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New Frontiers in Practical Risk Management

English edition Issue n. 3 - Summer 2014

Iason ltd. and Energisk.org are the editors of Argo newsletter. Iason is the publisher. No one is al-lowed to reproduce or transmit any part of this document in any form or by any means, electronicor mechanical, including photocopying and recording, for any purpose without the express writtenpermission of Iason ltd. Neither editor is responsible for any consequence directly or indirectly stem-ming from the use of any kind of adoption of the methods, models, and ideas appearing in the con-tributions contained in Argo newsletter, nor they assume any responsibility related to the appropri-ateness and/or truth of numbers, figures, and statements expressed by authors of those contributions.

New Frontiers in Practical Risk ManagementYear 1 - Issue Number 3 - Summer 2014

Published in June 2014First published in October 2013

Last published issues are available online:www.iasonltd.comwww.energisk.org

Summer 2014

NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT

Editors:Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Andrea RONCORONI (ESSEC Business School, Paris)

Executive Editor:Luca OLIVO (Iason ltd)

Scientific Editorial Board:Alvaro CARTEA (University College London)Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Mark CUMMINS (Dublin City University Business School)Gianluca FUSAI (Cass Business School, London)Sebastian JAIMUNGAL (University of Toronto)Fabio MERCURIO (Bloomberg LP)Andrea RONCORONI (ESSEC Business School, Paris)

Iason ltdRegistered Address:6 O’Curry StreetLimerick 4Ireland

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Iason ltd and Energisk.org are registered trademark.

Articles submission guidelinesArgo welcomes the submission of articles on topical subjects related to the risk management. Thetwo core sections are Banking and Finance and Energy and Commodity Finance. Within these twomacro areas, articles can be indicatively, but not exhaustively, related to models and methodologiesfor market, credit, liquidity risk management, valuation of derivatives, asset management, tradingstrategies, statistical analysis of market data and technology in the financial industry. All articlesshould contain references to previous literature. The primary criteria for publishing a paper are itsquality and importance to the field of finance, without undue regard to its technical difficulty. Argois a single blind refereed magazine: articles are sent with author details to the Scientific Committeefor peer review. The first editorial decision is rendered at the latest within 60 days after receipt of thesubmission. The author(s) may be requested to revise the article. The editors decide to reject or acceptthe submitted article. Submissions should be sent to the technical team ([email protected]). LaTex orWord are the preferred format, but PDFs are accepted if submitted with LaTeX code or a Word file ofthe text. There is no maximum limit, but recommended length is about 4,000 words. If needed, forediting considerations, the technical team may ask the author(s) to cut the article.

Summer 2014


Table of Contents

Editorial pag. 5

energy & commodity finance

Security of Supply and the Cost of Storing Gas pag. 7

Alvaro Cartea, James Cheeseman and Sebastian Jaimungal

Monetary Measurement of Risk:A Critical Overview - Part II: Coherent Measure of Risk pag. 13

Lionel Lecesne and Andrea Roncoroni

banking & finance

Towards a Theory of Internal Valuationand Transfer Pricing of Products in a Bank pag. 21

Antonio Castagna

special interview

Alessandro Mauro pag. 46LITASCO International Trading and Supply Company - Lukoil Group, Geneva

Front Cover: Ivo Pannaggi Treno in corsa, 1922.

EDITORIAL

Dear Readers,we reached the third issue of Argo newsletter with

great results. After the interesting contributions of theSpring publication, we are ready to reveal the latest de-velopments in the fields of banking and energy finance.

This time we start with the Energy & Commodities finance sectionthat raises with a contribution of three well-known personalities: Ál-varo Cartea, James Cheeseman and Sebastian Jaimungal put forwarda method devised to evaluate gas storage where a least-square MonteCarlo simulation method is used to solve the underlying optimizationproblem. It follows the second part of the crash course put forwardby Lionel Lecesne and Andrea Roncoroni: these authors presentthe celebrated class of "coherent" measures of risk and provide thereader with a critical view of the axioms underlying this framework

In this issue the Banking & Finance section is totally fo-cused on transfer pricing topic. In an interesting contribu-tion Antonio Castagna provides an innovative theory of in-ternal valuation and transfer pricing of products in a bank.

In a dedicated section, Andrea Roncoroni interviewsAlessandro Mauro (LITASCO company of Lukoil Group,Switzerland) about importance and evolution of finan-cial and physical risk management in the energy sector.

We conclude by encouraging submission of contributions forthe next issue of Argo Newsletter. Detailed information aboutthe process is indicated at the beginning. Thanks for download-ing Argo: feel free forwarding the publication’s web link to who-ever might be concerned with any of the subject dealt with.

Enjoy your reading!Antonio CastagnaAndrea Roncoroni

Luca Olivo

Summer 20145


Energy & CommodityFinance

Energy Storage Valuation

Crash Course Part II:Monetary Measurement of Risk

6

Security of Supply and theCost of Storing Gas

The article offers an interesting insighton the importance of building gas stor-age facilities. There is a clear relation-ship between the capacity to store gasand the stability of gas prices. The fo-cus of the research presented here ison the financial value of the gas stor-age: the authors show how to valuestorage facilities using Least SquaresMonte Carlo (LSMC) without excludingrealistic constraints from the valuation.

Álvaro CARTEAJames CHEESEMANSebastian JAIMUNGAL

In 2009 Europe saw how the Ukraine-Russiagas crisis unraveled. After protracted nego-tiations over natural gas contracts, Russia’senergy company Gazprom cut gas supplies

to the Ukraine which also affected large parts ofsoutheast Europe. During this period, Ukraine hadto survive on stored gas which would not last for-ever. These events highlighted the importance ofguaranteeing stability of gas flows and that duringtimes of duress, gas storage facilities play a decisiverole to meet demand.

The case for building gas storage facilities ismade on the back of both security of supply andto smooth prices at different time scales. Gas stor-age smooths prices across seasons because demandgenerally exhibits a strong seasonal pattern, andalso smooths variations in prices across days byserving as a buffer to absorb unforeseen changes indaily supply and demand. Clearly, the greater thecapacity to store gas, the more stable are prices.

However, there are two key questions. First,what is the cost of building a storage facility? Sec-ond, how much is a gas storage facility worth? Thefirst question addresses the actual cost to build stor-age, and the second question, which is the subjectmatter of this article, addresses the financial valuethat is obtained from operating the facility. There-fore, only when the financial value of the storage isgreater than the costs of building it, should storagebe built.

In this article we show how to value a storage fa-cility using Least Squares Monte Carlo (LSMC). Weshow how to incorporate realistic constraints in thevaluation including: the maximum capacity of thestorage, injection and withdrawal rates and costs,and market constraints such as bid-ask spread inthe spot market and transaction costs.

Valuation of Gas Storage

Central to the problem of gas storage is its finan-cial valuation not only for investment decisions,but also for operating and strategies, and hedgingdecisions for those wishing to trade around suchassets. There are two approaches to the valuationof storage: intrinsic and extrinsic.

The value of a storage facility, like any optionis its premium, which is made up of two parts, in-trinsic and extrinsic. Relating this to a standardAmerican option we have that the intrinsic value isthe payoff from immediate exercise and the extrin-sic value is the additional future exercise value ofthe option, the possibility of increasing the payoff.The simplest approach to valuing storage, but onewhich ignores future exercise rights, is to computethe intrinsic value given the futures curve at thevaluation date. This is the optimal combination offutures contracts which can be put in place, whilerespecting all the physical constraints of the storage

Summer 20147

ENERGY STORAGE VALUATION

facility, to extract value from the calendar spreadsof the futures curve.

The extrinsic value represents the optionality inthe physical asset and together with the intrinsicvalue this makes up the premium. The extrinsicvalue is that of the flexibility of the storage, theability to trade the daily volatility. An extrinsic val-uation can increase the value of storage from a fewpercent over the intrinsic to multiples of the intrin-sic value depending upon the physical constraintsof the facility and the dynamics of the underlying.

Least Squares Monte Carlo

Francis and Schwartz [5] originally applied theLSMC algorithm to valuing Bermudan options.Subsequently it has been extended and appliedas a method for pricing swing options by AlfredoIbáñez [1], and further to storage valuation by Car-mona and Ludkovski [3] and Boogert and de Jong[2].

Reviewing the algorithm as presented in [5] wesee that the intuition behind the least squares ap-proach is best appreciated through the original ap-plication, an American-style option. The holder ofsuch an option may exercise at any time prior tomaturity, and at every potential exercise time thedecision whether to exercise or not depends uponassessing the value of continuing without exercis-ing, the continuation value, against that of exercis-ing immediately, the intrinsic value. Longstaff andSchwartz use a simple least squares regression ofthe simulated future cashflows on to the currentsimulated stock prices to obtain this continuationvalue as a conditional expectation and hence theoptimal exercise price (at that time) that maximisesthe option value.

As in any Monte Carlo simulation of a pathdependent derivative the time to maturity is dis-cretised, into k discrete potential exercise times0 < t1 ≤ t2 ≤ . . . ≤ tk = T and the decision aswhether to exercise or not is taken at every pointby comparing the intrinsic value with the valueof continuation. Using the notation from [5] wehave that when C(ω, s; t, T) is the path of cashflowsgenerated by the option, conditional on the optionhaving not been exercised at or prior to time t andon the holder of the option following the optimalstopping strategy for all s (t < s ≤ T), and whereω is a sample path, the value of continuation attime ti, can be expressed as the risk neutral expecta-tion of the discounted future cashflows. We denotethis continuation value as Cont(ω; ti) and the above

argument is represented formally as

Cont(ω; ti) = EQ

[k

∑j=i+1

e−r(tj−ti)C(ω, s; t, T)∣∣∣∣Fti

]

where r is the risk free rate and Q the risk neutralpricing measure and Ft is the filtration at time trepresenting the current futures forward curve.

LSMC works recursively through the discretisedtimes using least squares regression to estimatethe conditional expectation at tk−1, tk−2, . . . , t1. TheLSMC is a recursive process as at each time the de-cision to exercise could change all the subsequentcashflows and therefore C(ω, s; ti, T) is not neces-sarily the same as C(ω, s; ti−1, T). The initial step isalways to define the cashflows at the final timesteptk and for an American option this is the intrinsicvalue of the option at maturity.

Longstaff and Schwartz give an example usingthe set of Laguerre polynomials as the basis func-tions for the regression and with these polynomials,the estimate of Cont(ω; ti) is then represented as:

ContM(ω; ti) =M

∑j=0

ajLj(X)

where aj and Lj are constant coefficients and the La-guerre polynomial terms respectively, M the num-ber of basis functions and X are the underlying spotprices. They report that if M > 2 the accuracy ofthe algorithm is not significantly dependent on thetype or number of basis functions used, but morerecent research show that accuracy does depend onthe choice of basis functions, see [7].

In [5] ContM(ω; ti) is estimated by regressingthe discounted values of C(ω, s; ti, T) onto the ba-sis functions for the paths which are in the money,since the decision on whether to exercise or not isonly required if the option has an intrinsic valuegreater than zero. Using only in-the-money pathsreduces the number of basis functions requiredfor an accurate estimation of the continuation val-ues.1 With the conditional expectation estimatedwe can decide if exercise is optimal at ti for eachin-the-money path by comparing the intrinsic valueagainst ContM(ω; ti). The algorithm continues re-cursively until we have the optimal decision foreach point. We then discount the cashflow fromeach exercise point and take the average to find thevalue of the American put option, thus:

CAt =

1n

p=n

∑p=0−e−r(τp−t)(K− Sτp),

where n is the number of paths.1In a storage problem we cannot make this simplifying assumption as all the cashflows which can be zero, positive or negative,

must be considered.

8energisk.org

It is important to note that when optimal ex-ercise is indicated at ti the subsequent cashflowsare altered to zero, as if exercise has taken place atti then by definition it cannot take place at timesti < t ≤ T. It is these altered cashflows whichfeed into the next iteration where ContM(ω; ti−1) isestimated.

Extending the LSMC to price Gas Storage

Here we discuss how to use the LSMC approach inthe valuation of gas storage, for other extensionsand the pricing of gas interruptible contracts see[4]. If for example, the LSMC is used to priceAmerican-style options, the algorithm would re-quire a matrix of two dimensions, time and numberof simulated paths, where on each path an estimateof the optimal exercise time, a stopping rule for theoption, is obtained. In application to multiple ex-ercise problems we must create a third dimension,for swing options this is the number of exerciserights. We then proceed as follows, suppose wehave a swing option with N up-swing rights thenwe create a standard two dimensional LSMC ma-trix corresponding to each N, N − 1, . . . , 1 rightsremaining and working recursively, our initial stepis to set the final cashflow conditions. These aredifferent from the American option as we havemultiple exercise opportunities and therefore if wehave N rights remaining the final N timesteps inthe Nth matrix must be assumed to be exercisepoints, intuitively if the holder of a swing optionis three days away from maturity and has threeexercise rights left he will try and exercise on everyday. The continuation values are estimated andthen we must make the optimal exercise decision,but we must compare values from across the thirddimension of our LSMC matrix.

Suppose we have N swing rights remaining, ifwe want to decide whether to exercise one of ourN rights at time ti, then we compare the value ofnot exercising and continuing with N rights withthe value of exercising and continuing with N − 1rights. Introducing ContN

M(ω; ti) as the continua-tion value with N rights remaining at time ti, andrecall that ω is a sample path, we exercise a swingoption with N rights remaining at time ti if:

ContNM(ω; ti) < I(ω; ti) + ContN−1

M (ω; ti)

where I(ω; ti) is the intrinsic value of exercising.That is, if the value of the exercising one of ourswing rights, I(ω; ti) in addition to the value ofcontinuing with one less right, ContN−1

M (ω; ti) ex-ceeds the value of not exercising and continuingwith the current number of rights, then we exercise

one right.

The many possible constraints of swing options,penalty functions, maximum number of upswingsand downswings, all add complexity to the prob-lem and change the way the algorithm proceeds.For instance, if a penalty is payable when the max-imum number of swings is not reached then thefinal cashflow may be negative as it may be optimalto pay the penalty rather than suffer a loss fromexercising, for other approaches see [6].

Storage LSMC

A storage facility is much like a swing option withan up- and down-swing option for every day ofthe contract, for instance, on any day we may buygas and inject it into storage or withdraw gas andsell it to the market. The injection and withdrawallimits are governed by ‘ratchets’ which are volumedependent and unique to each type of storage fa-cility. These occur as the inventory level increasesand it becomes harder to inject more gas and as theinventory decreases it becomes harder to withdrawgas.

Our action is also governed by minimum andmaximum inventory levels that apply across theduration of the storage contract, and initial andfinal conditions which stipulate the inventory statein which the facility is obtained and must be re-turned. There may also be a penalty, dependent onthe volume level and spot price for breaking anyof these constraints. The type of facility may alsodictate fixed injection and withdrawal costs whichpay for the additional energy required to operatein injection or withdrawal mode. In addition to thephysical constraints we are also governed by mar-ket constraints such as a bid-ask spread, transactioncosts and of course the discount rate.

To value a storage facility we again set up athree dimensional LSMC matrix with the third di-mension being the volume of gas in the storagefacility. In this way we can accommodate the aboveconstraints by making the appropriate comparisonswith continuation values across the different vol-ume levels.

So working recursively from maturity back totime zero, at inventory level V, we compare threepossible actions, inject volume vi and continue atinventory level V + vi, take no action and continueat inventory level V, or withdraw volume vw andcontinue at inventory level V − vw. Obviously thephysical constraints above, ratchets, max and mininventory levels and the initial and terminal con-straints define the allowed sets of vi and vw fromwhich we can choose.

Summer 20149

ENERGY STORAGE VALUATION

20 40 60 80 100 120 140 160 1800

2

4

6

8

10x 10

7

Day Index (1 Jan 08 − 30 Jun 08)

Sto

rage

Val

ue

(p)

20 40 60 80 100 120 140 160 18030

40

50

Fw

d P

ricePremium with Jumps

Premium w/o jumps

Intrinsic

Fwd Curve

FIGURE 1: Storage valuation in pence, intrinsic and extrinsic with and without jumps, LSMC valuation with 1,000 paths.

To summarise we value a storage facility subjectto the following constraints. Vmax: The maximumcapacity of the storage facility. Vi: The inventorylevel at which the storage contract is initiated. Vf :The inventory level at which the storage contractmust be returned at the end of the contract. vinj(V)& vwdraw(V): The ratchets, i.e. the inventory leveldependent injection and withdrawal rates. Cinj &Cwdraw: Fixed injection and withdrawal costs inpence per therm. BA, Ctrans & r: The market con-straints, bid-ask spread, transaction costs and thediscount rate respectively.

Storage Value

We value a contract on a storage capacity of 1 mil-lion therms with daily injection and withdrawalrates of 100,000 therms, we receive and must returnthe facility empty. We value a contract durationof 1 year, with a daily trading frequency and zerointerest rates, across a range of contract start datesfrom 1st January to June 30th 2008. In figure 1 weshow the intrinsic value and the premium with andwithout jumps added to the price process, we alsoshow the futures curve to which the spot price wascalibrated.

In Figure 1 we see that the most basic indicatorof the storage value, the intrinsic value, is constantacross the contract start dates. The intrinsic value isthe optimal combination of futures contracts givena static view of the futures curve on a particular,and subject to the physical constraints of the storagefacility. As the gas futures curve is dominated by arepeating seasonal pattern the relative difference inprices across the year is constant as we change the

start date of our storage contract. This results in aconstant intrinsic value regardless of any contangoor backwardation inherent in the market.

In the range of Spring start dates, where we seea higher spot price, the premia (with and withoutjumps) are lower than spring and summer startdates. When we start a storage contract with zeroinventory our first action can only be to inject gas.If we start the contract in Spring we wait until thelow summer prices to fully fill our facility then sell-ing the following Spring. If, in contrast, we starta contract in the summer the situation is reversedand we immediately fill our storage and sell thefollowing Spring. Throughout the duration of boththese cases we trade not only the overall calendarspread but the daily volatility as well. The calendarspread is constant regardless of when the contractstarts as it depends only the relative difference ofsummer and Spring prices. This implies that thedifference in storage premia across the year mustarise from the way in which the daily volatility istraded, specifically from the increased flexibility ofhaving more gas at an earlier date in the case of asummer start contract.

In respect of the jumps our path simulation withjumps give spot prices in the range of 150-200 penceper therm, and approximately 7 jumps per year.This is in comparison to normal prices in the rangeof 7-100 pence per therm. With this comparisonin mind we see that the majority of the extrinsicvalue comes from the volatility of the price pro-cess rather than the jumps, as the extrinsic valueshows an increase of 2.5 × 107 over the intrinsicvalue, whereas the extrinsic with jumps shows only

10energisk.org

a further 1× 107 increase.

ABOUT THE AUTHORS

Álvaro Cartea University College London

Email address: [email protected]

James Cheeseman BP plc


Sebastian Jaimungal University of Toronto


ABOUT THE ARTICLE

Submitted: January 2014.Accepted: February 2014.

References

[1] Alfredo Ibáñez. Valuation by simulation of contin-gent claims with multiple early exercise opportunities.Mahtematical Finance, v.14, n.2, pp. 223-248. April 2004.

[2] Boogert, A. and C. de Jong. Gas Storage Valuation Usinga Monte Carlo Method. The Journal of Derivatives, v.15,n.3, pp. 81-98. Spring 2008.

[3] Carmona, R. and M. Ludkovski. Gas Storage and Sup-ply Guarantees: An Optimal Switching Approach. Un-published. 2007.

[4] Cartea, Á. and T. Williams. UK Gas Markets: the MarketPrice of Risk and Applications to Multiple InterruptibleSupply Contracts. Energy Economics, v.30, n.3, pp. 829-846. May 2008.

[5] Francis A. L. and E. S. Schwartz. Valuing American Op-tions by Simulation: a Simple Least-Squares Approach.The Review of Financial Studies, v.14, n.1, pp. 113-147.Spring 2001.

[6] Jaimungal, S. and V. Surkov Lévy-Based Cross-Commodity Models and Derivative Valuation. SIAMJournal on Financial Mathematics, v.2, n.1, pp. 464-487.2011.

[7] Stentoft, L. Assessing the Least Squares Monte-Carlo Ap-proach to American Option Valuation. Review of Deriva-tives Research, v.7, n.2, pp. 129-168. 2004.

Summer 201411

ADVERTISING FEATURE

12

Monetary Measurement ofRisk: a Critical OverviewPart II: Coherent Risk Measures

In this second part of the course Li-onel Lecesne and Andrea Roncoroni in-troduce the subclass of coherent mea-sures of risk introduced by Artzner et al.(1999), providing with specific properties,definitions and examples from this class

Lionel LECESNEAndrea RONCORONI

In Lecesne and Roncoroni [4], we introduce thenotion of monetary measure of risk borne by anyfinancial claim. Our presentation moves from

general definitions to concrete instances, includingthe benchmark measure Value-at-Risk (VaR). PartII develops a treatment of the class of coherent (mon-etary) measures of risk put forward by Artzner et al.[2]. Our goal is to illustrate the main features ofthis class of measures in light of application to prac-tical cases. In this respect, we put our focus on theambiguity of the term “coherent” and show thatthe connotation of consistency it naturally embedsis more an issue of lexical interpretation than actualmeaning. As an example, we show that lack of sub-additivity of VaR (which prevents from it to beinga “coherent” measure of risk) is more a desirableproperty than a drawback, as is claimed in most ofexisting sources in the specialized literature.

Coherent Measures of Risk

At the end of the nineties, risk measurement theorywas suffering from lack of axiomatic foundations.

Artzner et al. [2] propose a number of “economi-cally desirable” axioms that a “coherent” monetarymeasure of risk ρ ought to satisfy. These propertieslead to the definition of a class of risk measuresreferred to as coherent by the authors.

Since “coherent” measures of risk constitute asubclass of monetary risk measures, they obviouslymeet the anti-monotonicity and translational anti-variance requirements. We reproduce their state-ment here for the reader’s convenience (Lecesneand Roncoroni [4]):

- Anti-monotonicity (AM) : for any pair ofpositions X, Y ∈ X , with X ≤ Y:

ρ(X) ≥ ρ(Y).

- Translational anti-variance (TA) : forany pay-off X ∈ X and cash amount m > 0:

ρ(X + m) = ρ(X)−m.

To these properties, one requires that combining po-sitions does not generate additional risk. Embrechtset al. [3] provide three qualitative arguments under-pinning this assumption, which is often labelled as“subadditivity”:

1. Diversification effect. The property reflectsa well-known principle whereby risk can bereduced through diversification;

2. Merging incentive. If a regulator imposes ameasure of risk which might entail risk in-crease upon merging, then financial institu-tions might get an incentive to break up intosubsidiaries in order to reduce regulatory cap-ital requirements stemming from risk assess-ment.

Summer 201413

CRASH COURSE

3. RM decentralization. Subadditivity ensuresthat the risk borne by any financial institu-tion is superiorly bounded by the sum of riskfigures of composing desks. Therefore, lim-iting these latter offers control to the overallrisk, thus making of risk decentralization aneffective risk management policy.

If we denote a coherent measure of risk as ρcoh :X → R, the previous property can be stated asfollows:

Subadditivity (SUB) : for any pair of posi-tions X, Y ∈ X ,

ρcoh(X + Y) ≤ ρcoh(X) + ρcoh(Y);

One further asks for factoring the size of a po-sition out of the corresponding risk assessment.In particular, doubling a financial position exactlydoubles the initial risk.

Positive homogeneity (PH) : for anyX ∈ X , λ ≥ 0,

ρcoh(λX) = λρcoh(X).

We are led to the following:

Definition. A Coherent Measure of Risk ρcoh :X → R is a subadditive and positively homogenousmonetary measure of risk.

Spectral Measures of Risk)

Expected Shortfall (or, Conditional VaR)

Expected Shortfall (ES) is probably the most usedrisk measure after VaR. It is anti-monotone,translational-antivariant, and positively homoge-neous. The main theoretical advantage over VaRis that it is subadditive, hence coherent. From apractical point of view, a major benefit over VaR liesin the kind of information it provides to the user.Recall that VaR delivers an assessment about theminimal size of losses a portfolio may experienceat a defined level of confidence. Consequently, noinformation is provided about the severity of theactual losses exceeding a VaR threshold. Given aconfidence level, ES provides a first estimate of thisseverity by delivering the expected value of lossesconditional to exceeding the VaR threshold.

VaR may be defined either as the α-quantile ofa loss distribution, i.e., F−1

−X(α) := inf{x ∈ R :P [−X ≤ x] ≥ α} or as the (1 − α)-quantileof the P&L distribution, i.e., −F−1

X (1 − α) :=

−inf {x ∈ R : P [X ≤ x] ≥ 1− α}, where α ∈ [0, 1]is generally taken to be close to 1. In what follows,we will retain the second of these definitions.

Definition. For a random P&L X ∈ X with con-tinuous distribution function and a confidence levelα ∈ [0, 1], the 100α% Expected Shortfall of X is de-fined as the expected loss (i.e., minus P&L) conditionalto it exceeding the 100α% Value-at-Risk:

ES(X) := −E [X|X ≤ −VaRα(X)] , (1)

where VaRα(X) = −F−1X (1 − α). Equation (1)

clearly shows the role played by VaR in the def-inition of ES, which is sometimes referred to asConditional Tail Expectation or Conditional VaR.A more general, although less immediate to grasp,definition of ES is the following one:

ESα(X) := − 11− α

∫ 1

αF−1

X (1− u)du

=1

1− α

∫ 1

αVaRu(X)du,

(2)

where the link to VaR is ever more evident. Thissecond definition is less restrictive since it does notrequire the distribution function to be continuous.

Next, we provide an example based on a simpleposition showing the way ES varies across confi-dence level and volatility of the underlying asset.

An Illustrative Example

A trader buys 1,000 stock shares at a spot priceS0 = $100. He wishes to assess the regulatorycapital associated to his $100,000 investment overa management horizon of T time units. Estima-tions based on past data indicate that absolute pricechanges ∆S := ST − S0 follow a Normal law withzero mean and variance given by σ2. Position’sP&L’s X := 1000∆S are Normal as well, with zeromean and variance equal to (1000σ)2. If capital re-quirement is determined by ES, then formula (2)reads as:

ESα,σ (X) = − 11− α

∫ 1

α1000F−1

∆S (1− u)du,

= −1000σ

1− α

∫ 1

αΦ−1(1− u)du,

where F∆S is the cumulative probability distributionof ∆S, Φ denotes the standard normal distribution,and the subscripts explicitly indicate dependenceon confidence level α and volatility σ.

Figure 1 exhibits a 3-dimensional plot of ESacross varying levels of confidence threshold andasset volatility. We note that:

14energisk.org

FIGURE 1: ES of a normal centered P&L with varying volatility (sigma) and confidence level (alpha)

FIGURE 2: VaR, ES and SMRs of a normal centered and reduced P&L

Summer 201415

CRASH COURSE

1. ES is positive, and so is capital requirement.In other terms, cash withdrawal is never au-thorized, as opposed to what might be thecase if one uses VaR assessments.

2. ES increases along with confidence level α.For a standard deviation σ = 1, we haveESX(α = 0.98, σ = 1) = $2, 401.1 andESX(α = 0.99, σ = 1) = $2, 625.63.

3. ES is proportional to asset volatility. For aconfidence level α =0.95, an increase of σfrom 1 to 5 entails an economic capital risefrom $2,054.8 to $10,274.

General Spectral Measures of Risk

Acerbi [1] generalizes the notion of ES and putsforward the class of Spectral Measures of Risk (SMR).The idea is to compute an appropriate weightedaverage of all possible payoffs stemming from agiven position. Weighing is performed in a wayto emphasize worst outcomes over best ones. Theresulting figure turns out to be “coherent” in thesense of Artzner et al. [2].

Definition. For a random P&L X ∈ X and general-ized inverse cumulative distribution function F−1

X (u) :=inf {x ∈ R : FX(x) ≥ u}, u ∈ [0, 1], a Spectral Mea-sure of Risk of X is defined as:

SMRw(X) := −∫ 1

0w(u)F−1

X (u)du (3)

for each positive weighing function w : [0, 1] → R

satisfying:

1.∫ 1

0 w(u)du = 1 and

2. w(u1) ≥ w(u2) for any 0 < u1 < u2 ≤ 1.

Remark. The weighing function w is called “riskspectrum” and properties 1 and 2 ensure that SMRare “coherent” (Acerbi [1]).

Example. A flexible class of risk spectra is given bythe parametric set of functions:

wγ(u) :=exp(−u/γ)

γ(1− exp(−1/γ)), (4)

where γ > 0 is the parameter allowing for tuningweight assignment to severe losses (the lower γ, thestronger the weight).

Example. Risk spectrum wα(u) = (1/1− α)1u≤1−α

delivers ES:

SMRwα(X) = − 11− α

∫ 1−α

0F−1

X (u)du =: ESα(X).

While ES averages losses beyond VaR threshold, ageneral SMR scans the whole set of quantiles. Akey practical feature of SMR is that the user (orregulator) may tune the weight assignment acrossthe set of potential losses to reflect his own aversionto risk.

A Numerical Example

We now put forward a pictorial comparison of VaR,ES, and some other SMR’s. Figure 2 exhibits fig-ures corresponding to VaR and ES across varyinglevels α of confidence over the interval [0, 1]. Thesegraphs are then superimposed to those related totwo SMR’s. These latter stem from adopting weigh-ing functions defined through formula (4) with tailparameter γ = 0.1 and γ = 1. These paramet-ric instances correspond to great and, respectively,low importance assigned to high level of potentiallosses.

A few comments of this picture are worth ofnoticing:

- ES always exceeds VaR. This property israther obvious. The key point is the mainconsequence it entails: ES is more conserva-tive than VaR, in that it always provides ahigher capital requirement to the user.

- VaR is negative for some values of the confi-dence level α (here α ≤ 0.5), whereas ES is al-ways positive. Again, this observation reflectsthe idea that ES is more prudent than VaR.Indeed, assuming that the portfolio managerwould choose a such low confidence level, thenegative outcome of VaR would allow him towithdraw cash from the position (and henceincrease the risk due to the cash translationalanti-variance property) until VaR attains valuezero. On the contrary, ES is never negativewhich means that it does not authorize towithdraw cash from the position.

- The exact location of SMR relative to bothVaR and ES depends on the value of the tailparameter γ. As previously remarked, regu-latory capital stemming from SMR increasesas long as stronger weight is assigned to highlosses: this corresponds to values of the tailparameter γ near 0.

16energisk.org

On the Consistency of Risk Coherence

VaR may be superadditive. Hence it is not a “co-herent” measure of risk, in the sense of Artzner etal. [2]. An example helps at clarifying why risksuperadditiveness may well be acceptable.

Let banks A and B dispose of $10,000 availablefor lending. Assume they follow distinct creditlending policies: bank A lends the total amountto a single client, whereas bank B grants $1,000to each of ten individual clients. Let all debtorsbe identical, exhibit a probability of default equalto 0.02, and assume one’s default has no impactonto the default probability of the others. The paidback sum amounts to the borrowed cash plus a 5%-interest payment. For reader’s ease of comparison,we reproduce here the definition of VaR:

VaRα(X) := inf {m ∈ R : P [−X ≤ m] ≥ α} . (5)

Let XA denote the bank A random P&L resultingfrom lending. If the sole client defaults, they lose$10,000; otherwise, a benefit of10,000×5 % = $500 occurs upon repayment. Wehave:

{P [−XA = −500] = 0.98 (no default)P [−XA = 10, 000] = 0.02 (default)

(6)

and nesting (6) into formula (5) leads toVaR0,98(XA) = −$500 and VaR0,99(XA)= $10,000.

Let us now look at the risk borne by bank B.Let Xi denote the bank B random P&L stemmingfrom debtor i = 1, ..., 10. If individual i defaults, thebank loses the whole outstanding capital , henceXi = −$1, 000. If they honor their debt, then thebank’s profit equals interest and Xi = $50. Let Dibe a random variable assuming value 1 in case in-dividual i defaults and 0 otherwise. Equation (7)then synthesizes bank B’s P&L relative to the debtof client i:

Xi = 50× (1− Di)− 1, 000× Di. (7)

Since bank B has 10 clients, their total P&L equalsXB = ∑10

i=1 Xi. In regard to equation (7), we mayrewrite:

XB = 500− 10, 50010

∑i=1

Di, (8)

where ∑10i=1 Di follows a binomial law with a num-

ber of trials equal to 10 and probability of success(i.e., default) equals 0.022. Recall that formula (5) isbased on the loss distribution, we hence consider

−XB = 10, 500 ∑10i=1 Di − 500. By positive homo-

geneity and translational anti-variance of VaR, wehave

VaRα(XB) = 10, 500×VaRα(10

∑i=1

Di)− 500. (9)

VaRα(∑10i=1 Di) corresponds to the α-quantile of the

cumulative distribution of ∑10i=1 Di. From a bino-

mial probability table, we get VaR0,98(∑10i=1 Di) = 1

and VaR0,99(∑10i=1 Di) = 2. Finally, by replacing

these values into (9), we find that VaR0,98(XB) =$10, 000 and VaR0,99(XB) = $20, 500.

In agreement with the idea that diversificationreduces risk, bank B has, unlike bank A, cut the$10,000 in order to grant them to several clients.Paradoxically, VaR assessment yields greater riskto bank B than to bank A. This is a consequenceof non-subadditivity of VaR. Table 1 reports VaRoutcomes for this example.

Confidence level α = 0.98 α = 0.99VaRα(XA) -500 10,000VaRα(XB) 10,000 20,500

TABLE 1: VaR outputs for banks A and B with varying confi-dence level.

For both values of confidence level α, capitalrequirement for bank B is more than twice the onefor bank A. It is striking to observe that bank Anegative VaR corresponding to α = 0.98 implicitlyallows an increase of risk exposure by $500, inspite of the fact that bank B is required to reduceexposure by $10,000.

According to Artzner et al. [2], financial riskmetrics ought to be subadditive, among others.This property is grounded on the intuitive assump-tion whereby gathering two positions in a singleportfolio would eventually reduce risk through adiversification effect. In our view, this propertyshould not be universally accepted: although com-bination often reduces overall risk, it may increaseit as well. Examples ranging from corporate fi-nance to energy economics and liquidity theoryshow that a ’snowball’ effect resulting from combin-ing positions may lead to an increase of expectedlosses and the corresponding capital requirementsor reserves. The dependence structure betweenfinancial positions ultimately constitutes the reasonwhy subadditivity (and positive homogeneity) mayneed fail upon risk assessment. Liquidity relatedissue are another source of risk superadditivity:doubling the size of an illiquid portfolio more thandoubles the risk from holding the portfolio. An

2Formally, we denote it as: ∑10i=1 Di ∼ B (10; 0, 02).

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CRASH COURSE

example may help at clarifying the latter point

Consider an investor who needs to immedi-ately cash a portfolio of identical bond issues.Assume that the buy side of the bond order bookis described by the following bids: the best bidquotes $1,000 for each of the first 50 contracts;the second bid quotes $990 for each of the fol-lowing 100 contracts; the third bid quotes $970for each of the last 200 contracts. Assume thatthe risk borne by the portfolio holder is repre-sented by the difference between the value of theportfolio quoted at the best bid and the effectiveliquidation value. Let the investor hold 100 bondissues. Given the previous description, the ficti-tious value of this portfolio at best bid is equal to100 × 1, 000 = $100, 000. Its liquidation value isequal to 50× 1, 000 + 50× 990 = $99, 500. Hence,they would incur into a lost of $500 due to a liquid-ity issue. Now assume that the standing portfoliodoubles to 200 bonds. The value at best bid is now200 × 1, 000 = $200, 000, whereas the total liqui-dation value equals 50× 1, 000 + 100× 990 + 50×970 = $197, 500. Now the cost of liquidity is $2,500.We see that positive homogeneity fails. Moreover,risk assessment moves from $500 to $2,500, whichexceed twice the initial figure, that is $1,000. Hence,subadditivity fails too. This example shows liquid-ity issues may reasonably require a superadditive

risk assessment in other to appropriately representthe exposure of combined positions.

We may then conclude with a statement of a con-sistency principle: subadditivity and superadditiv-ity should be properties of the underlying positionsand their dependence structure as opposed to be-ing taken as an axiom fulfilled by the adopted riskmetrics. As long as VaR does not impose any con-dition about either subadditivity or superadditivity,we believe it complies with the mentioned consis-tency principle in sharp contrast to what “coherent”measures of risk do.

ABOUT THE AUTHORS

Andrea Roncoroni is full professor of Finance at ESSECBusiness School, Paris, and Visiting Fellow at BocconiUniversity, Milan. Research interests are related to energy andcommodity finance, risk monitoring and mitigation and appliedquantitative modelling.


Lionel Lecesne is Ph.D. candidate at University of Cergy,France. He was formerly student at Ecole Normale Supérieurein Cachan, France. Research interests include risk measurement,energy finance and financial econometrics.


ABOUT THE ARTICLE

Submitted: March 2013.Accepted: April 2014.

References

[1] Acerbi, C. Spectral Measures of Risk: A Coherent Repre-sentation of Subjective Risk Aversion. Journal of Bank-ing & Finance, 26, pp. 1505-1518. 2002

[2] Artzner, P., F. Delbaen, J. M. Eber and D. Heath. Co-herent Measures of Risk. Mathematical Finance, 9, pp.203-228. 1999.

[3] Embrechts, P., R. Frey and A. J. McNeil.QuantitativeRisk Management: Concepts, Techniques, and Tools.Princeton Series in Finance, Princeton University Press.2005.

[4] Lecesne, L. and A. Roncoroni. Monetary Measurement ofRisk - Part I. Argo, 1 (Summer 2014), pp. 67-72. 2013.

18energisk.org

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Summer 201419


Banking & Finance

Transfer Pricing

20

Towards a Theory ofInternal Valuation andTransfer Pricing of Productsin a BankFunding, Credit Risk and Economic Capital

In this article Castagna outlines a frame-work to theoretically identify the compo-nents of the value that a bank shouldattach to a deal. Moreover the frame-work suggests how to charge these com-ponents to the relevant departmentsand/or to the final counterparty (client)by an internal transfer pricing system.

Antonio CASTAGNA

The pricing of contracts in the financial in-dustry relies on theoretical results in manycases. Suffice to remember that the valuation

of derivative contracts hinges on the results of Op-tion Pricing Theory, mainly developed between the1970’s and the 1990’s.

In some cases the practice differs from the the-ory. For example, the inclusion of funding costsin the valuation of loans has been seen in strikingcontrast with the results proved by academicianssuch as Modigliani and Miller, awarded with Nobelprized for their achievements.

In the last few years, the debate has sparked alsoas far as derivative contracts are concerned: thereis not a widespread agreement on whether adjust-ments such as the Debit Value Adjustment (DVA)and Funding Value Adjustments (FVA) are justifiedor not. Some authors have proposed full pricingapproaches including every possible adjustment inthe value: we can refer to the most recent works inthis area by Brigo, Morini and Pallavicini [1] and

Crepy et al. [11]. It should be noted that these au-thors are more interested in finding out a valuationformula encompassing all the components of theprcicing as seen by both counterparties, but they donot investigate which components can be effectivelyreplicated so that they represent a recoverable pro-duction cost. In few words, these approaches aimat determining a fair tradable price on which bothparties may agree by acknowledging each others’risks and costs, but they cannot always be correctand consistent methods to evaluate derivative con-tract once they are traded and entered in the bank’sbooks.

The heart of the matter with the revaluation ofderivative contracts is in the assumptions the bankmakes about its hedging policies: they can be moreor less realistic and feasible, so that different evalu-ations can be obtained which have to be challengedby the ability to actually implement the assumedhedging strategies. In this work we will try to shedsome lights on how products (contracts) should beinternally evaluated by a bank, which are the coststhat should be transferred to the counterparty, theextent to which the theoretical results still appliesin practice and under which conditions they do so.To this end, we will refer to a stylised balance sheetin a basic multi-period setting.

Some of the results we will crop up in thepresent work are derived in Castagna [9], but withina framework that, although similar to the one wewill sketch below, was not capable to avoid anaporetic situation that was however acknowledgedin the final part of that paper. By modifying and ex-panding said framework, we will hopefully provide

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TRANSFER PRICING

a firmer theoretical ground to some of the claimsmade elsewhere in Castagna [6] and confirm theother results in Castagna [7].

Balance Sheet with a Single Asset

Let us start with the assumption that we are in aneconomy with interest rates set constant at zerolevel. An asset A1(t) has an initial price X1 attime t, terminal pay-off A1(T) = X1 × (1 + s1). Forsimplicity, we will assume that T − t = 1 in whatfollows. There is a probability PD1 that the asset’sissuer defaults between times t and T, in which casethe buyer of the asset will receive in T a stochasticrecovery Rec1 expressed as a percentage of the facevalue X1. Rec1 can take values Rec1j , for j = 1, ..., J,and each possible value can occur with probabilityp1j .

A1(t) = E [X1 × (1 + s1)]

= X1 × (1 + s1)(1− PD1) + X1Rec1PD1 = X1(1)

where Rec1 = ∑Jj=1 Rec1j p1j is the expected recov-

ery in the event of the issuer’s bankruptcy.Assume that an investor (with zero leverage)

buys the asset A1. In perfect markets, the spreads1 they require from the issuer of the asset A1 issimply the fair credit spread cs1 remunerating thecredit risk:

s1 = cs1 =Lgd1PD1

1− PD1(2)

where Lgd1 = (1−Rec1) is the loss given defaultrate (the complementary to 1 of the recovery rate).The spread s1 is set at the level that makes the ter-minal expected value of A1 equal to the presentvalue, X1.

Assume now that a bank invests in the sameasset A1. The bank uses leverage, which means thatit issues a bond to buy the asset. We denote thevalue of the debt at time t with D1(t) and of theequity with E. The equity is invested in a risk-freebank account, B(t) = E. The amount needed tofund the asset is D1(t) = X1: this amount is raisedby the bank with a bond issuance. The terminalpay-off of the bank’s debt is linked to the pay-offof asset A1. In fact, the debt D1 pays at maturityX1 × (1 + f1) if the asset’s issuer does not go de-faulted, or X1Rec1j + E (for each possible recoveryrate) if it does. In a perfect market, with perfectlyinformed agents, the fair bank’s funding spread, re-quested by the buyers of the bank’s bond, is derived

from the following equation:

D1(t) = E [X1 × (1 + f1)]

= X1 × (1 + f1)(1− PD1) +J

∑j=1

min[X1Rec1j

+ E; X1 × (1 + f1)]p1j PD1

= X1

(3)

The LHS of equation (3) is the expected value ofthe bank’s bond at expiry: if the issuer of theasset A1 survives, then the bank is able to reim-burse the notional of the bond plus the fundingspread X1 × (1 + f1). If the assets’ issuer goes bust,then the bond will be reimbursed with the fund-ing spread only if this amount is smaller than therecovery value of the asset plus the initial equitydeposited in the risk-free account: this is computedfor each possible value of Rec1j and weighted forthe corresponding probability. This expected valuemust equal the present value of the debt, which isthe face amount X1.

We assume that there is at least one case whenE < X1Lgd1j

+ X1 f1 (the equity is not able to fullycover the loss given the default of the asset’s issuerand the funding cost). The funding spread is thelevel of f1 equating the expected value at expirywith the present value:

f1 =Lgd

∗1PD1

1− PD1(4)

where Lgd∗1 = 1−∑J

j=1 min[X1Rec1j + E; X1× (1+

f1)]p1j /X1.3 The quantity Lgd∗1 can be also be seen

as the expected loss given default on the bank’sdebt, Lgd

∗1 = LgdD1

, and the corresponding re-

covery is RecD1 = ∑Jj=1 min[X1Rec1j + E; X1 × (1 +

f1)]p1j /X1. They are different from the loss givendefault and the recovery of the asset A1 since whenthe asset’s issuer defaults, a fraction of the loss iscovered by the equity E, which is also the maximumamount the shareholders of the bank are liable for.Since Lgd1 > Lgd

∗1 = LgdD1

, then s1 > f1, the bank“enjoys” (if an economic agent may ever enjoy fromlosing a part of its own capital) from the fact thatthe equity is contributing to the coverage of the po-tential future credit losses. If E > X1Lgd1j

+ X1 f1

for all possible Lgd1jthen Lgd

∗1 = LgdD1

= 0 andf1 = 0.

Assume now that the bank has a bargainingpower in setting the fair rate s1, so that it is able toset it at a level different from the fair level cs1 re-quired by a non-leveraged investor. In this way the

3It should be noted that equation (4) is formally the solution to equation (3), but it is not a closed-form formula, since the definitionof Lgd

∗1 includes f1. That means that a numerical search for f1 is needed in (3), starting by setting f1 = 0: for practical purposes and

for typical values of the involved variables, two steps are sufficient.

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bank tries to cover costs and losses other than thecredit losses. The fair mark-up spread ms1 the bankhas to charge on asset A1, given the limited liabilityof the shareholders, is obtained by the equation:

VB(t) = E [X1 × (1 + ms1) + E− X1 × (1 + f1)]

= [(ms1 − f1)X1 + E](1− PD1)

+J

∑j=1

max[X1Rec1j + E− X1 × (1 + f1); 0]p1j PD1 = E

(5)

Equation (5) is the net value of the bank at time t,which is equal to the expected value of the futurevalue in T of the bank’s total assets (the final as-set’s pay-off X1 plus the margin ms1X1, plus theequity amount deposited in the risk-free accountE), minus the bank’s total liabilities (the notionalof the debt X1 plus the funding costs f1X1). Sincethe bank’s shareholders invested the initial amountE, the expected net value VB(t) must equal E toprevent any arbitrage.

Assume E < X1Lgd1j+ X1 f1 for one or more

Lgd1j’s. Indicating with

R1 =J

∑j=1

max[X1Rec1j + E− X1 × (1 + f1); 0]p1j ,

the mark-up spread, from (5), is:

ms1 =E−R1

X1PD1

1− PD1+ f1 = cs∗1 + f1 (6)

This is the sum two components:

- the “adjusted” credit spread cs∗1 < cs1 onthe asset A1, due the loss given default(E − R1)/X1 lower than Lgd1. The smallerloss given default suffered by the sharehold-ers is generated by the leveraged investmentin the asset A1, and by the limited liability upto the equity amount E. In practice a share ofthe losses given default is taken by the debtholders, whence the lower credit spread;

- the funding spread f1 paid by the bank on itsdebt.

By some manipulations, it is easy to check thatin perfect markets where the credit spreads set byinvestors are fair and given in (2), we have:

ms1 =E−R1

X1PD1 + Lgd

∗1PD1

1− PD1=

Lgd1PD11− PD1

= s1 (7)

So the mark-up spread is just the credit spreadof the asset A1 required by a for a non-leveragedinvestor.

If E > X1Lgd1j+ X1 f1 for all possible Lgd1j

(the equity is large enough to cover the loss giventhe default of the asset’s issuer and the fund-ing costs in each possible case), then f1 = 0and ∑J

j=1 max[X1Rec1j + E− X1 × (1 + f1); 0]p1j =

X1Rec1 + E − X1 × (1 + f1). From equation (5)we have that the mark-up spread is ms1 =(Lgd1PD1)/(1−PD1), which is consistent with thestated results.

Proposition 1. If the bank holds only one asset, theleverage is immaterial in its internal pricing. Differentlystated, the bank can price the asset as it were an non-leveraged investor, and the assets’ price would dependonly on its expected future pay-off.

The bank’s default probability PD1 is linked tothe issuer’s probability of default PD1, and it de-pends on the amount of equity that can be used tolower the Lgd1j

in the different cases. In generalterms, the bank defaults when its value is belowzero: since the shareholders’ liability is limited, thebank’s value is floored at zero, so that the bankgoes bust when R1j = max[X1Rec1j + E− X× (1 +f1); 0] = 0, for one or more j’s. We have that :

PDB =J

∑j=1

1{R1j=0}p1j PD1 (8)

where 1{·} is the indicator function.The conclusion is that bank’s leverage and fund-

ing costs are immaterial in pricing the asset A1: therequested spread to make fair the contract is thesame both for a non-leveraged investor and for theleveraged bank. This result is definitely not new: ina different setting, we reached the same conclusionsas the well known works by Modigliani&Miller(M&M) [18] and Merton [16], whose results remainfully valid also in our setting.

When we are not in perfect markets, formula (6)provides the bank with a useful tool to price the as-set A1. In fact, it is possible that the debt holders donot have access to the information set of the bank,so that they do not know which is the correct levelof PD1 and Rec1 to apply in the pricing. For thisreason, the funding spread required on the bank’sdebt can be different from the fair spread appliedin a market where a perfect information is availableto all market participant.

Let f ∗1 6= f1 be the funding spread required bythe debt holders. The equivalence in equation (7)does not hold anymore (and so does not also theM&M theorem and its extension by Merton). Thebank should use equation (6), where f1 is set equalto the spread requested by the debt holders f ∗1 , andthe credit spread is the “enhanced” spread cs∗1 .

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TRANSFER PRICING

Formula (6) allows also to consider differentrisk-premia implicit in the preferences of the share-holders and bond holders: in perfect markets, inthe simple framework we are working in, the risk-premia are reflected in the PD of the asset’s issuer.In the pricing process, if the bond holders use adifferent PD from the one used by shareholders (orby the bank’s managers, if they operate mainly inthe interest of the latter), then the irrelevance ofthe leverage does not hold anymore and the correctformula to use. Different PDs are the consequenceof different premia due to a limited informationof the bond holders about the issuer of the assetbought by the bank.4

Including the Economic Capital

The economic capital is a concept used by finan-cial institutions to indicate the amount of equitycapital needed to keep the probability of default(i.e.: of ending the business activity) below a givenlevel, according to a model to measure the futureuncertainty of the assets’ value.5

We can embed a premium in the return on eq-uity (upon the risk-free rate) different from the oneimplicitly originated by the use of a PD containing arisk-premium fair to the bank. Let us indicate by πthis additional return on equity and with EC1 ≤ Ethe amount of economic capital “absorbed” by assetA1. Equation (5) modifies as follows:

VB(t) = E [X1 × (1 + ms1) + E− X1 × (1 + f1)]

= [(ms1 − f1)X1 + E](1− PD1)+

+J

∑j=1

max[X1Rec1j + E− X× (1 + f1); 0]p1j PD1

= EC1π + E

(9)

The margin spread including this additional pre-mium is:

ms1 =

E−R1X1

PD1 + EC1/X1π

1− PD1+ f1 = cs∗1 + f1 + cc1

(10)where cc1 = EC1/X1π

1−PD1is the cost of the economic

capital associated to asset A1.

Proposition 2. In internally evaluating an asset heldby the bank, the economic capital enters in the bank’sinternal pricing process through the return π requestedby the shareholders.

Including Interest Rates Different from Zero

It is relatively easy to introduce non-zero interestrates in the framework sketched above: we limitthe analysis to the case of a single, constant interestrate equal to r. In this case equation (1) modifies asfollows:

A1(t) = E[

11 + r

X1 × (1 + r + s1)

]=

X1 × (1 + r + s1)(1− PD1) + X1Rec1PD11 + r

= X1

(11)

After same manipulations (see also Castagna andFede [10], ch. 8), we have a formula for the creditspread s1 similar to formula (2), where the lossgiven default is defined as Lgd1 = 1 + r−Rec1; fortypical values of the interest rates, the two Lgd1’sdo not differ too much.

Similarly, when rates are different from zero, thebank’s debt equation (3) reads:

D1(t) = E[

X1 × (1 + r + f1)

1 + r

]=

X1 × (1 + r + f1)(1− PD1)

1 + r+

+∑J

j=1 min[X1Rec1j + E(1 + r); X1 × (1 + r + f1)]p1j PD1

1 + r= X1

(12)

so that the funding spread has still a formula as in(4), with:

Lgd∗1 = 1 + r−

J

∑j=1

min[X1Rec1j+

+ E(1 + r); X1 × (1 + f1)]p1j /X1

Finally, the fair mark-up spread ms1 the bankhas to charge on asset A1 when rates are differ-ent from zero, is given by solving the followingequation:

VB(t) =

E[

X1 × (1 + r + ms1) + E(1 + r)− X1 × (1 + r + f1)

1 + r

]=

[(ms1 − f1)X1 + E(1 + r)](1− PD1)

1 + r

+∑J

j=1 max[X1Rec1j + E(1 + r)− X1 × (1 + f1); 0]p1j PD1

1 + r= E

(13)4It is quite reasonable to assume that typically banks have a deeper knowledge of their clients, which are ultimately the issuers of

most of the assets. On the other hand, when the asset is a publicly traded security, it is more reasonable to assume that the informationis evenly distributed amongst market participants. This is the case implicitly assumed in the M&M’s and Merton’s works.

5The level is often set by national and international regulation, such as the Basel regulation for banks.

24iasonltd.com

Rec1j pj

75% 20%35% 70%5% 10%

Exp. Recovery Rec1 40%

TABLE 1: Possible recovery rates Rec1j and associated probabilities p1j for asset A1

The formula for ms1 is similar as (6), if E(1 + r) <X1Lgd1j

+ X1 f1 for one more Lgd1j’s:

ms1 =

E(1+r)−R1X1

PD1

1− PD1+ f1 = cs∗1 + f1 (14)

where R1 = ∑Jj=1 max[X1Rec1j + E(1 + r) − X1 ×

(1 + f1); 0]p1j .If E(1 + r) < X1Lgd1j

+ X1 f1 in all cases, then

ms1 = (Lgd1PD1)/(1− PD1) with Lgd1 = 1 + r−Rec1.

In conclusion, the introduction of interest ratesinto the analysis does not significantly affect results.Let the total yield of the asset A1 be i1: it can bewritten as the sum of the following components:

i1 = r + ms1 = r + cs∗1 + f1 + cc1 (15)

We present now an example to illustrate in prac-tical terms what we have stated in this section.

EXAMPLE 1 Assume that the bank starts its ac-tivity in t = 0 with an amount of equity capitalE = 35, which is deposited in a bank accountB(0) = 35. The bank wishes to invest in an as-set whose price is A1(0) = 100 and it issues debtD1(0) = 100 to buy it. We will suppress the timereference in the following to lighten the notation.A sketched bank’s balance sheet is the following:

Assets LiabilitiesB = 35 D1 = 100

A1 = 100——————–

E = 35

We assume interest rates are zero and that theissuer of the asset A1 can default with proba-bility PD = 5%. Moreover, upon default, therecovery rate for the bank is stochastic: possi-ble outcomes and the associated probabilities arein table 1. Clearly, Lgd1 = 1 − Rec1 = 60%Firstly, let us determine which is the fair(credit) spread s1 = cs1 requested by a non-leveraged investor. This is given by equation (2):

s1 = cs1 =60%× 5%

1− 5%= 3.158%

so that at the expiry the asset has a ter-minal value of A1 = 100 × (1 + 3.158%).The bank is a leveraged investor, since it issuesan amount of debt sufficient to buy the asset. As-suming we are in a market where perfect infor-mation is available to all participants, then thecreditors of the bank know that it will buy theasset A1 and consequently they set a credit spreadon the debt D1, which is a funding spread forthe bank, by applying equation (4), by meansof a two-step iterative procedure (the first stepis computed by setting f1 = 0), so that we get:

f1 = 1.406%

We can now compute the fair margin ms1 that thebank should charge on asset A1, by means of for-mula (6). We start with computing the differentR1j s: they are shown in the table below. Sincewith these quantities we can compute also thebank’s default probability, by means of formula(8), this is shown in the table below as well.

R1j PDB

1.719 0.000%- 3.500%- 0.500%

————– ————–1.719 4.000%

We have all we need to calculate the “adjusted”credit spread cs∗1 :

cs∗1 =35−1.79

100 5%1− 5%

= 1.752%

which plugged in (6):

ms1 = cs∗1 + f1 = 1.752% + 1.406% = 3.158% = s1

thus confirming (7).The equity should be used to prevent a defaultof the bank occurring with a probability higherthan a given level. The economic capital is theamount of equity necessary to keep the probabilitybelow the targeted level. In this example, assum-ing that the target probability is 4%, we can seefrom the calculations above that actually PDB = 4%.

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It is easy to prove that the amount of economic cap-ital to achieve this target is just above 26.75, not 35,so EC1 = 26.75 and the remaining is equity capitalthat could be invested in other risky assets. Wesuppose that the stockholder require a return oneconomic capital of π = 5%. The cost of capital is:

cc1 =26.75100 5%

1− 5%= 1.408%

The total ms1 to charge on A1, by formula (10), willthen become

ms1 = cs∗1 + f1 + cc1 =

1.752% + 1.406% + 1.408% = 4.556%

Balance Sheet with Two Assets (Uncorre-lated Defaults)

We now move on to a multi-period setting, wherethe bank, after the investment in asset A1, de-cides to further invest in a new asset A2, whoseinitial price is X2 and terminal pay-off A2(T) =X2 × (1 + s2). Without a great loss of generality,we assume that the expiry of the asset A2 is thesame as asset A1, in T, and that the investment inthe new asset occurs in t+, just an instant after theinitial time t; to lighten the notation, we will sett+ = t in what follow, even though they are twodistinct instants.

Also for asset A2 there is a probability PD2 thatthe asset’s issuer defaults, in which case the buyerof the asset will receive a stochastic recovery Rec2expressed as a percentage of the face value X2. Rec2can take values Rec2l , for l = 1, ..., L, and each possi-ble value can occur with probability p2l . We assumefor the moment that the defaults of the issuers ofassets A1 and A2 are uncorrelated.

Similarly to asset A1, for a (non-leveraged) in-vestor

A2(t) = E [X2 × (1 + s2)]

= X2 × (1 + s2)(1− PD2) + X2Rec2PD2 = X2(16)

where Rec2 = ∑Ll=1 Rec2l p2l is the expected recov-

ery in the event of the issuer’s bankruptcy. The faircredit spread is derived as n the case of asset A1and it is:

s2 = cs2 =Lgd2PD2

1− PD2(17)

where Lgd2 = (1−Rec2).Assume now that the bank buys the asset

A2(t) = X2 by issuing new debt: the total debtis D(t) = D1(t) + D2(t) = X1 + X2 = X, i.e.: theleverage increases as well. Let f2 be the fundingspread paid on debt D2(t).

The funding spread requested by bank’s bondholders on the new debt D2(t) is derived in a waysimilar to equation (3):

D2(t) = E [X2(1 + f2)]

= X2(1 + f2)(1− PD1)(1− PD2)

+ ∑Jj=1 min

[X1Rec1j+X2(1+s2)+E

X ; (1 + f2)

]∗ X2 p1j PD1(1− PD2)

+ ∑Ll=1 min

[X1(1+s1)+X2Rec2l +E

X ; (1 + f2)

]∗ X2 p2l PD2(1− PD1)

+ ∑Jj=1 ∑L

l=1 min[

X1Rec1j+X2Rec2l +EX ; (1 + f2)

]∗ X2 p1j p2l PD2PD1 = X2

(18)

The spread s2 can be either the credit spread orthe mark-up spread set by the bank. The fundingspread f2 can be found by solving equation (18):also in this case, as for the case in the previous sec-tion for a single asset balance sheet, the solution f2is found by a very quick numerical search, startingby setting f2 = 0. It should be noted that in thisstylised multi-period setting, the funding spread f1on the first debt D1 enters in the pricing of the newdebt D2 as an input: the bank cannot change thisparameter that is contractually set until the expiryin T and this is consistent with the real world. Itshould be also noted that this spread is no morefair and it should be revised, since the new recoveryratio for the total debt D is now different from theoriginal one set before the investment in the assetA2. This anyway cannot happen, as the outstandingdebt pays a contract rate fixed until the expiry.

The mark-up margin, set by the bank on thesecond asset, is such that the expected net valueof the bank is still the amount of equity posted byshareholders:

VB(t) = E [X1(ms1 − f1) + X2(ms2 − f2) + E]

= [X1(ms1 − f1) + X2(ms2 − f2) + E]

∗ (1− PD1)(1− PD2)

+J

∑j=1

max[

X1(Rec1j − (1 + f1)) + X2(ms2 − f2) + E; 0]

∗ p1j PD1(1− PD2)

+L

∑l=1

max[X1(ms1 − f1) + X2(Rec2l − (1 + f2)) + E; 0

]∗ p2l PD2(1− PD1)

+J

∑j=1

L

∑l=1

max[

X1(Rec1j − (1 + f1))

+ X2(Rec2l − (1 + f2)) + E; 0p1j p2l PD2PD1

= E(19)

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Let

R∗1 =J

∑j=1

max[

X1(Rec1j − (1 + f1))

+ X2(ms2 − f2) + E; 0p1j ,

R∗2 =L

∑l=1

max [X1(ms1 − f1)

+ X2(Rec2l − (1 + f2)) + E; 0p2l

and

R∗1,2 =J

∑j=1

L

∑l=1

max[

X1(Rec1j − (1 + f1))+

+ X2(Rec2l − (1 + f2)) + E; 0p1j p2l .

We will rewrite the (19) in a lighter notation:

VB(t) = E [X1(ms1 − f1) + X2(ms2 − f2) + E]

= [X1(ms1 − f1) + X2(ms2 − f2) + E]

∗ (1− PD1)(1− PD2)

+ R∗1PD1(1− PD2) + R∗2PD2(1− PD1)

+ R∗1,2PD2PD1 = E

(20)

The solution ms2 to (20) can be found by a quicknumerical procedure, by initialising ms2 = 0. It ispossible, anyway, to distinguish two cases that willshed some light on which is the value that ms2 cantake.

Before analysing the two distinct cases, we pro-vide the probability of default of the bank once thenew asset is bought. As before, we need to checkfor all the cases when the bank’s value drops be-low zero, in which case the limited shareholders’liability floors the value at zero. We have that :

PDB =J

∑j=1

1{R∗1j=0}p1j PD1(1− PD2)

+L

∑l=1

1{R∗2l=0}p2l PD2(1− PD1)

+J

∑j=1

L

∑l=1

1{R∗1j ,2l=0}p1j p2l PD1PD2

(21)

where 1{·} is the indicator function and R∗abis the

b-th addend in the summation in each R∗a .

The Default of the Asset A2 Does not Imply theDefault of the Bank

First, we assume that the default of the asset A2does not imply the default of the bank. This mayhappen for a variety of reasons, but mainly becausethe quantity X2 of the asset bought by the bankis small compared to the entire balance sheet. So,even in the event of bankruptcy of the issuer of A2,the bank is able to cover losses and to repay all itscreditors without completely depleting its equity

capital E and hence declaring default. On the otherhand, the default of the asset A1 still causes thedefault of the bank as before.

If X2 is much smaller than the quantity X1 ofasset A1 already included in the assets of the bank’sbalance sheets, then we have the following approxi-mations:

R∗1 ≈ R∗1,2 ≈ R1

and

R∗2 ≈ X1(ms1 − f1) + X2(Rec2 − (1 + f2)) + E

After substituting the values above, equation(20) can be written as:

VB(t) = E [X1(ms1 − f1) + X2(ms2 − f2) + E]

= [X1(ms1 − f1)) + X2(ms2 − f2) + E]

∗ (1− PD1)(1− PD2)

+ [X1(ms1 − f1) + X2(Rec2 − (1 + f2)) + E]

∗ PD2(1− PD1)

+ R1PD1 = E

(22)

Let [(ms1− f1)X1 + E] = H; we can re-write (22) as

VB(t) = H[(1− PD1)(1− PD2) + PD2(1− PD1)]

+ R1PD1 + [X2(ms2 − f2)] (1− PD1)(1− PD2)

+ [X2(Rec2 − (1 + f2))]PD2(1− PD1)

= E

(23)

By formula (5) and the definition of ms1:

[(ms1 − f1)X1 + E](1− PD1) + R1PD1 = E

then the first term on the RHS of the first line in(23) is:

H[(1− PD1)(1− PD2) + PD2(1− PD1)] + R1PD1

= H(1− PD1) + R1PD1 = E

so that:

VB(t) = [X2(ms2 − f2)(1− PD1)(1− PD2)] +

+ [X2(Rec2 − (1 + f2))]PD2(1− PD1) = 0(24)

Recalling that X2(1− Rec2) = X2Lgd2, equation(24) simplifies:

VB(t) = [X2(ms2 − f2)(1− PD2)−(X2Lgd2 + f2)PD2(1− PD1) = 0

(25)

which holds if[X2ms2(1− PD2)− X2 f2 − X2Lgd2PD2

]= 0

or

ms2 =Lgd2PD2 + f2

1− PD2= cs2 +

f21− PD2

(26)

The fair margin spread ms2 is no more equal tothe credit spread cs2, as it was the case for asset A1(see equation (7)). It is worth noting that the totalmargin spread is made of two components:

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- the credit spread cs2, equal to the spread re-quested by a non-leveraged investor. This isdifferent from the credit spread required onasset A1 by the bank, in equation (6). Thereason is that when the asset A1 issuer goesdefaulted, this will cause also the the bank’sdefault and the shareholders are liable onlyup to the amount E to the debt holders, hencea reduced credit spread cs∗1 can be applied;but the default of the issuer of the asset A2does not cause the bank’s default, so that theloss is fully covered by the equity, so in thiscase the limitation of the liability up to E is im-material, and the entire (expected) loss givendefault Lgd2 must be born by the share hold-ers. As a consequence, the credit spread ap-plied on the asset A2 in the mark-up marginis equal to cs1, in the non-leveraged investorcase;

- the funding spread f2, conditioned to the sur-vival of the issuer of the asset A2. Also in thiscase there is a difference with the case of theasset A1: the funding spread in (6) is not di-vided by the survival probability of the issuerof A1 since the bank is interested in receivingthe amount f1X1 without considering the is-suer’s default. In fact, when the issuer of A1goes defaulted, the margin spread m1 is notcashed in by the bank and it cannot repay thefunding costs on the debt D1. But in the eventof default also the bank goes bankrupt, so thatshareholders are not interested in the miss-ing payments since they are anyway limitedup to the amount E. In case of asset A2, themissing payment of f2 has to be covered bythe shareholders, which will pay f2X2 on thedebt D2 in any case. The expected loss dueto this cost must equal the expected returnembedded within the mark-up spread ms2:E[ f ∗2 X2] = f ∗2 X2(1 − PD2) = f2X2, whichmeans that f ∗2 = f2/(1− PD2).

We can state the following result:

Proposition 3. When pricing an asset that representsa small percentage of the bank’s total assets and whosedefault does not affect the bank’s default, the correct (ap-proximated) and theoretically consistent mark-up mar-gin to apply includes the issuer’s credit spread, fair to anon-leveraged investor, plus the bank’s funding spreadconditioned to the issuer’s survival probability.

This result is an accordance with the result inCastagna [5], where the fair spread that an issuerwould pay to a non-leveraged investor includesonly the credit spread (cs2 in this case), whereas

the mark-up margin charged by the leveraged bankincludes also the funding costs (see also chapter11 in Castagna and Fede [10]). In Castagna [5] theirrelevance of the default of the bank was postu-lated in an axiom based on the going concern princi-ple. In this work on the contrary we are explicitlyconsidering the case when such irrelevance arises.An example is given by an asset representing asmall percentage of the total bank’s assets: whenits issuer defaults, this bankruptcy would not affectthe survival of the bank. In this case the resultsare re-derived and confirmed in a much theoreti-cally sounder framework extending the simplifiedeconomies in Modigliani& Miller and Merton, thusdelimiting the cases when their results remain fullyapplicable.

The results just shown confirm also that thepractice of including the funding valuation adjust-ment (FVA) in the valuation (i.e.: internal pricing)process of a contract is fully justified: this thesiswas supported in Castagna [6] (arguing against theopposite view in Hull&White [13] and [14]) but notproved analytically. A first attempt to extend theModigliani&Miller [18] and Merton [16] frameworkin a multi-period setting was in Castagna [9], butthere the simplified structure of the bank’s balancesheet did not allow to achieve a fully convincing jus-tification of the all-inclusive pricing rules followedby practitioners. We here showed that the mainpoint is to consider also a richer balance sheet witha more complex set of interrelations between assets’issuers and the bank.

If we consider also the economic capital that theasset A2 entails to ensure that the default probabil-ity of the bank is equal or smaller than given level,the margin spread can be easily modified:

ms1 =Lgd2PD2 + EC2/X2π + f2

1− PD2

= cs∗2 +f2

1− PD2+ cc2

(27)

where EC2 is the economic capital required by theinvestment in asset A2 and cc2 = EC2/X2π is theassociated cost.

EXAMPLE 2 We reprise the Example 1 and weassume that an instant after the start of the ac-tivities in t = 0, i.e.: in t′ (which, by an ap-proximation, we set equal to t) the bank investsin an asset A2, whose issuer can default be-tween times 0 and 1 with probability PD2 = 6%.There are three recovery scenarios, with associ-ated probability, as shown in the following table:

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Rec2l pl75% 0.235% 0.75% 0.1

Exp Rec Rec2 40%

We can immediately compute the fairspread s2 = cs2 that a non-leveragedinvestor would require on this asset:

s2 = cs2 =60%× 5%

1− 6%= 3.83%

The price in t′ = t = 0 is A2(0) = 10 and thebank issues new debt for an equivalent amountD2(0) = 10: the amount represents a rela-tively small fraction of the entire assets held bythe bank; moreover, when the issuer of A2 de-faults, the bank will not go bankrupt in any re-covery scenario. As in the first example, wesuppress all references to time from now on.The bank’s balance sheet will now be:

Assets LiabilitiesB = 35 D1 = 100

A1 = 100 D2 = 10A2 = 10

——————–E = 35

The spread requested by the creditors ofthe bank can be found be recursivelysolving equation (18), starting by settingf2 = 0. We come up with the result:

f2 = 1.318%

The fair spread ms2 that the bank has to chargeon asset A2, can be found by means of (26)

ms2 = cs2 +f2

1− PD2= 3.830%+

1.318%1− 6%

= 5.229%

The simple rule often followed by banks toset the spread (under the hypothesis that in-terest rates are zero) is to charge in themargin the credit spread referring to theissuer of the asset and the funding cost:

ms2 = 3.830% + 1.318% = 5.148%

in case of low issuer’s probability ofdefaults this is an approximation ofthe fair margin that should be applied.The probability of default of the bank, after theinvestment in the asset A2, is calculated from equa-tion (50) and it is PDB = 4.01%. This means thatthe equity is almost fully sufficient to keep theprobability of default at the chosen level of 4%we mentioned in example . To avoid unneeded

complication, we take the new PDB as compli-ant with the limit in practice, which means thatthe economic capital absorbed by the new asset isEC = 8.25. If we want to include also the remunera-tion for EC2 in the margin, then from (27) we have:

cc2 =8.2510 5%

1− 6%= 4.145%

so that the total margin is ms2 = 5.148%+ 4.125% =9.273%. In this example the economic capital re-quested to keep PDB ≤ 4% is equal to amountinvested in A2: this is due to the simplified bal-ance sheet we are considered and to the smallnumber of scenarios for the recovery of the twoassets. In the real world, in a much more complexbalance sheet, the economic capital for an addi-tional investment is typically smaller than the in-vested amount and consequently also the cost ofcapital component of the margin is much smaller.

The Default of the Asset A2 Implies the Defaultof the Bank

Assume now that the default of the asset A2 im-plies the default of the bank: this is typically thecase when the asset A2 is, in percentage terms, agreat share of the bank’s total assets. The marginspread that the bank has to apply on the secondasset is derived numerically by solving equation(23). It is possible to write down a formally closedform formula, to assess which are the parametersaffecting the spread.

By rewriting equation (20), we have:

VB(t) = E [X1(ms1 − f1) + X2(ms2 − f2) + E]

= [X1(ms1 − f1) + X2(ms2 − f2) + E]

∗ (1− PD1)(1− PD2)

+ ε = E

(28)

where ε = R∗1PD1(1− PD2) + R∗2PD2(1− PD1) +R∗1,2PD1PD2. Let PD1,2 = 1− (1− PD1)(1− PD2);we solve for ms2 and we get:

ms2 =E

X2PD1,2

(1− PD1)(1− PD2)+ f2

−X1(ms1− f1)(1−PD1)(1−PD2)+ε

X2

(1− PD1)(1− PD2)= cs∗2 + f2 − db

(29)

The margin spread ms2 resembles the marginspread ms1 derived for asset A1: it is made of threeconstituent parts:

- an “adjusted” credit spread cs∗2 , due to thelimited liability of the shareholders, referringto the asset A2: the loss given default sufferedby the bank, E/X2, is smaller than the lossgiven default of the asset A2, Lgd2, sufferedby a non-leveraged investor. This lower loss

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X2 f2 ms2 db PDB10 1.318% 5.229% -0.081% 4.01%20 1.215% 5.108% -0.063% 4.05%30 1.128% 5.007% -0.050% 4.05%40 1.059% 4.904% -0.015% 4.63%50 1.038% 4.762% 0.105% 4.63%60 1.135% 4.570% 0.395% 8.62%70 1.281% 4.274% 0.837% 8.62%80 1.411% 4.073% 1.168% 8.62%90 1.527% 3.932% 1.425% 8.62%100 1.632% 3.832% 1.630% 8.62%

TABLE 2: Funding spread, margin, diversification benefit and bank’s probability of default as the amount invested in asset A2increases.

given default multiplies the probability thateither the issuer of the asset A2 or the issuerof asset A1 goes bust, to get the expectedloss,6 and then divided by the survival prob-ability of the bank (i.e.: the joint probabilitythat both issuers of assets A1 and A2 survive).This means that the margin spread shouldequal the expected loss given default, giventhe bank’s survival;

- the funding spread f2 paid by the bank onthe new debt issued to buy the asset A2. Thefunding spread is not divided by the survivalprobability of issuer since when it defaults,the bank will go bankrupt as well and it willnot pay the spread to its creditors.

- a diversification benefit db: all the quantitiesentering in db are positive, so that its effecton the margin is to abate m2. Basically db ac-counts for the fact that a part of the terminalvalue of the equity is made by the terminalvalue of A1, provided that its issuer does notdefault; moreover, when the default of eitherissuers, or of both, occurs then the total recov-ery value for the bank considers both terminalvalues A1 and A2 (the quantity ε). These twoquantities are divided by the survival prob-ability of the bank (i.e.: the joint probabilitythat both issuers of assets A1 and A2 survive).

Even if the capital markets are perfect, the totalmargin spread m2 differs from the market creditspread cs2 and it will typically be greater than thelatter. When the amount of asset A2 held by thebank increases, the diversification benefit db tendsto abate the margin ms2 and it can be also lowerthan cs2 for sufficiently large A2. In any case ms2will not be the sum of the credit spread cs2 and

the bank’s funding spread f2, as in the case abovewhen the default of the asset A2 did not imply thebank’s default.

Proposition 4. When evaluating an asset whose de-fault implies the bank’s default, the correct and theoret-ically consistent mark-up margin to apply includes and“adjusted” issuer’s credit spread, the funding spreadpaid of the new debt to buy the asset, and a diversi-fication benefit due to the different recoveries of assetsand on the non-simultaneous defaults, since they areuncorrelated.

EXAMPLE 3 We continue from the Example 1 andwe assume that the bank wishes to invest in A2 anamount that is big enough to trigger its own defaultwhen the issuer goes bust, in at least one recoveryscenario. In this case formula (29) is computed bynumerically solving (20) by a numerical procedure.In table 2 we show the effect of increasing the invest-ment in A2 by issuing new debt. The probabilityof default will increase because the shareholderwould pour more equity in the bank to preservethe desired target level (it was set at 4% before).Consequently, also the funding spread f2 increases,since the expected losses suffered by bank’s credi-tors will be higher due to the higher PDB and thelower recoveries. The diversification benefit willbe negative for relatively small amount of A2. Wehave also a confirmation that for small amounts ofA1, (as X2 = 10 of the example before) formula (26)is a good approximation, since we are getting thesame result 5.229%. For larger amounts of A2 thebenefit becomes positive, thus lowering the marginapplied on the asset. The margin will approachthe level required by a non-leveraged investor forA2 = 100. The diversification benefit db startsbeing positive when the amount of X2 > 40, or

6When the default of A1 or A2 can both trigger the default of the bank, the loss is suffered when either of the two issuers gobankrupt, whence the weighting by the probability PD1,2.

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when the default of the issuer of A2 implies, in atleast one recovery scenario, the default of the bank.

Balance Sheet with Two Assets (PerfectlyCorrelated Defaults)

Assume now that the defaults of the issuers for as-sets A1 and A2 are perfectly correlated: let 1A1 theindicator function equal to 1 when the default ofthe issuer of A1 occurs, and 1A2 similarly defined.Moreover let PD be the probability of default ofboth issuers. We have:

Pr[1A1 = 1∩ 1A2 = 1] = PD

Pr[1A1 = 0∩ 1A2 = 0] = 1− PD

Pr[1A1 = 1∩ 1A2 = 0] = Pr[1A1 = 0∩ 1A2 = 1] = 0

The fair funding spread requested by the cred-itor of the bank on the new debt D2, in a perfectmarket, is derived by solving the following equa-tion:

D2(t) = E [X2(1 + f2)] = X2(1 + f2)(1− PD)

+J

∑j=1

L

∑l=1

min[

X1Rec1j+X2Rec2l +EX ; (1 + f2)

]X2 p1j p2l PD

= X2

(30)

The margin the bank has to apply on the asset A2,provided it has enough bargaining power, is ob-tained by a modified version of equation (31) thataccounts for the perfect correlation between theissuers of the two assets:

VB(t) = E [X1(ms1 − f1) + X2(ms2 − f2) + E]

= [X1(ms1 − f1) + X2(ms2 − f2) + E] (1− PD)

+ R∗1,2PD = E

(31)

where the notation is the same used before.If we further assume that the bankruptcy of the

two issuers triggers, in at least in one of the pos-sible cases of loss given default, the default of thebank then the formula for the fair spread ms2 canbe formally written as:

ms2 =E

X2PD

(1− PD)+ f2 −

X1(ms1− f1)(1−PD)+R∗1,21PDX2

(1− PD)

= cs∗∗2 + f2 − db

(32)

Formula (32) is similar to (29) for the case whendefault are correlated. The main difference is thatthe joint probability of default of the two issuers(1− PD1)(1− PD2) has been replaced by (1− PD),the probability of the issuer of asset A2 is PD, and

in the diversification benefit db, the quantity ε isreplaced by a smaller amount R∗1,2: this is due tothe impossibility to exploit the value of the nondefaulting asset to cover the losses of the defaultingone, since this situation can never occur if the twodefaults are correlated.

Proposition 5. When pricing an asset whose defaultimplies the bank’s default and that is perfectly corre-lated with the default of the other asset(s) held by thebank, the correct and theoretically consistent mark-upmargin to apply includes and “adjusted” issuer’s creditspread, the funding spread paid of the new debt to buythe asset, and a limited diversification benefit due to thedifferent possible recoveries on assets whose defaults arecorrelated.

Balance Sheet with a Derivative Contract

We have analysed how to bank should evaluate (i.e.:internally price) assets when it decides to invest inthem. Assume now that the bank, after investingat time t = 0 in asset A1, enters in a derivativecontract, instead of buying another asset. Actually,entering in a contract can be seen as buying an assetor issuing a liability, or in some cases doing bothdepending on contingent evolution of (typicallyfinancial) variables (as for example when a swapcontract is closed). In this case, the valuation ofthe contract can be operated as it were an asset (ora liability) with stochastic intermediate cash flowsand terminal pay-off, whose expected amounts arediscounted at the reference date.

We would like here to stress that a bank canalso offer a service of market-making for deriva-tive contracts: in this case it does not really try tobuy an asset or issue a liability; on the contrary,the bank is selling a product manufactured with agiven technology. For example, the bank can be amarket-maker for options on a given asset: it is notreally interested in buying or selling options basedon some expectation of the future evolution of theunderlying asset, simply it sells a product that canbe a long or a short position in the option. Onceone of two positions is sold to the client, the bankhas the internal skills to use the available technol-ogy to manufacture the product so that its profitderives from the ability to sell at a margin over theproduction cost.7

The Black&Scholes (B&S) model, for example, isa technology to manufacture (or replicate, to use thefinancial term) an option contract. More generally,the replication hinges on the idea to set up a (possi-bly continuously rebalancing) trading strategy that

7In Castagna [9] we adopt this “industrial” metaphor in a rather discursive argument to justify the inclusion of funding cost in theinternal evaluation of contracts by the bank. In this work we will expand in a more quantitative way the ideas presented in there.

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satisfies the following conditions:

1. Self financing condition, that is: no other in-vestment is required in operating the strategybesides the initial one;

2. Replicating condition, that is: at any time t thereplicating portfolio’s value equals the valueof the contract.

Keeping the analysis in a discrete time setting,we work in the classical binomial setting by Coxand Rubinstein [12]. Assume that the underlyingasset is S(t) = S at time t, and that it can go up toSu = Su or down to Sd = Sd, with d < 1, u > 1and u× d = 1 in next period T, with a probabil-ity, respectively, equal to q and 1− q. Let V(t) bethe price of a derivative contract at time t, and Vuand Vd its value when the underlying jumps to,respectively, to Su and Sd. Besides, let B(t) = B bethe value in t of a risk-free bank-account depositearning the risk-free rate. We want to replicate along position in the derivative contract.

To do so, we have to set the following equalitiesin each of the two state of the world (i.e.: possibleoutcomes of the underlying asset’s price):

Vu = αSu + βB(1 + r) (33)

andVd = αSd + βB(1 + r) (34)

Equations (33) and (34) are a system that can beeasily solved for quantities α and β, yielding:

α = ∆ =Vu −Vd(u− d)S

(35)

andβ =

uVd − dVu

(u− d)B(1 + r)(36)

We have indicated α = ∆ because it is easily seenin (35) that it is the numerical first derivative ofthe price of the contingent claim with respect to theunderlying asset, usually indicated so in the OptionPricing Theory.

If the replicating portfolio is able to mimic thepay-off of the contract, then its value at time t isalso the arbitrage-free price of the contract:

V(t) = ∆S + βB =Vu −Vd(u− d)

+uVd − dVu

(u− d)(1 + r)(37)

It is possible to express (37) in terms of dis-counted expected value under the risk neutral mea-sure:

V(t) =1

1 + r[pVu + (1− p)Vd] (38)

with p = (1+r)−du−d . V(t) is the expected risk neutral

terminal value of the contract, discounted with therisk-free rate.

This is the original setting by Cox and Rubin-stein [12] that assumed a perfect and friction-lessmarket, where default risk is absent (hence, thebank cannot go bankrupt and there are no differ-ential rates for borrowing or lending). But if thebank is a defaultable agent, it is no more guaran-teed that the replication argument works exactly asdescribed above. We try to investigate this matternow.

Let us go back to our setting above, after thebank started its activities and invested in asset A1,at time t′. As before, for sake of simplicity of thenotation, we set t′ = t in practice, even if they aretwo distinct instants. At this time, the bank closes aderivative contract denoted by φV, where φ = ±1depending on whether the bank has, respectively, along or short position in it. The replication strategyinvolves trading in a quantity −φα in the underly-ing asset S and −φβ in a bank deposit: in the realworld being long a risk-free bank account meansfor the bank lending money to a risk-free coun-terparty8 whereas being short means for the bankborrowing money.

The standard replication argument does not con-sider the default of the replicator (the bank) so thatborrowing and lending money occurs at the sameinterest rate. In reality, the replicator can borrowmoney at a rate which possibly includes a spreadfor the risk of its own default: this is the fundingspread that we analysed above.9 In the followinganalysis we assume that the default of the coun-terparty of the derivative contract (occurring withprobability PDV) is independent from the evolutionof the underlying asset and that the issuer of thelatter cannot default. The possible recoveries of thefinal pay-off of the derivative contract are RecVl , forl = 1, ..., L, each with occurring with probabilitypVl .

First we need to know if the bank is able toborrow money (if this is prescribed by the repli-cation strategy) at the risk-free rate, as supposedby the theory. If the bank needs to borrow money(i.e.: −φβ < 0), it will issue new debt D2 = |φβB|:in a market with perfect information the creditorswill set a credit (funding, from bank’s perspective)spread to remunerate the credit risk. If the bankhas to invest positive cash-flows in a risk-free bank

8It can be alternatively assumed that money is lent to a defaultable counterparty, at a rate including a credit spread compensatingfor the risk. In this case the expected interest rate yielded by the loan is the risk-free rate, when accounting for the expected lossesgiven default.

9See also Castagna [8] and Castagna and Fede [10] for more details on the replication of contract including funding and liquiditycosts. For an extended formal treatment of the replication of a contract in a world with differential rates, see Mercurio [17].

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account (i.e.: −φβ > 0), it will earn the risk-freerate (set equal to 0 in the current analysis).

The credit spread required by the creditor of thebank (alternatively said, the funding spread f2 thebank has to pay) is determined by the followingequation (recall we are setting r = 0, the extensionto non-zero rates is straightforward):

D2(t) = E [|φβB| (1 + f2)]

= [|φβB| (1 + f2)] (1− PD1)(1− PDV)

+J

∑j=1

min[

X1Rec1j+φ(V−∆S)+EX1+|φβB| ; (1 + f2)

]|φβB|

∗ p1j PD1(1− PDV)

+L

∑l=1

min[

X1(1+s1)+EPEV RecVl−φ∆S+EX1+|φβB| ; (1 + f2)

]∗ |φβB| p2l PDV(1− PD1)

+J

∑j=1

L

∑l=1

min[

X1Rec1j+EPEV RecVl−φ∆S+EX1+|φβB| ; (1 + f2)

]∗ |φβB| p1j pVl PDVPD1

= |φβB|

(39)

where EPEV = φV+(T) = φ[pVu1{φVu>0} + (1 −p)Vd1{φVd>0}] is the expected positive exposure (thebank looses on the counterparty’s default only ifthe contract has a positive value); V = E[V] andS = E[S]. Basically the first second line of equa-tion (39) is the amount the creditors of the bankreceive if the bank survives (i.e.: if no default of theissuer of asset A1 or counterparty of the derivativecontract) occurs; the third, fourth and fifth lines isthe amount that the creditors of the bank expectto recover if, respectively, the issuer of asset A1,the counterparty of the derivative contract or bothdefault. The total expected amount returned to thecreditors equals the present amount borrowed bythe bank, in the last line.

As we have seen above when adding a new assetA2, if we focus on the case when the new contractis such that the default of the counterparty doesnot imply the default of the bank, than we shouldbe able to exclude also the cases when the residualvalue of the assets is negative. Although this cannothappen when replicating plain vanilla options, itcannot be excluded for some exotic options. Wewill not deal with this very specific situations, butthey can be dealt with by introducing a floor at zeroin the arguments of the min[] functions.

It should be also manifest that the fundingspread f2 is above zero for non trivial cases, evenassuming a perfect market. This is due to the factthat the composition of the existing balance sheet,

when closing the contract and starting its replica-tion, does affect the ability of the bank to borrowmoney at a given spread. Although theoretically thebank is adding a risk-free operation in its balancesheet (i.e.: a derivative contract and its associatedreplication strategy10), the funding spread is not nilbecause the spread on the already issued debt D1 isfixed and it cannot be updated. Burgard and Kjaer[3] theoretically justify a funding spread equal tozero in the replication strategy by showing that, ina simplified balance sheet, the total spread paidafter the start of the replication strategy would besuch that it implies a risk-free rate for the incre-mental debt needed in it. A similar view has beenproposed also in Nauta [20]. Nonetheless, this jus-tification relies on the fact that all the outstandingdebt is renewed at the moment of the replication’sinception, and in any case every time a change inthe total outstanding amount occurs: in this waythe overall lowering of the bank’s riskiness is re-distributed on the old and new debt, so that theincremental funding spread would be zero (i.e.: thenew debt marginally costs the risk-free rate). But, ifthe existing debt cannot be freely renewed, becauseit has its own expiry (in T in our setting) and thecontract funding spread cannot be changed, thenthe marginal funding spread on the new debt willnot be the risk-free rate.

Hull&White [13], [14] and [15] also think thatthe inclusion of funding costs is not justified: theirargument hinges on the M&M theorem and, in themore recent works, on some example reproducingpossible real cases of derivative pricing. We showedabove that the funding costs do exist when some ofthe assumptions made by M&M are not matchedin the real world, namely when considering a com-plex banking activity in a multi-period setting; assuch funding costs have to considered, as it willbe apparent in what follows. Besides, the authorsseem to be more worried about the fact the fundingcan be seen also a benefit in the price of the deriva-tive contracts: this may originate some arbitrageopportunities. We will show that we tend to agreewith them as far as the inclusion of funding benefitis concerned, but we disagree with them on thatfunding cost should not be included as well. Butmore on this later on.

After having determined the funding costs thatthe creditors of the bank would rationally set onthe debt issued by the bank to set up the replicationstrategy, let us see how this modifies, with respectto the framework outlined above, when we knowknow that the bank needs to borrow money (oth-

10We exclude here the model risk inherent to the replication strategy: the model can be actually a partially correct representationof the reality, so that the replication is not perfect. An example of model risk can be using the B&S model in a world where theunderlying asset follows a dynamics with stochastic volatility instead of a deterministic one.

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erwise no change applies). Equations (33) and (34)can be generalised as follows:

Vu = αSu + βFB(1 + r + f21{βF<0}) (40)

and

Vd = αSd + βFB(1 + r + f21{βF<0}) (41)

The funding spread appears only if the bank needto borrow money (βF < 0).

Equations (33) and (34) are a system that can beeasily solved for quantities α and βF, yielding:

α = ∆ =Vu −Vd(u− d)S

(42)

andβF =

uVd − dVu

(u− d)B(1 + r + f21{βF<0})(43)

The value at time t of the contract is then:

V(t) = ∆S + βFB

=Vu −Vd(u− d)

+uVd − dVu

(u− d)(1 + r + f21{βF<0})

(44)

After some manipulations, it is possible to ex-press (44) in terms of discounted expected valueunder the risk neutral measure:

VF(t) =1

1 + r[pVu + (1− p)Vd] + FVA (45)

where FVA the funding value adjustment, equal to:

FVA = B[βF − β] (46)

So the value of the contract including funding isequal to the otherwise identical contract valuedin an economy where the counterparty risk is ex-cluded (and funding costs are nil) plus the fundingvalue adjustment. When βF > 0, then βF = β andFVA = 0, otherwise |β| > |βF| (since f2 is a positivequantity) and the FVA is a positive quantity as well.This means that a bank that wishes to replicate along position after going short in the contract, itsreplication cost is higher than the cost paid in aperfect market where default risk is excluded andf2 = 0 always.

It is worth noting that if the bank wishes toreplicate as short position in the contract, after en-tering a long position with the counterparty, thenthe contract is worth, in absolute value:

|V(t)F| =∣∣∣∣− 1

1 + r[pVu + (1− p)Vd] + FVA

∣∣∣∣where FVA is defined as before with βF modifiedas

βF =uVd − dVu

(u− d)B(1 + r + f21{−βF<0})

FVA, for the same considerations above, is still apositive number. This means that the short replica-tion cost entails a lower value for the long positionin the contract for the replicator. We can summarisethese results and write the value to the replicatorfor a long/short position in the contract as:

φVF(t) = φ1

1 + r[pVu + (1− p)Vd]− FVA

= φV(t)− FVA(47)

where φ = 1 if the replicator is long the contractand φ = −1 if it is short. Equation (47) impliesa buy/sell price that the replicator would quote(even in absence of bid/ask spreads), since thevalue when going long the contract is lower thenthe theoretical risk-free price (equation (38)), whilethe value when going short is higher in absoluteterms (i.e.: more negative) then the same theo-retical risk-free price. In the Option Pricing The-ory no-arbitrage bounds are identified typically bytransaction costs, such as pure bid/ask spreads.We are here adding another component widen-ing these bounds, which is given by the fundingcosts due to the default-riskiness of the replicator(see also Castagna [7] , in the final remarks, andCastagna&Fede [10], chapter 10, for a discussionon this topic).

Let us now see how the contract is evaluatedby the bank. Let Π(t) = φ[VF(t)− αS(t)− βFB]: ifthe bank correctly applies the replication strategyprescribed by the model, given that this is a perfectrepresentation of the real world, then we shouldhave that Π(s) = 0 for any s ∈ [t, T]. We are sup-posing here that the bank is considering also thefunding costs in the replication strategy. Nonethe-less, taking into account also the arguments of thosethat tend to exclude the FVA, we want to see whathappens when the contract is inserted in the bank’sbalance sheet and ascertain whether the marginalfunding cost is actually the risk-free rate. The eval-

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uation process relies on the following equation:

VB(t) = E [X1(ms1 − f1) + Π(T) + E]

=[X1(ms1 − f1) + Π + E

](1− PD1)(1− PDV)

+J

∑j=1

max[

X1(Rec1j − (1 + f1)) + Π + E; 0]

∗ p1j PD1(1− PDV)

+L

∑l=1

max[X1(ms1 − f1) + RecVl EPEV + ENEV

− φ(∆S− βFB(1 + f21{−φβF<0})) + E; 0]

∗ p2l PDV(1− PD1)

+J

∑j=1

L

∑l=1

max[X1(Rec1j − (1 + f1)) + RecVl EPEV

+ ENEV − φ(∆S− βFB(1 + f21{−φβF<0})) + E; 0]

∗ p1j pVl PDVPD1 = E

(48)

where Π = E[Π(T)], S = E[S] and ENEV =φV−(T) = φ[pVu1{φVu<0} + (1− p)Vd1{φVd<0}] isthe expected negative exposure.

The loss is computed only on the EPEV , other-wise it is nil: the bank is unaffected by the coun-terparty’s default when the value of the contractis negative and the expected value of the contractfully enters in the equation.

Let

R∗1 =J

∑j=1

max[

X1(Rec1j − (1 + f1)) + Π + E; 0]

p1j ,

R∗2 =L

∑l=1

max[X1(ms1 − f1) + RecVl EPEV + ENEV

− φ(∆S + βFB(1 + f21{−φβF<0})) + E; 0]pVl

and

R∗1,2 =J

∑j=1

L

∑l=1

max[X1(Rec1j − (1 + f1)) + RecVl EPEV

+ ENEV − φ(∆S + βFB(1 + f21{−φβF<0})) + E; 0]p1j pVl .

We rewrite the (48) in a lighter notation:

VB(t) = E [X1(ms1 − f1) + Π(T) + E]

=[X1(ms1 − f1) + Π + E

](1− PD1)(1− PDV)

+ R∗1PD1(1− PDV) + R∗2PDV(1− PD1) + R∗1,2PDVPD1

= E(49)

The probability of default of the bank once thederivative contract is closed, can be computed byconsidering all all the cases when the bank’s valuedrops below zero, in which case the limited share-holders’ liability floors the value at zero. We have

that :

PDB =J

∑j=1

1{R∗1j=0}p1j PD1(1− PDV)

+L

∑l=1

1{R∗2l=0}p2l PDV(1− PD1)

+J

∑j=1

L

∑l=1

1{R∗1j ,2l=0}p1j p2l PD1PDV

(50)

where 1{·} is the indicator function and R∗abis the

b-th addend in the summation in each R∗a .

The Default of the Counterparty Does not Implythe Default of the Bank

If the notional of the derivative contract is is muchsmaller than the asset A1 already included in the as-sets of the bank’s balance sheets, then (analogouslyto what have seen above for the asset A2) we havethat:

R∗1 ≈ R∗1,2 ≈ R1

and

R∗2 ≈ X1(ms1 − f1) + RecVl EPEV + ENEV

− φ(∆S + βFB(1 + f21{−φβF<0}) + E

The notation is the same as above.After substituting these values, equation (49)

can be written as:

VB(t) = E [X1(ms1 − f1) + Π(T) + E]

=[X1(ms1 − f1)) + Π + E

](1− PD1)(1− PDV)

+ [X1(ms1 − f1) + RecVl EPEV + ENEV

− φ(∆S + βFB(1 + f21{−φβF<0})) + E]

∗ PDV(1− PD1) + R1PD1

= E(51)

By repeating the same reasoning we have presentedin the case of two assets, let [(ms1− f1)X1 + E] = Hso that equation (51) can be written

VB(t) =

H[(1− PD1)(1− PDV) + PDV(1− PD1)]

+ R1PD1 + Π(1− PD1)(1− PDV) + [RecVl EPEV

+ ENEV − φ(∆S + βFB(1 + f21{−φβF<0}))]

∗ PDV(1− PD1) = E(52)

By formula (5) and the definition of ms1:

[(ms1 − f1)X1 + E](1− PD1) + R1PD1 = E

The first term on the RHS of the first line in (52) is(by equation (5) and the definition of ms1):

H[(1− PD1)(1− PDV) + PDV(1− PD1)] + R1PD1

= H(1− PD1) + R1PD1 = E

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So that:

VB(t) = Π(1− PD1)(1− PDV)+

+ [RecVl EPEV + ENEV − φ(∆S + βFB(1 + f21{−φβF<0}))]

∗ PDV(1− PD1)

= 0(53)

or:

VB(t) =[Π(1− PDV) + [RecVl EPEV + ENEV

− φ(∆S + βFB(1 + f21{−φβF<0}))]PDV](1− PD1) = 0

(54)

Since Π = φV − φ∆S − φβFB(1 + f21{−φβF<0}),and φV = φV+(T) + φV−(T),11 equation simpli-fies as:

VB(t) =[Π− LgdVl

EPEVPDV

](1− PD1) = 0

(55)

where we have set

− LgdVlEPEV = RecVl EPEV − φV+(T)

It is worth noting that by the definition of creditvalue adjustment:

LgdVlEPEVPDV = CVA

1 We also know that by definition of the replicationstrategy Π = 0 so that (55) holds only if we add anadditional sum of cash at the inception such thatwe set the initial cost of the replication strategy at:

φV∗(t) ≡ φVF(t)−CVA = φV(t)− FVA−CVA (56)

As we have seen before for the replicator, thevalue of the contract depends on the side (long orshort), on the funding costs (the FVA) and on theexpected loss given the default of the counterparty(the CVA). Both quantities operate in the same way:they lower the value to the bank when it is long thecontract, and they increase the negative value (i.e.:make it more negative) when it is short the contract.The two VBid bid and VAsk ask values at which thebank can fairly trade the contract, expressing themin absolute terms, are:

VBid = V − FVA−CVA ≤ V

andVAsk = V + FVA + CVA ≥ V

which confirms the results in Castagna [7].We can make the following considerations:

- in a multi-period setting, where existing debtcannot be renewed continuously, the produc-tion activity of derivative contracts by thebank, although theoretically risk-free,12 doesnot imply a marginal funding cost equal tothe risk-free rate, but on the contrary a posi-tive funding spread, if the replication strategyprescribes to borrow cash;

- when internally evaluating a derivative con-tract, if the banks pays for whatever reasona funding spread, then the standard replica-tion theory that assumes a unique lendingand borrowing risk-free rate has to be accom-modated to include also a funding valuationadjustment (FVA);

- a credit valuation adjustment (CVA) entersalso in determining the value to the bank;

- eventually the default of the bank does notmatter. This means that the value to the bankis determined as it were not subject to default,even if it is taken into account when startingthe evaluation process through PD1 and PDV .As a consequence the debit value adjustmentDVA is not present in the evaluation equation(56);

- the DVA is the CVA seen from the counter-party’s point of view. When, in the bargainingprocess, the bank has to yield the DVA as acompensation to the counterparty, it repre-sents a cost and it cannot be compensated bythe FVA, as it is usually affirmed nowadays inmany works. On the other end, the FVA andthe DVA are not two ways to call the samequantity, if not in the particular case of a loancontract (see Castagna [8] for a discussion onthis point).

- there is no double counting of DVA and FVA:they operate with different signs on the valueof the contract. The bank should try not topay the first one, since it cannot be replicatedunder realistic assumptions;13 the second onecan be included in the value of the contractand should be considered as a productioncost.

Proposition 6. When evaluating a derivative contractwhose counterparty’s default does not imply the bank’sdefault, the correct and theoretically consistent value isgiven by the otherwise risk-free theoretically fair price

11We are aware of that the pricing is much more involved as the one we are presenting, since each component of the price is affectedby the others and a simple decomposition as the one we are presenting here is possible only by disregarding them. Anyway, it is alsotrue that the error we make in doing this is not substantial.

12Again, disregarding possible model risks.13See Castagna [8] for a formal proof.

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plus the credit and funding valuation adjustments. Thesign of the adjustments depend on the long or short po-sition taken into the contract by the bank. The bank’sdefault does not enter in the evaluation process, so thatno debit valuation adjustment is considered.

The Default of the Counterparty Implies the De-fault of the Bank

Assume now that derivative contract has a notionalsuch that the default of the counterparty impliesthe default of the bank. The valuation cannot beoperated by pretending that the contract is a stan-dalone operation without any regard to the remain-ing parts of the balance sheet. We need to resortto numerical procedures, but at least formally wecome up woith a valuation formula.

By rewriting equation (49), we have:

VB(t) = E [X1(ms1 − f1) + Π(T) + E]

=[X1(ms1 − f1) + Π + E

](1− PD1)(1− PDV) + ε

= E(57)

where ε = R∗1PD1(1− PDV) + R∗2PDV(1− PD1) +R∗1,2PD1PDV . Let PD1,2 = 1− (1−PD1)(1−PDV).As before, we solve for the fair value of V and weget:

φV∗(t) = φV(t)− FVA− EPD1,2

+ [X1(ms1 − f1)(1− PD1)(1− PDV) + ε]

= φV(t)− FVA− EPD1,2 + dbV

(58)

which is the value of the risk-free fair value plus thecash needed to compensate the other adjustments.

It is likely useful to calculate the debit value ad-justment that the bank can be reasonably expectedto be requested by the counterparty, to compensatethe bank’s default risk it bears (the DVA from thebank’s point of view is the CVA from the counter-party’s point of view).

Assuming we are still in a market where perfectinformation is available to all agents, the expectedrecovery RecC

V on the derivative contract that thecounterparty calculates, is:

RecCV =

=J

∑j=1

min

[X1Rec1j+φ(∆S1{−φ∆>0}+βF B1{−φβF>0})+E

X1+∣∣∣ENE|+|∆S1{−φ∆<0} |+|βF B1{−φβF<0}

∣∣∣ ; 1

]p1j

(59)

Equation (60) is worth a few comments: firstly, thederivative contract is excluded from the calcula-tions, since it is considered for its expected liabilityaspect (the ENE), so that it cannot be part of the(possibly residual) value of the assets upon the

bank’s default (this is why the recovery on the EPEnever appears at the denominator). Secondly, ifthe bank went bust, that means that the issuer ofthe asset A1 defaulted, so that only a fraction ofthe face value X1 is recovered. Thirdly, the repli-cating portfolio can be an asset and/or a liability,depending on the signs of the quantity entering init; this explains the indicator functions at the numer-ator of fraction which make the components of thereplication portfolio an asset when their quantity ispositive. If their quantity is negative, the absolutevalue adds to the total amount of the bank’s liabil-ity, which is at the denominator of the fraction. Thetotal residual assets are divided by total liability tohave the average recovery that each of them willcollect (we are not considering priority rules in thedistribution of residual assets).

The DVA is then:

DVA = (1−RecCV)× ENE× PDB

= LgdCV × ENE× PDB

(60)

which is the expected loss suffered by the counter-party on the expected exposure (ENE) when thebank defaults. The probability of the bank’s defaultdepends the probability of the counterparty’s de-fault PDV : this is a circular relationship that causesno problem when the contract is small comparedto the total assets of the bank, so that we fall backin the first case examined. If the notional amountis big enough to produce the bank’s default whenthe counterparty goes bankrupt, PDB should beadjusted by excluding the effect of PDV . When thecounterparty’s default triggers the bank’s default,the value of the contract is positive to the bank,since it suffers a loss of the EPE such that the totalassets are not able to cover the outstanding liabili-ties. But this means that the cases when the bankdefaults after the counterparty’s default can neverbe applied to the ENE.

All this discussion on the DVA is to show thatthe claim usually stated in theoretical works (and,alas, often also in practice) are not very soundlygrounded: the DVA is always a very difficult quan-tity to deal with and it may cause many inconsis-tencies when taken into account not in a properfashion. As a general rule, the DVA is simply a costfor the bank corresponding to the counterparty’sCVA. The DVA can be computed by the usual toolsonly when the amount of the deal is such that thecounterparty’s default does not trigger the bank’sdefault. In any case we would like to stress thefact that DVA is quite a different quantity from thequantity dbV defined before.

The following considerations are in order:

- when the default of the counterparty can trig-

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ger the default of the bank, then the evalua-tion of the derivative contracts must considerthe entire balance sheet of the bank: the pro-cess is quite cumbersome;

- in very general terms, the value of the contractis made of the components:

1. the theoretical value of an otherwiseidentical risk-free contract;

2. minus the FVA, computed as in the caseexamined before, depressing the value ofa long position, or making more negativethe value of a short position;

3. the maximum loss given the default ofboth the issuer of asset A1 and of thecounterparty (and, hence, of the bank):this equals the equity capital E times thejoint probability of default PD1,2. Themaximum loss is limited to the equitydue to the limited liability of the share-holders. This quantity can be roughly as-similated to the CVA component of thefirst case examined above and it oper-ated in the same way on the initial valueof the contract;

4. diversification benefit dbV , which ac-counts for the ability of asset A1 to con-tribute to the final value of the equity,if its issuer survives, and for the jointrecovery when both the counterparty ofthe contract and the issuer default. In allcases the limited liability is considered,so that the total loss can never exceed theequity capital. This part can be roughlyassimilated to the DVA, although someparts have to be more correctly assignedthe the CVA, since ε contains also caseswhen the loss suffered by the bank doesnot generate its default. Moreover, it isapparent that there is no reference to theloss given default suffered by the coun-terparty, because the evaluation is per-formed from the bank’s point of viewand it is interested only in residual netvalues of the assets for itself, not in re-covery values for the counterparty. Inany case, the quantity dbV , as the DVA,increases the value of the contract whenthe bank has a long position in it, andreduces the negative value when it has ashort position.

- The justification adduced by some authors(see Hull&White [13], [14] and [15], for exam-

ple) that the DVA is a benefit upon defaultdue to the limited liability of the shareholders,is partially confirmed only in the cases whenthe derivative contract has a big impact onthe balance sheet, otherwise the DVA shouldbe considered as a cost (if paid). Also theidentification of the FVA with the DVA (or apart of it) is not justifiable.

Proposition 7. When evaluating a derivative contractwhose counterparty’s default can trigger the bank’s de-fault, the correct and theoretically consistent value isgiven by the otherwise risk-free theoretically fair priceplus the funding valuation adjustments, the maximumloss suffered on the joint default and the diversificationbenefit due to the possibility to cover losses with othernon defaulting assets and the limited shareholders’ lia-bility in case of bank’s default. The sign of the adjust-ments depend on the long or short position taken intothe contract by the bank. The bank’s default enters inthe evaluation process only as far as the maximum lossis considered. The diversification benefit, although play-ing a similar role, is only roughly ascribed to the DVA.

Balance Sheet Shrinkage, Funding Benefit andNo Arbitrage Prices

Some authors (Burgard&Kjaer [2] andMorini&Prampolini [19], for example) recur tothe “funding benefit” argument to justify the inclu-sion of the DVA into the valuation process of thederivative contracts by the bank. On the other hand,they see the DVA strictly related to the the cashreceived when entering in the derivative contracts,so that DVA clashes with the −FVA, in the sensethat the former should be seen as a funding benefit.

For some types of derivative contracts, such asan uncollateralised short position in a plain vanillaput option, the quantity −FVA may be the same asthe DVA. This happens when the underlying assetis repo-able, i.e.: can be bought via a repo transac-tion so that the actual financing cost is the impliedrepo rate, which is very near to the risk-free rateand the related cost of funding can be neglected.Hence, the only cost of funding refers to the cashthe bank needs to pay the option’s premium (in thiscase it is zero, since the premium is received): inCastagna [8] we identified this part of thetotal FVAas FVAP, to distinguish it from the other compo-nent of the total funding value adjustment, FVAU,due to the financing costs of the position in theunderlying, if any.14

On the other hand, limiting the FVA to the onlyFVAP component can be incorrect for all those con-tracts that can have a double sided value for either

14Hull&White [13], [14] define these two quantities as DVA1 and DVA2, respectively.

38iasonltd.com

counterparties and that generally start with zerovalue for both, so that no premium is exchangedbetween parties. In this case the FVA refers to pos-sible future cash-flows and it definitely differs fromthe DVA.15

We have seen above, confirming the resultsachieved in Castagna [5] and [7], that the DVA andthe FVA operate with opposite signs on the valueto the bank of the contract: in case the bank, whengoing long (short) the contract, is forced to pay theDVA, it makes the value higher (lower) than thelevel the bank would be willing to pay (receive)at the inception; at the same time, the acceptablevalue for the bank for a long (short) position is de-creased (increased) by the FVA. If we accept theinclusion of −FVA into the valuation of derivativecontracts, then there could be some overlapping ofthe DVA and the FVA, seen as a funding benefitand accounted for with the opposite sign.

It seems here that we have an opposite view onsomething upon which an agreement could (andshould) be reached. The difference between ourview and the one originated by the “funding ben-efit” argument lies in the balance sheet shrinkage:basically, this activity consists in the reduction ofthe outstanding debt as soon as a positive cash flowis received by the bank. In this way, both assets(the cash) and liabilities (debt) decrease and thebank pays less funding costs.16. We have shown inthe two cited works that the “funding benefit” isa badly posed concept, and it would be better totalk about balance sheet shrinkage instead; more-over, the funding benefit can be used to replicatethe DVA only under very specific (and very likelyunrealistic) conditions.

When the valuation of a derivative contract re-lies on the balance sheet shrinkage policy, an im-mediate consequence is that any cash-flow, eitherpositive or negative, is present valued by addingfunding costs. The reason is simple: if the cash-flowis negative, the bank must finance it by borrowingmoney and pay at the expiry of the loan also thefunding costs (the difference between the actual andthe risk-free interests); if the cash-flow is positive,the bank can buy back some outstanding debt, thuspartially saving the future expenditure of fundingcosts, which then add to the present value of thereceived cash.

The assumptions, implicit in the possibility toimplement such balance sheet shrinkage policy, arethat the bank has some outstanding debt to buyback, preferably with the same expiry as the deriva-

tive contract (a fairly realistic assumption) and thatthe buy back can be done relatively easily with lowtransaction costs (this could hardly be met in prac-tice, since not all outstanding debt is liquid anddealing with tight bid/ask spreads in the market).

Assume the shrinkage policy is feasible and letus modify the evaluation process of contract by thebank by including it. In practice, if −φβF > 0 wemodify equation (49) as follows:

VB(t) =

E[

X1ms1 − (X1 − |φβFB|)(1 + f1) + φV(T)− φ∆S + E]

=[

X1ms1 − (X1 − |φβFB|)(1 + f1) + φV(T)− φ∆S + E]

∗ (1− PD1)(1− PDV) + R∗1PD1(1− PDV)

+ R∗2PDV(1− PD1) + R∗1,2PDVPD1

= E(61)

where

R∗1 =J

∑j=1

max[

X1Rec1j − (X1 − |φβFB|)(1 + f1)

+ φV(T)− φ∆S + E; 0p1j ,

R∗2 =L

∑l=1

max[

X1ms1 − (X1 − |φβFB|)(1 + f1)

+ RecVl EPEV + ENEV − φ∆S + E; 0pVl

and

R∗1,2 =J

∑j=1

L

∑l=1

max[X1Rec1j − (X1 − |φβFB|)(1 + f1)

+ RecVl EPEV + ENEV − φ∆S + E; 0]p1j pVl .

In equation (61) the positive cash-flows related tothe long bank account position −φβFB > 0, isdeducted from the outstanding amount of debtD1 = X1. This will generate a saving equal to thesmaller interests paid on the reduced notional ofdebt. To simplify things, we did not analysed howthe creditors of the bank would update f2 whenincluding the balance sheet policy; moreover wedid not consider the fact the the outstanding debtwould probably deal in the market at a price lowerfrom X2, which is the level we implicitly assumed.

In the standard replication argument presentedabove, when −φβF > 0 we have βF = β andFVA = 0. If the positive cash-flows are not investedin a risk-free bank account, but they are used toshrink the balance sheet, they will implicitly yieldf1 instead of the risk-free rate (set equal to zero inthis analysis). If we limit the analysis to the casewhen the notional of the derivative contract is smallcompared to the total assets, we set f2 = f1, and

15See Castagna&Fede [10], chapter 12, for the case of uncollaterlised swap contracts.16If the equity capital (E) is constant in time, balance sheet shrinkage is the same as the activity currently denoted as deleveraging. It

should be noted, though, that the balance sheet delevarage can be alternatively achieved by only increasing the equity capital, thussimply decreasing the amount of debt capital as a percentage of the total liabilities

Summer 201439

TRANSFER PRICING

we repeat the same reasoning as that one presentedabove, then it is easy to realise that the value of thecontract to the bank will be:

φV∗(t) ≡ φVF(t)−CVA = φV(t)− φFVA−CVA (62)

If the bank is long, the value will be lower, as inthe previous case when the shrinkage policy is notadopted; when the bank is short, the value of thecontract will be less negative to the bank, so thatit will be ready to accept a lower premium fromthe counterparty to enter in the short position ofthe contract. In contracts where the value is alwayspositive to one of the two parties, when the bankis long it will accept to pay the FVA but not theDVA; when it is short, it will yield the DVA to thecounterparty, thus accepting a lower premium. Butthis is theoretically the same quantity as the (neg-ative) of the FVA17 considering the balance sheetshrinkage policy, or the funding benefit as it isgenerally mentioned. The problem related to thedouble counting of the DVA and the FVA stemsfrom this equivalence.

The balance sheet shrinkage policy seems aneffective way for the bank to be aggressive in of-fering derivative contracts to the counterparties: itjustifies the inclusion of the DVA, seen as a fundingbenefit, or −FVA. Although the two quantities maynot fully coincide in any case, they are very similarin most of cases.

Nonetheless, this policy has many drawbacksthat make it less enticing than it may appear at firstlook. We will briefly list some related problems:

- as mentioned above, it is not always possibleto find suitable bonds (outstanding debt) tobuy back in the market at fair prices;

- the policy produces counter-intuitive results,pushing the bank to progressively willinglylower the selling price of derivative contractwhen its creditworthiness worsens. It can beforced to do so because the counterparty canhave enough bargaining power to ask for theinclusion of the DVA in the traded price, butthis has to be seen as a cost and not an adjust-ment to lightheartedly accept;

- the shrinkage is possible only when there isa premium that must be paid by the coun-terparty (the asset can be financed in mostof cases by a repo transaction, although thisis not always feasible). In contracts suchas uncollateralised forwards and swaps thebank still will pay funding costs related to the

replication strategy and possibly also a DVA,but they act with opposite signs, as we haveshown before;

- the funding benefit argument is basically aclaim on the replicability of the DVA morethan the justification of the inclusion of theFVA with reverse sign. The DVA can be repli-cated only under very restrictive conditions.

Proposition 8. If the balance sheet shrinkage policy isincluded in the evaluation a derivative contract,the cor-rect and theoretically consistent value is given by theotherwise risk-free theoretically fair price plus the creditand funding (cost) valuation adjustments and the debitvalue adjustment (or alternatively the funding (benefit)value adjustment). The sign of the adjustments dependon the long or short position taken into the contract bythe bank.

We already supported elsewhere18 the idea thatthe balance sheet shrinkage should not be consid-ered a sound assumption when evaluating deriva-tive contracts, which means that the DVA is treatedalways as a cost (if paid) and that the FVA neverenters in the internal value of the contract with thenegative sign (i.e.: as a funding benefit). This meansthat the internal value is the one we have definedin equation (56), when no balance sheet shrinkagewas considered.

EXAMPLE 4 We sketch here the effectsof the assumptions made by the bank re-garding the replication strategy (i.e.: fea-sible or not feasible balance sheet shrink-age) when a derivative contract is closed.To make things concrete, consider an option Von an asset S, expiring in T. Assume also thatgoing short and long the underlying asset S canbe done with repo/reverse-repo transactions, pay-ing a repo rate (approximately) equal to the risk-free rate. In this case the funding is due only tothe payment of the premium. The bank evaluatesthe “production” costs to sell it (i.e.: enter in ashort position) and, applying equation (56), it gets:

- −V(0) = −10, value of the risk-free optionevaluated with the chosen model (e.g.: B&S);

- −FVA = 0, since the premium is received;

- −CVA = 0, since no counterparty risk exitsfor a short position in an option;

- V∗(0) = −1017In practice there are some differences between the FVA and the DVA, due to the fact the different recoveries are attached to the

outstanding debt and the exposure of the derivative contract.18See Castagna [7].

40iasonltd.com

The bank should sell the option at a price P = 10: inthe is case the P&L = P−V∗ = 0 at the inception.The counterparty of the bank in this contract isanother bank with an equal bargaining power, sothat it manages to charge DVA in the option price,since it wants to be remunerated for the coun-terparty risk related to the bank’s default proba-bility (the DVA is the CVA from its standpoint).Given the bank’s default PDB and the loss givendefault of the option, the counterparty calculatesthe DVA = 1. So the traded price of the contract is:

- −V(0) = 10

- +DVA = 1

- −FVA = 0

- P = −10 + 1 = 9

The bank sells the option at the price P = 9, sothat the P&L = P − V∗ = −1 at the inception,given the value the bank attached to this contract.Consider now the case when the bank assumesthat it is able to operate the balance shrinkageas an ordinary and routine liquidity policy. Theoption “production” cost (i.e.: the value to thebank) is in this case derived by equation (62):

- −V(0) = −10

- +FVA = 1, the funding benefit, originated byown bonds’ buy-back;

- −CVA = 0, since no counterparty risk exitsfor a short position in an option;

- V∗(0) = −10 + 1 = −9

The bank is happy to sell the option at P = 9 andthe P&L in the trade above would be P−V∗ = 0 atthe inception. Holding the P&L = 0 up to theexpiry (or, the end of the replication) dependson the ability to actually and effectively imple-ment the balance sheet shrinkage. In practice,even if the counterparty has not a great bargain-ing power, it will be compensated for the DVAby the bank all the same, since it includes thefunding benefit in its value. In the first case(no balance sheet shrinkage), during the bargain-ing process, the bank will reluctantly yield theDVA because of the counterparty’s strength; inthe second case (balance sheet shrinkage) the bankwill include the DVA (seen as a funding benefitf va) even if not requested by the counterparty.Consider now the case when the bank goes longthe option. In this case the counterparty riskis relevant; the bank measures the risk and setsCVA = 0.75. The option “production” cost (valueto the bank) is given, again, by equation (56):

- V(0) = 10

- −FVA = −1 since the bank pays the option’spremium, which needs to be funded;

- −CVA = −0.75, the compensation for thecounterapry risk;

- V∗(0) = 10− 1− 0.75 = 8.25

The bank should buy the option at a price P = 8.25,so that P&L = V∗ − P = 0 at the inception.Let us assume that the counterparty is a bank witha great bargaining power, so that it does not ac-cept to pay the CVA (i.e.: to compensate the bankfor the counterparty risk) and the FVA (i.e.: thecosts the bank pays to fund the premium pay-ment above a risk-free agent). The price of thecontract the counterparty manages to receive, is:

- V(0) = 10

- −CVA = 0

- −FVA = 0

- P = 10

The bank buys the option at P = 10, the so it marksa loss since P&L = V∗− P = −1.75 at the inception.If the bank thinks that the balance sheet policycan be hardly implemented in practice, the selland buy prices at which it is willing to trade, are

VBid = 8.25 < V(0) = 10 < VAsk = 10.

If the bank evaluates contracts under theassumption that the of the balance sheetshrinkage is a feasible liquidity policy, reg-ularly and effectively run in reality, then

VBid = 8.25 < VAsk = 9 < V(0) = 10.

In this case the bank is willing to sell the op-tion at a price which is lower than the other-wise risk-free value, relying on the funding ben-efit it receives. This point may also raise someeye brows, besides the considerations on the fea-sibility of the balance sheet shrinkage policy.

Conclusion

We have proved that the classical results on theevaluation of investments by Modiglian&Miller [18]and Merton [16] hold only when the bank holdsa single asset: in this case the value of the invest-ment is independent from the capital mix (i.e.: theamount of debt and equity capital used to fundit). In a multi-period setting, for an additionalinvestment the results do not hold (or they holdpartially), so that the way it is financed does matter

Summer 201441

TRANSFER PRICING

and the funding cost enter in the evaluation pro-cess. The main point in this framework is that theexisting debt cannot be updated, until its expiry,to reflect the new risk of the bank’s total assets,so the total financing cost is higher than the levelthat would prevail if all the debt could be freelyrenewed. Moreover, in the real world the equity isused to cover losses generated indistinctly by allassets: the limited shareholders’ liability producesits effects on the evaluation process only if the heldamount of the asset is large enough to trigger thebank’s default when the issuer goes bankrupt. Thisresult gives a theoretical support to the well es-tablished practice in the financial industry to takeinto account funding costs and to charge them inthe value of the asset when the bank’s bargainingpower allows to do so.

For derivative contracts the results we derivedare similar: the counterparty’s credit risk and thefunding costs have to be included in the value ofthe contract to the bank, whereas the bank’s owndefault does not play any role, thus the debit valueadjustment should be excluded. When the bankembeds the balance sheet shrinkage policy in theevaluation process, then the negative of the fund-ing cost, meant to be a funding benefit, is includedin the value, producing the same (often identical)effect of the inclusion of the DVA. Since in ourframework the DVA never enters in the value, thepossible inclusion of the funding benefit will nevergenerate any double counting effects. Anyhow, wethink that relying on the balance sheet policy israther incautious, since we think that in reality itcannot be implemented on the base required by thederivative market-making activity.

It is maybe interesting to note that the conclu-sion we have reached is the same as in Hull&White[13], [14] and [15], although partially and for adifferent reason. In fact, the two authors excludethe FVA in any case, either it enters in the valueas a cost or as a benefit, mainly relying on the

classical investment evaluation theory’s result ofModigliani&Miller. Besides, they show in [15] thepossibility of arbitrages when the funding benefit−FVA is accounted for in the value of the contract.In any case, their aim is to support the theoreticalresult of one price dealing in the market.

In our framework, the FVA increases the valueof short positions (makes it more negative) anddecreases the value of long positions. So the FVAis always a positive or nil value. When the bal-ance sheet shrinking is considered as a feasible andever implementable policy, then the FVA can alsoenter with a negative sign (as a funding benefit)thus decreasing the value of both short and longpositions.

We believe that, although theoretically possible,the balance sheet policy should not be consideredwhen evaluating derivative contracts. Firstly, thepolicy can be hard to be followed in practice andsecondly it is not feasible when the contract is twosided (such as swaps) and it is dealt with no cashexchanged between parties. If and when it is actu-ally implemented in practice, then a windfall forthe bank occurs. If the buy-back of own bonds isnot operated strictly following the rule, the bankshould evaluate derivative contracts as we have in-dicated in formula (56), hence agreeing with H&Wat least as far as the funding benefit is concerned.The funding benefit, if any, should be referred tothe smoothing of the future cash-flows’ profile ofthe derivatives’ book, but this will be the object offuture research.

ABOUT THE AUTHOR

Antonio Castagna is Senior Consultant and co-founder at Iasonltd and CEO at Iason Italia srl.Email address: [email protected]

ABOUT THE ARTICLE

Submitted: January 2014.

Accepted: February 2014.

42iasonltd.com

References

[1] Brigo, D., M. Morini and A.Pallavicini. Counterparty CreditRisk, Collateral and Funding: WithPricing Cases For All Asset Classes.Wiley. 2013.

[2] Burgard, C. and M. Kjaer. PDERepresentations of Options withBilateral Counterparty Risk andFunding Costs. Available athttp://papers.ssrn.com. 2010.

[3] Burgard, C. and M. Kjaer. In thebalance. Risk. November, 2011.

[4] Castagna, A., F. Mercurio and P.Mosconi. Analytical Credit VaRwith Stochastic Probabilities ofDefault and Recoveries. Work-ing Paper, available online at:http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1413047.2009.

[5] Castagna, A. Funding, Liquid-ity, Credit and CounterpartyRisk: Links and Implications. Ia-son research paper. Available athttp://iasonltd.com/resources.php.2011.

[6] Castagna, A. Yes, FVA is a Cost forDerivatives Desks. A Note on IsFVA a Cost for Derivatives Desks?by Prof. Hull and Prof. White. Iasonresearch paper, available online at:http://iasonltd.com/resources.php.2012.

[7] Castagna, A. On the DynamicReplication of the DVA: Do

Banks Hedge their Debit ValueAdjustment or their Destroy-ing Value Adjustment?. Iason re-search paper, available online at:http://iasonltd.com/resources.php.2012.

[8] Castagna, A. Pricing of Deriva-tives Contracts under CollateralAgreements: Liquidity and Fund-ing Value Adjustments. Iason re-search paper, available online at:http://iasonltd.com/resources.php.2012.

[9] Castagna, A. Funding ValuationAdjustment (FVA) and Theory ofthe Firm: A Theoretical Justifi-cation of the Inclusion of Fund-ing Costs in the Evaluation ofFinancial Contracts. Iason re-search paper, available online at:http://iasonltd.com/resources.php.2013.

[10] Castagna, A. and F. Fede. Measuringand Management of Liquidity Risk.Wiley. 2013.

[11] Crépey, S., R. Gerboud, Z. Grbacand N. Ngor. Counterparty Riskand Funding: The Four Wings of theTVA. International Journal of The-oretical and Applied Finance, n.2,v.16. 2013.

[12] Cox, J.C., S. A. Ross and M. Rubin-stein. Option Pricing: A SimplifiedApproach. Journal of Financial Eco-nomics, n. 7, pp 229-263. 1979.

[13] Hull, J. and A. White. Is FVA a Costfor Derivatives Desks?. Risk. July,pp 83-85. 2012.

[14] Hull, J. and A. White. ValuingDerivatives: Funding Value Adjust-ments and Fair Value. Working Pa-per, forthcoming in Financial Ana-lysts Journal. 2013.

[15] Hull, J. and A. White. Collateraland Credit Issues in DerivativesPricing. Working Paper. 2013.

[16] Merton, R. On the Pricing of Cor-porate Debt: The Risk Structure ofInterest Rates.Journal of Finance, n.2, pp 449-471. 1974.

[17] Mercurio, F. Bergman, Piterbargand Beyond: Pricing Deriva-tives under Collateralization andDifferential Rates. BloombergL.P., working paper, available athttp://papers.ssrn.com. 2013.

[18] Modigliani, F. and M.H. Miller. TheCost of Capital, Corporation Fi-nance and the Theory of Investment.The American Economic Review, n.3, v. 48, pp 261-297. 1958.

[19] Morini, M. and A. Prampolini.Risky Funding with Counterpartyand Liquidity Charges. Risk. March,2011.

[20] Nauta, B-J. On Funding Costs andthe Valuation of Derivatives. Dou-ble Effect, working paper, availableat http://papers.ssrn.com. 2012.

Summer 201443

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44

Special Interview

Alessandro MauroLITASCO, International Trading and Sup-ply Company of Lukoil Group, Geneva

45

SPECIAL INTERVIEW

A talk with...

Alessandro Mauro

In this section Andrea Roncoroni isinterviewing Alessandro Mauro, headof risk management for LITASCO, theInternational trading and supply com-pany of the Lukoil Group, in Geneva.Starting from his personal experienceAlessandro will provide a clear overviewof energy risk management, denotingits differences with respect to financialrisk management and its future devel-opments. For a matter of clearness, R.and M. stands for Andrea Roncoroniand Alessandro Mauro respectively.

R.: Hi Alessandro, let’s start with your personal experi-ence. Why did you get interested in risk management?

M.: Hi Andrea, thanks for your question. In the 90ssecond half risk management was an underdevel-oped discipline, in general and in the energy sectorspecifically. To give you an idea the Global Associa-tion of Risk Professionals (GARP), now the leadingprofessional organization in this discipline, was cre-ated in 1996. I felt this was a place where I couldbe creative and innovative. There were no books onthe topic and it looked like a learn-by-doing sort ofjob, which I liked.

R.: How did you start?

M.: I was working in Scuola “Enrico Mattei” of ENIas a researcher in energy matters. In those years en-ergy was becoming “financialized”, i.e. energy andfinance were started having something in common.One day I found some articles about Value-at-Risk(“VaR”), at that time representing a new approachto risk measurement, and I started studying howVaR could be applied in the oil sector, and laterto the energy sector in general. When I thought Ihad a decent understanding of the subject, I wrote

an article. Those were the years of the VaR modelcreated and marketed by Riskmetrics, a companyfounded by JP Morgan. It was a good product forfinancial institutions, but not for commodities pro-ducers and traders. It is enough to say that it wasmodelling exposure to energy commodities usingonly four US-based risk factors, where an oil traderwill normally have exposures to dozens or evenhundreds of risk factors. Hence, I started from riskmeasurement, which is only one of the three pillarsof risk management.

R.: What are the other two pillars?

M.: Well they are risk identification and risk treat-ment. If you do not properly identify risks, youcannot measure them and you cannot “treat” them.Risk treatment is often understood as “hedging”,even if speculation is a form of risk treatment. Atthat time hedging in energy markets was taking itsfirst steps, at least in Italy. ENI was starting, fol-lowing advices coming from London-based banks,selling natural gas under innovative pricing formu-lae. At that point in time, even selling at fix pricewas to be considered innovative, after decades ofoil-prices gas indexation. I was called on to helpin performing analysis aimed at assessing how thisnew exposures could be hedged. In fact the projectwas called "hedging project".

R.: Is energy risk management different from risk man-agement in financial institutions?

M.: Yes, I am totally convinced about that. Riskidentification, and partially risk measurement, aredifferent. Risk identification is much more diffi-cult, there is nothing as simple as ‘I am short 100shares’. In fact exposure in energy markets doesnot depend uniquely on conscious choices made bytraders. Many external actors, with their actions,have an impact on exposure taking and value cre-ation. You may say that even risk treatment usingfinancial derivatives is a quite specialized one and

46

adapted to the energy markets peculiarity, due tothe specific elements underpinning the underlyingassets of those derivatives.

R.: What one can do to properly address the specificityof energy risk management?

M.: You need to have excellent resources, whichmeans primarily excellent IT systems and humanresources. As far as the latter are concerned, thechoice is not simple. The talent pool is not vastand there is big competition in attracting talents,in a situation in which the request for energy riskanalysts is increasing. The alternative is trainingyour employees, either on the job or by selectingproper external training. In this respect, profes-sional certifications in risk management are goodchoices.

R.: Which is currently the biggest challenge for energyrisk management?

M.: I choose two. Having IT infrastructure and

resources to support a complex business, under-standing and mitigating other risks. InformationTechnology is crucial because nowadays simply youcannot run any business process without it. Riskmanagement is a business process, but a very spe-cialised one, and consequently it needs customizedIT software. It is so specialized that the name of“ETRM” (Energy Trading and Risk Management)was coined. Being risk management a general busi-ness process, we need to make sure that we identify,measure and mitigate different risks, not just mar-ket risk. This part is often neglected, many peoplelike to introduce more and more complexities inmarket risk models, and they overlook that at thesame time they are increasing other typologies ofrisks (operational, fraud, etc).

R.: Thank you, Alessandro, for this interesting talkabout energy risk management.

M.: Thank you too, Andrea.

Summer 201447


in the previous issue

Spring 2014

banking & finance

Fast Monte Carlo pricing of Nth-to-defaultswaps

Analytical Credit VaR Stress Tests

energy & commodity finance

Stochastic Optimization for the Pricing ofStructured Constracts in Energy Markets

Pricing Spark Spread Option withCo-Dependent Threshold Dynamics

special interview to Marco Bianchetti

special interview

last issues are available at www.iasonltd.com and www.energisk.org

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new frontiers in practical risk anagementenglish...new frontiers in practical risk management year 1...

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