Selasa, 14 Juli 2009


This paper presents a theoretical model of default risk n the context of the “market model approach to interest rate dynamics. We propose a model for tinite-inteival interest rates (such as LIBOR) which explicitly takes into account the possibility of default through the influence of a point process with deterministic intensity. We relate the defaultable interest rate to the non-defaultable intecest rate and ta the credit risk chaiacteristics default intensity and recovery rate. We find that the spread between defaultable and non-dethultable rate depends on the non-defaultable rate even when the default intensity is deterministic. Prices of a cap on the defaultable rate and of a credit spread option are derived. We consider swaps with unilateral and bilateral default risk and derive the fair fixed swap rate in both cases. Under the condition that both counterparties are of the samo risk class. show that for a monotonously increasing term structure the swap rate for a defaultable swap will lie below the swap rate for a swap without default risk. © 2000 Elsevier Science By. All rights reserved.

JEL classification: G30; G33: E43

Kerwords: Market model; Forward rate agreements; Swaps: Credit risk

1. Introduction

Traditionally, models of credit risk have focused on the valuation of defaultable bonds. However, bonds are not the only fixed income instrument

‘Corresponding author. Tel.: ±49-228-739-268: fax: +49-228-735050.

E-mail address: lotz@addifinastouni-bonnde (C. Lotz).

0378-4266/00/S - see front rnaar 2000 Elsevier Science By. AU rights reserved.

P11: SO 3 7 8 - 4 2 6 6 1 9 9 ) 00 I) I – S

302 C. Lotz, I.. Schlogi / Journal of Banking & Finance 24 (2000) 301—327

subject to credit risk. For example, LIBOR, which is the underlying of many interest rates and money market derivatives, is not a rate for a riskiess issuer, but is assumed to apply to a party of A A-rated quality. Therefore, this interest rate is influenced not only by the general behaviour of riskiess interest rates, but also by the risk level of A A-rated banks.

In addition, every derivative written on this interest rate, such as future. forward or option, is itself subject to default risk by the counterparties involved in the derivative. The situation is especially complicated for an interest rate swap, which can become either an asset or a liability to each counter- party depending on the random evolution of interest rates. Our aim with the present paper is to analyse the effect of default risk on these contingent claims.

For a credit risk model which focuses on LIBOR and its derivatives, the natural choice of the undcrlyng interest rate model is a model of the simple forward rate. in the recent past, such models have been proposed in the default-free term structure literature. They directly model simple forward rates (LIBOR) as the primary process (cf. Jamshidian, 1997; Miltersen et al., 1997; Musiela and Rutkowski, 1997). as opposed to deriving them from instantaneous rates. One motvation for this, which is also useful in the credit risk setting, is to show how the market practice of using Back’s formula tc price caps can be placed in the context of an arbitrage-free interest rate model. Black’s formula assumes that the interest rate underlying the option coniract is lognormal!y distributed, and then va1ue the option similar to the well-known Black—Scholes formula for options on stocks. Because models of the simple forward rhte justify using black’s formula, they are termed ‘Market Models’. They also have the desirable property of generating nonnegative itercst rates while avoiding the non-integrability problems associated with lognormal instantaneous rates. Contrary to the assumptions in these interest-rate models, however, the LIBOR we observe in the real world is not a default-free rate, but incorporates default risk, typically comparable to that of an A A-rated bank.

Existing credit risk models show us two ways in which default risk can be introduced into an interest rate model: On the one hand, there is the classical firm-value based approach, initiated by Black and Scholes (1973) and Merton (1974) and later adopted by Longstaff and Schwartz (1995) and Zhou (1997). Here default occurs when the firm value process hits a possibly stochastic boundary, which may be prespecified or endogenous.

On the other hand, there is the intensity approach, where the time of default is modelled in more reduced form as the first jump time of a point process with deterministic or stochastic intensity, and this jump time is totally unpredictable. This approach was adopted by Jarrow and Turnbull (1995), Duflie and Singleton (1997), and others. In most models of this type, the intensity of the point process as well as the payout ratio after default are imposed exogenously.

C Lot:, L. Schlôgl / Journal of Banking & Finance 24 (2000) 301—327 303

Duffee (1996) shows how the default intensity can be fitted observed term structure of corporate bond prices. The main advantage of the intensity approach is its tractability, and for this reason we will also use it in the present paper.

The aim of this paper is to study default risk in the context of LIBOR models. We propose an extension of a market model for LIBOR which explicitly takes into account the possibility of default. In our model, the time of default is determined by the first jump time of a point process with deterministic intensity, and the payoff after default is assumed to be a constant fraction of the value of a non-defaultable, but otherwise equivalent asset. By considering defaultable forward agreements on credit, we derive the arbitrage- free term structure of the defaultable simple rate and find prices for several types of options on this rate. Also, we consider interest rate swaps where one or both counterparties are subject to default risk. The paper is strutured as follows. In Section 2, we describe the market model used to characterize the evolution of non-defaultabic simple forward rates and the mechanism governing the occurrence of default and the corresponding payoffs. Section 3, we derive the arbitrage-free vaiue of the simple defaukable forward rate and the prices of some related derivatves. Section 4 deals with interest rate swaps. Section 5 concludes and some of the proofs have been relegated to Appendix A.

2. The basic model

This section outlines the general principles and building blocks of our credit risk model. We give an intuitive explanation of the construction of the term structure of interest rates in a market model and show how tht. risk of default enters our model.

Throughout this paper, we will confine our discussion to a discrete tenor model, that is we consider only contracts with maturities in the set

i={T0.T1 =T()TT+2 T.To+(N—l)1},

where is a fixed accrual period. The final date is denoted by T* TN = I’0 + N. For each T E f there exists one simple spot interest rate L(T, T) for the interval T. T + ] which is not susceptible to default risk, and which we will call the non-defaultable spot LIBOR. Consequently, we have a forward LIBOR process (LQ, T)), for every T E .Y. We assume that nondefaultable zero-coupon bonds are traded for each maturity T E .9 and T*, the price at time t of the bond with maturity T is denoted by BQ, T). Bond prices and forward rates are related via the usual formula

1 + L(t, T) = B(t, T)

B(t, T + )

304 C Lot:, L Schlögl / Journal of Banking & Finance 24 (2000) 301-327

We assume a LIBOR market model, i.e. the stochastic set-up is as follows. We are given a stochastic basis (Q. ., f. F) which satisfies the usual hypotheses and supports a d-dimensionai Brownian motion W. For each T E 3, there exists the (T ± )-forward measure pT and the dynamics of L(., T) are

dLQ, T) a’(r. T)L(t, T)dJT’,

where o(., T) is a deterministic function and WT is a d-dimensional Brownian motion under pT÷, For a given set of volatilities oL such a model can be constructed by, for example, backward induction (cf. Musiela and Rutkowski, 1997; Jamshidian, 1997), and we can assume that P is the terminal forward measure. When dealing with swaps, we also use a swap market model, i.e., assume that non-defaultable forward swap rates are lognormal martingales under the appropriate martingale measure.

Default is described by the first jump time of a point process. We assume the jump processes involved are Cox processes with a deterministic intensity under the terminal forward measure P. In particidar, this implies that the intensity is invariant under changes from one forward measure to another: Under every forward measure, the jump process has the same intensity. The generalization of the results of the paper to a stochastic iinensity which is independent of the non-defaultable interest rates is straightforward.

We distinguish between the cases of unilateral and of bilateral default. In the first case, we use:

Assumption I (Unhlaieral default). There exists on (Q. . f. P) a jump process

N(t) with deterministic intensity .(t). Default occurs at the first jump time of

N(r’), formally

r := inf{1 0 I NQ) l}.

The recovery rate A is constant. The probability of no default event occurring between time t and T is given by

P[N(T)-N(:) =o =exp{ _f.(s)ds}. (1)

This implies that the distribution of the default time is given by

I ,pT

P[t T] = I — exp — J i(s) ds


and its density is

fQ) exp { - f (s)ds}(i).

In the case where both counterparties are subject to default risk, we use the following assumption instead:

C. Lotz. L. SchlOgl I Journal of Banking & Finance 24 (2000) 301—327 305

Assumption 2 (Bilateral default). For i 1. 2, there exists a jump process ]V(t) with deterministic intensity (t). The two jump processes are assumed to be independent. Default of counterparty i occurs at the first jump time z of N(i). so that

P[r > tj=P[(t) =O]=exP{_fi(s)ds}. i= 1,2.

There exist two constant recovery rates A1 and A2. so that A determines the payoff after default of counterparty i.

Due to Brémaud (1981, Ch. 11l.2, T7, p. 238), we have the following proposition (see also Last and Brand, 1997, Ex. A5.19. p. 441).

Proposition I. (Jonsider the marked point process associa ted with the two jump processes N1 (t) and NQ) and the mark space {z1 ,z2}. This has the compensator

(t,2) =(t)( 1j} ±l{z,}).


— i.,Q) ± 2(t).

This proposition means that the time r = r1 A r2, at which the first default occurs, is the first jumli timc of a point process with the intensity

= ..j(t) + 2(t). Conditional on there having been a default event at time 1, the probability that it was counterparty i who defaulted is given by

PEz=zIe=t]=. (2)

Within the class of intensity-based credit risk models, different specifications of the payoff after default have been used. Duffie and Singleton (1997) assume that the asset pays back a certain percentage of its immediate pre-default value. Jarrow and Turnbull (1995) and also Hull and White (1995) follow the convention that the payoff is in terms of a non-defaultable, but otherwise equivalent security. For example, a defaultable zero-coupon bond would pay off a fraction A of a non-defaultable zero-coupon bond immediately in the event of default. This is equivalent with assuming that the face value of the defaultable bond drops from 1 to A. which is then paid at maturity of the bond. It is quite clear that the second convention simplifies the model significantly, because a security’s payoff after default does not depend on the default process any more. In this paper, we will use the second method because of its advantages in terms of tractability. With this convention, the occurrence of a default becomes equivalent to a drop in the face value of the security, for example a zero-

306 C. Lotz, L. Schiogi/Journal of Banking & Finance 24 (20(X)) 301-327

coupon bond, from I to 4. Therefore, the following shorthand will simplify notation later on.

Definition 2. Consider a security with maturity T + which pays its nominal face value 1 if no default has occurred and 4 in the case of default. The expected payoff of this security, given that no default has occurred before time 7’, is

it(T, T + ) : ETl {r>T+} + 4l{T±}Ir].

Using Eq. (1), it is easy to see that


Here, as throughout the paper, the superscript T ± on the conditional expectation denotes that it is to b taken under the (T ± )-forward measure.

3. DefultabIe credit agreements

We will use credit agreements (CAs) as central building blocks to generalize the notion of forward LIBOR to a setting with default risk. As we ‘.i1l see, it is quite easy tc calculate the simple interest rate at time Tapplying to a loan from T to T -- to a borrower subject to default risk. The definition of forward defäuitabie LIBOR is based on this SpOt defaultable UBOR and is analoguous to the usoal definition of forward LIBOR. The forward deJbultahle LIBOR will be the answer to the ouestion: What is the “ftir” simply compounded fixed rate a default-risky borrower can contract at time t for a loan during the time interval

{T. T + }?

The fundamental contract underlying the notion of a forward LIBOR is the Forward Rate Agreement (FRA), and before we start with the default-risky case, we want to make some general remarks on FRAs when there is no default risk, it is important to note that the cash flow in an FRA is not uniquely defined, but may depend on the individual contract. Below, three different cash flow patterns will be described. If there is no default risk, all patterns lead to the same value of the FRA and consequently to the same forward LIBOR. This is no longer the case in the presence of default risk, which is the reason for our discussion.

With FRAs. it is usual to speak about the buyer and the seller of the contract, where the buyer is the one who pays the predetermined interest rate, and the seller receives the predetermined interest rate. Because we will consider swaps later on, where the cash flows are very similar, we have chosen to use the swap convention (payer and receiver) throughout the whole paper: therefore,

C. Lot:, L Schlogl / Journal of Banking & Finance 24 (2000) 301—327 307

we will call the FRA buyer the payer side (it pays the predetermined and fixed interest rate). and thc seller of the FRA is called the receiver.

Assume an FRA which is signed at time 0 for the interval fT, T ± }, with the fixed rate K. in this case, three cash flow patterns are possible:

1. All payments may be made at time T: in this case, the receiver side will at time T receive the amount

1 ±L(T,T)

Of course, if the amount is negative, the receiver side will have to pay

• something to the payer side.

2. Another possibility is that the exchange of payments is scheduled for time T ± x. in this case, the receiver side will at time T + receive the amount


Here, the FRA can also be interpreted as a one-period swap.

3. Finally, the payments can be split up between the dates Tand T + : at time T, the receiver side will pay 1, and at time T + , it will receive 1 + in this situation. the payer side takes out a loan from the receiver side at time Tand pays back the loan with interest at tme T + .

in the last case, we can interpret the fixed rate K in the following way: it s the inlerest rate which can be fIxed at time 0 in a contract for a loan for the time period [7’, T ± J. It is this interpretation we rely upor when we define the notion of “defaultable LIBOR”, and therefore we will define FRAs in terms of cash flow number 3).

Although traded FRAs are usually settled in cash at time T, the underlying interest rate applies to default-risky borrowers. Our specification allows us to capture in one contract the risk between T and T + , which is usually incorporated in real-world LIBOR, and the counterparty default risk before T. From now on, we will use the term “Credit Agreement” (CA) to emphasize that we use an FRA with cash flow number 3).

Definition 3. The non-defaultable CA with the fixed interest rate K an

agreement to the following cash flows:

• At time T, the receiver pays I.

• At time T ± z, the receiver gets I + zc.

Assume first that a set of zero-coupon bonds, B(t, T) is given for all maturities T € .9T and for T*.

Proposition 4. The price of CA at time rfor the period [T. T + ] with the fixed interest rate K is given by

308 C. Lot:, L Schlogl / Journal of Banking & Finance 24 (2000) 301—327

CA(t, T. K) B(t. T ± )(l 4- K) B(1. T).

Definition 5. The non-defaultable forward LIBOR at time t for the interval {T, T + j is the fixed interest rate for which a new, non-defaultable CA at time t for the interval [T. T ÷ r.] has zero value. This particular fixed interest rate is denoted by LQ, T) and is determined via

1 + L(t, T) = B(t. T)

B(t. T + )

L(T. T) is then the spot LIBOR at time T for the interval T, T ± j.

3.1. The unilateral case

Throughout this section, we will always assume that only the payer of the fixed rate (the borrower in the contract) is in danger of default. All contract values will be given from the perspective of the recever of the fixed rate.

First, we deal with the simplest case of a spot interest rate on a credit with default risk in the intervil {T. T —i- . To this end, we consider the following contract

Definition 6. An agreement to a defaultable credit with fixed rate K, starting

date T, and accrual period x is specified by the following cash flows:

• At time T, the receiver of the fixed ratc pays 1.

• At time T H- , two things can happen: -

Either there was no default in [T. T ± ]. Then the receiver gets I ± K. Or a default occurred. Then the receiver gets A(l + ic).

We denote the value of this agreement at time T by CA(T, T, K).

Proposition 7. The value at time T of the credit agreement described in the previous definition is given hi’

CA(T,T,K)=BçT.T){(1K)(T,T±)—(l--aL(T.T))} (3)

with ?z(T, T + ) as given in Definition 2.

Proof. The value of the credit agreement at time T is determined by the following equation:

CA(T, T. K) (1 + K)B(T. T ± + Al{(T}j.T] — 1.


By inserting the definitions of (T. T + ) and of spot LIBOR L(T. T) we obtain

C. Loiz, L Schlögl / Journal of Banking & Finance 24 (2000) 301—327 309

CA(T. T. ic) = (1 B(T, T ± a)z(T. T ± ) — 1

= B(T, T - ){(1 + i)it(T. T + z) — (I + L(T, T))}.

Just as in the non-defaultable case, the defaultable spot LIBOR rate L’(T, T) is the value of the fixed rate ,c which leads to a value of zero for the credit agreement in Definition 6. From Eq. (3) we see:

Proposition 8. The defaultable LIBOR rate Ld(T, T) at (line Tfor the interval [T, T + J is determined by the relation

i +(T,T)= (5)

Eq. (3) also lets us easily determine the value at time t <>

CAQ, T. ic) B(t, T -4- r){(1 -4- ri)t(T. T - l .- (1 + iL(t. T))}. (6) By the construction of CA(t. I’, K), the process

(CA(t, T. ic)

B(t. T + ) )

is a martingale uader the (T + r)-forward measure.

It is important to note that CA(t. T. ic) would not be the correct value for a forward agreement on a credit starting at time T v;ith a default-risky payer, as no allowance is made for the risk of default between time t and T. Instead we need to consider the fcflowing contract:

Definition 9. Suppose that t <>

• If no default occurs up to time T, the contract then becomes an agreement to a defaultable credit as described in Definition 6.

• If default occurs at time r T, the position is closed at time t according to a mark-to-market procedure as follows: if the value CA(r. T, K) of the equivalent non-defaultable agreement is negative to the (non-defaulting) receiver, she must meet her obligation in full. If the value to the receiver is positive at time r, then she receives only a fraction A of this value from the (defaulting) payer.

We denote the value at time t of this contract to the receiver by CA’(t. T, ic).

The cash flows to the receiver resulting from this agreement can be summarized as follows:

3W C. Lot:, L. Schtogi/JournatoJ Banhing & Finance 4 (OOO) 301—327

• If no default occurs until time T, receive CA(T, T. ) at time T

• If default occurs at a time r T. receive CAQr. 7’. i) (1 — zl)[CA(r. 7’. i)1 at the random time ‘r.

We will now calculate CAd(t. T, K). We need the following lemma, which tells us how to value payments made at the random time ‘r. Its proof can be found in Appendix A.

Lemma 10. Let Z be a predictable process on (2, , . P). Suppose that a payment of Z will be made at time t f’t 7’. Let t <>

B(t. T ) (u)e_J,tET I] du. (7)

Proposition 11, The value at time t of the defriultahie forward agreement is

CAd(I, 7’ K) = CA(t, T, ) -- (I )B(t. T + )

x J .(u)c ;(r) dtET+ 1J du. (8)

P&oof. The fonuula can bc proved by calculation while applying Lemma 10 to the process Z, where

4 = CA(u.T.) —(1

However, the following approach is more intuitive. As we have mentioned above, the value of CAd(t. T. K) is generated by a payment of CA(T. T. ic) at time T if default does not occur before T, and a payment of CA(t. T. K) — (1 — )[CA(r. T. ic)] at time t if r T. We recall that (CA(u. T. 1c))UEIO] is the price process of a non-defaultable claim with maturity T and terminal payoff CA(T, 7’. ic). The receiver gets this claim in any case ai the time min(t. T). Since this is a European claim without any interim payments. the time at which the claim is actually received is irrelevant. Therefore, this component of the price has the value CA(t. 7’. ic). The remaining payment of —(1 — i){CA(z, 7’, )J at time t if r T is valued by applying Lemma 10 to Z, where

Z, —(1 — )[CA(u, T, K)j

We note that the expectation appearing inside the integral in Eq. (8) has the form of a put option on non-defaultable LIBOR, i.e., of a European floorlet. We define

318 C. Lot:, L. Schlog/IJournalofBan.king & Finance 24 (2000) 30T.327

4.]. Unilateral default risk

We start oil by limiting the possibility to default to only one counterparty. in this case the counterparty which receives the fixed rate. However, the method shown is very general and can easily be extended to the case where both counterparties are in danger to default.

Definition 21. The swap with receiver default risk is an agreement to the following cash flows:

• Before default occurs, on the dates 7) the payer side of the contract pays the predetermined fixed interest c. whereas the receiver side pays the floating rate

o At the time when default of the receiver party occurs, the value of a non-defauitable swap contract with the same fixed interest rate K to the payer side is determined. If this value s positive, the receiver side pays a certain fraction of it to the payer side. However, if the value is negative, it is paid in full by the payer side.

in order to obtain a nice representation of the value of a defaultable swap, we introduce some notation

Definition 22. We denote the index of the last time of exchange of interest payments before default vih

In other words, the time of defauh is in me interai r Ej T. 7jt+. T is the last time of an exchange of interest payments before defai’lt occurs.

We define the price of a swap contract as its value to the fixed-rate paying party, and as a direct result of Definition 21 we have the following proposition, the proof of which is contained in Appendix A.

Propositin 23. The price of a de/iu1tahle swap contract is given by the following expression:

For ease of notation, here and in the folio si’ing propositions expectations are taken under the spot measure.

Observe that in our setup, the fixed rate , will be lower than in a swap with no default risk, because the payments of the default-risky floating-rate counterparty are insecure, so that it can obtain only a smaller fixed interest rate in return than a floating-rate paying counterparty without default risk.

The value of a defaititable swap consists of two parts: The first is a defaultfrce swap with the same rate as the defaultable swap, the second is a forward payer swaption, but with a random exercise time. This can be interpreted as follows: The writer o the swap earns the amount given by the price of a defaiih-free swap, if no default occurs. However, if default occurs, he loses a fraction A — 1 of this amount, wluch is represented in the payer swaption part of the formula.

To find the fixed swap rate foi- the defaultable swap, its price at time 0 must be set to zero, FSd(0. ) = 0, and this equation has to be solved for the fixed swap rate. Unfortunaely, under the lognormal model of forward LIBOR, no clos’rd form solution is available for the swaption price which occurs in this formula, although, as Brace et al. (1997) show, a closed form approximation still exists.

However, if forward swap rates instead of forward LIBOR are assumed to be lognormally distributed, as is common when pricing swaptions, the problem can again be reduced to the one encountered in the case of defaultable LIBOR. We obtain the following result, also proven in Appendix A.

Proposition 24. The fair fixed interest rate K for the defaultable swap contract can be approximated fri solving the f11o wing equation:

320 C Lot:, L. Schlögl / Journal of Banking & Finance 24 (2000) 30I-327 where

As in the case of credit agreements with unilateral default risk, this formula contains only an integral over normal distributions and is therefore easy to implement numerically.

4.2. Bilateral default risk

We generalize the valuation formulas from the previous subsection to a setting where both parties are subject to default risk. We assume that both default premia are incorporated in the fixed rate, ard that the floating rate paid is the riskless LIBOR L(T1. 7)).

Similar to the part on (‘As with bilateral default risk, we make the assuinption 2. With the aid of Proposition 1, we can state the price of a swap contract with two-sided default risk.

Proposition 25. The price of a shop contract with ni’o-sided defliult risk is gien

Again, this swap contract can be rewritten in the following form:

C. Lotz, L. Schlögl / Journal of Banking & Finance 24 (2000) 301—327 321

Intuitively, this formula is easy to explain: The first line represents the value of the swap without default risk. The second line represents the loss to the fixed rate payer side due to default of the floating rate party, if the va’ue of the swap to the payer was positive. The third line represents the gain of the fixed rate payer side if she defaults and does not have to pay the full value of the swap which wa negative to her.

The problems in finding the fixed swap rate in this case are the same as in the case where only one paity is subject to default risk, and they can be treated similarly.

Let us assume now that both counterparties are of the same risk class, i.e.,

= zi2 = A and = 2.2. In th!s case. we have the following result (for the proof see Appendix A:

Proposition 26. The price of a swap contract with bilateral and ‘qzal defiult risk is give;; by

There is an intuitive reason for this result: because both counterparties are of the same risk class, the swaption parts, which represent the option to default, cancel out. Therefore, this result is independent of our assumptions of log- normally distributed forward LIJ3OR or swap rates.

From the formula it can be seen that future interest payments (L(7)_1, 7,.) — ic) are discounted with the expression

This is the value of a default-risky zero-coupon bond with maturity T. default intensity 2. and loss ratio (l — A). This result can be explained intuitively:

Because now two counterparties instead of one are subject to default risk, the default intensity 2. = 2.i + 2.2 is double that of the unilateral default risk case.

322 C. Lot:, L. Schlogl / Journal of Banking & Finance 24 (2000) 301—327

However, the expected loss due to a default is halved because the probability that it was a default of each counterparty is exactly 50%.

We are now interested in the relationship between the swap rate of a swap with bilateral and equal default risk, ic1, and the swap rate of a swap without default risk, K.

We have the following result:

Corollary 27. Assume that the forward LIBOR curve is monotone. Then

• for a monotonously increasing Jrward LIBOR curve, the deJàulrable swap rate is smaller than the non-defaultable swap rate.

• for a mono tonousli’ decreasing forward LIBOR curve, the defaultable swap rate is greater than the non-defaultable s4ap rate.

Minton (1997) and Sun et al. (1994) compare swap rates and par bond yields based on LIBOR with OTC swap rates. In particular, Monton finds that swap rates derived form Eurodollar futures are higher than those observed in the OTC market. Two possible explanations for this result are given: firstly, the practice of daily re.ettlement in the future5 market leads to credit enhancements absent in plain vanilla swaps. Secondly, Burghardt and Hoskins (1994) have shown numerically that, even without default risk, Eurodollar futures prices introduce a bias towards higher swap rates when the yield curve is upward-sloping. In our setting, default risk has the same efects as was shown in the preceding corollary.

Similar to Minton (19Q7), Sun et al. (1994) find that par bond yields based

LIBOR are higher than OTC’ swap rates. As was noticed before (cf. the discussion after Definition 19), constructing the yield of a defaultable par bond frcm LIBOR introduces an overcompensation for default risk, leading to a higher yield. Therefore, our model is able to explain these empirical phenomena through no-arbitrage arguments.

5. Conclusion

By analysing the impact of default risk on credit agreements and swaps in a market model, we have derived equations determining the arbitrage-free values of simple defaultable forward rates and swap rates. The time of default is given by the first jump of a time-inhomogenous Poisson process with deterministic intensity, and the payoff after default is a constant percentage of the value of a non-defaultable, but otherwise equivalent asset. Under the assumption of log- normality of non-defaultable forward LIBOR, we derive equations for the fixed rate in forward credit agreements which compensates for (unilateral or bilateral) default risk. These equations involve nothing more complicated than a one-dimensional integral over normal distributions and therefore numerical

C. Lot:, L. Schlógl / Journal of Banking & Finance 24 (2000) 301-327 323

implementation is straightforward. In addition, we are able to obtain a rigorous definition of LIBOR which incorporates default risk in analogy to the non-defaultable case. Furthermore, we value swaps with unilateral and bilateral default risk. We show that a defaultable swap can be written as the sum of a non-defaultable swap and a swaption, and we present an approximation for the fair swap rate in the presence of unilateral and bilateral default risk. Finally, we analyse the impact of the term structure of forward LIBOR on the defaultable swap rate. The results allow lenders and money market dealers to find the correct interest rate for forward credit agreements and swaps which take into account the differential credit standings of lender and borrower. This allows them to incorporate the credit standing into the price of the contract. A further application of the results presented here would be the valuation of credit facilities or credit lines, which correspond to the option of entering a credit agreement and, therefore, can be valued in a similar way.


Financial assistance by the German Academic Exchange Service, DAAD HSP 111, is gratefully acknowledged. The authors would like to thank Dieter Sondermaun and seminar participants at the CEPR conference on Credit Risk, the Bank of England and Bonn University for helpful comments and discussions. All errors are the authors’ responsibility.

Appendix A

Since Z and the bond price process are predictable, we can rewrite this conditional expectation using the intensity of H:

324 C. Loiz. L. Schlogl / Journal of Banking & Finance 24 (2000) 301—327

Since N is a Cox process (cf. Lando, 1998), we can use iterated conditional expectations to obtain

By inserting this relation into Eq. (24), we obtain the formula that was claimed. U

Proof of Proposition 24. Foi this result, we have rewritten the swap part of Eq. (22) as a weighted sum of zero-coupon bond prices: -

C. Liii:, L Schlögl / Journal of Banking & Finance 24 (2000) 301-327 325

For the swaption part of Eq. (22). we use first the default intensity as the distribution of default times

and now the expectation inside the integral is similar to the value at time 0 of a

standard European rwaption with exercise time t. In terms of the forward swap

rate, this can be written as (see Musiela and Rutkowski, 1997b)

Proof of Proposition 26. The last two lines of Eq. (23) can be rewritten in the following way:

326 C. Lot:, L Schldgl / Jouriwl of Banking & Finance 24 (2000) 301-327

This gives the desired result.

Proof of Corollary 27. The two swap rates are determined by the expressions


Black. F., Scholes.M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 631—654.

Brace. A.. Gatarek. D.. Musiela. M.. 1997. The market model of interest rate dynamics. Journal of Mathematical Finance 7. 127-155.

Brémaud, P., 1981. Point Process and Queues. Springer, Berlin.

Burghardt. G., Hoskins. B.. [994. The Convexity Bias in Eurodollar Futures. Dean Witter Institutional Futures. Chicago.

Cooper, l.A., Mello, A.S.. [991. The default risk of swaps. Journal of Finance 46, 597—620.

DufTee, G., 1996. Estimating the price of default risk. Working paper. Federal Reserve Board, Washington, DC.

C Lot:, C. Schlogllfournal of Banking & Finance 24 (2000) 301—327 327

Duffie, D., Huang, M., 1996. Swap rates and credit quality. Journal of Finance 51. 921—948.

Duffie, D., Singleton, K.J.. 1997. Modelling term structures of defaultable bonds. Working paper, Graduate School of Bu.siness, Stanford University, Stanford, CA.

H all. J.C, 1997. Options, Futures and Other Derivatives. Prentice-Hall, Englewood Cliffs, NJ.

Hull, J.C.. White. A.. 1995. The impact of default risk on the prices of options and other derivative securities. Journal of Banking and Finance 19. 299—322.

Jamshidian. F., 1997. LIBOR and swap market models and measures. Finance and Stochastics 1,


Jarrow. R., Turnbull, SM.. 1995. Pricing derivatives on financial securities subject to credit risk. The Journal of Finance 50, 53—85.

Lando. D.. 1998. On Cox processes and credit risky securities. Working paper, Institute of Mathematical Statistics. University of Copenhagen.

Last, A.. brand, C.. 1997. Marked Point Processes on the Real Line. Springer, Berlin.

LongstatT. F.A., Schwartz, ES., 1995. A simple approach to valuing risky fixed ana floating rate debt. Journal of Finance 50, 789—819.

Merton. R.C.. 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29, 449-470.

Miltersen, K.t., Sandmann, K., Sondertnann, 0., 1997. CLosed form solutiona for term stiucture d:rivatives with log-normal interest rates. Journal cf Finance 52, 409—430.

Minion. BA.. 1997. An empirical examination of basic valuation models for plain vanilla US interest rate swaps. Journal of Financial Economics 44, 251—277.

Musiela. M.. Rutkowski. M.. 1997. Continuous-time tcr.n structure models: Fonvard measure approach. Finance end Stochastics I, 261 -291.

Musiela, M.. Putkowski, M.. 1997. Martingale Methods in Option Pricing. Springer, Berlin.

Sorensen. E.H., Bottler, T.F. 1994. Prtcing swap default risk. Financial Analysts Journal, 23—33.

Sun. T.S.. Sundaresar, S.. Wang, C.. 1994. Interest rate swaps: An empirical investigation. Journal of Financial Economies 34, 7799.

Zhou, C.. 1997. A jump-diffusion approach to modeling credit risk and valuing defaultable securities. Working paper Federal Resen’e Board, Washington, DC.


Bank internal ratings of corporate ciients are intended to quantify the expected likelihood of future borrower defaults. This paper develops a comprebensive framework for evaluating the quality of standard rating systems. We suggest a number of principles that ought to be met by “good rating pcacticc”. These “generalty accepted rating principlcs’ are potentially rele’vant for the improvement of existing rating systems. They arc also relevant for the development of certification standards for internal rating systems, as cur-ently discussed in a consultati\e paper issued by the Bank for International Settlements in BasIc, entitled “A new capital adequacy framework”. We would very much appreciate any comments by readers that help to develop these rating standards further. © 2001 Elsevier Science B.V. All rights reserved.

JEL clus.sUication: G21

Keywords: Corporate rating; Credit risk management: Capital adequacy: Banking supervision

1. Objectives of the paper

The rating of borrowers is a widespread practice in capital markets. It is meant to summarize the quality of a debtor and, in particular. to inform the

*Corrcsponding author. leL: +49-69-79822568: fax: +49-69-7982895!.

E-mail addresses: (J P. Krahnen). (M. Weber).

0378-4706/0/8 - ee front IndOer 2001 E1sc er Scienc.e By. A1! iights reserved. Pll:S0378-4266(00)U0l 15-I

4 J.P. Krahnen, Al. Weber / Journal of Banking & Finance 25 (2001) 3-23

market about repayment prospects. Apart from so-called external ratings by agencies, there are also internal ratings by banks and other financial intermediaries providing debt finance to corporates. While external ratings by agencies are available since many years, in fact since 1910 for Moody’s, the oldest agency, internal ratings by commercial banks are a more recent development. Their history in most cases does not exceed 5—10 years.

This paper is a first attempt to answer a simple question: What are criteria for good rating practice? We will propose a consistent set of rules that an appropriate rating system should meet. These rating standards, presented below, are not only a collection of “best practice”-rules. Instead, the standards will also be motivated from a decision-theoretic and a statistical perspective, and by examining internal ratings systems currently used in Germany. We have derived insights on properties of actual rating systems from the investigation of two special data sets that contain detailed information on corporate ratings, see Elsas et a!. (1998), Elsas and Kralinen (1998) and Machauer and Weber (1998) for the first data set. Wcber et a!. (1998) and Brunner eta!. (2000) for the second. By itself, these standards will provide a guideline for the development of new rating systems, and they will help to improve existn2 systems. Furthermore, they will help to evaluate established systems, as it is already practiced by auditing firms, rating agencies and, occasionally, by supervisory authorities.

It is common among practitioners to distinguish between borrower rating and facility rating. The tormer reiates to the borrower as a legal entity. while the latter relates to a specific loan-cuin--collateral. In this paper, we concentrate on borrower ratings alone. The empirical basis lbr our analyses and suggestions is derived from internal rating systems common among major German universal banks. Internal rating systems are therefore the primary fields of application for our principles; at this stage we leave open the question of their applicability to external ratings.

On a broader level, the paper also wants to contribute to the economics of ratings. In particular, we will discuss the ability of ratings to establish credibility vis-à-vis external observers as, for instance, supervisory authorities, and market participants. Of course, credibility of rating information is closely related to acceptable rating reqbirements. A consultative paper by the Bank for international Settlements in Basle (1999) has put the discussion about the proper role of ratings, notably internal ratings, in the forefront of financial policy debate. Under the title “A New Capital Adequacy Framework”, the Basle commission issued a report on how to modify the current international standards on capital adequacy of financial institutions. The current standards, dating back to 1988, require banks to put a 8% equity position against its risky assets, in particular its corporate loans. No consistent distinction is made between high risk and low risk assets.

In the proposed new equity standards, the capital to be held against assets should match implied default risk. A variety of ways how to account for

J.P. Krahnen, M Weber / Jouriwl of Bankmg & Finance 25 (2001) 3—23 5

differences of default risk in the loan book can be thought of. Information provided by rating agencies is one way how to deal with different risk categQries in a bank’s loan book, ratings by lending institution’s own internal models is another one. This paper attempts to propose a set of rules sound rating practice should respect. In doing so, we mainly rely on our experiences with internal rating systems in Germany.

Its remaining parts are organized as follows. Section 2 outlines the economic background for an understanding of rating methodologies. Section 3 contains our main contribution. It presents and discusses a list of 14 rating principles that, in our opinion, every rating system should fulfill. Section 4 discusses further implications for the credibility of ratings, and points at a number of open research questions.

2. Economic background

2.1. Why ratings mai ter

Rating categories, typically letter labded (AAA or Aaa for prime quality), or simply numbered (1 to 10, say), are a shorthand to quantify credit risk. On the basis of historiaI data, ratings can be related to the relative frequency of defaults (default-mode paradigm), or they become the basis for the valuation of an asset (mark-to-model paradigm). The most prominent application relates to corporate asset-liability management, where risk-adjusted return on capital (RAROC) numbers are used to benchmark divisional performance. Ratings allow to measure credit risk. and to manage consistently a bank’s credit portfolio, i.e. to alter the banks’ exposure with respect to type of risk. In particular, ratings are useful for the pricing of a bond or a loan, reflecting an intended positive relation between expected credit risk and nominal return.

For all these reasons, the quality of a financial institution’s rating system has attracted attention from many parties. Auditing firnis discuss the risk reporting systems of a corporation in the annual report, rating agencies evaluate the risk assessment system of a borrower who wants to issue asset backed securities, and supervisory authorities are expected to start soon to certify institutional rating systems and credit risk models.

A final remark is in order about the differences between two types of ratings:

internal and external. External ratings are generated by rating agencies. These agencies specialize in the production of rating information about corporate or sovereign borrowers, they do not engage in the underwriting of these risks. The rating information is made public, while the rating process itself remains nondisclosed. Internal ratings, in contrast, are produced by financial intermediaries (notably banks) to evaluate the risks they take into their own books. The rating information is seen as a source of competitive advantage, because it is believed

6 J.P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23

to contain proprietary information, and is therefore not made public. Even the firm being rated is typically not informed about its current internal rating.

While there is a-growing empirical literature on the validity and the reliability of external ratings (see notably Ederington et al., 1987; Blume et a!., 1998), and on the informational content of external rating changes (see Hand et al., 1992; Liu et aL, 1999), there is still very little published work on the methodology and the empirics of internal ratings. A notable exception that relies on data from the US is Treacy and Carey (1998) and Carey (1998). We will base our subsequent discussion on our experiences and insights derived primarily from internal ratings of major German banks.

2.2. Ratings and defiu1t risk

We define a rating of a corporate as the mapping of the POD, the expected probability of default, into a discrete number of quality classes, or rating categories. The POD is a continuous variable, bounded by zero from below and by one from above.

POD: Companies —‘ O, 1]. (1)

A POD is the expected relative frequency of a credit event, where the latter is

de!ned as a non-payment of principal or interest due (over a period of at least

30 days, say). The POD is one component of a lenders’ expected loss, as in (2).

E(L) POD . E(LGD). (2)

Here, E(L) is expected loss, and E(LGD) is the expected loss given default. The expectations ar taken over a common time interval, usually one year in the future. Expected loss is thus the average amount a lender is expecting to loose over the next twelve months.

Fig. I exemplifies the calculation of expected loss on the assumption that default probability is 0.05, and recoveries vary discretely between 80% and 20%, with equal probability. All values are expressed as percentages of the loan outstanding at the time of default.

This definition is less innocent as it may first appear. In particular, if rating captures POD (expected probability of default), but not LGD (loss given default), then in general there will be no direct relation between rating and credit spread. To see this, consider the following simple example of two firms A and B with an identical POD: Assume firm A to have a low LGD. while B’s LGD is high. In equilibrium, the observable spreads for A loans and B loans have to be set such that the creditor breaks even in expected values. Therefore, the B-spreads have to be larger than the A-spreads. Note the tradeoff: Either we define ratings to measure expected default probability, or we let ratings proxy for expected loss. While the former definition is in line with the interpretation given by credit officers, and by the agencies as implicit in their historical default rates tables (see Moody’s. l999b; Standard & Poor’s, 1998), it does not allow to relate statistically ratings to spreads.

J. P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23 7

Fig. I. Graphical representation of expected loss calculation assuming independence between default probabdity and severity.

Expression (2) and Fig. 1 essentially assume that the incidence of default and the severity of a given default are independently distributed random variables. Thus, losses will typically vary between zero and one hundred percent (in our example: between 07 and 80%). A more general expression allows for a nonzero covariance between POD and LGD:

E(L) = E(POD . LGD). (3)

The distribution of losses around tteir expected value is an important measure of overall (institutional) value at risk. The unexpected risk is the number of standard deviations a given quantile (99%) lies away from its expected value. Unexpected losses are addressed by recent value-at-risk tools. 2

Though in theory, PODs are mapped in rating classes, in practice it is the other way round. Rating classes are mapped into PODs on the basis of’ historical data. The established agencies. notably S&P and Mond’s. use historical deThult rate to afibrate their model. The default rate is tñe percentage utah bond issues outstanding at t that will have a credit event between t and t + I, e.g., a 12 months period. Conceptually, there is no simple direct relation (linear or log-linear, say) between ordinal rating notches and cardinal PODs. Empirical studies using studies of S&P and Moody’s have found an exponential relation between POD and rating notch.

in this paper, we focus exclusively on PODs as the objective of rating systems. The typical client we have in mind is a corporate entity. a firm, not an

2 See Saunders ((999) and W’ahrenburg (1999).

8 J. P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23

individual borrower, nor a financial instrument, like an asset backed transaction. ‘ A distinct set of principles may has to be developed in order to deal with LGD, which is beyond the scope of this paper.1n differentiaUng between POD and LGD, and in clarifying the objective of rating systems, we are in line with the ideas expressed in the BIS consultative paper (Bank for International Settlements, 1999, Annex 2).

2.3. Rating models

There are a variety of procedures to arrive at a rating, i.e., a discretized POD-measure. The typical procedure used today is the scoring method. It relies on a well-defined set of criteria, each of which is scored separately. The individual scores relating to the set of criteria are weighted and then added up, yielding the overall score. This score is translated in one of the rating classes, defined as an interval on the real line that extends from minimum o’erall score to its maximum.

A well known example is the z-score proposed by Edward Altman in 1977 (see Altman and Saunders (1996) for a survey and further references). This author has suggested to regress historical default experience on a set of accounting variables (mostly balance sheet and P&L) in order to determine an optimal separating function between issuers that defaulted later on and those that survived. The weights of the estimated function are then used to predict default piobabilitv for an individual firm, called the s-score. Ttiis ;-score may again be translated into a rating class (see Caouette et al.. 1998, Chapter 10).

A different approach to rating is exemplified by KMV’s publc firm model. Building on Gption pricing theory KMV, a data vendor, derives dcfault estimates from expected movements of stock prices over a specified pesiod of time. typically one year. In contrast to the scoring approach, there is no need here to collect a variety of firm-related, fundamental information, nor is there any weighting function needed. It only requires a time series of observable stock mat ket prices and an estimate of firm indebtedness.

We will next turn to internal rating systems, as exemplified by the systems in place at major German banks. A recent study contains a more detailed description of their individual rating models, and presents an empirical analysis of the determinants of ratings. (see Brunner et al., 2000). All institutions included in the study apply the scoring methodology, as defined in (4). It specifies a number of distinct criteria a. an equal number of value functions v, and an aggregation rule, typically linear, with weights k.

Though our principles might apply to individuals and financial instruments as well, we will confine our discussion in this paper exclusively to corporate loans.

J. P. Krahnen, Al. Weber / Journal of Banking & Finance 25 (2001) 3—239

Differences across institutions refer to the list of criteria, particularly the mlportance given to so-called soft factors, or qualitative criteria. This includes the assessment of management quality, or a general forecast of the prospects of the firm in its market. Table I gives a summary assessment of these criteria. It can be seen that the banks typically draw on a list of criteria comparable to the one utilized by S&P, or Moody’s.

Table I suggests that internal and external rating systems are relying on a similar set of explanatory variables. With respect to the number of rating classes, Standard & Poor’s and Moody’s each have 22 rating notches (excluding the watchiist), whereas internal rating systems of commercial banks typically have less, e.g. 6-10 rating classes.

10 J. P. Kralinen, Al Weber / Journal of Banking Finance 25 Y2001} 3—23

Though we have no information about the aggregation process by which agencies derive their final ratings from the underlying criteria, we proceed under the assumption that general accepted rating requirements may appJyto external agencies and internal models alike.

3. Rating requirements: What should a good rating system be like?

In the following, we will derive properties good rating systems should obey. These properties can be a foundation for what we propose to be “generally accepted rating principles”. We will call these principles occa- ionally “requirements”. Altogether, we conie up with 14 requirements, some of which are formally derived, some of which are empirically founded, some of which are inspired by the recent publication of the Basle Committee on Banking Supervision, and some of which we learnt from talking to high level practitioners.

3.1. A ruling system is a nwpping

Rating systems arc what is mathematically called a function:

R: {companies) .‘ {Ratiiig-values}.

meaning that the rating system R is a function which assigns each element of the set of companies to a rating value. These rating values, or short ratings, can be categories, i.e. {A,B--.B.8—,.. .}, or values of an interval rmjn.r. R(compan X) 0.67 means that the rating system R assigns the rating value of 0.67 to company X. We will assume that rating categories and values can be ranked, i.e., A >— B means, that rating category A is better, in the sense of a lower default probability, than rating category B. The symbol ““ means that both ratings are identical.

This simple mathematical definition of rating systems as functions allows us to define the first requirements without specifyuig at this point, what rating really means.

Requirement 1 (comprehensiteness). A bank’s rating system should be able to rate all past, current and future clients.

This requirement defines the potential set of companies to be rated. A bank’s rating system should be able to cope with all clients possible. Of course, this requirement is quite general, and hard to meet. There may be future clients, and risk criteria, a given bank may not even imagine. There

J. P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23 11

may be past clients who do not exist any more. However, a bank should make any effort possible to ensure that its rating system is flexible enough to cope with all foreseeable types of risk. It should not happen, e.g., that foreign companies cannot be rated or that the rating system is not able to handle certain industries.

Requirement 2 (Completeness). A bank should rate all current clients and keep on rating its past clients.

The requirement states that a bank should rate all its current clients. This is rather trivial and will in most cases be current management practice. In addition we require that a bank should keep on rating its past clients. This might not be easy and in certain cases it might not even be possible. Accounting data as well as qualitative data from talking to the companies’ management might not be available for past clients. Nevertheless, we think that a bank should put effort in maintaining its rating database. It is of central importance for any type of back-testing and further development of the bank’s rating that the bank has an ongoing set of rating data. If the bank stops rating clients which, e.g., defaulted, the set of companies which are in the rating database can be biased. Such a batik would know nothing about the probabilities of events that happen after a default: how likely is the success of restructuring, etc. The survivorship bias (to consider “surviving companies” only) is well known fiom empirical work in capital markets.

Requirement 3 (GornpltxTh’). A bank should have as many different rating systems as rocessary and as few as possible. The leasons for choosing the jiumber of rating systems should be made transparent.

We have to ask, if there should be one function R or if we allow for different functions. From a mathematical perspective it does not matter. We can make one function R so complex that it can he applied to all companies or we can “split” this function up into different functions. In practice. however, there are different aspects to be considered. One function would be a rating system which could be applied to foreign real estate companies as well as to medium sized companies in Southern Germany. The complexity of such a system, however, would make it difficult to use in an organization. Quite a number of aspects are important in evaluating real estate companies which are of no interest if a manufacturer is considered. On the other hand, one should not divide the set of companies into too many subsets. i.e., construct too many different rating systems. Certain companies might fall into more than one system. too many rating systems might ask too much from the credit oflicers. and the rating systems might be difficult to backtest due to relatively small data-pools. It is for this reason that we recommend to balance both aspects. In addition, we suggest the reason for choosing a certain number of rating systems to be made transparent.

12 J P. Kra/men, M. Wither / Journal of Banking & Finance 25 (2001) 3—23

3.2. Rating systems map probabilities of default

In Section 2 we have argued that the probability of default is the central variable to be considered when a bank wants to judge the risk of a single loan. In this section, we will define requirements that link rating systems to probabilities of default (POD).

Requirement 4 (POD-definition). Probabilities of default have to be well defined.

This requirement states that a bank has to have a proper definition what its PODs mean. The bank has to define what it considers to bc a default event. We found that financial institutions rely on a variety of definitions of a default event, e.g. loan loss provision, or failure to pay interest, or principal, over a specified time span. Note that without a harmonization of default definitions, it will prove difficult to pooi POD-data across banks. We therefore suggest that the industry works towards a common definition of POD, which is both transparent and reasonable. In addition, financial institutions have to state the time horizon within which a default is considered. Some banks just consider one time horizon (mostly one year), some other consider multiple time horizons which lead to different sets of PODs. Still other institutions, notably ratings agencies, estimate PODs by averaging over a complete business cycle. The ultimate goal should be a term structure of ratings or, for that matters, PODs that capture default risk beyond the one year horizon. For example, a company might have a small POD over the next two years, and a large POD for year three (when a patent will have expired).

Requirement 5 (Mono!oniciij’).

(i) POD(company X) = POD(company Y) R(company X) R(company Y).

(ii) POD(company X) <>.- R(company Y) POD(company X) <>

This requirement defines the relation between ratings and expected default frequencies. As discussed before, we take POD as the primitive and derive rating from there on. If two PODs are identical, the ratings also have to be identical (case (i)). If the POD of company X is smaller than that of company Y (case (ii)), the rating of company X has to be at least as good as that of company Y. To illustrate the weak inequality for ratings, let us consider a bank which only has two rating categories {good, bad} with good >- bad. This might

The averaging of default estimates of a cycle is. in our view, problematic if the objective of a POD-assessment lies in specifying minimum equity requirements.

J. P. Kralmen, M. Weber I Journal of Banking & Finance 25 (2001) 3—23 13

be a bank which only wants to know if a credit should be given or not. Case (ii) allows two different PODs to yield the same rating, if the rating of a company is better than that of another company (case (iii)), the POD of the first company should be smaller than the POD of the second one. Note that (iii) is implied by (i) and (ii).

Requirement 6 (Fineness). The rating system can vary in the degree of fineness. It should always be as fine as necessary.

Looking back at Requirement 5, the central question for the definition of a rating system now remains, how fine a rating system should be, i.e., how many categories it should have. It could be as fine as the POD itself, being basically identical to POD, or it could map PODs into a finite number of categories. Of course, a rating system which models POD ould be the most exact one. However, for quite a number of situations a less fine rating system would be sufficient and more appropriate in an organizational context. The fineness of a rating system cannot be considered independently from back- testing (sec Requirement 8). There is no use in defining a large number of rating categories, if a bank ri not able to back-test consistently, due to lack of data.

Thus the fineness of the rating system is a function of its intended use. It is therefore that one should allcw rating systems to communicate PODs in different degrees of fineness. For pricing the rating system should be finer than for defining credit limits. Some banks, e.g., use traffic lights (three categories: red, yellow, green) to attract the attention of the credit officer to more or less risky credits. Knowing the conversion of raJog into POD will always allow us to transform one way of ccmmunication into the other.

Requirement 7 (Reliabiliiv). The rating system should be reliable.

Suppose, that a company has some true POD. Then the rating should he identical regardless of the person who rates, or the point in time when the rating is done. Note, that this requirement does not assume that the rating does not change. The rating might change with the creditworthiness of the client, or along the economic cycle. However, it should siay constant, if’ the creditworthiness does not change. An example to test for the stationarity property of the data set is explained in Bluine et al. (1998).

3.3. Do raring systems really map probabilities of default?

Now that we have defined some first key requirements for rating systems, the question remains how a bank or even a supervisory agency makes sure that the rating system is correct. Thus it is required that ratings (or PODs) are rational forecasts on the basis of all available information, being the best ex-ante predictor of credit risk.

14 J. P. Kruhnen, M. Weber I Journal of Banking & Finance 25 (2001) 3—23

Credit ratings can be technically incorrect, i.e., even if applied properly their values do not correspond to the (ex-post) number of realized defaults. In addition. rating systems which are technically correct can be used in a way that the resulting ratings do not mimic PODs anymore. We will discuss the first class of problems first.

A POD is based on an ex-ante point of view. It states that a company with an POD of 0.7% has a 7 in a 1000 chance to default within a given time period. We know from research on capital markets that testing expectations is always tricky. In order to relate (ex-ante) expectations to (ex-post) observed data, we have to assume that the structure of the problem under consideration remains constant from the date where expectations are formed to the date where observations are taken. This assumption is called the stationarity assumption. We will assume stationarity for the next requirement. Nevertheless, we are aware that in the future, statistical methods will have to be introduced that account for possible non-stationarities.

Requirement 8 (Back-testing). The (ex-ante) probability of default should not be significantly different from the (ex-post) realized default frequency.

Requirement 8 basically states that what you expect is what you should get. It also stresses the need of a data-base to fulfil! this requirement. Back-testing in eedit-management is especially difficult necause first, there are no market prices for most types of credits and second. Lhere arc so few historical data of credit defaults. As we will argue ii1 more detail in Section 4, it might be useful to pool resOurces across different banks to create a better database which aliows for an improved hack-testing.

Since back-testing is central for validating a ratiiig, the need for it yields some important implications for the design and use of ratings. As already mentioned, a bank should not have too many rating systems (i.e. define to many subsets of companies) and it should not change the rating system too often.

There are numerous ways of testing rating systems, and apparently a number of them are already used in the industry. Test procedures are related to back-testing and they may be seen as defining necessary conditions for the appropriateness of the rating system:

• Ex-post default rates within any given rating category should be larger than that of a higher (i.e. better) rating category.

• Even if we do not know whether a cardinal relation between rating and POD can be assumed, the above condition will test ordinality.

• Ex-post default rates should increase with the time horizon.

• It is obvious that the default rates of companies based on a time horizon of five years have to be equal or greater than those based on a time horizon of one year.

• For companies with corporate bonds outstanding, credit spreads may be compared to internal credit ratings.

J. P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23 15

• Across companies, the bank will he able to compare the risk-ordering implied by the market with the risk-ordering implied by credit-ratings. Besides back-testing, credit ratings have to obey certain structural and technical necessities (see Weber et al., 1998 for further details).

Requirement 9 (Informational efficiency). Ratings should be informationally efficient, i.e., it should not be possible to predict rating changes based on rating history. All the available information should be modeled correctly in the rating. The rating system should cope with biases known from the general literature on rating (splitting bias, range bias. etc.).

As mentioned before, a rating should correctly inccrporate all information available to the bank, both public and plivate, i.e., it should be efficient. This requirement is idcntiëal to the use of the Lerm “information efficiency” in financial markets. Today’s rating should be the best predictor for tomorrow’s rating, i.e., it should not be possible to get information about tomorrow’s rating by knowing which rating the company had yesterday (ot in earlier periods). In addition, quite a number of biases known from the psycholocal literature on judgment have to be taken care of when designing a rating system. Credit officers may. e.g., have the tendency to rate qualitative criteria of a rating system bctter than quantitative ones and thcy tend to change qualitative variables less than quantitative (Brunner et al., 2000).

Requirement 10 (Svttcm dc’celopment). A rating system has to be improved over time.

It might sound trivial hut after a bank has seen deficiencies in its rating, it should be willing to change it. Such a change can result from back-testing and from ex-ante management insight. Management might know that the structure and the aggregation of variables to estimate creditworthiness have changed, i.e., stationarity is violated. One should not wait until (ex-post) back-testing forces system modifications, provided that ex-ante insights had suggested these changes already. A modification of the system has to be carefully considered. There are large costs (back-testing is more difficult, education of credit staff, etc.) and in some cases uncertain benefits.

Requirement 11 (Data management). Past and current rating data should be easily available.

A modern data management is a prerequisite for successful back-testing as well as successful system development. Any type of statistical analysis requires data to be (easily) available. Even if the fulfillment of this requirement seems to

Even on an efficient market, due to the categorial nature of ratings, first differences (i.e. rating changes) will not necessarily be distributed like independent random variables.

16 J. P. Krahnen, M. Weber I Journal of Banking & Finance 25 (2001) 3—23

be easy on a first glance, we are well aware of problems which can arise in practice. The change of a bank’s computer system, the further development of an existing rating system. the introduction of a finer rating system, a change in the organizational structure of the rating process, a merger of two banks are just examples to demonstrate that the requirement can pose a serious challenge. However, without a well maintained data management, no testing of a rating system will be possible.

3.4. Good rating syslcms account for incentice problems

Ratings compile objective and subjective information. The higher the share of subjective, or soft information, the more difficult is the detection of untruthful reporting. This may be a considerable problem, because credit officers in charge of rating a particular client may have an incentive to underestimate the risk of a loan. e.g. to overestimate the quality of a panicular management. For instance, in some institutions. loan responsibility migrates from the credit officer to a special work-out group, once the rating falls below a critical value. fhis organizational rule may induce the credit officer to adjust his or her risk assessment to the point where control over the customer is not migrating. Another example of how an organizational rule may affect reporting incentives relates to bonus systems. where performance measures depend on ratings.

Requirement 12 (Incentic compatibility). The rating process has to be embed- dccl in the organization of credit business such that the risk of misrcpresen- tation by credit officers is minimized.

We know of no simple test of organizational incentive compatibility, but several rules of thumb are available. First, and inspired by the above example, possible critical values” of rating assessments that trigger action have to be recorded and followed up. In particular, measures of statistical similarity and significance may help to identify unusual frequencies of specific rating decisions, oi. rating migrations. Second, the internal reward system of the institution may or may not be related to past rating performance of loan officers. As a rule, an officer’s rating history should “stick to him”. For example. a significant, above average frequency of rating revisions after the officer in question has moved from his post, or authority for certain loans been moved away from him, could have a predictable (and negative) impact on his overall evaluation. The fulfillment of Requirement 12 can be checked by asking to what extent management has thought about possible incentive conflicts caused by the organizational design of the lending process, and what it has done to control for its behavioral consequences.

16 iF Krahnen, Al Weber / Journal of Banking & Finance 25 (200)) 3—23

be easy on a first glance, we are well aware of problems which can arise in practice. The change of a bank’s computer system, the further development of an existing rating system, the introduction of a finer rating system, a change in the organizational structure of the rating process, a merger of two banks are just examples to demonstrate that the requirement can pose a serious challenge. However, without a well maintained data management, no testing of a rating system will be possible.

3.4. Good rating systems account for mcentwe problems

Ratings compile objective and subjective information. The higher the share of subjective, or soft information, the more difficult is the detection of untruthful reporting. This may be a considerable problem, because credit officers in charge of rating a particular client may have an incentive to underestimate the risk of a loan, e.g. to overestimate the quality of a panicular management. For instance, in some institutions, loan responsibility migrates from the credit officer to a special work-out group, once the rating falls below a critical value. This organizational rule may induce the credit officer to adjust his or her risk assessment to the point where control over the customer is riot migrating. Another example of how an organizational rule may affect reporting incentives relates to bonus systems, where performance measures depend on ratings.

Requirement 12 (Incentive compatibility). The rating process has to be embedded in the organization of credit business such that the risk of misrepresentation by credit officers is minimized.

We know of no simple test of organizational incentive compatibility, but several rules of thumb are available. First, and inspired by the above example, possible “critical values” of rating assessments that trigger action have to be recorded and followed up. In particular, measures of statistical similarity and significance may help to identify unusual frequencies of specific rating decisions, or rating migrations. Second, the internal reward system of the institution may or may not be related to past rating performance of loan officers. As a rule, an officer’s rating history should “stick to him”. For example, a significant, above average frequency of rating revisions after the officer in question has moved from his post, or authority for certain loans been moved away from him, could have a predictable (and negative) impact on his overall evaluation. The fulfillment of Requirement 12 can be checked by asking to what extent management has thought about possible incentive conflicts caused by the organizational design of the lending process, and what it has done to control for its behavioral consequences.

J.P. Krahnen, Al. Weber / Journal of Banking & Finance 25 (2001) 3 —23 17

However, we do not advocate the minimization of discretionary decisions in the rating process. because the specific- value added (in terms of incremental information) by internal ratings mainly consist of aggregating “soft”. or subjective information produced by the loan officer. A certain degree of consistency check may help to improve incentives, and to establish credibility of the overall rating process. This is, summarized in the following requirement: -

Requirement 13 (Internal compliance). The distribution of rating outcomes is

constantly monitored by controllers, assisted by random inspections.

In order to identify systematic biases in the evaluations of loan officers, all ratings and their histories are to be kept in a back-testing file (see Requirement 11). Rating quality maintenance has to develop (and, of course, to aoply’i statistical test routines that are capable of identifying significant variations in rating decisions over time, or across firms. The task resembles a statistical quality control as it is common in, e.g. production management. ‘The follow- up to these statistical tests could be a partial or complete replication of past ratings.

Fulfillment of Requirement 13 would not only aHcw the detection of specific behaviora! patterns, but also would strengthen the Incentive compatibility (Requirement 12). In order to have some deterrent effect, the algorithms of the sampling plan must not be completely transparent to loan officers. Again, outside rating quality assessment would try to clarify to what extent sampling plans havc been developed, and are applied consistently.

Requirement 14 (External compliance). The adherence of a bank’s management to its agreed rating standards is monitored by neutral (uninterestud) outside controllers, either on a continuous, or on a random basis.

Requirement 14, though similar in nature to the preceding Requirement 13, is the keystone for establishing credibility to rating data produced by an interested party. Here, interested party refers to, e.g. banks as providers of internal ratings. A bank’s “interest” derives from the underwriting of credit risk vis-à-vis the customer that has been rated. Requirement 14 involves an evaluation by an outside party. e.g. a supervisory authority. Past ratings have to he shown to be without biases, or deliberate misrepresentation. Therefore. external compliance is not about the informational value of any particular rating. but rather it is about the consistency of its use. The methodology

Building on well established methodologies of random quality inspections. a continuous sampling p]an may prove helpful (see Shirland, 1993; Krishnaiah and Rao, 1985). Such a plan specifies a set of algorithms that would analyze the similarity of specific rating subsets pertaining to, e.g.. a cross section of ratings given within an industry, or a time series of ratings given by a particular officer.

18 J. P. Krahnen, AL Weber / Journal f Banking & Finance 25 (2001) 3—23

applied to control external compliance is likely to be similar to the one used in Requirement 13.

4. Policy considerations and agenda

In the concluding section we want to address two questions. First, is regulation of the rating process really needed? We will argue that indeed some type of outside regulation is required to safeguard credibility of internal ratings. Second, we will point out additional needs in two areas which are of great importance to the future acceptance of rating as a risk measurement instrument, namely a need for research, and a necd for better, and larger data-bases.

4.1. Is there a iwedJr external supL’reision of internal ratings?

The BIS consultative paper as of June 1999 giver sonic consideration to ratings as a basis for the assessment of bank capital requirements. The regulatory importance of ratings do apply not only for external ratings, but also for internal ratings. To answer the question of whether or not internal ratings should he certified and constantiy superviscd by a regulator. or an auditor, we will first compare the processes by which internal and external ratings earn credibility. There are basicay two models: In the first, professional (external) rating age’ncies produce public rating information without doing any underwriting; their credibility derives from reputation in the market place. lii the second model, bank loan depat tments produce private (or internal) rating information on the basis of an underwriting business. Here, credibility derives from the shareholder value interest of bank management. and hence its credit department, in a proper loan repayment.

Let us start with external ratings. Default probability estimates by specialized agencies (S&P, Moody’s, Fitch JBCA, a.o.) draw on the agency’s reputation as a provider of accurate default predictions. Reputational value stems from Ihe impact ratings exert on credit spreads. Thus, reputational value is high (low), if a change of published ratings has a significant (an insignificant) influence on corporate cost of capital. This means that ratings have to he accepted as a proxy for true fundamental information in order to be valuable in the market.

The market value of a rating agency, its franchise value, is therefore directly related to the discounted stream of cost of capital effects that are due to its corporate ratings. Firms are willing to pay a fee to an agency up front in order

The methodological question of reliability of rating decisions is not trivial even if it comes to large data sets, as those assembled by the agencies.

J. P. Krahnen, ji4 Weber I Journal of Banking & Finance 25 (2001) 3—23 19

to receive public rating. By the same line of argument faulty ratings will, if detected, eventually destroy the franchise value of a rating agency.

Thus, the reputational argument developed in the preceding paragraph claims that agencies have a proper incentive to produce true and unbiased corporate ratings. Of course, the reputational model of incentive compatibility is subject to an important caveat. It relies on the market being able to detect faulty ratings ex-post. What is needed, therefore, is a statistical methodology to spot changes of the distribution function (or, for that matter, of the rating behavior of an Agency) relatively soon after its onset.

Note that the reputational argument for external ratings will have to take into account the fact that rated companies usually pay the Agency for pro.. viding the rating label. There is a natural incentive problem here, and ratings probably derive much of their value from the Agency’s reputation for being unwavering in their high standards.

Let us now turn to internal ratings and possible determinants of their credibility. The basic credibility explanation is simple: Internal ratings are private information, typically not even communicated to the rated finn itself. Of great importance for internal ratings is their ability to incorporate all types of information accessible that may contribute to a good default forecast. In particular, a relationship-based financial system may be in a position to exploit not-easy-to-measure qualitative information, and thereby improve estimates. This includes insider information due to. e.g.. account surveillance, and advisory business. The comparadvc advantage of internal ratings in our view refers precisely to this fact, the incorporation of soft information. Jf one assumes, fof simplicity, no incentive conflicts within the financial institution itself, then there is good reason to believe in the unbiasedness of internal ratings. Since a bank underwrites credit risk, she essentially takes a bet on the creditworthiness of any particular borrower. Any bias in POD-estimations would harm the banks competitive position and would eventually impair equity value. True and fair private ratings arc therefore in the proper interest of the bank.

However, there is a caveat here as well. The proposed new equity standards outlined in the BIS consultative paper attach, in fact, a sort of shadow price to the internal ratings of bank borrowers. Since the amount of equity required to be held against a given structure of bank assets will then be affected by their internal ratings, there may well be pressure to accommodate rating decisions in the future. Once ratings fulfill a regulatory task, they have a dual function, measuring risk and triggering equity charges. These two functions are likely to have opposite incentive effects.

A within-finn conflict of inwrest may arise when. e.g. a loan officer tries to avoid a shift of responsibility from himself to a work-out group, They may then accept a better-than-justified rating.

20 J. P. Kralmen, M. Weber I Journal of Banking & Finance 25 (200!) 3—23

To sum up: In the light of the emerging new equity standards both external and internal ratings constantly have to prove their unbiasedness, and their neutrality. While there is a market test for external ratings (which, in fact, has been effective for many years already), there is no external check for internal ratings to date. One way to test for neutrality of internal ratings is a serious test of rating methodology and rating performance, see Requirement 13. Both may be elemcnts in a certification process carried out by a supervisory authority (which, for that matter, may also be delegated to a specialized entity or auditing firm, say).

In the future, the mandatory issuance of non-secured, short-term, subordinated debt may produce additional market based information on overall loan quality. Somc authors argue that cost-of-capital effects of these short-term debt instruments will be able to enforce proper internal rating standards. A discussion of subordinated debt is beyond the scope of this paper. In our opinion, however, subordinated debt may help to establish credibility of internal ratings hut it will, for various reasons, not solve the above mentioned incentive problem altogether. Therefore, we believe that external supervision will have to play an important role in the future, as outlined in the paragraphs above (see requirements 12—14).

4.2. What is needed: Data and research

As pointed out a couple of tunes, data i key to successfully maintain anu develop rating systems. Due to the number of rating classes, the long time periods and the small probabilities of failure, statistical analysis at the level of a single bank might be limited. We advocate to think about the need of a shared data base. Such a database could aggregIte the ratings for a company across different banks (of course, with full confidentiality of each bank’s private rating). Based on the joint data, each bank might be able to analyze and validate its internal rating system against some average rating. .A combined database would allow for a more elaborate back-testing thereby preparing the ground for an official recoition of internal rating systems.

In addition, for companies which are rated by quite a number of banks, the aggregate risk rating would reflect something like a “market opinion” of the default risk of a particular company, a specific industrial sector or even the whole economy of a country (thus creating a default risk index of certain entities). A joint database would also allow to derive correlations between the credit risks of companies, industry sectors and countries as well. Such inforination is of great importance for the development of credit portfolio models.

Finally, we want to point out some research needs. We have tried to state some requirement, a good rating system should obey. Nevertheless, we have said very little (on purpose) on how to construct a good rating system. Which factors should be elements of the scoring rule? How should the weights for each

J. P. Krahnen, M Weber / Journal of Banking & Finance 25 (2001) 3—23 2

factor within the scoring model be derived9 With respect to the value function, what number should be attached to an average vs. an excellent management? Today,—most banks use a mixture of mathematical models and management intuition to construct their systems. We do think this is a good approach, but we would like to know more. Along these lines, it would be interesting to analyze in greater detail how LGD (loss given default) depends upon the state of default (see Section 2). Furthermore, methods for hack-testing rating systems are not yet well developed. Sophisticated statistical sampling plans are needed to check for internal and external compliance. Equally, statistical tools can be used to correct for any trends (like industry cycles) and biases (like survivorship bias) that are to be found in the raw data. Finally, one would like to see more research being undertakan on the validation of ratings. Given the low data frequency, and thus the long duration for a detection of faulty ratings, the reputational justification of true and fir ratings would also benefit from this exercise. Finally, it may be of interest to use credit rating for optimizing portfolio risk. For this end, estimates of corrclatwns across borrowers, and over time are needed. The correlation structure of corporate loans will help to model risk migration, and the dependence of risk ratings on the economic cycle.

We conclude with a general disclaimer concerning Generally Accepted Rating Principles: Their development is seen as work in progress. And we expect it to remain work in progress for quite some time, since such principies have to be developed jointly by regulators. researchers and experts in the field.


In developing the above rating principles we were able to draw on inspiralion and discussion from many sources. First, the CFS-based joint research project on Bank Risk Management gave us basic insights into the rating practice of the leading German banks, with emphasis on methodology and statistical testing. Project partners were the Chief Credit Officers, as well as their staff, of Deutsche Bank, Dresdner Bank AG, Commerzbank AG, HypoVerejusbank, DG Bank. and West LB. Second. we profited immensely from comments received at the CFSroundtable on Generally Accepted Rating Principles, November 11, 1999 at the Center for Financial Studies in Frankfurt. The participants who contributed to this roundtable were expert from the above listed major German banks and from the DSGV (Deutscher Sparkassenund GiroVerband), SGZ-Bank, Fitch IBCA Ltd.. RS Rating Services AG, Rating Cert e.V., URA (Unternemensrating Agentur). Bundesaufsichtsamt für das Kreditwesen, as well as the Bundesbank, We are grateful for helpful comments from Ed Altman, Mark Carey. Tony Saunders and conference

22 J.P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23

participants at NYU’s Stern School of Business, and the University of Frankfurt’s CFS. The Generally Accepted Rating Principles, as outlined in this paper, have benefitted from these discussions. It goes without saying that their contents must not be interpreted as a consensus view. Rather, these GARPs solely reflect the authors’ view after thorough consideration of the comments and, on several occasions, diverging viewpoints expressed by roundtable participants.


Altman, El., Saunders. A.. 1996. Credit risk measurement: Developments over the last 20 years. Journal of Banking and Finance 21. 1721—1742.

Bank for International Settlements, 1999. A new capital adequacy framework. Consultative paper issued by the Basle Committee on Banking Supervision. Basle. Jane 1999.

Blume, ME., Lim, F., MacKinlay, A.C., 1998. The declining credit 9uality of US corporate debt:

Myth or reality. Journal of Finance 53. 1389—1413.

Brunrier, A., Krahnen, J.P., Weber, M., 2000. Information production in lending relationships: On the role of corporate talings in commercial banking. Working Paper, (‘US. FrankfurtIMan, in progress.

Cacuette. JR.. Aliman, El.. Narayanars. p.. 1998. Managing credit rsk: The next great financial challenge. Wiley Frontiers in Finance.

(‘arey. M S., 1998. CredO risk in private debt portfolios. Journal of Finance 53. a63—l387.

Ederinton, LII.. Yawitz, J.B.. Roberts, bE., 1987. The informational Content of bond ratings. Journal of Financial Research 19, 211—261.

Elsas, R., Henke, S., Machauer. A.. Rott. R., Schenk, G.. 1998. Empirical analysis of credit relationships n small firms financing: Sampling design and descriptive statistics. Working Paper 98/14, Center for Financial Studies, FrankfurtlMain.

Elsas. R . Ktahncn J.P.. t998. is reationship lending special? Evidence from credit-ijIe data in Germany. Journal of Banking and Finance 22 (lO&l 1). 1283 1316.

hand, J.R.N., Hoithausen. RB., Leftwich. RB.. 1992. The died of bond rating agency announcements on bond and stock prices. Journal of Finance 47. 733 752.

International Monetary Fund, 1999. International capital markets: Developments, prospects, and key policy issues. September (www: iinf.orexternal/pubs/ft/icm/l 999/index him).

Krjshnaiah. P.R., Rao, C.R. (Eds.). 1988. Handbook of Statistics, vol. 7: Quality Control and Reliability. North-Holland. Amsterdam.

Liii. P., Scyyed. F.J., Smith. S.t).. 1999. The independent impact of credit rating changes . the case of Moody’s rating refinement on yield premiums. Journal of Business Finance and Accounting

26, 337 363.

Machauer, .. Weher. M.. 1Q98. Bank behavior based on internal credit ratings of borrowers. Journal of Banking and Finance 22 (tO&l II. 1355—1383.

Moody’s Investor Service. l999a. Measunng private firm default risk. Special comment, June.

Moody’s Investor Service. l999b. Historical default rates of corporate bond issuers, 1920 1998. Special comment. January.

Saunders, A.. 1999. C’redit Risk Measurement. New Approaches to Valuc at Risk and other Paradigms. Wile’. Ness York.

Shirhand, L.E.. 1993. Statistical Quality Control with Microcomputer Applications. Wiley, New York.

Standard & Poor’s. 1999. 1999 Corporate ratings criteria, wwsv: standardandpoors.corn.

J. P. Krahnen, M. Weber / Journal of Banking & Finance 25 (2001) 3—23 23

Treacy, W.F., Carey, M.S., 1998. Credit risk ratings at large US banks, Board of Governors of the Federal Reserve System. Federal Reserve Bulletin (November), 897—921.

Wahrenburg, M., Niethen, S., 2000. Vergleichende Analyse alternativer Kreditrisikomodefle. Kredit mid Kapital 33, 235—257. — Weber, M.. Krahnen, J.P.. Volmann, F., 1998. Risikomessung im Kreditgeschäft: Eine empirische

Analyse bankintcrner Ratingverfahren. Zeitschrift für betriebswirtschaftliche Forschung, Sonderheft 41, S. 117—142.

1 komentar:

  1. Terimakasih atas informasinya, sangat membantu.
    salam kenal dari saya mahasiswa fakultas Ekonomi :)