Discussion Paper No.

228, Volkswirtschaftliche Diskussionsreihe,

Institut für Volkswirtschaftslehre, Universität 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 affects 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 confirmed. In case of basis risk, however, we find a weaker notion of
separation.

Keywords: credit risk, credit derivatives, banking firm, risk aversion

JEL classification: G21

∗
Department of Economics, Saarland University, D-66041 Saarbrücken, email: u.broll@mx.uni-
saarland.de
†
Faculty of Business Administration and Economics, University of Augsburg, D-86135 Augs-
burg, email: thilo.pausch@wiwi.uni-augsburg.de
‡
Corresponding author. Faculty of Business Administration and Economics, University of Augs-
burg, D-86135 Augsburg, email: peter.welzel@wiwi.uni-augsburg.de
 Credit Risk and Credit
Derivatives in Banking

Udo Broll, Thilo Pausch and Peter Welzel

Saarland University and University of Augsburg

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 affects 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 confirmed. In case of basis risk, however, we find
a weaker notion of separation.

Keywords: credit risk, credit derivatives, banking firm, risk aversion

JEL classification: G21

1 Introduction
Credit risk is one of the oldest and most important forms of risk faced by banks as
financial intermediaries. The risk of borrower default — on interest and/or prin-
cipal — carries the potential of wiping out enough of a bank’s capital to force it
into bankruptcy. Managing this kind of risk through selecting and monitoring bor-
rowers and through creating a diversified 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 financial 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 financing,
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 difficult to securitize loans or
sell them individually.
Credit Risk and Credit Derivatives 2

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 bank’s uncertain
loan portfolio at a deterministic price affects bank behavior in deposit and loan
markets.
The framework we use for our analysis is sometimes called the industrial orga-
nization approach to the microeconomics of banking (for a brief survey see Freixas
and Rochet, 1997, chapt. 3). It is focused on the bank’s role as intermediary, but
abstracts from informational aspects of banking — adverse selection and moral haz-
ard — 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
sufficiently important to justify the use of this approach.1 To our knowledge, Wong
(1997) was the first author to add aspects of uncertainty and risk aversion to the
industrial organization approach to the bank. We supplement Wong’s 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 firm’s 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 firm 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.

2 The model
Consider a large banking firm in a one–period 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 α ∈ (0, 1) of its deposits
1
For an analysis combining aspects of market structure and asymmetric information see e.g.
Gehrig and Stenbacka (2001).
Credit Risk and Credit Derivatives 3

as non–interest bearing reserves. It faces operational costs C(D, L) with strictly

0
positive marginal costs CD and CL0 . 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 − α)D − L (1)

The bank’s 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 θ̃ of the loan
volume will turn out to be non–performing. The random variable θ̃ follows a distri-
bution function defined on the interval [0, 1]. A loan is defined as non–performing,
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 off completely, leading also to a loss on
the principal. Extending the model to the case of write–offs poses no difficulty, but
offers no additional insights and leads to some more complicated formal expressions.
Given credit uncertainty, the random profit of the bank is defined as

Π̃ = (1 − θ̃)rL (L)L + rM − rD (D)D − C(D, L). (2)

Π̃ 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, financial markets today offer new financial instru-
ments 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 bank’s opti-
mal 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
real–world forms of credit derivatives and model a most simple hedge instrument
which corresponds to a total return swap. The credit derivative offers an exchange
of an uncertain future cash flow against a certain cash flow. By selling a volume
H of the derivative the bank agrees to exchange a stochastic claim H θ̃ against a
deterministic claim H θ̄ at the end of the period. θ̄ is the forward rate for one unit
of credit risk. Seen from the beginning of the period, hedging therefore contributes
H(θ̃ − θ̄) to the bank’s profit. In this section we assume a perfect negative correla-
tion between credit risk exposure and the gain or loss H(θ̃ − θ̄) from hedging. This
absence of basis risk assures that credit risk can completely be traded away.
Credit Risk and Credit Derivatives 4

Substituting the balance constraint (1) for M in (2) and taking account of hedging
leads to a modified profit function of the bank:

Π̃ = ((1 − θ̃)rL (L) − r)L + ((1 − α)r − rD (D))D + rK − C(D, L) + H(θ̃ − θ̄). (3)

In (3) we have used the fact that the bank’s balance sheet constraint has not changed
due to participation in the market for derivatives since derivatives contracts only
define 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 bank’s owners or managers maximize a von Neumann–Morgenstern utility
function U (Π), U 0 > 0, which exhibits risk aversion, i.e., U 00 < 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

max E[U (Π̃)] (4)

D,L,H

where Π̃ is defined by (3) above.

3 Hedging without basis risk

The first order necessary conditions for (4) are given by

E[U 0 (Π̃∗ )(1 − α)r − rD (D∗ ) − rD

0
(D∗ )D∗ − CD
0
(D∗ , L∗ )] = 0 (5)
h 
0 ∗ ∗
E U (Π̃ ) (1 − θ̃)(rL (L ) + rL0 (L∗ )L∗ ) −r− CL0 (D∗ , L∗ ) )] = 0 (6)
E[U 0 (Π̃∗ )(θ̃ − θ̄)] = 0 (7)

Examination of (5), (6) and (7) leads to the following

Proposition 1 Given a credit derivative with perfect negative correlation with the
bank’s exposure to credit risk, (a) the bank can separate its decision on risk manage-
ment 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 0 (Π̃∗ )θ̄] for E[U 0 (Π̃∗ )θ̃] 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 − α)r − rD (D∗ ) − rD
0
(D∗ )D∗ − CD
0
(D∗ , L∗ ) = 0 (8)
(1 − θ̄) (rL (L∗ ) − rL0 (L∗ )L∗ ) − r − CL0 (D∗ , L∗ ) = 0 (9)
Credit Risk and Credit Derivatives 5

(b) If the derivative market is unbiased, i.e., E(θ̃) = θ̄, Cov[U 0 (Π̃∗ ), θ̃] = 0, which
implies a deterministic Π∗ . 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 well–known separation prop-
erty 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 θ̄ (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 re–written as

(1 − α)r − rD (D∗ ) − CD 0
(D∗ , L∗ ) 1
= (10)
rD (D∗ ) ∗D
(1 − θ̄)rL (L∗ ) − r − CL0 (D∗ , L∗ ) 1
∗
= ∗ (11)
(1 − θ̄)rL (L ) 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 − α)r − rD (D), just equals
0
marginal operating costs CD . This holds analogously for the loan market.

4 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 non–tradeable risk as basis risk. The most important
causes of basis risk discussed in the literature are differences in the maturities of the
hedging instrument and the bank’s risky position, and differences in the stochastic
properties between the underlying of the hedging instrument and the risk the bank
faces. In the case of credit risk the first 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 6

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 diversified loan portfolio. However, banks may find it difficult 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 specific sectors of the economy. However, this non–diversifiable
unsystematic risk is also non–tradeable 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 non–tradeable 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 modification: The market uses no
longer the share of non–performing loans θ̃, but a share g̃ as underlying of the
derivatives contract. g̃ can be interpreted as the share of loans non–performing
due to systematic risk. From this definition 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.,

θ̃ = b + βg̃ + s̃ (12)

where b ≥ 0, β > 0 and s̃ is a zero mean noise term stochastically independent

from g̃. For each unit of the credit derivative sold the bank receives a deterministic
payment ḡ in exchange for the stochastic amount g̃.
We assume unbiasedness of the derivatives market, i.e., E(g̃) = ḡ, with ḡ denoting
the market price of the underlying chosen by the contracting parties. This implies
θ̄ = βḡ, where we assume b = 0 without loss of generality.
The bank’s profit can now be re–written as
 
Π̃ = (1 − θ̃)rL (L) − r L + ((1 − α)r − rD (D)) D + rK − C(D, L) + H(g̃ − ḡ). (13)

Maximizing (4), where Π̃ 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 h i
E U 0 (Π̃∗ )(g̃ − ḡ) = 0 (14)
Inspection of the first order conditions leads us to
Proposition 2 (a) In the presence of basis risk the bank hedges a portion β of
the uncertain interest payment rL L∗ (beta–hedge 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 7

Proof (a) Unbiasedness of the derivatives market implies that (14) can be written
as Cov[U 0 (Π̃∗ ), g̃] = 0. Replacing Π̃∗ by (13) and using (12) yields

Cov [U 0 (s̃rL L∗ − g̃(βrL L∗ − H ∗ ) + const.) , g̃] = 0 (15)

Due to the stochastic independence of s̃ and g̃ this can only be true, if

H∗
β= (16)
rL L∗
(b) Inserting (12) and the optimal hedge rule (16) into the first 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 beta–hedge. While the bank may
find 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 bank’s hedging decision. q.e.d.

5 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 confirmed
for the banking firm. 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 beta–hedge 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 8

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 Fu-
tures 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 Capi-
tal 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 Reg-
ulation, 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.

————————————