Discussion Paper No.

228, Volkswirtschaftliche Diskussionsreihe, Institut fr Volkswirtschaftslehre, Universitt Augsburg

Credit Risk and Credit Derivatives in Banking

Udo Broll
Saarland University

Thilo Pausch
University of Augsburg and

Peter Welzel
University of Augsburg

July 2002
Abstract Using the industrial economics approach to the microeconomics of banking we analyze a large bank under credit risk. Our aim is to study how a risky loan portfolio aects optimal bank behavior in the loan and deposit markets, when credit derivatives to hedge credit risk are available. We examine hedging without and with basis risk. In the absence of basis risk the usual separation result is conrmed. In case of basis risk, however, we nd a weaker notion of separation.

Keywords: credit risk, credit derivatives, banking rm, risk aversion JEL classication: G21

Department of Economics, Saarland University, D-66041 Saarbrcken, email: u.broll@mx.uniu saarland.de Faculty of Business Administration and Economics, University of Augsburg, D-86135 Augsburg, email: thilo.pausch@wiwi.uni-augsburg.de Corresponding author. Faculty of Business Administration and Economics, University of Augsburg, D-86135 Augsburg, email: peter.welzel@wiwi.uni-augsburg.de

Credit Risk and Credit Derivatives in Banking

Using the industrial economics approach to the microeconomics of banking we analyze a large bank under credit risk. Our aim is to study how a risky loan portfolio aects optimal bank behavior in the loan and deposit markets, when credit derivatives to hedge credit risk are available. We examine hedging without and with basis risk. In the absence of basis risk the usual separation result is conrmed. In case of basis risk, however, we nd a weaker notion of separation.

Keywords:

credit risk, credit derivatives, banking rm, risk aversion

JEL classication: G21

Introduction

Credit risk is one of the oldest and most important forms of risk faced by banks as nancial intermediaries. The risk of borrower default on interest and/or principal carries the potential of wiping out enough of a banks capital to force it into bankruptcy. Managing this kind of risk through selecting and monitoring borrowers and through creating a diversied loan portfolio has always been one of the predominant challenges in running a bank. Since the 1980s a number of new risk sharing markets and nancial instruments have become available which make credit risk more manageable (see Neal, 1996, and Bank for International Settlements, 2001). Banks can pool assets with credit risk and sell parts of the pool. This asset securitization or creation of asset backed securities has seen considerable growth in areas such as home mortgages or automobile loans, where underlying loan contracts and payment schedules are fairly standardized and risk characteristics are similar. Loan sales play a role, e.g., in takeover nancing, where a bank originates a loan and sells it in smaller shares to other banks. More recently, credit derivatives such as credit swaps, credit options, and credit-linked notes have gained importance as instruments to manage risk in situations, where the diversity of loan types and credit risks makes it dicult to securitize loans or sell them individually.

Credit Risk and Credit Derivatives

In the sequel we will use the term credit derivatives both for securities originating from loan securitization and for more advanced instruments such as credit options. Our objective is to examine how the possibility to sell part or all of a banks uncertain loan portfolio at a deterministic price aects bank behavior in deposit and loan markets. The framework we use for our analysis is sometimes called the industrial organization approach to the microeconomics of banking (for a brief survey see Freixas and Rochet, 1997, chapt. 3). It is focused on the banks role as intermediary, but abstracts from informational aspects of banking adverse selection and moral hazard which have dominated banking theory throughout the last two decades. We consider the potential of the industrial organization approach to analyze banking under a variety of market structures ranging from perfect competition to monopoly suciently important to justify the use of this approach.1 To our knowledge, Wong (1997) was the rst author to add aspects of uncertainty and risk aversion to the industrial organization approach to the bank. We supplement Wongs analysis of credit risk by adding a hedging instrument which may or may not carry basis risk. Since the seminal work of Froot et al. (1993) hedging is known to contribute to a rms market value. In our treatment of deposits, we deviate from Wong (1997) by assuming a deterministic deposit rate and modelling an explicit deposit taking decision of the bank. The plan of the paper is as follows. In section 2 we present the model of a large banking rm under credit risk, when a credit derivative is available. Section 3 examines loan, deposit and hedging decisions for a credit derivative without basis risk. Section 4 adds basis risk to our analysis. Section 5 concludes the paper.

The model

Consider a large banking rm in a oneperiod framework. The bank is a classical intermediary, taking deposits D and making loans L. By large we mean that the bank faces a downward sloping inverse demand rL (L) for loans with rL denoting the interest rate on loans and an upward sloping supply rD (D) of deposits with rD denoting the interest rate on deposits. Both demand for loans and supply of deposits are assumed to be deterministic. The case of perfect competition can easily be considered in this framework. However, by making the assumption of a single large bank we deliberately neglect the strategic interactions among banks under an oligopolistic market structure. An analysis of a banking duopoly with credit uncertainty and hedging will be reserved for future research. The bank is required by regulation to hold a portion (0, 1) of its deposits
For an analysis combining aspects of market structure and asymmetric information see e.g. Gehrig and Stenbacka (2001).
1

Credit Risk and Credit Derivatives

as noninterest bearing reserves. It faces operational costs C(D, L) with strictly positive marginal costs CD and CL . Assumptions on second derivatives of the cost function will be discussed later when they are needed to derive results on optimal behavior. Equity capital, K, of the bank is taken as given. The balance sheet constraint of the bank can be written as M = K + (1 )D L (1)

The banks interbank market position, M , can take a positive or a negative value, implying lending or borrowing in the interbank market at an interest rate r assumed to be deterministic and given. To motivate the existence of an interbank money market, imagine our bank being one of a large number of local monopolists or a central bank providing liquidity to the banking system at a rate r. The bank faces credit risk in the sense that a stochastic portion of the loan volume will turn out to be nonperforming. The random variable follows a distribution function dened on the interval [0, 1]. A loan is dened as nonperforming, if the borrower does not pay interest in the period under consideration, i.e., we do not assume that the loan has to be written o completely, leading also to a loss on the principal. Extending the model to the case of writeos poses no diculty, but oers no additional insights and leads to some more complicated formal expressions. Given credit uncertainty, the random prot of the bank is dened as = (1 )rL (L)L + rM rD (D)D C(D, L). (2)

consists of the uncertain interest earned on loans plus the positive or negative interest on the interbank position minus interest paid on deposits and operational costs. As noted in the introduction, nancial markets today oer new nancial instruments which alleviate risk management. The creation of instruments to manage credit risk may be one of the most important steps towards complete risk sharing markets. In the sequel we analyze the impact of credit derivatives on a banks optimal deposit and loan decisions and its risk management. We assume the existence of a market for credit derivatives. As noted before, we neglect the huge variety of realworld forms of credit derivatives and model a most simple hedge instrument which corresponds to a total return swap. The credit derivative oers an exchange of an uncertain future cash ow against a certain cash ow. By selling a volume H of the derivative the bank agrees to exchange a stochastic claim H against a at the end of the period. is the forward rate for one unit deterministic claim H of credit risk. Seen from the beginning of the period, hedging therefore contributes H( ) to the banks prot. In this section we assume a perfect negative correla tion between credit risk exposure and the gain or loss H( ) from hedging. This absence of basis risk assures that credit risk can completely be traded away.

Credit Risk and Credit Derivatives

Substituting the balance constraint (1) for M in (2) and taking account of hedging leads to a modied prot function of the bank: = ((1 )rL (L) r)L + ((1 )r rD (D))D + rK C(D, L) + H( ). (3) In (3) we have used the fact that the banks balance sheet constraint has not changed due to participation in the market for derivatives since derivatives contracts only dene payments to be made at the end of the period. Further, notice that the volume H of contracts sold is not constrained. This means for H > 0 the bank sells credit derivatives, whereas in the case of H < 0 it is a buyer of the hedging instrument. The banks owners or managers maximize a von NeumannMorgenstern utility function U (), U > 0, which exhibits risk aversion, i.e., U < 0 (for a theoretical basis of the assumption of risk aversion see Froot and Stein, 1998, and in the framework of the industrial organization approach to banking Pausch and Welzel, 2002). This leads to the expected utility maximization problem
D,L,H

max E[U ()]

(4)

where is dened by (3) above.

Hedging without basis risk

The rst order necessary conditions for (4) are given by E[U ( )(1 )r rD (D ) rD (D )D CD (D , L )] = 0 E U ( ) (1 )(rL (L ) + rL (L )L ) r CL (D , L ) )] = 0 E[U ( )( )] = 0 Examination of (5), (6) and (7) leads to the following Proposition 1 Given a credit derivative with perfect negative correlation with the banks exposure to credit risk, (a) the bank can separate its decision on risk management from its decisions on deposit and loan volumes, (b) the bank fully hedges its credit risk exposure, if the hedge instrument is unbiased. Proof (a) Substituting E[U ( )] for E[U ( )] from (7) in (5) and (6) yields two deterministic equations in D and L which can be solved for the optimal values D and L : (1 )r rD (D ) rD (D )D CD (D , L ) = 0 (1 ) (rL (L ) rL (L )L ) r CL (D , L ) = 0 (8) (9) (5) (6) (7)

Credit Risk and Credit Derivatives

(b) If the derivative market is unbiased, i.e., E() = , Cov[U ( ), ] = 0, which implies a deterministic . This in turn implies that the bank has no exposure to risk, i.e. there is a full hedge H = rL L . q.e.d. Part (a) of the proposition is an example for the wellknown separation property in the presence of a hedging instrument without basis risk. As a consequence the bank will choose the same volumes of deposits and loans as in the case of a deterministic rate (certainty equivalence). Introducing the elasticity of supply of deposits D = (dD/drD )(rD /D) and the elasticity of loan demand, L = (dL/drL )(rL /L), (8) and (9) can be rewritten as (1 )r rD (D ) CD (D , L ) = rD (D ) (1 )rL (L ) r CL (D , L ) = (1 )rL (L ) 1
D

(10) (11)

1
L

These are the familiar equalities between a Lerner index (price minus marginal cost divided by price) and an inverse elasticity adapted to the case of banking (cf. Freixas and Rochet, 1997, p. 58). Greater market power in the market for deposits, i.e., a smaller value of D , implies a higher Lerner index and a higher intermediation margin. For D the model leads to the limiting case of perfect competition in the deposits market where the interest margin, (1 )r rD (D), just equals marginal operating costs CD . This holds analogously for the loan market.

Hedging with basis risk

In the previous section we considered a market for credit derivatives which permitted the bank to perfectly avoid exposure to risk. In reality selling all credit risk may not be possible. We refer to the nontradeable risk as basis risk. The most important causes of basis risk discussed in the literature are dierences in the maturities of the hedging instrument and the banks risky position, and dierences in the stochastic properties between the underlying of the hedging instrument and the risk the bank faces. In the case of credit risk the rst problem arises when the derivatives contract matures at an earlier date than the underlying loan contract. As an example for the second cause of basis risk consider the case of an underlying of the credit derivative which is not perfectly correlated with the credit risk. The latter aspect appears to be of minor importance since credit derivatives are usually traded over the counter which should imply that the contracting parties search for an underlying with a very high correlation to the risk at hand. In addition, we can think of the risk of giving loans in perfect analogy to the risk of holding shares. Part of the risk is systematic (market risk), part of it is unsystematic (idiosyncratic risk) (cf. Diamond, 1984). In the case of a loan, systematic risk is primarily driven by macroeconomic conditions,

Credit Risk and Credit Derivatives

whereas unsystematic risk is caused by characteristics of the debtor and his project. Systematic risk is tradeable. It contributes most of the total risk of a loan (cf. Wilson, 1998). Unsystematic risk should be avoided by the bank itself through creating a diversied loan portfolio. However, banks may nd it dicult to fully diversify this idiosyncratic risk, because they face institutional constraints, such as credit unions in the U.S. or cooperative banks and savings banks in Germany, or are focused on specic sectors of the economy. However, this nondiversiable unsystematic risk is also nontradeable due to the information problems attached to the loan contract: A potential buyer is at an informational disadvantage compared to the bank willing to sell. We conclude from this discussion that nontradeable credit risk may exist and should therefore be analyzed as basis risk in the framework of our model. Consider a market for total return swaps as described in the previous section. To model basis risk we introduce the following modication: The market uses no longer the share of nonperforming loans , but a share g as underlying of the derivatives contract. g can be interpreted as the share of loans nonperforming due to systematic risk. From this denition it is apparent that the two risks are not necessarily independent. We assume regression dependence between the two random variables (cf. Benninga et al., 1984), i.e., = b + + s g (12)

where b 0, > 0 and s is a zero mean noise term stochastically independent from g . For each unit of the credit derivative sold the bank receives a deterministic payment g in exchange for the stochastic amount g . We assume unbiasedness of the derivatives market, i.e., E() = g , with g denoting g the market price of the underlying chosen by the contracting parties. This implies = , where we assume b = 0 without loss of generality. g The banks prot can now be rewritten as = (1 )rL (L) r L + ((1 )r rD (D)) D + rK C(D, L) + H( g ). (13) g Maximizing (4), where is now given by equation (13), yields (5) and (6) as in the case without basis risk. Condition (7) for the optimal hedge volume, however, is replaced by g E U ( )( g ) = 0 (14) Inspection of the rst order conditions leads us to Proposition 2 (a) In the presence of basis risk the bank hedges a portion of the uncertain interest payment rL L (betahedge rule). (b) The usual separation property no longer exists. Instead, a weaker notion of separation holds. (c) In the absence of economies or diseconomies of scope, the optimal volume of deposits D can be determined as in the case of certainty.

Credit Risk and Credit Derivatives

Proof (a) Unbiasedness of the derivatives market implies that (14) can be written as Cov[U ( ), g ] = 0. Replacing by (13) and using (12) yields Cov [U (rL L g (rL L H ) + const.) , g ] = 0 s Due to the stochastic independence of s and g this can only be true, if = H rL L (16) (15)

(b) Inserting (12) and the optimal hedge rule (16) into the rst order condition (6) for loans shows that L still depends on probabilities and risk preferences, even if D were known. This in turn implies from (5) that D also cannot be determined without knowledge of probabilities and risk preferences. More than market data is required to decide the optimal loan and deposit volumes, which prevents the traditional notion of separation of production and risk management. Notice, however, that the optimal hedge rule derived holds for any pair (D, L). We can therefore imagine a bank choosing loan and deposit volumes randomly and still minimizing its risk exposure by applying the betahedge. While the bank may nd it impossible to determine the optimal values of D and L in the presence of basis risk, it can still separate its hedging decision from its production decisions. We call this a weak notion of separation. (c) Inspection of (5) shows that for CDL = 0 (neither economies nor diseconomies of scope) D can be determined on the basis of market data alone, i.e., without knowledge of probabilities, risk preferences, or the banks hedging decision. q.e.d.

Conclusion

Using the industrial organization approach to the microeconomics of banking, we analyzed the implications of credit risk and credit derivatives without and with basis risk for optimal bank behavior under risk aversion. Under perfect correlation between credit risk and derivative, the familiar separation property was conrmed for the banking rm. A full hedge turned out to be optimal, if the market for derivatives is unbiased. The usual separation result no longer holds in the presence of basis risk, i.e., optimal loan and deposit volumes depend on risk preferences, expectations etc. However, the betahedge rule derived for this case is optimal irrespectively of the loan and deposit volumes chosen. In this sense, there is still a separation of production decisions and risk management.

Credit Risk and Credit Derivatives

References
Bank for International Settlements, 2001, Triennial Central Bank Survey. Foreign Exchange and Derivatives Market Activity in 2001, Basle. Benninga, S.R. Eldor, I. Zilcha, 1984, The Optimal Hedge Ratio in Unbiased Futures Markets, Journal of Futures Markets 4, 155-161. Diamond, D.W., 1984, Financial Intermediation and Delegated Monitoring, Review of Economic Studies 51, 393-414. Freixas, X. and J.-C. Rochet, 1997, Microeconomics of Banking, Cambridge, MA: MIT Press. Froot, K.A., D.S. Scharfstein, J.C. Stein, 1993, Risk Management: Coordinating Corporate Investment and Financing Policies, Journal of Finance 48, 1629-58. Froot, K.A., Stein, J.C., 1998, Risk Management, Capital Budgeting, and Capital Structure for Financial Institutions: An Integrated Approach, Journal of Financial Economics 47, 55-82. Gehrig, Th., R. Stenbacka, 2001, Information Sharing in Banking: A Collusive Device?, Discussion Paper, University of Freiburg. Neal, R.S., 1996, Credit Derivatives: New Financial Instruments for Controlling Credit Risk, Federal Reserve Bank of Kansas City Economic Review, 2nd Quarter 1996, 15-27. Pausch, Th., P. Welzel, 2002, Credit Risk and the Role of Capital Adequacy Regulation, Discussion Paper in Economics, No. 224, University of Augsburg. Wilson, Th.C., 1998, Portfolio Credit Risk, Federal Reserve Bank of New York Economic Policy Review 4, 71-82. Wong, K.P., 1997, On the Determinants of Bank Interest Margins Under Credit and Interest Rate Risk, Journal of Banking and Finance 21, 251-271.