruistic motive to provide financial as-sistance to firms that face negative externalities There he found that the scheme

encouraged moral hazard and that the amount of loans provided was negatively

correlated with the risk associated with high risk firms43 In a two period model

Glazer and Kondo (2010) use contract theory to show that moral hazard will be

abated if a governmentlsquos altruistic transfer scheme is such that it does not reallo-

cate money from savers to comparative dissavers They predict a Pareto-superior

outcome over a Nash equilibrium with no taxes or transfers

More on point (Merton 1992 Chapter 20) extended Merton (1977) put option

model to ldquoshow[] that the auditing cost component of the deposit insurance pre-mium is in effect paid for by the depositors and the put option component is paid

for by the equity holders of the bankrdquo In order not to overload this paper we did

not include that model here Suffice to say that by virtue of the put option charac-

teristic of the model used to price transfers ie compensating shocks to minority

banks in order to mitigate moral hazard theory predicts that minority banks will

bear costs for any regulation scheme ostensibly designed to compensate them for

the peculiarities of the risk structure of altruistic projects they undertake This

gives rise to another minority bank paradoxndashminority banks must bear the cost of transfer schemes intended to help them in order to mitigate moral hazard The

literature on minority banks is silent on mitigating moral hazard in any transfer

scheme designed to assist themndashor for that matter distribution of the burden and

regulation costs of such schemes

43See eg (Laffont and Matrimint 2002 Chapters 4 5)

40

832019 SSRN-id1711002


54 Minority banks exposure to systemic risk

An independently important consequence of the foregoing compensating relation-

ships is that when σ S is a random variable we have the following decomposition

of compensating risk factor

cov(ε lowastmε ∆) = γ mcov(σ Sε ∆) + cov(ε mε ∆) gt 0 (548)

Assuming that cov(σ Sε ∆) = 0 and γ m = 0 we have the following scenarios for

cov(ε lowastmε ∆) gt 0

Scenario 1 Negative exposure to systemic risk (γ m lt 0) Either cov(σ Sε ∆) lt 0 and

cov(ε mε ∆) gt 0 or cov(σ Sε ∆) gt 0 and cov(ε mε ∆) gt γ mcov(σ Sε ∆)

Scenario 2 Positive exposure to systemic risk (γ m gt 0) Either cov(σ Sε ∆) gt 0 and

cov(ε mε ∆)gt 0 or (a) if cov(σ Sε ∆)lt 0and cov(ε mε ∆)gt 0 then cov(ε mε ∆)gt

γ mcov(σ Sε ∆) and (b) if cov(σ Sε ∆)gt 0 and cov(ε mε ∆)lt 0 then cov(ε mε ∆)lt

γ mcov(σ Sε ∆)

As a practical matter the foregoing analysis is driven by exposure to systemic

risk (γ m) From an econometric perspective since ε lowastm ε lowastn we would expect that

on average the residuals from our factor pricing model are such that elowastm elowastn

From which we get the two stage least squares (2SLS) estimators for systemic

risk exposure

ˆγ m =cov(elowastmσ S)

σ 2S(549)

ˆγ n =cov(elowastnσ S)

σ 2S(550)

ˆγ m ˆγ n (551)

This is an empirically testable hypothesis

More on point Scenario 1 plainly shows that when minority banks aresubject to negative exposure to systemic risks they need a rdquogoodrdquo ie positive

compensating shock ε ∆ to idiosyncratic risks That is tantamount to an infusion

of cash or assets that a regulator or otherwise could provide The situations in

Scenario 2 are comparatively more complex In that case minority banks have

positive exposure to systemic risks ie there may be a bull market low unem-

ployment and other positive factors affecting bank clientele We would have to

41

832019 SSRN-id1711002


rule out sub-scenario (a) as being incongruent because a good shock to idiosyn-

cratic risk would be undermined by its negative correlation with systemic risk to

which the bank has positive exposure So that any sized good shock to idiosyn-

cratic risk is an improvement A similar argument holds for sub-scenario (b) be-

cause any sized negative correlation between good shocks and idiosyncratic risksis dominated by the positive exposure to systemic risks and positive correlation

between good shocks and systemic risks Thus Scenario 2 is ruled out by virtue

of its incoherence with compensatory transfers In other words Scenario 2 is the

ldquorising-tide-lifts-all-boatsrdquo phenomenon in which either sufficiently large good id-

iosyncratic or systemic shocks mitigate against the need for compensatory shocks

The foregoing analysis gives rise to the following

Theorem 516 (Compensating minority banks negative exposure to systemic risk)

Let

ε lowastm = γ mσ S + ε m and

∆ = 2cov(ε lowastmε ∆)minusσ 2ε ∆

be a compensating risk factor for minority banks relative to nonminority banks

Then a coherent response to compensating minority banks for negative exposure

to systemic risk is infusion of a good shock ε ∆ to idiosyncratic risks ε m

That is to generate similar IRR payoffs given similar managerial skill at mi-

nority and nonminority banks minority banks need an infusion of good shocks to

idiosyncratic risks in projects they undertake That is under Brimmerrsquos surmise a

regulator could compensate minority banks with good ε ∆ shocks for trying to fill a

void in credit markets By contrast non-minority banks do not need such shocks

because they enjoy a more balanced payoff from both systematic and idiosyncratic

risks in the projects they undertake

6 Minority Banks CAMELS rating

In this section we provide a brief review of how a regulatorlsquos CAMELS rat-

ingscore may be derived In particular the choice set for CAMELS rating are

determined by regulation

Under the UFIRS each financial institution is assigned a composite rat-

ing based on an evaluation and rating of six essential components of an

42

832019 SSRN-id1711002


institutionrsquos financial condition and operations These component fac-

tors address the adequacy of capital the quality of assets the capability

of management the quality and level of earnings the adequacy of liq-

uidity and the sensitivity to market risk Evaluations of the components

take into consideration the institutionrsquos size and sophistication the na-ture and complexity of its activities and its risk profile

Source UFIRS Policy Statement (eff Dec 20 1996) Composite ratings re-

flect component ratings but they are not based on a mechanical formula That

is ldquo[t]he composite rating generally bears a close relationship to the component

ratings assigned However the composite rating is not derived by computing an

arithmetic average of the component ratingsrdquo op cit Moreover CAMELS scores

are assigned on a scale between 1 (highest) to 5 (lowest) Given the seemingly sub-

jective nature of score determination analysis of CAMELS rating is of necessityone that falls under rubric of discrete choice models

61 Bank examinerslsquo random utility and CAMELS rating

The analysis that follow is an adaptation of (Train 2002 pp 18-19) and relies on

the random utility concept introduced by Marschak (1959) Let

C 12345 the choice set ie possible scores for CAMELS rating

U n j utility regulatorexaminer n obtains from choice of alternative j ie CAMELSscore from choice set C

xn j attributes of alternatives ie CAMELS components faced by regulator

sn be attributes of the regulatorexaminer

V n j = V n j( xn j sn) representative utility of regulator specified by researcher

ε n j unobservable factors that affect regulator utility

ε n = (ε n1 ε n5) is a row vector of unobserved factors

U n j = V n j + ε n j random utility of regulator

43

832019 SSRN-id1711002


The researcher does not know ε n j but specifies a joint distribution f (ε n) So that

the probability that she chooses to assign a CAMELS score i instead of an alter-

native CAMELS score j is given by

Pni =

Pr(U

ni gtU

n j)(61)

= Pr(V ni + ε ni gt V n j + ε n j forall j = i) (62)

= Pr(ε n jminus ε ni ltV niminusV n j forall j = i) (63)

This probability is ldquoknownrdquo since V niminusV n j is observable and the distribution of

ε n was specified In particular if I is an indicator function then

Pni =

ε nI (ε n jminus ε ni ltV niminusV n j forall j = i) f (ε n)d ε n (64)

In Lemma 41 we derived a probit model for bank specific factors Thus weimplicitly assumed that ε n has a mean zero multivariate normal distribution

ε n simN (0ΣΣΣ) (65)

with covariance matrix ΣΣΣ Suppose that for some unknown parameter vector β β β

V n j = β β β xn j so that (66)

U n j = β β β xn j + ε n j (67)

The bank regulatorlsquos CAMELS rating is given by

Pr(CAMELSratingsim 1 2 3 4 5)

= Pr(U n1 gtU n2 gt U n3 gtU n4 gtU n5)(68)

Let

U n jk = U n jminusU nk (69)

˜ xn jk = xn jminus xnk (610)ε n jk = ε n jminus ε nk (611)

44

832019 SSRN-id1711002


So that by subtracting adjacent utilities in reverse order ie low-high we trans-

form

Pr(U n1 gtU n2 gt U n3 gtU n4 gtU n5) = Pr(U n21 lt U n32 lt U n43 lt U n54) (612)

The random utility transformation is based on a design matrix

M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)

In which case for the vector representation of random utility

U n = V n + ε n (614)

we can rewrite Equation 612 to get the probit generated probability for the regu-

lator


= Pr(U n21 lt U n32 lt U n43 lt U n54)(615)

= Pr( M M MU n lt 0) (616)

= Pr( M M M ε n ltminus M M MV n) (617)

Whereupon we get the transformed multivariate normal

M M M ε n simN (000 M M M ΣΣΣ M M M ) (618)

62 Bank examinerslsquo bivariate CAMELS rating function

The analysis in Section section 4 based on Vasicek (2002) model suggests the

existence of a hitherto unobservable bivariate CAMELS rating function U (middot) ie

random utility function for a bank examinerregulator which is monotonic de-creasing in cash flows K and monotonic increasing in exposure to systemic risk

γ In other words we have an unobserved function U (K γ ) where

Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)

However under Vasicek (2002) formulation K is a proxy for bank specific fac-

tor Z and γ is a proxy for systemic factor Z S In the context of our factor pricingLemma 59 the bivariate CAMELS function reveals the following regulatory be-

havior For r = m n substitute

K r sim Z r =sgn(β r )β r σ r

σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)

Since dZ r prop d σ r we have

U σ r =

part U

partσ r sim sgn(β r )β r

σ

T (1minusρr )U K r (623)

According to Equation 619 this implies that for minority banks

U σ m =minusβ m

σ T (1minusρm)U K m gt 0 (624)

That is the regulatorlsquos CAMELS rating is increasing in the risk σ m for minority

bank projects44 Thus behaviorally regulators are biased in favor of giving mi-

nority banks a high CAMELS score by virtue of the negative risk factor price for

NPV projects undertaken by minority banks By contrast under Assumption 61

U σ n =β n

σ

T (1minusρn)U K n lt 0 (625)

44According to (US GAO Report No 07-06 2006 pg 37) bank examiners frown on cash transaction by virtue

of the Bank Secrecy Act which is designed to address money laundering However traditionally in many immigrant

communities there are large volumes of cash transaction A scenario which could cause an examiner to give a lower

CAMELS score

46

832019 SSRN-id1711002


which implies that regulators are biased in favor of giving a low CAMELS score

for nonminority peer banks by virtue of the positive risk factor price for their NPV

projects Continuing with the behavioral analysis we have

U σ S = minusσ radic

T ρr Z S

σ 2SU γ (626)

According to the UFIRS policy statement on ldquosensitivity to market riskrdquo small

minority and nonminority peer banks are primarily exposed to interest rate risk

Heuristically other things equal a decrease in interest rates could increase small

bank capital because they could issue more loans against relatively stable deposits

In which case the Z S market wide variable and hence γ is positive Behaviorally

under Assumption 62 this implies that

U σ S = minusσ radicT ρr Z S

σ 2SU γ lt 0 (627)

That is regulators would tend to decrease CAMELS scores for minority and non-

minority peer banks alike when interest rates are declininng ceteris paribus The

reverse is true when interest rates are rising In this case the extent to which the

CAMELS score would vary depends on the pairwise correlation ρr for the distri-

bution of risk factors driving the bankslsquo ROA

Another independently important ldquosensitivity analysisrdquo stems from the reg-

ulators behavior towards unconditional volatility of loan portfolio value σ in

Equation 42 based on bank specific factors There we need to compute

U σ =part U

partσ =

part U

part K r

part K r

partσ (628)

= minusU K r sgn(β r )β r

σ 2

T (1minusρr )(629)

Once again bank regulator attitude towards σ is affected by their attitudes towards

K r and sgn(β r ) In particular type m banks tend to get higher CAMELS scoresfor their loan portfolios because of their asset pricing anomaly sgn(β m) lt 0

The foregoing results give rise to the following

Lemma 63 (Behavioral CAMELS scores) Let r = mn be an index for minority

and nonminority banks and U (K r γ r ) be a bivariate random utility function for

bank regulators Let σ be the unconditional volatility of loan portfolio value σ r be

47

832019 SSRN-id1711002


the risk in NPV projects undertaken by type r banks β r be the corresponding risk

factor price and γ r be exposure to systemic risk Then bank regulators CAMELS

scores are biased according to

U σ r =part U

partσ r simsgn(β r )β r

σ

T (1minusρr )U K lt 0 (630)

U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U

partσ S=minusσ radicT ρr Z S

σ 2SU γ gt 0 (632)

In particular bank regulators are biased in favor of giving type m banks high

CAMELS scores by virtue of the negative risk factor price for their NPV projects

and biased in favor of giving type n banks low CAMELS scores by virtue of their positive factor price of risk Bank regulators CAMELS scores are unbiased for

type m and type n banks exposure to systemic risk

Remark 61 The random utility formulation implies that there is an idiosyncratic

component to CAMELS scores That is bank regulators commit Type I [or Type

II] errors accordingly

As a practical matter we have

V n j = β β β xn j where (633)

xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)

where j corresponds to the CAMELS score assigned to bank specific factors as

indicated It should be noted that the ldquomanagementrdquo component of CAMELS con-templates risk management in the face of systemic risks Ignoring the n j subscript

we write U instead of U n j to formulate the following

Proposition 64 (Component CAMELS rating of minority banks) Let the sub-

scripts m n pertain to minority and nonminority banks Let σ S be a measure of

market wide or systemic risks K mt (σ m| σ S) and K nt (σ n| σ S) be the cash flows

48

832019 SSRN-id1711002


and σ n σ m be the risks associated with projects undertaken by nonminority and

minority banks respectively Let γ m γ n be the exposure of minority and nonmi-

nority banks to systemic risk Let U (K γ ) be a bivariate CAMELS rating function

Assume that

part U

part K lt 0 and

part U

partγ gt 0

so the marginal densities U |K prop (K )minus1 and U |γ prop γ Assume that the IRR and

managerial ability for minority and nonminority banks are equivalent Let Γ be

the space of market exposures X be the space of cash flows κ (middot) is the probability

density function for state contingent cash flows and φ (

middot) is the probability density

for state contingent market exposure Then we have the following relations

1 The probability limit for minority bank exposure to systemic risk is greater

than that for nonminority banks

Pminus lim ˆγ m Pminus lim ˆγ n (636)

2 As a function of cash flow generated by NPV projects the expected compo-

nent CAMELS rating for minority banks is much higher than the expected

CAMELS rating for nonminority banks

E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)

3 As a function of exposure to systemic risk the expected component CAMELS

rating for minority banks is much higher than that for nonminority banks

E [U |γ (γ m)] = E [

X U (K mγ m)κ (K m)dK m] (639)

E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

Using microfoundations this paper provides a menu of criteria which could be

used for performance evaluation regulation and pricing of minority banks We

provide formulae for project comparison of minority and nonminority peer banksceteris paribus And show how synthetic options on those cash flows are derived

Additionally starting with dynamics for a banklsquos return on assets we introduce a

simple bank failure prediction model and show how it relates to the probit class

That dynamic model also laid the foundation of a two factor asset pricing model

Furthermore we introduced an augmented behavioral mean-variance type asset

pricing model in which a measure for compensating risks for minority banks is

presented Whereupon we find that isomorphy in that asset pricing model is func-

tionally equivalent to a regulatorlsquos put option on minority bank assets And we

show how to determine an optimal strike price for that option In particular despitethe negative beta phenomenon minority bankslsquo portfolios are viable if not inef-

ficient provided that the risk free rate is less than that for the minimum variance

portfolio of the portfolio frontier Finally we provide a brief review of how a reg-

ulatorlsquos seemingly subjective CAMELS score can be predicted by a polytomous

probit model induced by random utility analysis And we use that to show how

bank examinerregulator behavior can be described by a bivariate CAMELS rating

function The spectrum of topics covered by our theory suggests several avenues

of further research on minority banks In particular security design motivated bysynthetic cash flow relationship between minority and nonminority banks optimal

bidding in the pricing of failing minority banks real options pricing in lieu of NPV

analysis of projects undertaken by minority and nonminority peer banks econo-

metric estimation and testing of seemingly anomalous asset pricing relationships

behavioral analytics of bank examiner CAMELS rating and regulation costs asso-

ciated with providing assistance to minority banks all appear to be independently

important grounds for further research outside the scope of this paper

50

832019 SSRN-id1711002


A Appendix of Proofs

A1 PROOFS FOR SUBSECTION 21 DIAGNOSICS FOR IRR FROM NPV PROJECTS

OF MINORITY AND NONMINORITY BANKS

Proof of Lemma 210

Proof Since the summand comprises elements of a telescoping series the inequal-

ity holds if δ gt 1 + IRRm See (Apostol 1967 pg 386) for details on convergence

of oscillating and or telescoping series

Proof of Lemma 216

Proof Let F (d m) be the [continuous] distribution function for the discount rated m For a present value PV (d m) on the Hilbert space L2

d mdefine the k -th period

cash flow by the inner product

K k =lt PV (d m)Pk (d m) gt =

d misin X

PV (d m)dF (d m) (A1)

Orthogonality of Pk implies

lt Pi(d m) P j(d m) gt =

1 if i = j0 otherwise

(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that

lt PV (d m)Pk (d m) gt = 0 (A4)

rArr ltinfin

sumk =0

PV (d m)Pk (d m) gt= 0 (A5)

rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218

Proof Consider the filtered probability space (Ω F F t t ge0 P) in Corollary

212 By an abuse of notation for some ε small and k discrete let F k be a

σ -field set such that the set

Ak = ω | K mt (ω )minusK nt (ω )ge ε isinF k (A7)

P( Ak ) lt 2minusk (A8)

So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)

be the set of ε -convergence In which case we have the set of convergenceinfinq=1 M q and the corresponding set of divergence given by

A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that

P( A) = limk rarrinfin

P( Ak ) = 0 (A12)

Since the set has probability measure 0 only when k rarrinfin it implies that for k ltinfin

there exist n0 such that

P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that

P( Ak ) le P( A)le P(infin

k =n0

Ak ) ltinfin

sumk =n0

2minusk = 2minusn0+1 (A14)

Whereupon

P( Ack ) ge P( Ac)ge 1minus2minusn0+1 (A15)

rArr P(ω | K mt (ω )minusK nt (ω ) lt ε ) ge 1minus2minusn0+1 = 1minusη (A16)

which has the desired form for our proof when η = 2minusn0+1 That is in the context

of our theory there are finitely many times when the cash flow of a minority bank

exceeds that of a nonminority bank But ldquocountablyrdquo many times when the con-

verse is true That is the number of times when nonminority banks cash flows aregreater than that for minority banks is much greater than when it is not

Proof of Corollary 219

Proof Since E [K mt ] le E [K nt ] it follows from (Myers 1977 pg 149) that nonmi-

nority banks pass up positive cash flow projects with low return in favor of those

with higher returns in order to maximize shareholder wealth To see this assume

that the random variables

K mt = K nt +ηt (A17)

where ηt has a truncated normal probability density That is

φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt

+σ 2ηt + 2σ ηt K nt

(A19)

53

832019 SSRN-id1711002


where σ 2(middot) is variance and σ ηt K nt the covariance term is negative for any positive

cash flow stream K nt We have two scenarios for

σ ηt K nt lt 0 (A20)

i If σ ηt K nt is sufficiently large ie 2|σ ηt K nt

| σ 2ηt then σ 2K mt

lt σ 2K nt That is

the risk in the cash flows for minority banks is lower than that for nonminor-

ity banks In that case nonminority banks have a higher expected cash flow

stream for more risky projects So that [risk averse] nonminority banks are

compensated for the additional risk by virtue of higher cash flows For exam-

ple under Basel II they may be hold more mortgage backed securities (MBS)

or other exotic options in their portfolios In which case minority banks

are more heavily invested in comparatively safer short term projects The

higher expected cash flows for nonminority banks implies that their invest-ment projects [first order] stochastically dominates that of minority banks In

either case minority bank projects are second best

ii If σ 2K mt gt σ 2K nt

then the risk in minority bank projects is greater than that for

nonminority banks In this case the risk-return tradeoff implies that minority

banks are not being compensated for the more risky projects they undertake

In other words they invest in risky projects that nonminority banks eschew

Thus they invest in second best projects

Thus by virtue of the uniform dominance result in Lemma 217 it follows that

the expected cash flows for minority banks E [K mt ] must at least be in the class

of second best projects identified by nonminority banks-otherwise the inequality

fails

A2 PROOFS FOR SUBSECTION 31 MINORITY BANK ZERO COVARIANCE FRON-

TIER PORTFOLIOS

Proof of Proposition 33

Proof Following (Huang and Litzenberger 1988 Chapter 3) let wwwr r = mn be

the vector of portfolio weights for minority (m) and nonminority (n) banks frontier

54

832019 SSRN-id1711002


portfolios So that for some vectors ggg and hhh we have

wwwm = ggg +hhhE [ IRRm] (A21)

wwwn = ggg +hhhE [ IRRn] (A22)

According to the two fund separation theorem there existβ and a portfolio ( E [ IRRq]σ q]such that upon substitution for E [ IRRm] and E [ IRRn] we get

E [ IRRq] = minusβ E [ IRRm] + (1 +β ) E [ IRRn] (A23)

= ggg +hhh[(1 +β ) E [ IRRn]minusβ E [ IRRm]](A24)

From Lemma 32 write

IRRr = α r 0 +β r σ r + γ r σ S +ε r r = mn

(A25)

And let

ε lowastm = γ mσ S + ε m (A26)

ε lowastn = γ nσ S +ε n (A27)

Without loss of generality assume that

cov(ε mε n) = 0 (A28)

To get the zero covariance portfolio relationship between minority and nonminor-

ity banks we must have

cov( IRRm IRRn) = E [( IRRmminus E [ IRRm])( IRRnminus E [ IRRn])](A29)

= E [ε lowastmε lowastn ] (A30)= γ mγ nσ

2S + γ mcov(σ Sε n) + γ ncov(σ Sε m) + cov(ε m

(A31)

= 0 (A32)

55

832019 SSRN-id1711002


Divide both sides of the equation by γ mγ nσ 2S to get

1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)

Since each of γ m and γ n prices the bankslsquo exposure to economy wide systemic risk

we can assume concordance in this regard So that

γ n gt 0 and γ m gt 0 (A35)

For Equation A34 to hold we must have

either β Sm lt 0 or β Sn lt 0 or both (A36)

But by hypothesis in Lemma 32 β m lt 0 Therefore it follows that for internal

consistency we must have

β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)

to complete the proof

Remark A1 Each of the 4-parameters β Sm β Sn γ m γ n can have +ve or -ve signs

So we have a total of 24 = 16 possibilities The zero-covariance relation in Equa-

tion A34 rules out the case when the signs of all the parameters are -ve or +ve

Thus we are left with 14-possible scenarios that satisfy Equation A34 In fact

Meinster and Elyasini (1996) found that minority and nonminority banks hold in-

56

832019 SSRN-id1711002


efficient portfolios In which case its possible that β Sm and β Sn each have negative

signs If so then either γ m lt 0 or γ n lt 0 Thus our assumption in Equation A35

can be weakened accordingly without changing the result

A3 PROOFS FOR SECTION 4 PROBABILITY OF BANK FAILURE WITH VA-SICEK (2002) TWO-FACTOR MODEL

Proof of Lemma 41

Proof Rewrite

Φminus1( p)minus( X minus Z radic

1minusρ)radic1minusρ as (A41)

β 0 + X β 1 + Z β 2 where (A42)β 0 =

Φminus1( p)Φ(1minus p) β 1 = minus1

Φ(1minus p) and β 2 = 1 (A43)

In matrix form this is written as X X X β β β where 1 X Z are the column vectors in the

matrix and the prime signifies matrix transposition so

X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)

is a column vector In which case

p( Z | X ) = Φ( X X X β β β ) (A46)

which is a conditional probit specification

A4 PROOFS FOR SECTION 5 MINORITY AND NONMINORITY BANKS ROAFACTOR MIMICKING PORTFOLIOS

PROOFS OF LEMMA 511

57

832019 SSRN-id1711002


Proof Assuming that σ K = σ (Cadogan 2009 Prop IIB1) showed how Arrow-

Pratt risk measure r ( x) can be embedded in a Markov process X = X (t ω )F t 0let ltinfin by virtue of the infinitesimal generator

L =σ 2

2

part 2

part x2 + r ( x)part

part x (A47)

which for Brownian motion B(t ω ) defined on F t produces

dX (t ω ) = r ( X (t ω ))dt +σ dB(t ω ) (A48)

In our case we assume that risky cash flow K = X follows a Markov process

K = K (t ω )F t 0 le t ltinfin Cadogan considered a simple dynamical system

K (t ω ) = t

0eminus Rs H (sω )ds (A49)

dK (t ω ) = C (t )dU (t ω ) (A50)

dK (t ω ) = r (K )dt +σ (K )dB(t ω ) (A51)

(A52)

where K (t ω ) is the discounted cash flow H (t ω )F t 0 le t lt infin and U (t ω )and B(t ω ) are Brownian motions After some simplifying assumptions motivated

by Kalman-Bucy filter theory and (Oslashksendal 2003 pp 99-100) he derived the

solution

r (K (t ω )) =σ

cosh(σ ct )

t

0sinh(σ cs)dK (sω ) (A53)

=σ

cosh(σ ct )

t

0sinh(σ cs)eminus Rs H (sω )ds (A54)

Assuming that minority and nonminority banks risky cash flows follows a Markov

process described by the simple dynamical system then under the proviso that

R = IRR other things equal ie σ K = σ and H (sω ) is the same for each typeof bank substitution of IRRn = IRRm + δ in the solutions for r (K m(t ω ) and

comparison with the solution for r (K n(t ω )) leads to the result r (K m(t ω )) gtr (K n(t ω )) The proof follows by application of (Gikhman and Skorokhod 1969

Thm 2 pg 467) for dyadic partition t (n)k

k = 01 2n in C ([01]) and Roth-

schild and Stiglitz (1970) deformation of finite dimensional distribution functions

58

832019 SSRN-id1711002


for r (K r (t (n)n1 ω )) r (K r (t

(n)nk

ω )) nk isin 0 2n to shift probability weight

to the tails

59

832019 SSRN-id1711002


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Alexis M (1971 May) Wealth Accumulation of Black and White Families The

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Apostol T M (1967) Calculus One-Variable Calculus with an Introduction to

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Artzner P F Delbaen J M Eber and D Heath (1999) Coherent measures of

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Barclay M and C W Smith (1995 June) The Maturity Structure of Corporate

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Barnes M L and J A Lopez (2006) Alternative Measures of the Federal Reserve

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Bates T and W Bradford (1980) An Analysis of the Portfolio Behavior of Black-

Owned Commercial Banks Journal of Finance 35(3)753-768 35(3) 753ndash768

Black F (1972 July) Capital Market Equilibrium with Restricted Borrowing

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Black F and M Scholes (1973) The Pricing of Options and Corporate Liabilities

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Brealey R A and S C Myers (2003) Principles of Corporate Finance (7th ed)New York N Y McGraw-Hill Inc

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Cadogan G (2009 December) On Behavioral Arrow-Pratt Risk Process With

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Cadogan G (2010a October) Bank Run Exposure in A Paycheck to

Paycheck Economy With Liquidity Preference and Loss Aversion To De-

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Cadogan G (2010b June) Canonical Representation Of Option Prices and

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Cole J A (2010b October) Theoretical Exploration On Building The Capacity

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Cole R and J W Gunther (1998) Predicting Bank Failures A Comparison of On

and Off Site Monitoring Systems Journal of Financial Services Research 13(2)

103ndash117

Dahl D (1996 Aug) Ownership Changes and Lending at Minority Banks A

Note Journal of banking and Finance 20(7) 1289ndash1301

Damodaran A (2010 Sept) Risk Management A Corporate Governance Man-

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De Marzo P and D Duffie (1999 Jan) A Liquidity-Based Model of Security

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Variability Journal of Finance 25(4) 869ndash879

Diamond D and R Rajan (2000) A Theory of Bank Capital Journal of Fi-nance 55(6) 2431ndash2465

Diamond P A and J E Stiglitz (1974 July) Increases in Risk and in Risk

Aversion Journal of Economic Theory 8(3) 337ndash360

Dixit A V and R Pindyck (1994) Investment under Uncertainty Princeton N

J Princeton University Press

Duffie D and R Rahi (1995) Financial Market Innovation and Security Design

An Introduction Journal of Economic Theory 65 1ndash45

Elton E M J Gruber S J Brown and W N Goetzman (2009) Modern

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Wiley amp Sons Inc

Fishburn P (1977 March) Mean Risk Analysis with Risk Associated with

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Flood M D (1991) An Introduction To Complete Markets Federal Reserve

Bank-St Louis Review 73(2) 32ndash57

Friedman M and L J Savage (1948 Aug) The Utility Analysis of Choice In-

volving Risk Journal of Political Economy 56 (4) 279ndash304

Gikhman I I and A V Skorokhod (1969) Introduction to The Theory of Random

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832019 SSRN-id1711002


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Greene W H (2003) Econometric Analysis (5th ed) Upper Saddle Rd NJ

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Hasan I and W C Hunter (1996) Management Efficiency in Minority and

Women Owned Banks Economic Perspectives 20(2) 20ndash28

Hays F H and S A De Lurgio (2003 Oct) Minority Banks A Search For

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Henderson C (2002 May) Asymmetric Information in Community Bankingand Its Relationship to Credit Market Discrimination American Economic Re-

view 92(2) 315ndash319 AEA Papers and Proceedings session on Homeownership

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Henderson C C (1999 May) The Economic Performance of African-American

Owned Banks The Role Of Loan Loss Provisions American Economic Re-

view 89(2) 372ndash376

Henriksson R D and R C Merton (1981 Oct) On Market Timing and Invest-ment Performance II Statistical Procedures for Evaluating Forecasting Skills


Hewitt R and K Stromberg (1965) Real and Abstract Analysis Volume 25

of Graduate Text in Mathematics New York N Y Springer-Verlag Third

Printing June 1975

Huang C and R H Litzenberger (1988) Foundations of Financialo Economics

Englewood Cliffs N J Prentice-Hall Inc

Hylton K N (1999) Banks And Inner Cities Market and Reg-

ulatory Obstacles To Development Lending Boston Univer-

sity School Of Law Working Paper No 99-15 Available at

httpswwwbuedulawfacultyscholarshipworkingpapersabstracts1999pdf˙filesh

63

832019 SSRN-id1711002


Iqbal Z K V Ramaswamy and A Akhigbe (1999 Jan) The Output Efficiency

of Minority Owned Banks in the US International Review of Economics and

Finance 8(1) 105ndash114

Jensen M C and H Meckling (1976) Theory of the Firm Managerial Behavior

Agency Costs and Ownership Structure Journal of Financial Economics 3

305ndash360

Johnson H (1994 Spring) Prospect Theory in The Commercial Banking Indus-

try Journal of Financial and Strategic Decisions 7 (1) 73ndash88

Karatzas I and S E Shreve (1991) Brownian Motion and Stochastic Calculus

(2nd ed) Graduate Text in Mathematics New York N Y Springer-Verlag

Khaneman D and A Tversky (1979) Prospect Theory An Analysis of DecisionsUnder Risk Econometrica 47 (2) 263ndash291

King M R (2009 Sept) The Cost of Equity for Global Banks A CAPM

Perspective from 1990 to 2009 BIS Quarterly Available at SSRN


Kmenta J (1986) Elements of Econometrics (2nd ed) New York NY Macmil-

lan Publishing Co

Kuehner-Herbert K (2007 Sept) High growth low returnsfound at minority banks American Banker Available at

httpwwwminoritybankcomcirm29html

Kupiec P H (2009) How Well Does the Vasicek-Basel Airb Model Fit

the Data Evidence from a Long Time Series of Corporate Credit Rat-

ing Data SSRN eLibrary FDIC Working Paper Series Available at

httpssrncompaper=1523246

Kwan S (2009 Dec) Capital Structure in Banking Economic Letter 37 1ndash4

Federal Reserve BankndashSan Francisco

Laffont J-J and D Matrimint (2002) The Theory of Incentives The Principal

Agent Model Princeton N J Princeton University Press

Lawrence E C (1997 Spring) The Viability of Minority Owned Banks Quar-

terly Review of Economics and Finance 37 (1) 1ndash21

64

832019 SSRN-id1711002


LeRoy S F and J Werner (2000) Principles of Financial Economics New York

N Y Cambridge Univ Press

Lopez J A (2004 April) The Empirical Relationship Between Average Asset

Correlation Firm Probability of Default and Asset Size Journal of Financial

Intermediation 13(2) 265ndash283

Lucas R E (1983) Econometric Policy Evaluation A Critique In K Brunner

and A Metzer (Eds) Theory Policy Institutions Papers from the Carnegie-

Rochester Series on Public Policy North-Holland pp 257ndash284 Elsevier Sci-

ence Publishers B V

Markowitz H (1952a Mar) Portfolio Selection Journal of Finance 7 (1) 77ndash91

Markowitz H (1952b April) The Utility of Wealth Journal of Political Econ-omy 40(2) 151ndash158

Marschak J (1959) Binary Choice Constraints on Random Utility Indi-

cations In K Arrow (Ed) Stanford Symposium on Mathematical Meth-

ods in the Social Sciences pp 312ndash329 Stanford CA Stanford Uni-

versity Press Cowles Foundation Research Paper 155 Available at

httpcowleseconyaleeduPcpp01bp0155pdf

Martinez-Miera D and R Repullo (2008) Does Competition Reduce The

Risk Of Bank Failure CEMFI Working Paper No 0801 Available atftpftpcemfieswp080801pdf published in Review of Financial Studies

(2010) 23 (10) 3638-3664

Meinster D R and E Elyasini (1996 June) The Performance of Foreign Owned

Minority Owned and Holding Company Owned Banks in the US Journal of

Banking and Finance 12(2) 293ndash313

Merton R (1981 July) On Market Timing and Investment Performance I An

Equilibrium Theory of Value for Market Forecasts Journal of Business 54(3)363ndash406

Merton R C (1977) An Analytic Derivation of the Cost of Deposit Insurance

And Loan Guarantees An Application of Modern Option Pricing Theory Jour-

nal of Financial Economics 1 3ndash11

65

832019 SSRN-id1711002


Merton R C (1992) Continuous Time Finance (Revised ed) Cambridge MA

Blackwell Publishing Co

Modigliani F and M H Miller (1958) The Cost of Capital Corporation Finance

and the Theory of Investment American Economic Review 48(3) 261ndash297

Myers S C (1977) Determinant of Corporate Borrowing Journal of Financial

Economics 5(2) 147ndash175

Nakamura L I (1994 NovDec) Small Borrowers and the Survival of the Small

Bank Is Mouse Bank Mighty or Mickey Business Review Federal Reserve

Bank of Phildelphia

Oslashksendal B (2003) Stochastic Differential Equations An Introduction With

Applications (6th ed) New York N Y Springer-VerlagRajan U A Seru and V Vig (2010) The Failure of Models that Predict Failure

Distance Incentives and Defaults Chicago GSB Research Paper No 08-19

EFA 2009 Bergen Meetings Paper Ross School of Business Paper No 1122


Reuben L J (1981) The Maturity Structure of Corporate Debt Ph D thesis

School of Business University of Michigan Ann Arbor MI Unpublished

Robinson B B (2010 November 22) Bank Creation A Key toBlack Economic Development The Atlanta Post Available at

httpatlantapostcom20101122bank-creation-a-key-to-black-economic-

development

Ross S A R W Westerfield and B D Jordan (2008) Essentials of Corporate

Finance (6th ed) McGraw-HillIrwin Series in Finance Insurance and Real

Estate Boston MA McGraw-HillIrwin

Rothschild M and J E Stiglitz (1970) Increasing Risk I A Definition Journal

of Economic Theory 2 225ndash243

Ruback R S (2002 Summer) Capital Cash Flows A Simple Appproach to

Valuing Risky Cash Flows Financial Management 31(2) 85ndash103

66

832019 SSRN-id1711002


Schwenkenberg J (2009 Oct) The Black-White Income Mobil-

ity Gap and Investment in Childrenrsquos Human Capital Working Pa-

per Department of Economics Rutgers University-Newark Available at

httpjuliaschwenkenbergcommobility˙jschwenkenberg˙oct09pdf

Stiglitz J E and A Weiss (1981 June) Credit Rationing in Markets with Imper-

fect Information American Economic Review 71(3) 393ndash400

Tavakoli J M (2003) Collateralized Debt Obligation and Structured Finance

New Developments in Cash and Synthetic Securitization Wiley Finance Series

New York N Y John Wiley amp Sons Inc

Tobin J (1958) Liquidity Preference as Behavior Towards Risk Review of Eco-

nomic Studies 25(2) 65ndash86

Train K E (2002) Discrete Choice Methods with Simulation New York N Y

Cambridge University Press

Treynor J and K Mazuy (1966) Can Mutual Funds Outguess The Market Har-

vard Business Review 44 131ndash136

Tversky A and D Khaneman (1992) Advances in Prospect Theory Cumulative

Representation of Uncertainty Journal of Risk and Uncertainty 5 297ndash323

U S GAO Report No 08-233T (2007 October) MINORITY BANKS Reg-ulatorslsquo Assessment of the Effectiveness of Their Support Efforts Have Been

Limited Technical Report GAO 08-233T United States General Accountability

Office Washington D C Testimony Before the Subcommittee on Oversight

and Investigations Committe on Financial Services U S House of Represen-

tatives

US GAO Report No 07-06 (2006 October) MINORITY BANKS Regulatorslsquo

Need To Better Assess Effectiveness Of Support Efforts Technical Report GAO

07-06 United States General Accountability Office Washington D C Reportto Congressional Requesters

Varian H (1987 Autumn) The Arbitrage Principle in Financial Econommics

Journal of Economic Perspectives 1(2) 55ndash72

Vasicek O (1977) An Equilibrium Characterization of The Term Structure Jour-


67

832019 SSRN-id1711002


Vasicek O (2002) The Distribution Of Loan Portfolio Value mimeo published

in Risk Magazine December 2002

Walter J R (1991 JulyAugust) Loan Loss Reserves Economic Review 20ndash30

Federal Reserve Bank-Richmond

Williams W E (1974 AprilJuly) Some Hard Questions on Minority Businesses

Negro Educational Review 123ndash142

Williams-Stanton S (1998 July) The Underinvestment Problem and Patterns in

Bank Lending Journal of Financial Intermediation 7 (3) 293ndash326

Wu L and P P Carr (2006 Sept) Stock Options and Credit Default Swaps

A Joint Framework for Valuation and Estimation Working Paper CUNY

Zicklin School of Business Baruch College Available at SSRN eLibraryhttpssrncompaper=748005 Published in Journal of Financial Econometrics

8(4)409-449

832019 SSRN-id1711002


Abstract

This paper introduces a comprehensive theory of performance evaluation and asset pricing for

minority (type-m) banks incuding but not limited to CAMELS rating bank failure prediction

compensating risks and risk adjusted internal rates of returns in a complete market Our the-ory predicts and explains several empirical regularities of minority banks First we provide a

novel mechanism for generating a synthetic cash flow stream in Hilbert space by comparing NPV

projects selected by minority and nonminority (type-n) peer banks Thus laying a foundation for

security design for sound type-m banks Second even when internal rates of return and managerial

ability are equal for type-m and type-n banks the expected cash flows for type-n banks uniformly

dominate that for type-m banks by a factor greater than 2 Third a behavioral mean-variance

analysis of liquidity preference explains type-m bankslsquo seeming zero covariance frontier portfolio

and asset pricing anomalies like concavity of their internal rates of return in project risk Such

anomalies affect computation of unlevered betas imply undercapitalization and inefficiency that

crowds out positive NPV projects and suggest that type-m banks overinvest in residual [second

best] projects eschewed by type-n banks See eg Ruback (2002) Jensen and Meckling (1976)

Myers (1977) and Williams-Stanton (1998) Even so the returns on minority bankslsquo portfolios are

viable provided that the risk free rate is less than that for the minimum variance [frontier] portfo-

lio Fourth we introduce a compensating risk factor for type-m banksndashpriced by a two-factor asset

pricing model induced by Vasicek (2002) for loan portfolio value ie return on assets (ROA) Our

theory predicts that that compensating transfer scheme is tantamount to a put option on type-m

bank assets in accord with Merton (1977) And we use Deelstra et al (2010) model to identify the

optimal strike price for that option According to Merton (1992) and Glazer and Kondo (2010)

to mitigate moral hazard in such schemes either (1) type-m banks must bear the associated costs

or (2) the costs must not be bourne by other banks Arguably our put option prices type-m banks

under FIRREA sect 308 (1989) Fifth on the fixed income side Vasicek (1977) interest rate modelpredicts that the term structure of bonds issued by type-m banks is a floor for the term structure of

bonds issued by type-n peer banks Moreover if risky cash flows follow a Markov process then

under Rothschild and Stiglitz (1970) type mean preserving spread analysis type-m bank bonds are

riskier provide lower yield and need to be issued over shorter durationndashaccording to Barclay and

Smith (1995) empirical findings on asymmetric information and the ldquocontracting-cost hypothesisrdquo

Sixth we extend Vasicek (2002) to an independently important conditional probit representation

for bank failure predictionndashmore granular than Cole and Gunther (1998) ad hoc probit model Sev-

enth we establish a nexus between random utility analysis the conditional probit and a simple

bivariate CAMELS rating function that (1) predict bank examiners attitudes towards type-m banks

and (2) identifies root causes of their bias towards high CAMELS scores for type-m banks In par-

ticular we show how bank examinerslsquo CAMELS scores for return on assets risk is predicted bybank specific variables

Keywords minority banks synthetic cash flows internal rates of return behavioral asset pric-

ing CAMELS rating random utility options pricing bank regulation return on assets

JEL Classification Code D03 D81 G11 G12 G21 G28 G31 G32 M48

832019 SSRN-id1711002


Contents

1 Introduction 2

2 The Model 7


nority Banks 8

















7 Conclusion 49







1

832019 SSRN-id1711002




References 59

2

832019 SSRN-id1711002


1 Introduction
































3

832019 SSRN-id1711002







































4

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


Contents

1 Introduction 2

2 The Model 7


nority Banks 8

















7 Conclusion 49







1

832019 SSRN-id1711002




References 59

2

832019 SSRN-id1711002


1 Introduction
































3

832019 SSRN-id1711002







































4

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




References 59

2

832019 SSRN-id1711002


1 Introduction
































3

832019 SSRN-id1711002







































4

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


1 Introduction
































3

832019 SSRN-id1711002







































4

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002







































4

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002








in Henderson (2002)





















5

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




























Reserve Banks









6

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





































7

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002










2 The Model




(21)














8

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




cash flows

Proof Let P(


isin 12345

be a





rating











of generality


Banks



NPV m = NPV n = 0 (25)



9

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






probability space19


Definition 21


E [K r t (ω )] =

E


Remark 22







middot]









bank is different



10

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




parisonT

minus1


T

minus1
























11

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






T minus1

sumt =0


sumt =0


=T minus1

sumt =0

E [K nt ]infin

sumq=0

(minus1)q

t + qminus1

q


=

T

minus1

sumt =0

E [K nt ]

infin


uminus

1

uminus t




T minus1

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


(213)



infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm












12

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002












infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0








u

minust )(1+ IRRm

δ )u

be the




E

max


T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u

0

F t

(215)




13

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



T

sumu=t

(minus1)uminust

uminus1

uminus t

1+ IRRm

δ

u




























14

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002







infin

sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm

δ )u

infint =1

(216)

is complete in X



that


sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u χ t gt0 = 0

(217)



let







als




15

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






infin

sumt =0


sumu=t

(minus1)uminust

uminus1

uminus t

(1+ IRRm


infin

sumk =0

E [K mk ]minus

E [K nk ]infin

sumu=k

(minus

1)uminusk uminus1

uminus k (1+ IRRm

δ

)u

Pk (d m) = 0

(221)




sumu=k

(minus1)uminusk

uminus1

uminus k

(1+ IRRm

δ )u (222)

Then

infin

sumk =0

K k Pk (d m) = 0 (223)

If and only if




this implies that


sumu=k

(minus1)uminusk

uminus1

uminusk

(1+ IRRm

δ )u

= 0 (225)

16

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




1 + IRRm

δ lt 1 (226)

Thus

δ gt 1 + IRRm (227)






factor

DF t =infin

sumu=t

(minus1)(uminust )

uminus1

uminus t

(1+ IRRm

δ )u) (229)




Whereupon

E [K mt ] le E [K nt ] (231)











17

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




following








DF t =infin

sumu=t

(

minus1)uminust

uminus1

uminus

t (1+ IRRm

δ )u)

le1





ing series


sumu=t


(1+ IRRm

δ )u = 0 (232)










18

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




















E [ f (K mt )] le E [ f (K nt )]


t ] le E [K n



equality

19

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



nority banks







dr (t ω ) = α (r







α + cT φ 2(T α ) (32)


minuseminusα T )





deterministic



2(T n)] (33)

where


φ (T n)(34)

with the proviso


20

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


and

limT rarrinfin

φ (T ) = 0 (36)







φ (T α )

α + cT φ 2

(T α )



and c = σ 2





deterministic




(T nα )]









21

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





1minusθ (T nT m)







this result in







φ (T n)(37)


a0 =1

φ (T n)(39)

22

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



φ (T mα ) gt1

a0(310)

minusβ m = φ

2


(311)

=φ 2(T mα )


σ m = cT m (313)

σ n = cT n (314)

γ m =ϑ (T m)

1minusϑ (T m)

1

a20

(315)

α m = R(infin) + ε m (316)


















23

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






model




dA(t ω )





2σ 2T +σ

radicT X (42)


X = Z Sradicρ + Z

1minusρ (43)










24

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




2 σ 2T

σ radic

T (45)



L =

1 if bank fails


E [ L| Z S] = p( Z S) = Φ


radicρradic

1minusρ

(47)



p( Z | X ) = Φ


radic1minusρradic

1minusρ

(48)


variable


p( Z | X )φ ( X )dX (49)



models




opined




25

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002













Portfolios

















26

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




IRR = WACC +ϕ (51)


V = D + E (52)

WACC = DV

r debt +E V

r equity (53)











β U m =

DV β Dm + E

V β E m

(55)







φ geminus

DV β Dm + E

V β E m



27

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





|β E m|gt V

E

R f

˜ R p

(58)




|β Dm|gtV

D

R f

˜ R p (510)





equity ratio D E



|β

E m||β Dm| propD

E (511)



|β E m| =V

E

R f

˜ R p

+ b E (512)

|β Dm| = V D

R f ˜ R p

+ b D (513)




minority banks

28

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





D E (514)



(515)


β U m = |

β Dm

|[c0 + (1

minusτ )]

[1 + (1minus τ ) D E ]

D

E (516)












following





risk is given by


29

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




whenever λ gt λ c

























30

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





















risk exposure



















31

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




tradeoffs38


(519)











γ r σ S = σ

T ρr Z S (522)








paradox Let







32

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002








where

α m0 = α n0 (528)

Thus of necessity


and consequently













33

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





σ

2

ε lowastm σ

2

ε lowastn




by




that

r (K (t ω )) =σ K

cosh(σ K ct )

t






r (K m(t ω )) gt r (K n(t ω )) (533)



flows) Let K =

K r (t ω )F t 0

let ltinfin








r (K n(t (n)1 ω )) K n(t


34

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002










∆gt 0 (534)



















35

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


In that case





That is








following










36

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002













following






term






37

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




















dW (t ω )





38

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


where

x1 =log

Bε ∆V ε lowastm


2 )T

σ V ε lowastm

radicT

(543)


T (544)














minhK




et al (1999)

39

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



relation


part K = 0 (547)




purposes























40

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002














γ mcov(σ Sε ∆)





risk exposure


σ 2S(549)


σ 2S(550)

ˆγ m ˆγ n (551)








41

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002













Let



















42

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002

























43

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





Pni =

Pr(U

ni gtU

n j)(61)





Pni =






U n j = β β β xn j + ε n j (67)




Let



44

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



form



M M M =

minus1 1 0 0 0

0 minus1 1 0 0

0 0 minus1 1 0

0 0 0 minus1 1

(613)


U n = V n + ε n (614)


lator



= Pr( M M MU n lt 0) (616)









Assumption 61

U K =part U

part K lt 0 (619)

45

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


Assumption 62

U γ =part U

partγ gt 0 (620)





σ

T (1minusρr )(621)

γ r sim σ radic

T ρr Z S

σ S(622)


U σ r =

part U


σ



U σ m =minusβ m






U σ n =β n

σ





CAMELS score

46

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






T ρr Z S

σ 2SU γ (626)








σ 2SU γ lt 0 (627)









U σ =part U

partσ =

part U

part K r

part K r

partσ (628)


σ 2

T (1minusρr )(629)







47

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





U σ r =part U


σ


U σ =part U

partσ = minusU K r

sgn(β r )β r

σ 2

T (1minusρr )(631)

U σ S ==part U


σ 2SU γ gt 0 (632)










xn j = (K γ ) β β β = (β 0 β 1) (634)

U n j = β β β xn j + ε n j (635)







48

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





Assume that

part U

part K lt 0 and

part U

partγ gt 0













E [U |K (K m)] = E [

Γ

U (K mγ m)φ (γ m)d γ m] (637)

E [U |K (K n)] = E [

Γ

U (K nγ n)φ (γ n)d γ n] (638)



E [U |γ (γ m)] = E [


E [U |γ (γ n)] = E [

X

U (K nγ n)κ (K n)dK n] (640)

49

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


7 Conclusion

























50

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002





Proof of Lemma 210




Proof of Lemma 216





d misin X





(A2)

Moreover

PV (d m) =infin

sumk =0

K k Pk (d m) (A3)

So that


rArr ltinfin

sumk =0


rArr K k = 0 (A6)

51

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


Prooof of Lemma 218






So that

limk rarrinfin

P( Ak ) = 0 (A9)

Let

M q = Ωinfin

k =q

Ak (A10)


A = Ω

infinq=1

M q =

infin p=1

infink = p

Ak (A11)

So that


P( Ak ) = 0 (A12)



P( A) = P(infin

p=n0

infink = p

Ak ) (A13)

52

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


So that


k =n0

Ak ) ltinfin

sumk =n0


Whereupon














φ (ηt ) =

2radicπ

eminusη2

t

2 ηt le 0

0 ηt gt 0(A18)

Furthermore

σ 2K mt = σ 2K nt


(A19)

53

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002







lt σ 2K nt That is



















fails


TIER PORTFOLIOS




54

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002








From Lemma 32 write


(A25)

And let




cov(ε mε n) = 0 (A28)






(A31)

= 0 (A32)

55

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



1

γ m

cov(σ Sε m)

σ 2S+

1

γ n

cov(σ Sε n)

σ 2S+ 1 = 0 so that (A33)

β Sm

γ m+ β Sn

γ n+ 1 = 0 (A34)








β Sm lt 0 (A37)

β Sn gt 0 (A38)

That isβ Sn

γ n+ 1 =minusβ Sm

γ m(A39)

In Equation A23 let

β Sm

γ m=minusβ (A40)







56

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






Proof of Lemma 41

Proof Rewrite



β 0 + X β 1 + Z β 2 where (A42)β 0 =





X X X = [1 X Z ] (A44)

And

β β β = (β 0 β 1 β 2) (A45)


p( Z | X ) = Φ( X X X β β β ) (A46)



PROOFS OF LEMMA 511

57

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




L =σ 2

2

part 2


part x (A47)





K (t ω ) = t


dK (t ω ) = C (t )dU (t ω ) (A50)


(A52)



solution

r (K (t ω )) =σ

cosh(σ ct )

t


=σ

cosh(σ ct )

t







k = 01 2n in C ([01]) and Roth-


58

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002



(n)nk


to the tails

59

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002


References


























60

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002























103ndash117








61

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002




767ndash800


Hill Inc















Wiley amp Sons Inc









62

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002






Prentice-Hall Inc









ties








Printing June 1975







63

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002







305ndash360










lan Publishing Co













64

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002










ence Publishers B V










(2010) 23 (10) 3638-3664









65

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002










Bank of Phildelphia










development








66

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002























tatives








67

832019 SSRN-id1711002













8(4)409-449

832019 SSRN-id1711002













8(4)409-449

ssrn-id1711002

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