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Journal of Financial Economics

Journal of Financial Economics 99 (2011) 333–348

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Diversification disasters$

Rustam Ibragimov a,1, Dwight Jaffee b,2, Johan Walden b,�

a Department of Economics, Harvard University, Littauer Center, 1875 Cambridge St., Cambridge, MA 02138, USAb Haas School of Business, University of California at Berkeley, 545 Student Services Building #1900, CA 94720-1900, USA

a r t i c l e i n f o

Article history:

Received 20 November 2009

Received in revised form

8 February 2010

Accepted 10 March 2010Available online 17 September 2010

JEL classification:

G20

G21

G28

Keywords:

Financial crisis

Financial institutions

Systemic risk

Limits of diversification

5X/$ - see front matter & 2010 Elsevier B.V.

016/j.jfineco.2010.08.015

thank Greg Duffee, Paul Embrechts, Vito

participants at the Wharton Symposium on t

obability Events in the context of Financial

6–17, 2009, the Second Annual NHH Symp

Bergen Norway, May 16th 2009, the 2010 F

, Insurance and Intermediation, and at the 20

Finance Association. Ibragimov and Walden

ment Institute for support. Ibragimov also g

artial support provided by the National Scienc

0124 and a Harvard Academy Junior Faculty D

responding author. Tel.: +1 510 643 0547; fax

ail addresses: [email protected] (R. Ib

aas.berkeley.edu (D. Jaffee), [email protected]

den).

l.: +1 203 887 1631; fax: +1 617 495 7730.

l.: +1 510 642 1273; fax: +1 510 643 7441.

a b s t r a c t

The recent financial crisis has revealed significant externalities and systemic risks that

arise from the interconnectedness of financial intermediaries’ risk portfolios. We

develop a model in which the negative externality arises because intermediaries’

actions to diversify that are optimal for individual intermediaries may prove to be

suboptimal for society. We show that the externality depends critically on the

distributional properties of the risks. The optimal social outcome involves less risk-

sharing, but also a lower probability for massive collapses of intermediaries. We derive

the exact conditions under which risk-sharing restrictions create a socially preferable

outcome. Our analysis has implications for regulation of financial institutions and risk

management.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

It is a common view that the interdependence offinancial institutions, generated by new derivative pro-ducts, had a large role in the recent financial meltdown. Aprimary mechanism is that by entering into sophisticated

All rights reserved.

Gala, Todd Keister,

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Risk Management,

osium on Extreme

IRS Conference on

10 meetings of the

thank the NUS Risk

ratefully acknowl-

e Foundation grant

evelopment grant.

: +1 510 643 1420.

ragimov),

rkeley.edu

derivative contracts, e.g., credit default swaps (CDS) andcollateralized debt obligations (CDO), financial institu-tions became so heavily exposed to each others’ risks thatwhen a shock eventually hit, it immediately spreadthrough the whole system, bringing down critical partsof the financial sector. Since it created this systemicfailure, the interdependence was extremely costly from asocial standpoint.1

There is an opposite viewpoint, however, namely thatsuch interdependence plays a much more functionalrole—that of diversification. To motivate this in a simpleway, start with a case in which each financial firm holds aparticular risk class with its unique idiosyncratic risk.Now allow the firms to form a joint mutual marketportfolio, with each firm contributing its risky portfolio to

1 This view is, e.g., expressed in Warren Buffet’s 2009 letter to the

shareholders of Berkshire Hathaway in which Mr. Buffet argues that the

result is that ‘‘a frightening web of mutual dependence develops among

huge financial institutions.’’ A similar view is expressed in the academic

literature in Jaffee (2009).

www.elsevier.com/locate/jfecdx.doi.org/10.1016/j.jfineco.2010.08.015mailto:[email protected]:[email protected]:[email protected]:[email protected]/10.1016/j.jfineco.2010.08.015

4 See Mandelbrot (1997), Fama (1965), Ross (1976), Ibragimov and

Walden (2007), Ibragimov (2009b), the review in Ibragimov (2009a), and

references therein.5 As discussed in, e.g., Lux (1998), Guillaume, Dacorogna, Davé,

R. Ibragimov et al. / Journal of Financial Economics 99 (2011) 333–348334

the total and receiving back its proportional share of thetotal. Given a sufficient variety and number of risk classes,the firms may succeed in eliminating the idiosyncraticrisks embedded in their individual portfolios. This is inline with the classical view in finance, that risk-sharing—i.e., diversification—is always valuable (see,e.g., Samuelson, 1967). Therefore, interdependence isvaluable and, indeed, what we should expect. In practice,we may expect both effects to be present, i.e., by sharingrisks, intermediaries decrease the risk of individual fail-ure, but increase the risk of massive, systemic failure.2

Which factors determine the risks of systemic failuresof financial institutions and the benefits of diversification?When do the risks outweigh the benefits? What are thepolicy implications of such a trade-off? In this paper, weanalyze these questions by introducing a parsimoniousmodel that combines the two aspects of diversification inan integrated analysis. Along the lines of the previousviewpoints, while individual institutions may have anincentive to diversify their risks, diversification creates anegative externality in the form of systemic risk. If allintermediaries are essentially holding the same diversi-fied portfolio, a shock may disrupt all the institutionssimultaneously, which is costly to society, since it maytake time for the financial system, and thereby theeconomy, to recover. Specifically, the slow recovery timecreates a significant and continuing social cost becausethe unique market-making and information analysisprovided by banks and other intermediaries3 is lost untilthey recover; see Bernanke (1983). Indeed, Bernanke’sconcern with the social cost created by bank failuresappears to have motivated many of the government bankbailouts.

We show that the costs and benefits of risk-sharing arefunctions of five properties of the economy. First, thenumber of asset classes is crucial: The fewer the numberof distinct asset classes that are present, the weaker thecase for risk-sharing. Second and third, the correlationbetween risks within an asset class, and the heavy-tailedness of the risks are important. The higher thecorrelation and the heavier the tails of the risk distribu-tion, the less beneficial risk-sharing is. Fourth, the longerit takes for the economy to recover after a systemicfailure, the more costly risk-sharing is and, fifth, lowerdiscount rates also work against risk-sharing. We definethe diversification threshold to be the threshold at whichthe cost to society of systemic failure begins to exceed theprivate benefits of diversification, and we derive a formulafor the threshold as a function of these five properties.

The distributions of the risks that intermediaries takeon are key to our results. When these risks are thin-tailed,risk-sharing is always optimal for both individual inter-mediaries and society. But, with moderately heavy-tailedrisks, risk-sharing may be suboptimal for society,although individual intermediaries still benefit from it.

2 That the risk of massive failure increases when risks are shared

was noted within a finance context by Shaffer (1994).3 Our analysis applies to banks, but more broadly to general

financial intermediaries, like pension funds, insurance companies, and

hedge funds.

In this case, the interests of society and intermediaries areunaligned. For extremely heavy-tailed risks, intermedi-aries and society once again agree, this time that risk-sharing is suboptimal.

One can argue that the focus of our study, moderatelyheavy-tailed distributions, is the empirically interestingcase to study. It is well-known that diversification may besuboptimal in the extremely heavy-tailed case,4 and somerisks may indeed have extremely heavy tails (e.g.,catastrophic losses, in which case individual insurersmay withdraw from the market precisely because thebenefits of diversification are unavailable; see Ibragimov,Jaffee, and Walden, 2009). However, arguably mostfinancial risks are moderately heavy-tailed, as shown byseveral empirical studies in recent years. Specifically, therate at which a distribution decreases for large values, theso-called tail exponent, a (which we rigorously define inthe paper), provides a useful classification of heavy-tailedness. Risks with ao1 are extremely heavy-tailed,whereas risks with 1oao1 are moderately heavy-tailed, and when a¼1, they are thin-tailed. Many recentstudies argue that the tail exponents in heavy-tailedmodels typically lie in the interval 2oao5 for financialreturns on various stocks and stock indices (see, amongothers, Jansen and de Vries, 1991; Loretan and Phillips,1994; Gabaix, Gopikrishnan, Plerou, and Stanley, 2006;Gabaix, 2009). Among other results, Gabaix, Gopikrishnan,Plerou, and Stanley (2006) and Gabaix (2009) providetheoretical results and empirical estimates that supportheavy-tailed distributions with tail exponents a� 3 forfinancial returns on many stocks and stock indices indifferent markets.5

Our analysis has implications for risk management andpolicies to mitigate systemic externalities. We show thatvalue at risk (VaR) considerations lead individual inter-mediaries to diversify, as per incentives similar to those inthe Basel bank capital requirements. Within our frame-work, however, the diversification actions may lead tosuboptimal behavior from a societal viewpoint. It thenbecomes natural to look for devices that would allowindividual firms to obtain the benefits of diversification,but without creating a systemic risk that could topple theentire financial system. In Section 4, we provide aframework to develop such solutions and provide specificproposals.

Our paper is related to the recent, rapidly expandingliterature on systemic risk and market crashes. The closestpaper is Acharya (2009). Our definition of systemic risk issimilar to Acharya’s, as are the negative externalities ofjoint failures of intermediaries. The first and foremost

Müuller, Olsen, and Pictet (1997), and Gabaix, Gopikrishnan, Plerou, and

Stanley (2006), tail exponents are similar for financial and economic

time series in different countries. For a general discussion of heavy-

tailedness in financial time series, see the discussion in Loretan and

Phillips (1994), Rachev, Menn, and Fabozzi (2005), Embrechts, Klüuppel-

berg, and Mikosch (1997), Gabaix, Gopikrishnan, Plerou, and Stanley

(2006), Ibragimov (2009a), and references therein.

R. Ibragimov et al. / Journal of Financial Economics 99 (2011) 333–348 335

difference between the two papers is our focus on thedistributional properties of risks and the number of riskclasses in the economy, which is not part of the analysis inAcharya (2009). Moreover, the mechanisms that generatethe systemic risks are different in the two papers.Whereas the systemic risk in Acharya (2009) arises whenindividual intermediaries choose correlated real invest-ments, in our model the systemic risk is introduced whenintermediaries with limited liability become interdepen-dent when they hedge their idiosyncratic risks by takingpositions in what is in effect each others’ risk portfolios.Such interdependence may have been especially impor-tant for systemic risk in the recent financial crisis. Thisleads to a distinctive set of policy implications, as wedevelop in Section 4.

Wagner (2010) independently develops a model offinancial institutions in which there are negative extern-alities of systemic failures, and diversification thereforemay be suboptimal from society’s perspective. Wagner’sanalysis, however, focuses on the effects of conglomerateinstitutions created through mergers and acquisitions andthe effects of contagion. Furthermore, the intermediarysize and investment decisions are exogenously given inWagner’s study, and only a uniform distribution of assetreturns is considered. Our model, in contrast, emphasizesthe importance of alternative risk distributions and thenumber of risks in determining the possibly negativeexternality of diversification. The two studies thereforecomplement each other.

A related literature models market crashes based oncontagion between individual institutions or markets.A concise survey is available in Brunnermeier (2009).Various mechanisms to propagate the contagion havebeen used. Typically it propagates through an externality,in which the failure of some institutions triggers thefailure of others. Rochet and Tirole (1996) model aninterbank lending market, which intrinsically propagatesa shock in one bank across the banking system. Allen andGale (2000) extend the Diamond and Dybvig (1983) bankrun liquidity risk model, such that geographic or industryconnections between individual banks, together withincomplete markets, allows for shocks to some banks togenerate industry-wide collapse. Kyle and Xiong (2001)focus on cumulative price declines that are propagated bywealth effects from losses on trader portfolios. Kodes andPritsker (2002) use informational shocks to trigger asequence of synchronized portfolio rebalancing actions,which can depress market prices in a cumulative fashion.Caballero and Krishnamurthy (2008) focus on Knightianuncertainty and ambiguity aversion as the common factorthat triggers a flight to safety and a market crash. Mostrecently, Brunnermeier and Pedersen (2009) model acumulative collapse created by margin requirements anda string of margin calls.

The key commonalities between our paper and thisliterature is the possibility of an outcome that allows asystemic market crash, with many firms failing at thesame time. Moreover, as in many other papers, in ourmodel there is an externality of the default of anintermediary—in our case, the extra time it takes torecover when many defaults occur at the same time. The

key distinction between our paper and this literature is,again, our focus on the importance of risk distributionsand number of asset classes in an economy. Thus, aunique feature of our model is that the divergencebetween private and social welfare arises from thestatistical features of the loss distributions for the under-lying loans alone. This leads to strong, testable implica-tions and to distinctive policy implications. In our model,these effects arise even without additional assumptionsabout agency problems (e.g., asymmetric information) orthird-party subsidies (e.g., government bailouts). Nodoubt, such frictions and distortions would make theincentives of intermediaries and society even less aligned.

The paper is organized as follows. In the next sectionwe introduce some notation. In Section 3, we introducethe model, and in Section 4 we discuss its potentialimplications for risk management and policy making.Finally, some concluding remarks are made in Section 5.Proofs are left to a separate appendix. To simplify thereading, we provide a list of commonly used variables atthe end of the paper.

2. Notation

We use the following conventions: lower case thinletters represent scalars, upper case thin letters representsets and functions, lower case bold letters representvectors, and upper case bold letters represent matrices.The ith element of the vector v is denoted (v)i, or vi if thisdoes not lead to confusion, and the n scalars vi,i¼ 1, . . . ,nform the vector [vi]i. We use T to denote the transpose ofvectors and matrices. One specific vector is

1n ¼ ð1,1, . . . ,1|fflfflfflfflfflffl{zfflfflfflfflfflffl}n

ÞT , (or just 1 when n is obvious). Similarly,

we define 0n ¼ ð0,0, . . . ,0|fflfflfflfflfflffl{zfflfflfflfflfflffl}n

ÞT .

The expectation and variance of a random variable, x,are denoted by E½x� (or simply Ex) and varðxÞ (or s2ðxÞ),respectively, provided they are finite. The correlation andcovariance between two random variables x1 and x2 withEx21o1, Ex

22o1, are denoted by covðx1,x2Þ and

corrðx1,x2Þ, respectively. We will make significant use, inparticular, of the Pareto distribution. A random variable, xwith cumulative distribution function (c.d.f.) F(x) is said tobe of Pareto-type (in the left tail), if

Fð�xÞ ¼ cþoð1Þxa

‘ðxÞ, x-þ1, ð1Þ

where c,a are some positive constants and ‘ðxÞ is a slowlyvarying function at infinity:

‘ðlxÞ‘ðxÞ -1

as x-þ1 for all l40. Here, f ðxÞ ¼ oð1Þ as x-þ1 meansthat limx-þ1f ðxÞ ¼ 0. The parameter a in (1) is referred toas the tail index or tail exponent of the c.d.f. F (or therandom variable x). It characterizes the heaviness (therate of decay) of the tail of F. We define a distribution tobe heavy-tailed (in the loss domain) if it satisfies (1) withar1. It is moderately heavy-tailed if 1oao1, and it is

7 For review and discussion of models with common shocks and

modeling approaches for spatially dependent economic and financial

data, see, among others, Conley (1999), Andrews (2005), Ibragimov and

Walden (2007), and Ibragimov (2009b).8 A Toeplitz matrix A¼ ToeplitzN ½a�Nþ1 ,a�Nþ2 , . . . ,a�1 ,a0 ,a1 , . . . ,

aN�2 ,aN�1�, is an N � N matrix with the elements given by ðAÞij ¼ ai�j ,1r irN, 1r jrN. A Toeplitz matrix is banded if ðAÞij ¼ 0 for large ji�jj,corresponding to ai ¼ 0 for indices i that are large by absolute value.


thin-tailed if a¼1, i.e., if the distribution decreases fasterthan any algebraic power.

3. Model

Consider an infinite horizon economy, t 2 f0,1,2, . . .g, inwhich there are M different risk classes. Time value ofmoney is represented by a discount factor do1 so thatthe present value of one dollar at t¼ 1 is d. There is a bondmarket in perfectly elastic supply, so that at t, a risk-freebond that pays off one dollar at t+1 costs d.

There are M risk-neutral trading units, each trading ina separate risk class. We may think of unit m as arepresentative trading unit for risk class m. Henceforth,we shall call these trading units intermediaries, capturing alarge number of financial institutions, like banks, pensionfunds, insurance companies, and hedge funds. We thusassume that each trading unit, or intermediary, specia-lizes in one risk class. Of course, in reality, intermediarieshold a variety, perhaps a wide variety of risks. The keypoint here is that the intermediaries are not initiallyholding the market portfolio of risks, so that they mayhave an incentive to share risks with each other.

We think of the M risk classes as different risk lines or‘‘industries,’’ e.g., representing real estate, publicly tradedstocks, private equity, etc. Within each risk class, in eachtime period t, there is a large number, N, of individualmultivariate normally distributed risks, xt,mn , 1rnrN. Wewill subsequently let N tend to infinity, whereas M will bea small constant, typically less than 50, as is typical infinancial and insurance applications (the results in thepaper also hold in the case when the number of risks, N, inthe mth class depends on m, as long as we let the numberof risks in each risk class tend to infinity). For simplicity,we assume that risks belonging to different risk classesare independent, across time t, and risk class, n i.e., xt,mn isindependent of xt

0 ,m0

n0 if mamu or tuat. This is not a crucialassumption; similar results would arise with correlatedrisk classes.

For the time being, we focus on the first risk class, intime period zero. We therefore drop the m and t super-scripts. Per assumption, the individual risks have multi-variate normal distributions, related by

xiþ1 ¼ rxiþwiþ1, i¼ 1, . . . ,N�1, ð2Þ

for some r 2 ½0,1Þ.6

Here, wi are independent and identically distributed(i.i.d.) normally distributed random variables with zeromean and variance s2ðwiÞ ¼ 1�r2. Each xi representscross-sectional risk, with local dependence in the sensethat covðxi,xjÞ quickly approaches zero when ji�jj grows,i.e., the decay is exponential. The risks could, for example,represent individual mortgages and the total risk classwould then represent all the mortgages in the economy.For low r, the risks of these mortgages are effectively

6 We focus on multivariate normal risks, for tractability. Similar

results arise with other, thin-tailed, individual risks, e.g., Bernoulli

distributions, although the analysis becomes more complex, because

other distribution classes are not closed under portfolio formation so the

central limit theorem needs to be incorporated into the analysis.

uncorrelated, except for risks that are very close. ‘‘Close’’here could, for example, represent mortgages on houses inthe same geographical area. If r is close to one, shocks arecorrelated across large distances, e.g., representing coun-try-wide shocks to real estate prices. This structure thusallows for both ‘‘local’’ and ‘‘global’’ risk dependencies in asimple setting.7

For simplicity, we introduce symmetry in the riskstructure by requiring that

x1 ¼ rxNþw1, ð3Þ

i.e., the relationship between x1 and xN is the same as thatbetween xiþ1 and xi, i¼ 1, . . . ,N�1. We can rewrite therisk structure in matrix notation, by defining x¼ ½xi�i,w¼ ½wi�i. The relationship (2), (3) then becomes

Ax¼w:

Here, A is an invertible so-called circulant Toeplitzmatrix,8 given by

A¼ ToeplitzN½�r,1,0TN�2,�r�:

Thus, given the vector with independent noise terms, w,the risk structure, x, is defined by

x¼def A�1w:The symmetry is merely for tractability and we would

expect to get similar results without it (although at theexpense of higher model complexity). In fact, for large N,the covariance structure in our model is very similar tothat of a standard AR(1) process, defined by

x̂0 ¼ ŵ0,

x̂i ¼ rx̂iþŵiþ1, i¼ 2, . . . ,N,

where the ŵi’s are independent, s2ðŵ1Þ ¼ 1, s2ðŵiÞ ¼1�r2, i41 (although, of course, i does not denote a timesubscript in our model, as it does in a standard AR(1)process). In this case, the matrix notation becomesÂx̂ ¼ ŵ, where Â ¼ ToeplitzN½�r,1�. The only differencebetween A and Â is that A1N ¼ 1, whereas Â1N ¼ 0. Thecovariance matrix Ŝ ¼ ½covðx̂i,x̂jÞ�ij has elements

Ŝ ij ¼ rji�jj:

Thus, in the AR(1) setting, the covariance between risksdecreases geometrically with the distance ji�jj. As weshall see in Theorem 1, this property carries over to thecovariances in our symmetric structure when the numberof risks within an asset class, N, is large.

When ai ¼ 0 if io�k or i4m, for koN�1, moN�1, we use the notationa�k ,a�kþ1 , . . . ,a0 , . . . ,am�1 ,am to represent the whole sequence generat-

ing the Toeplitz matrix. For example, the notation A¼ ToeplitzN ½a�1 ,a0 �then means that Aii ¼ a0, Ai,i�1 ¼ a�1, and that all other elements of A arezero. For an N � N Toeplitz matrix, if aN�j ¼ a�j , then the matrix is, inaddition, circulant. See Horn and Johnson (1990) for more on the

definition and properties of Toeplitz and circulant matrices.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.10.20.30.40.50.60.70.80.9

1

y

Cum

ulat

ive

dist

ribut

ion

func

tion,

P (ρ

t=1: - Risks, x, realized- Dividend, D=k+cTx+Q, paid- If D=0, bank defaults - If D>0, game repeated

t=0: - Reserve capital, δk- Trade in portfolio, c- Total investment: -δE[k+cTx+Q]+dcT1

t=2,3,...

Fig. 2. Intermediary’s cash flows in model. At t=0, the intermediaryreserves capital k and trades in a risk portfolio c. At t ¼ 1 risks arerealized. If the intermediary defaults, it is out of business, otherwise, the

game is repeated, i.e., at t=1, the intermediary reserves capital and

trades in risk, at t=2 risks are realized, etc.


Now, if the intermediary survives, which it does withprobability

q¼defPð�cT xrkÞ,

the situation is repeated, i.e., at t=1 the intermediarytakes on new risk, reserves capital, and then at t=2 risksare realized, etc. If the intermediary defaults at any pointin time, it goes out of business and its cash flows are zerofrom there on. In the infinite horizon case, the value of theintermediary in recursive form is therefore

VB ¼ dðcT 1ÞþdqVB,

implying that

VB ¼ dðcT 1Þ

1�dq : ð8Þ

We note that the counter-party of the x risk transactionunderstands that the intermediary may default, and takesthis into account when the price for the risk is agreedupon. Therefore, since the price of the contract takes intoaccount the risk level of the intermediary, the counter-party does not need to impose additional covenants.

Eq. (8) describes the trade-off the individual inter-mediary makes when choosing its portfolio. On the onehand, a larger portfolio increases the cash flows per unittime—increasing the numerator—but on the otherhand, it also increases the risk of default—increasingthe denominator. It is straightforward to show that if theintermediary could, it would take on an infinitely largeportfolio. This is shown as a part of the proof of Theorem 1in Appendix A. In terms of Eq. (8), the numerator effectdominates the denominator effect.

A regulator, representing society, therefore imposesrestrictions on the probability for default, to counter-weight the risk-shifting motive. Specifically, the inter-mediary’s one-period probability of default is not allowedto exceed b at any point in time. In other words,the intermediary faces the constraint that the valueat risk at the 1�b confidence level can be at most k,VaR1�bðcT xÞrk. Therefore, since the realized dollar lossesare cTx and the capital reserved is k, the value at riskconstraint says that the probability is at most b that therealized losses exceed the capital. In our notation, this isthe same as to say that qZ1�b.

The VaR constraint is imposed in the model to reflectthe existing management and regulatory standards

through which most financial intermediaries currentlyoperate. Most major financial intermediaries apply VaR asa management tool and regularly report their VaR values.The Basel II bank capital requirements are based on VaR.European securities firms face the same Basel II VaRrequirements, while the major U.S. securities firms havebecome bank holding companies and thus also face theserequirements. European insurance firms are being rere-gulated under ‘‘Solvency II,’’ which is VaR-based, inparallel to Basel II. U.S. insurers are regulated byindividual states with capital requirements that varyby state and insurance line; these are generally consistentwith a VaR interpretation.

To be clear, VaR is not necessarily the optimal riskmeasure when financial intermediary behavior maycreate systemic externalities. Indeed, in this section weshow that VaR requirements can lead to diversificationdecisions by individual intermediaries that are inconsis-tent with maximizing the societal welfare. In Section 4,we consider alternative proposals for intermediary reg-ulation when systemic externalities are important.

It is natural to think of cT 1=k as a measure of theleverage of the firm, since cT1 represents the total liabilityexposure and k represents the capital that can be used tocover losses.

The assumption that there is an upper bound, K, thatthe intermediary faces on how much capital can bereserved is reduced form. In a full equilibrium modelwithout frictions, the investment level would be chosensuch that the marginal benefit and cost of an extra dollarof investment would be equal, and K would then beendogenously derived. In our reduced-form model, wherethe benefits d are constant, we assume that the marginalcost of raising capital beyond K is very high, so that Kprovides an effective hard constraint on the capital raisingabilities of the intermediary. We note that the bound maybe strictly lower than the friction-free outcome, becauseof other frictions on capital availability in imperfectfinancial markets (see, e.g., Froot, Scharfstein, and Stein,1993 for a discussion of such frictions). Given a choice ofcapital, krK , the VaR constraint imposed by the regulatorthen imposes a bound on how aggressively the inter-mediary can invest, i.e., on the intermediary’s size.The VaR constraint therefore also automatically imposesa capital requirement restriction on the intermediary.We next study the intermediary’s behavior in thisenvironment.

3.1. Optimal behavior of intermediary

In line with the previous arguments, the program forthe intermediary is

maxc,kdðcT 1Þ

1�dPð�cT xrkÞ s:t:, ð9Þ

k 2 ½0,K�, ð10Þ

c 2 Rn, ð11Þ

kZVaR1�bðcT xÞ: ð12Þ

0.2

0.25

0.3

Probability distribution function,f (y)


The following theorem characterizes the intermediary’sbehavior:

Theorem 1. Given a VaR constrained intermediary solving(9)–(12), where the risks are on the form (2,3), b is close tozero, and the distribution of correlations is of the form (4).Then, for large N:

0.15
(a) For a given r, the covariance covðxi,xjÞ converges to rji�jjfor any i, j.
0.1
(b)
-4 -2 2 4y

0.05

Fig. 3. Probability distribution function, f, of intermediary’s chosenportfolio of risk, for g¼ 1, corresponding to a heavy-tailed distributionwith infinite variance, and for g¼ 2:5, corresponding to a moderatelyheavy-tailed distribution with finite variance.

The payoff of the intermediary’s chosen risk portfolio,cTx, is of Pareto-type with the tail index 2g, and with theprobability distribution function (p.d.f.) f(y), where

f ðyÞ ¼ g2gb�gffiffiffiffipp �

G gþ12

,by2

2

!

jyj1þ2g, ð13Þ

for some constant, b40, for all y 2 R\f0g. Here, G is thelower incomplete Gamma function, Gða,xÞ ¼def

R x0 t

a�1e�t

dt, a40. Also, f ð0Þ ¼ 2g=ðbffiffiffiffiffiffi2ppð2gþ1ÞÞ.

(c)

Maximal capital is reserved, k¼ K , i.e., (10) is binding.

(d)

Maximal value at risk is chosen, i.e., (12) is binding.

(e)
If g41, then the variance of the portfolio is
s2 ¼ b2 gg�1 , ð14Þ

else it is infinite.

We note that, since Gða,xÞ ¼GðaÞþoð1Þ, as x-þ1,(where GðaÞ is the Gamma function, GðaÞ ¼

R10 t

a�1e�t dt,a40), we obtain that cT x has a Pareto-type distribution inaccordance with (1):

Pð�cT x4yÞ ¼2gb�gG gþ12

� �þoð1Þ

gffiffiffiffippjyj2g

, y-1:

Thus, the portfolio chosen by the intermediary whensolving (9)–(12) is moderately heavy-tailed, or evenheavy-tailed when go 12, although the individual risksare thin-tailed. We define the random variable x¼ cT x andthe constant c¼ cT 1, and show the p.d.f. of x for the twocases g¼ 1 (heavy-tailed distribution with infinite var-iance) and g¼ 2:5 (moderately heavy-tailed distributionwith finite variance) in Fig. 3.

We also note that the total portfolio risk changes withthe number of investments, N, in a very different way thanwhat is implied by standard diversification results, forwhich the size of the investment is taken for given. Ifportfolio size were given, then the portfolio risk wouldvanish as N grew. However, taking into account that theintermediary can change its total exposure with thenumber of risks, in our framework, the risk does notmitigate, but instead converges to a portfolio risk that ismuch more heavy-tailed than the individual risks.

Our approach of letting the number of risks within anasset class grow has some similarities with the idea of anasymptotically fine-grained portfolio, (see Gordy, 2000,2003), used for a theoretical motivation of the capitalrules laid out in Basel II. Gordy shows that under theassumptions of one systematic risk factor and infinitely

many small idiosyncratic risks, the portfolio invariant VaRrules of Basel II can be motivated. Similar to Gordy’ssetting, in our model each risk is a small part of the totalasset class risk when N is large. However, in our setting allrisk is not diversified away when the number of risksincreases, as shown in Theorem 1. In fact, the risk that isnot diversified away adds up to systematic risk in ourmodel (which turns into systemic risk if it brings downthe whole system), and since the number of risk classes,M41, there will be multiple risk factors. This is contraryto the analysis in Gordy (2000), where, after diversifica-tion, all institutions hold the same one-factor risk. There-fore, the capital rules in Basel II would not be motivated inour model.

3.1.1. The value to society

We next turn to the value to society of the markets forthe M separate risk classes. For the individual intermedi-ary, the game ends if it defaults. From society’s perspec-tive, however, we would expect other players to step inand take over the business if an intermediary defaults,although it may take time to set up the business, developclient relationships, etc. Therefore, the cost to society ofan intermediary’s default may not be as serious as thevalue lost by the specific intermediary. The argument isthat since there is a well-functioning market, it is quiteeasy to set up a copy of the intermediary that defaulted.We assume that an outside provider of capital steps in andsets up an intermediary identical to the one thatdefaulted. The argument per assumption, however, onlyworks if there is a well-functioning market. In the unlikelyevent in which all intermediaries default, the market isnot well-functioning and it may take a much longer timeto set up the individual copy. Thus, individual intermedi-ary default is less serious from society’s perspective, butmassive intermediary default (for simplicity, the casewhen all M intermediaries default at the same time) ismuch more serious. A similar argument is made inAcharya (2009).

13 This is not a critical assumption. Alternatively, we could have

assumed that the regulator anticipates whether the individual inter-

mediaries will trade risks, and adjusts the VaR requirements accordingly.


To incorporate this reasoning into a tractable model,we use a special numerical mechanism. When only oneindividual intermediary fails (or a few intermediaries), themarket continues to operate reasonably well. In suchcases, the replacement of a failed intermediary occursimmediately if an intermediary defaults in an even periodðt¼ 0,2,4, . . .Þ, and takes just one period if it defaults in anodd period ðt¼ 1,3,5, . . .Þ. We note that in this favorablecase without massive default, it takes, on average, half aperiod to rebuild after an intermediary’s default (zeroperiods or one period with 50% chance each). On the otherhand, when all intermediaries fail—a massive de-fault—we assume there is a 50% chance that it will takeanother 2T periods (2T +1�1) to return to normaloperations. Therefore, the expected extra time to recoverafter a massive default is ð50%Þ � ð2TÞþð50%Þ � ð0Þ ¼ T .Henceforth, we call T the recovery time after massivedefault.

Our distinction between defaults in odd and even timeperiods significantly simplifies the analysis by ensuringthat in even periods there are either 0 or M intermediariesin the market. Without the assumption, we would need tointroduce a state variable describing how many inter-mediaries are alive at each point in time, which wouldincrease the complexity of the analysis. Qualitativelysimilar results arise when relaxing the assumption, i.e.,when instead assuming that it always takes one periodto replace up to M�1 defaulting intermediaries andT periods if all M intermediaries default at the same time.

The assumption of longer recovery times after massiveintermediary defaults can be viewed as a reduced-formdescription of the externalities imposed on society byintermediary defaults. The simplicity of the assumptionallows us to carry out a rigorous analysis of the role of riskdistributions, which is the focus of this paper. Micro-foundations for such externalities have been suggestedelsewhere in the literature, e.g., liquidity risk as in Allenand Gale (2000). In Allen and Gale (2000), the externalityoccurs when failure of intermediaries triggers failure ofother intermediaries.

Society’s and individual intermediaries’ objectives maybe unaligned. From our previous discussion, it follows thatthe value to society of all intermediaries at t=0 is, inrecursive form:

VS ¼MdcþdqMdcþd2ð1�ð1�qÞMÞVSþd2Tþ2ð1�qÞMVS,ð15Þ

implying that

VS ¼ Mdcð1þdqÞ1�d2þd2ð1�d2T Þð1�qÞM

: ð16Þ

The first term on the right-hand side of (15) is the valuegenerated between t=0 and 1, and the second term is theexpected value between t=1 and 2. The third term is thediscounted contribution to the value from t=2 andforward if all M intermediaries do not default (whichoccurs with probability (1�(1�q)M)), and the fourth termis the contribution to the value if all M intermediariesdefault (which occurs with probability (1�q)M). We shallsee that this externality of massive intermediary default

in the form of longer recovery time significantlydecreases—or even reverses—the value of diversificationfrom society’s perspective and that the situations in whichdiversification is optimal versus suboptimal are easilycharacterized.

3.2. Risk-sharing

We next analyze what happens when the M differentintermediaries, with identically distributed risk portfolios,x1, . . . ,xM , get the opportunity to share risks. We recallthat the risk portfolios in the different risk classes areindependently distributed. Therefore, as long as theintermediaries do not trade, a shock to one intermediarywill not spread to others. If the intermediaries trade,however, systemic risk may arise when their portfoliosbecome more similar because of trades. In what follows,we explore this idea in detail.

We assume that the intermediaries may trade risks att¼ 12 ,

32 ,

52 , . . ., after they have formed their portfolio, but

before the risks are realized. We assume that the VaRrequirement must hold at all times. For example, even ifthe intermediaries trade with each other at t¼ 12, the VaRrequirement needs to be satisfied between t¼ 0 and 12.

13

Also, the counter-party of the risk trade correctlyanticipates whether trades between intermediaries willtake place at t¼ 12, taking the impact of such a trade on thedefault option, Q, into account when the price is decided.

We focus on symmetric equilibria, in which all inter-mediaries choose to share risks fully. We define

qM ¼P�PM

m ¼ 1 xmM

rK !

,

so qM is the probability that total losses in the market arelower than total capital. Thus, 1�qM, is the probability formassive intermediary default if intermediaries fully sharerisks. We note, in passing, that systemic risk is generatedthrough a different mechanism in our model than inAcharya (2009). In Acharya (2009), systemic risk ariseswhen intermediaries take on correlated real investments,and intermediaries do not trade risks. In our model, therisk portfolios of different intermediaries are independent,and systemic risk arises when intermediaries becomeinterdependent through trades. The latter type of sys-temic risk may have been especially important in thefinancial crisis.

All intermediaries hold portfolios of the same riskfeatures, albeit in different risk classes, so when theyshare risks, the net ‘‘price’’ they pay at t¼ 12 is zero.However, the states of the world in which default occursare different when the intermediaries share risks. In fact,defaults will be perfectly correlated in the full risk-sharingsituation: With probability 1�qM a massive intermediarydefault occurs and with probability qM no intermediarydefaults. Risk-sharing could, of course, be achieved not

Loss -ξ1

Loss -ξ2

K

K

-ξ1 -ξ2=2K

Diversified: Massive default

Separated: Massive default

Diversified: No default

Separated: No default


Separated: Single default







Fig. 4. Cost to society for diversified and separated cases, when there aretwo intermediaries. In the diversified case, massive default occurs if

�x1�x2 Z2K. In the separated case, massive default occurs if �x1 ZKand �x2 ZK , whereas single default occurs if �x1 ZK and �x2 oK , or if�x2 ZK and �x1 oK. Thus, massive default is rarer in the separated casethan in the diversified case, but it is more common that at least one

intermediary defaults in the separated case.


only by cross-ownership, but also by trading in deriva-tives contracts (credit default swaps, corporate debt, etc.).

For individual intermediaries, the value when sharingrisks (using the same arguments as when deriving (8))is then

VBM ¼dc

1�dqM, ð17Þ

which should be compared with the value of not sharing,(8). Therefore, as long as

qM 4q, ð18Þ

the intermediaries prefer to trade risks, since it leads to ahigher probability of survival, and thereby a higher value.

We note that there may be additional reasons forintermediaries to prefer risk-sharing. For example, ifintermediaries anticipate that they will be bailed out incase of massive defaults, if managers are less punished ifan intermediary performs poorly when all other inter-mediaries perform poorly too, or if the VaR restrictions arerelaxed, this provides additional incentives for risk-sharing, which would amplify our main result.

3.2.1. The value to society

From society’s perspective, a similar argument as whenderiving Eqs. (15) and (16) shows that

VSM ¼MdcþdqMMdcþd2qMV

SMþd

2Tþ2ð1�qMÞVSM , ð19Þ

so the total value of risk-sharing between inter-mediaries is

VSM ¼Mdcð1þdqMÞ

1�d2þd2ð1�d2T Þð1�qMÞ: ð20Þ

Therefore, via (16), it follows that society prefers risk-sharing when

1þdqM1þlð1�qMÞ

41þdq

1þlð1�qÞM, ð21Þ

where

l¼ d2ð1�d2T Þ1�d2

: ð22Þ

The value of l determines the relative trade-off societymakes between the costs in foregone investment oppor-tunities of individual and massive intermediary default.14

If T ¼ 0, there is no extra delay when massive defaultoccurs. In this case, l¼ 0, and society’s trade-off (21) isthe same as individual intermediaries’ (18). If, on theother hand, T is large and d is close to one, representing asituation in which it takes a long while to set up a marketafter massive default and the discount rate is low,l is large, and the trade-off between (1�qM) and(1�q)M in (21) becomes very important. In this case,society will mainly be interested in minimizing the riskof massive default and this risk is minimized whenintermediaries do not share risk, since 1�qM Zð1�qÞM .15

14 It is easy to show that l 2 ½0,T�, and that l is increasing in d and T.15 This follows trivially, since 1�qM ¼Pð

Pi�xi 4MKÞZ

Pð\if�xi 4KgÞ ¼ ð1�qÞM .

It is clear that society and individual intermediariesmay disagree about whether it is optimal to diversify.Specifically, this occurs when (18) holds, but (21) fails.The opposite, that (18) fails but (21) holds, can neveroccur.16 To understand the situation conceptually, westudy the special case when there are two risk classes,M=2. In Fig. 4, we show the different outcomes dependingon the realizations of losses in the diversified andseparated cases. When both �x14K and �x24K , it doesnot matter whether the intermediaries diversify or not,since there will be massive intermediary default eitherway. Similarly, if �x1rK and �x2rK , neither inter-mediary defaults, regardless of whether they diversify ornot. However, in the case when �x14K and �x2rK , theoutcome is different, depending on the diversificationstrategy the intermediaries have chosen. In this case, if�x1�x242K , then a massive default occurs if theintermediaries are diversified, but only a single default ifthey are not. If �x1�x2r2K , on the other hand, no defaultoccurs if the intermediaries are diversified, but a singledefault occurs if they are not. Exactly the same argumentapplies in the case when �x1rK and �x24K. Therefore,the optimal outcome depends on the trade-off betweenavoiding single defaults in some states of the world butintroducing massive defaults in others, when diversifying.The point that risk-sharing increases the risk for jointfailure is, of course, not new—it was made as early as inShaffer (1994). Our analyses of the trade-off betweendiversification benefits and costs of massive failures, theimportance of distributional properties, correlation un-certainty, and number of asset classes, as well as theimplications for financial institutions, however, are to thebest of our knowledge novel.

16 This also follows trivially, since 1�qM Zð1�qÞM and therefore, ifqZqM , then ð1þdqÞ=ð1þlð1�qÞMÞZð1þdqÞ=ð1þlð1�qMÞÞZ ð1þdqMÞ=ð1þlð1�qÞMÞ.

0

10

20

30

40

50

60

η

T=30

T=20

T=10

T=1


It is clear that the objectives of the intermediaries andsociety will depend on the distributions of the x�risks andit may not be surprising that standard results from thetheory of diversification apply to the individual inter-mediaries’ problem. Society’s objective function is morecomplex, however, since it trades off the costs of massiveand individual intermediary defaults. It comes as apleasant surprise that for low default risks (i.e., for a bclose to zero in the VaR1�b constraint) given the distribu-tion of the x�risk, we can completely characterize whenthe objectives of intermediaries and society are different.

Theorem 2. Given i.i.d. asset class risks, x1, . . . ,xM , ofPareto-type (1) with the tail index aa1, and d, d, and T aspreviously defined. Then, for low b,

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1δ
(a)
Fig. 5. The coefficient, Z¼ ð1�d2Tþ1Þ=ð1�dÞ, as a function of the discountfactor, d, for different recovery times after massive default, T=1,10,20,30.

From an intermediary’s perspective, risk-sharing isoptimal if and only if a41, regardless of the numberof risk classes, M.

All else equal, a higher Z leads to a higher diversification threshold, M� .
(b)
40

50

60sh

old,

M*

α=2

From society’s perspective, risk-sharing is optimal if andonly if a41 and M4M�, where M� is the diversificationthreshold

M� ¼1�d2Tþ1

1�d

!1=ða�1Þ: ð23Þ

0 10 20 30 40 50 600

10

20

30

η

Div

ersi

ficat

ion

thre

α=4α=3

α=2.5

Fig. 6. Diversification threshold, M*, for Pareto-type risk distributionswith tail indices a¼ 2,2:5,3,4, as a function of the coefficientZ¼ ð1�d2Tþ1Þ=ð1�dÞ, where d is the per period discount factor and T isthe recovery time after massive default. The diversification threshold is

the break-even number of risk classes needed for diversification to be

optimal for society. Results are asymptotically valid for VaR confidence

levels close to 100%, i.e., for b close to zero in (12).

Thus, the break-even for individual intermediaries,from (18), is a Pareto distribution with the tail index ofone, e.g., a Cauchy distribution. If the distribution isheavier, then (18) fails, and intermediaries and societytherefore agree that there should be no risk-sharing. Forthinner tails, intermediaries prefer risk-sharing. On thecontrary, for society, what is optimal depends on M. It iseasy to show that M� 2 ½1,ð2Tþ1Þ1=ða�1ÞÞ, and that Mn isincreasing in d and T. We note that for the limit case whend-0, a-1, or T=0, i.e., when M�-1, then society andintermediaries always agree.

It is useful to define

Z¼ 1�d2Tþ1

1�d, ð24Þ

so that M� ¼ Z1=ða�1Þ. This separates the impact on Mn ofthe tail behavior of the risk distributions from the otherfactors (d and T). It follows immediately thatZ 2 ½1,2Tþ1Þ. In Fig. 5, we show Z as a function of d forT ¼ 1, 10, 20, 30.

From Theorem 2, it follows that if there is a largenumber of risk classes, M, society agrees with theindividual intermediaries:

Corollary 1. Given i.i.d. Pareto-type risk classes with the tailindex a, if there is a large enough number of risk classesavailable, intermediaries and society agree on whether risk-sharing is optimal.

In Fig. 6, we show the break-even number of riskclasses, Mn, for Pareto-type risks with the tail indicesa¼ 2,2:5,3,4. With no externality of massive default(Z¼ 1), society prefers diversification when a41, justlike individual intermediaries. However, when Z41,

diversification is suboptimal up until Mn. For example,for a¼ 2, and Z¼ 10, at least 10 risk-classes are needed fordiversification to be optimal for society.

Theorem 2 provides a complete characterization of theobjectives of intermediaries and firms for VaR close to the100% confidence level, by relating the number of riskclasses, M, the discount rate, d, the tail distribution of therisks, a, and the recovery time after massive default, T. Italso relates to the uncertainty of correlations, g, throughthe relation a¼ 2g. Theorem 2 is therefore the funda-mental result of this paper. The theorem has severalimmediate empirical implications, since when (23) fails,we would expect there to exist financial regulationsagainst risk-sharing across risk classes:

Implication 1. Economies with heavier-tailed risk distribu-tions should have stricter regulations for risk-sharing

between risk classes.

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

η

Div

ersi

ficat

ion

thre

shol

d,M

*

α=5

α=4

α=3

Fig. 7. Diversification threshold, M� , i.e., sufficient condition for numberof risk classes, M4M� , to be large enough to guarantee that societyprefers diversification, for Pareto-type risk distributions with tail indices

a¼ 3,4,5, as a function of Z. Here, Z¼ ð1�d2Tþ1Þ=ð1�dÞ, d is the perperiod discount factor and T is the recovery time after massive default.

Results are valid for VaRs above the 99% confidence level.


Equivalently, using the relationship between uncertaintyof correlation structure and tail distributions,

Implication 2. Economies with more uncertain correlationbetween risk classes should have stricter regulations for

risk-sharing.

We also have

Implication 3. Economies with lower interest rates shouldhave stricter regulations for risk-sharing.

Implication 4. Economies with fewer risk classes shouldhave stricter regulations for risk-sharing.

Implication 5. Economies with risk classes for which ittakes longer to recover after a massive default should have

stricter regulations for risk-sharing.

Implication 4 may be related to the size of theeconomy, in that larger economies may have more riskclasses and therefore, society may be more tolerant torisk-sharing across these classes. Moreover, Implication 5may be related to the degree to which an economy is opento foreign investments in that economies that are openmay be faster in recovering after a massive default, andthereby allow more risk-sharing between risk classes. InSection 4, we apply these implications to derive policysolutions for controlling the social costs created whenactions by intermediaries to diversify create investmentsin highly correlated market portfolios.

Theorem 2 provides an asymptotic result and there-fore, does not enlighten us about what the break-evennumber of risk classes is when the default risks, b, issignificantly different from zero. For b significantlydifferent from zero, the analysis is less tractable. Becauseof the noncoherency of the value at risk measure, wesuspect that diversification may be less valuable forsociety in such cases. Indeed, we have the followingsufficient condition for diversification to be optimal fromsociety’s perspective, which holds for all b below somethreshold.

Theorem 3. Consider i.i.d. asset class risks, x1, . . . ,xM , ofPareto-type with the tail index a42, and Z as defined in(24). Define the tail probability FðxÞ ¼defPð�x14xÞ. Assumethat the variance of x1 is s2, that FðK0Þ ¼ b0, and that forK4K0,

c1Ka

rFðKÞr c2Ka

, ð25Þ

for constants 0oc1rc2. Then, for all brb0, risk-sharing isoptimal from society’s perspective, if the number of assetclasses, M, satisfies

c12aMa�CZMa=2�Mc2ZaaZ0, ð26Þ

where C ¼ ð2eas2Þa=2, and e is the constant e¼ 2:71828 . . ..

In Fig. 7 we show the break-even number of risks forthe Pareto-distributed risks of Section 3.1, for whicha¼ 2g. The bound is chosen so that the results are validfor all value at risk at confidence levels above 99%, i.e.,for br1%. We see that, compared with Fig. 6, M needs tobe significantly higher to ensure that society prefers

diversification. The bound gets even worse if the VaR is ata lower confidence level, e.g., at the 95% level.

As an example, assume that the average time it takesto replace a defaulted intermediary is six months if thedefault is individual (i.e., that one-period is six months),that the one period discount factor is d¼ 0:99 (so that theone-year discount rate is approximately 2%), and that ifthere is massive intermediary default, it takes five years torestore the whole market. In this case, Z¼ ð1�0:9910Þ=ð1�0:99Þ � 9:6. From Fig. 7, if the risk has a tail exponentof a¼ 4, there needs to be about M¼ 30 risk classes toguarantee that it is optimal to diversify at a VaR at the 99%confidence level in this case. Thus, diversification may besuboptimal from society’s perspective even with moder-ately heavy-tailed risks, as long as the number of riskclasses is not large.

3.3. Extensions to general distributions

We have analyzed the simplest possible framework,with specific independent identically distributed riskclasses. Here, we sketch how our results can be general-ized to more general distributions.

Similar to the proof of Theorem 3, one can obtain itsextension to the case of the distribution of K such that, forK4K0, c1=KarFðKÞrc2=Kz for constants 0oc1rc2 andaZz41. For such distributions, the risk-sharing outcomeis optimal from society’s perspective, if the number ofasset classes, M is sufficiently large compared to K so thatit satisfies 2zc1M

zKz�MZc2azKa�CZMz�a=2KzZ0, whereC ¼ 2z�a=2ðezs2Þa=2.

Using conditioning arguments, it is straightforward toshow that the results in this paper continue to hold for(possibly dependent) scale mixtures of normals, withwi ¼ ViZi, where Zi are i.i.d. normally distributed zero-mean random variables and Vi are (possibly dependent)random variables independent of Zi. This is a rather largeclass of distributions: it includes, for instance, theStudent’s t-distributions with arbitrary degrees of free-dom (including the Cauchy distribution), the double


exponential distribution, the logistic distribution, and allsymmetric stable distributions.

The bounds on the number of risk classes M providedby Theorem 3 can be sharpened using extensions andrefinements of probability inequality (36) provided byLemma 1 used in the proof of the theorem. In particular,one can use probability inequalities in terms of highermoments of the independent random variables x (see, forinstance, Theorem 1.3 and Corollaries 1.7 and 1.8 inNagaev, 1979).

4. Potential implications for risk management and forpolicy makers

Capital requirements provide the most commonmechanism used to control the risk of bank failure. Setat a high enough level, a simple capital-to-asset require-ment can achieve any desired level of safety for anindividual bank. Such capital requirements, however,impose significant costs on banks by limiting their useof debt tax shields, expanding the problem of debtoverhang, and creating agency problems for the share-holders.17 These costs have a negative impact on theoverall economy because they reduce the efficiency offinancial intermediation.18 In our model, it is clear thatincreasing capital requirements is an imperfect tool forthe regulator, since it cannot be used to specifically targetnegative externalities of systemic risk. In fact, there is aone-to-one relationship between VaR and capital require-ments in our model—increasing the capital requirementis equivalent to decreasing the VaR.

For this reason, new proposals to control systemic riskin the banking sector recommend focused capital require-ments based on each bank’s specific contribution to theaggregate systemic risk. Acharya (2009), for example,advocates higher capital requirements for banks holdingasset portfolios that are highly correlated with theportfolios of other banks. This follows from his model inwhich banks create systemic risk by choosing correlatedportfolios. In a similar spirit, Adrian and Brunnermeier(2009) advocate a ‘‘CoVaR’’ method in which banks face

17 The debt overhang problem arises in recapitalizing a bank

because the existing shareholder ownership is diluted while some of

the cash inflow benefit accrues as a credit upgrade for the existing

bondholders and other bank creditors. The agency problems arise

because larger capital ratios provide management greater incentive to

carry out risky investments that raise the expected value of compensa-

tion but may reduce expected equity returns; for further discussion, see

Kashyap, Rajan, and Stein (2008). While the tax shield benefit of debt is

valuable for the banking industry, it is not necessarily welfare-enhancing

for society.18 The efficiency costs of capital requirements can be mitigated by

setting the requirements in terms of contingent capital in lieu of balance

sheet capital. One mechanism is based on bonds that convert to capital if

bankruptcy is threatened (Flannery, 2005), but that instrument is not

particularly directed to systemic risk. Kashyap, Rajan, and Stein (2008)

take the contingent capital idea a step further by requiring banks to

purchase ‘‘capital insurance’’ that provides cash to the bank if industry

losses, or some comparable aggregate trigger, hits a specified threshold.

This mechanism may reduce or eliminate the costs that are otherwise

created by bankruptcy, but it does not eliminate the negative externality

that creates the systemic risk.

higher capital requirements based on their measuredcontribution to the aggregate systemic risk.

Focused capital requirements will be efficient incontrolling systemic risk, however, only if the source ofthe systemic risk is properly identified. In particular, themodel in this paper creates a symmetric equilibrium inwhich each of the M banks is responsible for precisely 1/Mof the systemic risk. Furthermore, systemic risk in ourmodel arises only as a byproduct—a true negativeexternality—of each bank’s attempt to eliminate its ownidiosyncratic risk. The risks that they take on areindependent, but the diversification changes the statesof the world in which massive default occurs. For thisreason, neither VaR constraints, nor the capital require-ment plans advocated by Acharya (2009) will be effectivein controlling the type of systemic risk that arises in ourmodel. The CoVaR measure in Adrian and Brunnermeier(2009) may also be imperfect, since it does not take tail-distributions into account. Of course, systemic risk in thebanking industry may reflect a variety of generatingmechanisms, so we are not claiming anything like amonopoly on the proper regulatory response. But it is thecase that even focused capital requirements will not beeffective if the systemic risk arises because banks hedgetheir idiosyncratic risk by swapping into a ‘‘market’’portfolio that is then held in common by all banks.

For the case developed in our model, where all bankswish to adopt the same diversified market portfolio, directprohibitions against specific banking activities or invest-ments in specific asset classes will be more effective thancapital requirements as a mechanism to control systemicrisk. For example, Volcker (2010) has recently proposed torestrict commercial banking organizations from certainproprietary and more speculative activities. While suchprohibitions may seem draconian, they would apply onlyto activities or asset classes in which moderately heavytails create a discrepancy between the private and publicbenefits of diversification. No regulatory action would beneeded for asset classes with thin-tailed risks, where thebanks and society both benefit from diversification, andfor severely heavy-tailed risks, where the banks andsociety agree that diversification is not beneficial.

It is also useful to note that direct prohibitions havelong existed in U.S. banking regulation. For one thing, U.S.commercial banks have long been prohibited frominvesting in equity shares. Even more relevant, the 1933Glass Steagall Act forced U.S. commercial banks to divesttheir investment banking divisions. Subsequent legisla-tion—specifically the 1956 Bank Holding Company Actand the Gramm-Leach-Bliley Act of 1999—provided moreflexibility, by expanding the range of allowed activities fora bank holding company, although commercial banks arethemselves still restricted to a ‘‘banking business.’’ Glass-Steagall thus provides a precedent for direct prohibitionson bank activities as well as an indication that theprohibitions can be changed over time as conditionswarrant.

Experience with regulating catastrophe insurancecounter-party risk suggests another practical and specificregulatory approach, namely ‘‘monoline’’ requirements.Monoline requirements have long been successfully


imposed on insurance firms that provide coverage againstdefault by mortgage and municipal bond borrowers; seeJaffee (2006, 2009). The monoline requirements prohibitthese insurers from operating as multiline insurers thatoffer coverage on multiple insurance lines. The monolinerestriction eliminates the possibility that large losses onthe catastrophe line would bankrupt a multiline insurer,thus creating a cascade of losses for policyholders acrossits other insurance lines. Such monoline restrictions docreate a cost in the form of the lost benefits ofdiversification, because a monoline insurer is unable todeploy its capital to pay claims against a portfolio ofinsurance risks. Nevertheless, Ibragimov, Jaffee, andWalden (2008) show that when the benefits of diversifi-cation are muted by heavy tails or other distributionalfeatures, the social benefits of controlling the systemicrisk dominate the lost benefits of diversification.

As a specific example, we reference the use of creditdefault swaps (CDS) purchased by banks and otherinvestors to provide protection against default on thesubprime mortgage securities they held in their portfolios.The systemic problem was that the CDS protection wasprovided by other banks and financial service firms actingas banks, i.e., American International Group, Inc. (AIG),with the effect that a set of large banks ended up holding avery similar, albeit diversified, portfolio of subprimemortgage risks.19 When the risks on the individualunderlying mortgages proved to be highly correlated, thisportfolio suffered enormous losses, creating the systemiccrisis. Capital requirements actually provided the invest-ing banks with incentive to purchase the CDS protection,so that higher capital requirements, per se, do not solvethe systemic problem. Instead, there must be regulatoryrecognition that moderately heavy-tailed risk distribu-tions create situations in which the social costs mayexceed the private benefits of diversification.

5. Concluding remarks

The subprime financial crisis has revealed highlysignificant externalities through which the actions ofindividual intermediaries may create enormous systemicrisks. The model in this paper highlights the differencesbetween risks evaluated by individual intermediariesversus society. We develop a model in which the negativeexternality arises because actions to diversify that areoptimal for individual intermediaries may prove to besuboptimal for society.

We show that the distributional properties of the risksare crucial: most importantly, with moderately heavy-tailed risks, the diversification actions of individualintermediaries may be suboptimal for society. We alsoshow that when there is uncertainty about correlationsbetween a large number of thin-tailed risks and inter-mediaries face value at risk constraints, this is exactly the

19 It is important to note that AIG wrote its CDS contracts from its

Financial Products subsidiary, which was chartered as a savings and loan

association and not as an insurance firm. Indeed, AIG also owns a

monoline mortgage insurer, United Guaranty, but this subsidiary was

not the source of the losses that forced the government bailout.

type of risk portfolios they will choose. Also, the numberof distinct asset classes in the economy, the discount rate,and the time to recover after a massive intermediarydefault influence the value to society of diversification.The optimal outcome from society’s perspective involvesless risk-sharing, but also creates a lower probability formassive intermediary collapses.

The policy implications of our model are very direct:banking regulation should be expanded in order to restrictthe ability of banks to swap their loan portfolios contain-ing idiosyncratic risk for market portfolios of loan risk.Such direct restrictions appear preferable to VaR con-straints, higher capital requirements, and to CoVaRmeasures when the goal is to deter the massive bankdefaults that may arise when banks diversify in thismanner. Monoline requirements, which are alreadyapplied to insurance firms that offer insurance coverageagainst default on mortgages and municipal bonds,provide an example of how such direct restrictions canbe implemented.

Appendix A. Proofs

Proof of Theorem 1. We first study the problem, givena fixed N and show that there exists a solution toEqs. (9)–(12). We replace the maximum by the supremumand show that the problem has a finite solution. Of course,c¼ 0, k¼ 0, is a feasible choice that satisfies Eqs.(10)–(12), immediately,immediately leading to a lowerbound of 0 for the function under the maximum sign in (9).

Now, if there is a finite upper bound on (9), then the

supremum problem has a finite solution. Given the

definition of S, it is clear that the variance of cTx isincreasing in r, and since, given r, the distribution of theportfolio is normal, the value at risk, VaR1�bðcT xÞ, for agiven c is also increasing in r. Therefore, the value at riskof cTx is greater than the value at risk for cTy, where y is a

vector of i.i.d. normally distributed random variables with

mean 0 and variance s2.Define the sets Cy ¼ fc 2 RNþ : VaR1�bðcT yÞrKg, and

Cx ¼ fc 2 RNþ : VaR1�bðcT xÞrKg. From the previous argu-ment, it follows that Cx � Cy . Now, Cy is defined by theelements c that satisfy

PiðciÞ

2rr, for some 0oro1, andsince this is a compact set, and the objective function is

continuous, the supremum of the relaxed problem

(over Cy) is bounded above, which obviously then is also

true for the intermediary’s problem. Moreover, it is clear

that Cx is a closed set, and (since Cx � Cy), therebycompact. By continuity, it follows that the supremum to

the intermediary’s problem is achieved.

We will show that the p.d.f. of a uniform portfolio,

c¼ ð1=NÞ1, takes the form (30). Therefore, given thatK40, it is always possible to choose k=K, find a strictly

positive e, and choose a uniform portfolio c¼ ðe=NÞ1, suchthat the constraint (12) is satisfied. Clearly, for such a

portfolio, (9) is strictly positive, so c¼ 0 is not a solutionto the optimization problem.


Given that there is a strictly positive solution to

Eqs. (9)–(12), we now analyze its properties. It is easy to

see that (10) must be binding with k¼ K , since theprogram is homogeneous of degree one in k, i.e., assuming

that koK , c is a solution with r¼ dcT 1=ð1�dPð�cT x4kÞÞ40, then choosing capital K, and portfolio c2 ¼ ðK=kÞcwill give ru¼ dcT21=ð1�dPð�cT2x4KÞÞ ¼ ðK=kÞdcT 1=ð1�dPð�cT x4kÞÞ ¼ ðK=kÞr4r, yielding a contradiction. Thus,property (c) is proved.

We next show that it is always optimal for the

intermediary to choose a uniform portfolio, c¼ c1 forsome c40. Given a normally distributed risk, y�Nð0,s2Þand a value at risk requirement of K0 at the 1�bconfidence level, it immediately follows that the max-

imum size of investment in this risk, c, and thereby the

maximum premium collected (that does not break the

VaR requirement) is

c¼ K0F�1ð1�bÞ

� 1s

, ð27Þ

where F is the standard normal c.d.f.: FðxÞ ¼R x�1 e

�t2=2=ffiffiffiffiffiffi2pp

dt. From (27) it follows that any

VaR-constrained portfolio manager, whose compensation

is monotonically increasing in scale, cT1—e.g., linear as

assumed—who chooses among portfolios of multivariate

normally distributed risks, will choose the portfolio with

the lowest variance, s.Given r, standard matrix algebra yields that

S¼ covðA�1x,A�1xÞ ¼ ð1�r2ÞA�1A�T . Also, for large N,standard theory of Toeplitz matrices (see, e.g., Gray,

2006) implies that Sij-Ŝ ij ¼ rji�jj. Therefore, property(a) holds.

Conditional on r, to choose the portfolio with thelowest variance, the manager solves minq2Sn qTSq, whereS depends on r. We have S�1 ¼ ð1=ð1�r2ÞÞAT A. Thesolution to the portfolio problem is

q¼ S�11

1TS�11¼ A

T A1

1T AT A1¼ s

sN1¼ 1

N1:

Here, the result follows, since 1 is an eigenvector to ATA

with nonzero eigenvalue, s, since A is invertible. The

manager thus invests uniformly, which is not surprising

since the risks are symmetric. This is the optimal strategy

regardless of the correlation, r, so even with uncertaincorrelations, it is always optimal to choose the uniform

portfolio.

We next study the distribution of the uniform portfolio.

We define xN ¼ ð1=ffiffiffiffiNpÞ1TNxN , where xN on the right-hand

side contains N elements. The c.d.f. of xN is

FNðyÞ ¼PðxN ryÞ. From the previous argument, it is clearthat the agent’s program with N risks simplifies to

maxcNdcN

1�dFNK

cN

� � s:t:, ð28Þ

FNK

cN

� �Z1�b: ð29Þ

Conditional on r, xN is normally distributed withvariance s2NðrÞ ¼ ðð1�r2Þ=NÞ1

T A�1A�T 1. However, since

ATA is a symmetric positive definite matrix and 1 is an

eigenvector with corresponding eigenvalue ð1�rÞ2, itfollows that 1 is an eigenvector to A�1A�T with

eigenvalue 1=ð1�rÞ2. Therefore, s2NðrÞ ¼ ðð1�r2Þ=NÞ1T

A�1A�T 1¼ ðð1�r2Þ=NÞ1T ð1=ð1�rÞ2Þ1¼ ð1þrÞ=ð1�rÞ.Using (5), we immediately obtain that the p.d.f. of xN is

f0ðyÞ ¼Z 1

1

1ffiffiffiffiffiffiffiffiffi2pvp e�y2=2v g

vgþ1dv

¼ g2gffiffiffiffipp �

G gþ12

,y2

2

� �jyj1þ2g

: ð30Þ

It is easy to show that f0(y) is decreasing in y 2 ð0,1Þ, thatis, f0(y) is unimodal and symmetric (about 0): this

conclusion may be obtained directly or by noting that

f0(y) is a scale mixture of normal distributions and

conditioning arguments (see, for instance, the proof of

Theorems 5.1 and 5.2 in Ibragimov, 2009b).

We study the optimization problem (28) and (29), and

rewrite it as

maxzKd

z

1

1�dFðzÞs:t:, ð31Þ

FðzÞZ1�b, ð32Þ

where FðyÞ ¼R y�1 f0ðzÞdz. The constraint that b is close to

zero now implies that the minimal feasible z, such that

(12) is satisfied, is large.

Clearly, ðKd=zÞð1=ð1�dFðzÞÞÞ is positive for all z. We notethat ðKd=zÞð1=ð1�dFðzÞÞÞ becomes arbitrarily large as z-0,so if there was no VaR constraint, it would be optimal to

take on an arbitrarily large amount of risk.

The derivative with respect to z is

dKd

z

1

1�dFðzÞ

� dz

¼ Kdz2ð1�dFðzÞÞ2

ðdFðzÞ�1þzdf0ðzÞÞ:

Now, since FðzÞr1, do1 and zf 0ðzÞ-0 for large z,dFðzÞ�1þzdf0ðzÞrd�1þzf 0ðzÞo0 for large z. Therefore,d½ðKd=zÞð1=ð1�dFðzÞÞÞ�=dzo0 for large z. The feasible setfor z is ½z,1Þ, for some z40, and the VaR constraint isbinding if z¼ z is chosen. When b is close to zero, z is largeand therefore d½ðKd=zÞð1=ð1�dFðzÞÞÞ�=dzo0 for all feasiblez. Therefore, the optimum is reached at z and the VaR

constraint, (32), is indeed binding. We note that this

argument does not depend on the distribution, so the VaR

constraint will also be binding when we study the

diversified case. We have shown property (d).

This binding VaR constraint corresponds to c¼K=F�1ð1�bÞ. Since there is a unique optimum, the samecN c in (28) will be chosen regardless of N, which via(30) immediately implies property (b) (with b¼ 1=cÞ.


Finally, it is easy to check that for g41,VarðxNÞ ¼

R1�1 y

2f0ðyÞ dy¼ g=ðg�1Þ, which immediatelyleads to property (e). We are done. &

Proof of Theorem 2. Define the tail probabilitiesFðxÞ ¼Pð�x14xÞ and FMðKÞ ¼Pð�

PMm ¼ 1 xm=M4KÞ.

From Feller (1970, pp. 275–279), it follows that since x1is of Pareto-type with F1ðKÞ ¼ ðð1þoð1ÞÞ=KaÞ‘ðKÞ, asK-þ1, for some slowly varying function, ‘ðKÞ,

FMðKÞ ¼M1þoð1Þ

MK

� �a‘ðMKÞ, ð33Þ

as K-þ1. For risk-sharing to be optimal from anintermediary’s perspective, (18) is therefore equivalent to

M1þoð1Þ

MK

� �a‘ðMKÞo 1þoð1Þ

Ka‘ðKÞ,

which implies

M1�ao1þoð1Þ,as K-þ1, which, for large K, holds if and only if ao1.We have proved the first part of the theorem.

For the second part, since q¼ 1�FðKÞ and qM ¼ 1�FMðKÞ,(21) and (33) together imply that risk-sharing from

society’s perspective is optimal if and only if

ð1þd�dFMðKÞÞð1þlFðKÞMÞ4 ð1þd�dFðKÞÞð1þlFMðKÞÞ,

as K-þ1, which is satisfied if and only if

dFðKÞ4dFMðKÞþlð1þdÞFMðKÞ ¼ dZFMðKÞ:

This is equivalent to

1þoð1ÞKa

‘ðKÞ4ZM 1þoð1ÞMK

� �a‘ðMKÞ,

as K-þ1, which, in turn, is satisfied if and only if, forlarge K,

Ma�14Z: ð34Þ

Since ZZ1, (34) is never satisfied for M41 and ao1, sorisk-sharing is never optimal from society’s perspective

when ao1. For a41, on the other hand, (34) is satisfied ifand only if

M4Z1=ða�1Þ: ð35Þ

We have proved the second part of the theorem. &

Proof of Theorem 3. Define FMðKÞ ¼Pðð�PM

m ¼ 1 xm=MÞ4KÞ. We first show that ZFMðKÞoFðKÞ is a sufficientcondition for (21). We have

ZFM oF3dFMþð1þdÞlFM odF) 1þd�dFþð1þdÞlFM�dFlFM o1þd�dFM3ð1þd�dFÞð1þlFMÞo1þd�dFM

31þd�dFo 1þd�dFM1þlFM

31þdqo 1þdqM1þlð1�qMÞ

) 1þdq1þlð1�qÞM

o 1þdqM1þlð1�qMÞ

:

To show that ZFMðKÞoFðKÞ, we use the following lemma.

Lemma 1 (Nagaev, 1979, Corollary 1.11). For independent

random variables Xi, with E½Xi� ¼ 0 and

PMi ¼ 1 E½X2i � ¼ BM , it

is the case that

PXMi ¼ 1

Xi4x

!rXMi ¼ 1

PðXi4yÞþex=yBMxy

� �x=y, x,y40:

ð36Þ

Using (36) for the x�risks, for which BM ¼Ms2, bychoosing x=MK and y¼ 2MK=a, we get

FMðKÞrMF2MK

a

� �þea=2sa 2Ma

� ��a=2K�a:

From (25), it then follows that, for K4K0,

FMðKÞrMc2aað2MKÞ�aþea=2sa2M

a

� ��a=2K�a,

and since FðKÞZc1K�a, it follows that the condition

Z Mc2aað2MKÞ�aþea=2sa2M

a

� ��a=2K�a

!rc1K�a ð37Þ

implies ZFMðKÞoFðKÞ. Now, by multiplying (37) byð2MKÞa, we get

Mc2ZaaþZea=2sað2MaÞa=2rc12aMa,

i.e.,

c12aMa�CZMa=2�MZc2aaZ0:

We are done. &

List of variables

M number of risk classesN number of risks within each risk classxt,mn individual risk n, within risk class m at time tr correlation between risks within risk class

(for large N)x vector of individual risks in first risk classc vector of investments of intermediary (in first

risk class)c total size of risk investment, c¼ cT 1x,xi risk portfolio invested in by intermediary

(in risk class i), x¼ cT xd one period discount factorK capital reserved by intermediaryb value at risk at 1�b confidence level may not be

higher that K, VaR1�bðxÞrKQ outcome of intermediary’s option to defaultMn diversification threshold, above which society

prefers risk-sharingT recovery time after massive defaultd premium intermediary collects per unit risk of

investmentg parameter governing uncertainty of correlations

between risks in same risk classa tail distribution of risk class (x). a¼ 2gf probability distribution function of portfolio

risk, x


F cumulative loss distribution of portfolio risk,FðxÞ ¼Pð�x4xÞ

q chance for individual intermediary of not de-faulting when risks are separated q¼ 1�FðKÞ

FM cumulative loss distribution of fully diversifiedportfolio FMðxÞ ¼Pð�

Pixi=M4xÞ

qM chance for intermediaries of not defaulting whenrisks are shared qM ¼ 1�FMðKÞ

l society’s trade off between risk-sharing andseparated outcomes (Eq. (22))

Z Z¼Ma�1� , a41VB value of one intermediary if separatedVS value of one intermediary if risk-sharingVBM value to society if separatedVSM value to society if risk-sharing

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Diversification disastersIntroductionNotationModelOptimal behavior of intermediaryThe value to society

Risk-sharingThe value to society

Extensions to general distributions

Potential implications for risk management and for policy makersConcluding remarksProofsList of variablesReferences

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