430205 Application of Fourier Inversion Methods to Credit Portfolio Models with Integrated Interest Rate and Credit Spread Risk First version: April 2004 Current version: September 2004 Abstract: Most credit portfolio models currently used by the banking industry rely on Monte Carlo simulations for calculating the probability distribution of the future credit portfolio value, which can be quite computer time consuming. Adding market risk factors, such as stochastic interest rates or credit spreads, as additional ingredients of a credit portfolio model, the computational burden of full Monte Carlo simulations even increases and the need for efficient methods for calculating credit risk measures becomes even more obvious. In this study, based on a version of the well-known credit portfolio model CreditMetricsTM extended by correlated interest rate and credit spread risk, it is analyzed whether the use of characteristic functions and inverse Fourier transformation can be an efficient tool for calculating risk measures in the context of integrated credit portfolio models. Unfortunately, the characteristic function of the credit portfolio value at the risk horizon can not be calculated in closed-form in all considered cases, but has to be computed by Monte Carlo simulations. However, this method can be still much faster than a full Monte Carlo simulation of the future credit portfolio distribution. The accuracy of the method depends on the composition of the portfolio. Keywords: credit risk, interest rate risk, credit spread risk, credit portfolio model, Value at Risk, characteristic function, inverse Fourier transforms JEL classification: C 63, G 21, G 33 430205 2 I. Introduction For calculating risk measures of credit portfolios, such as value at risk or expected shortfall, a range of models have been developed. Prominent examples are CreditMetricsTM by J.P. Morgan Chase, CreditPortfolioViewTM by McKinsey, Portfolio ManagerTM by KMV or CreditRisk+TM by CSFP. With the exception of CreditRisk+TM and one version of the Portfolio ManagerTM all these models rely on Monte Carlo simulations for calculating the probability distribution of the future credit portfolio value, which can be quite computer time consuming, especially for portfolios with many obligors and if percentiles corresponding to high confidence levels have to be estimated with sufficient accuracy. A typical shortcoming of most credit portfolio models currently used in the banking industry is that relevant risk factors, such as interest rates or credit spreads, are not modeled and hence are ignored during the revaluation of the credit sensitive instruments at the risk horizon. For example, fixed income instruments, such as bonds or loans, are revalued at the risk horizon using the current forward rates and (rating class specific) forward credit spreads for discounting future cash flows. Thus, the stochastic nature of the instrument's value in the future which results from changes in factors other than credit quality is ignored, and the riskiness of the credit portfolio at the risk horizon is underestimated. An additional consequence is that correlations between changes of the credit quality of the debtors and changes of market risk factors and hence the exposure at default cannot be integrated into the credit portfolio model. This is especially a problem for market-driven instruments, such as interest rate derivatives. Finally, correlations between the exposures at default of different instruments, which depend on the same or correlated market risk factors, cannot be modeled, too. Various studies1 showed that the missing stochastic modeling of market risk factors or credit spread risk causes a severe underestimation of economic capital, especially for high grade credit portfolios with a low stochastic dependence between the obligors' credit quality changes. 3 Adding market risk factors, such as stochastic interest rates or credit spreads, as additional ingredients of a credit portfolio model, the computational burden of full Monte Carlo simulations increases and the need for efficient methods for calculating credit risk measures becomes even more obvious. The aim of this study is to analyze whether the use of characteristic functions and inverse Fourier transformation can be an efficient tool for calculating risk measures in the context of a credit portfolio model with integrated market risk. This technique has already been successfully applied to market risk portfolio models.2 But there are only a few papers which are concerned with the application of this method to credit risk portfolio models, especially with integrated market or credit spread risk.3 However, it is certainly beyond the scope of this paper to present a full comparison of all methods which might improve the efficiency of risk measure calculations in credit portfolio models with integrated market risk factors. Possible further candidates would be saddle-point methods4, granularity adjustment techniques5 or Monte Carlo simulations combined with suitable variance reduction techniques6. This paper is structured as follows: In section II a short overview of the general approach and related papers is given. Then, in section III a version of the credit portfolio model CreditMetricsTM with integrated correlated interest rate and credit spread risk is presented. The use of characteristic functions and inverse Fourier transformation is explained when this modeling framework is applied to a portfolio of defaultable zero coupon bonds. Section IV contains a discussion of necessary changes of the approach when some of the assumptions of the previous section are modified. In particular, the case that the portfolio is composed of European call options with counterparty risk is analyzed. The differences between the percentiles of the future credit portfolio distribution when calculated either by a full Monte Carlo simulation or 4 by the method proposed in sections III and IV are presented within a numerical example in section V. Finally, in section VI the main results are summarized. II. General approach and related literature General approach The characteristic function of a continuous random variable X with density function f ( x) is a complex-valued function defined as:7 . X ( s) := E ei·s ·X = 8 -8 i ·s · x e · f ( x)dx = 8 -8 cos( s ·x) · f ( x)dx + i sin( s ·x) · f ( x)dx ,
-8 8 (2.1) where s . ! and i = -1 is the imaginary unit. Up to the real argument s which is replaced by i ·s the characteristic function equals the moment generating function, but it has the adx vantage that, as a consequence of the boundedness of ei·s · , it always exists. In a non- probabilistic context the characteristic function . X ( s ) is called the Fourier transform of the (density) function f ( x) . Two fundamental properties of characteristic functions are used in the following. First, the characteristic function of a sum of independent random variables equals the product of the characteristic functions of the individual random variables. Second, the characteristic function of a random variable uniquely determines its probability distribution, which is recovered from the characteristic function by the following inversion formula:8
P( X < x) = F ( x) = 1 1 + 2 2
8 x ei·s · . X (- s ) - e- i·s·.x i ·s 0 X (s) ds , (2.2) or9
8 e- i·s·x ·. X ( s ) 1 1 P( X < x) = F ( x) = - Im ds , 2 0 s (2.3) and, supposed . X ( s ) is integrable, the density function of the random variable X is given by 5
8 f ( x) = 1 2 -8 e - i · ·x t . X (t ) dt , (2.4) which is also called the inverse Fourier transform. Duffie and Pan (2001) Duffie and Pan (2001) model the vector R = (ln S1 ,... , ln S M ) of log-market price processes by a multidimensional jump-diffusion process where jumps arrive with constant intensity . and jump sizes are normally distributed. The random default time n of each counterparty (or borrower) n . {1,... , N } is assumed to have an intensity process of the following form . n (t ) =. I n (t ) + pn ·. C (t ) , (2.5) where pn is a constant between 0 and 1 and all intensity processes . nI (t ) ( n . {1,... , N } ) and . C (t ) are independent Cox, Ingersoll and Ross (CIR)-processes. Assuming homogeneous
borrowers in the sense that pn = p for all n and that the parameters appearing in all involved CIR processes coincide, the default intensity (2.5) is itself a CIR process: d . n (t ) =. ·(. nI + p ·.
C - . n (t )) ·dt + · . n (t ) ·dWn (t ) , (. ,. I n ,. C , . ! + ) (2.6) where Wn (t ) are independent standard Brownian motions. It is assumed that the individual default times n are modeled by the first jump of Cox processes with intensity processes . n (t ) as defined in (2.6). This means that, conditional on the realization of the intensity process . n (t , ) ( . . . . ), the default time n corresponds to the first jump time of a Poisson process with time-varying, but deterministic intensity . n (t , ) . As a consequence, the sur. vival probability up to time T > 0 is given by the following expression:10
T P ( n > T ) = E exp - . n (t )dt , 0 (2.7) 6 which can be easily calculated in closed-form due to the CIR process assumption. Stochastic dependence between the individual default times and between the credit quality changes of the obligors is induced by two mechanisms. The first one is based on the common factor . C (t ) , on which the default intensities of all obligors depend in (2.5). Hence, all default intensities . n (t ) are correlated, but conditional on the realization of all the intensity processes . 1I ,... . I N ,. C I the corresponding Poisson processes N1I ,... N N , N C are assumed to be inde- pendent. Second, whenever N C (t ) jumps, it is assumed that any obligor defaults with probability p and that these defaults at any common event time are conditionally independent. With regard to the loss given default, it is assumed that if obligor n defaults until the risk horizon, the portfolio value is reduced by a fraction Ln of the non-default market value Vn ( RH , . H ) of the instrument at the risk horizon H , but only if this market value is positive. The fraction Ln might be stochastic, but it is assumed to be independent of all other risk factors. If a counterparty defaults on a derivative, which represents a liability for the bank at
t = H , implying Vn ( RH , . H ) < 0 , nothing is subtracted from the portfolio value. Hence, the portfolio value . ( H ) at the risk horizon H is given by:
. (H ) = V ( R
n , . H ) -1{ n = H } ·Ln ·max{0;Vn ( RH ,. H )} . n =1 #$ %$$ #$$$$$ %$$$$$$ $ & n =1 $ &
H value component (VC ) default component ( DC ) N N (2.8) Next, Duffie and Pan (2001) compute the characteristic function of the random variable (2.8) and by numerical integration of a formula similar to (2.3) they obtain the probability distribution of . ( H ) , with which they can calculate risk measures, such as the value at risk. But, in order to compute the characteristic function . eral simplifying assumptions:
. (H ) ( s ) in closed-form, they have to make sev- 7 Assumption 1: max{0;Vn ( RH , . H )} max{0;Vn ( R0 ,. 0 )} . n . {1,... , N } , which means that changes of the stochastic risk factors are neglected when computing the loss given a default, Assumption 2: the stochastic dependence between . n , H and 1{ m = H } is ignored
. n, m {1,... , N } , . Assumption 3: the value component is "delta-gamma" approximated by a second order Taylor series expansion around the t = 0 -values of the market risk factors and the default intensities; for the risk factor vector ( R, . ) = (ln S1 ,... , ln S M ,. 1 ,... , . N ) a Gaussian approximation is used and, for simplicity, (ln S1 ,... , ln S M ) and (. 1 ,... ,. independent,
N ) are assumed to be Assumption 4: 1{ n = H } 1{ N I ( H )=1}+ . n ·1{ N C ( H ) =1} , where . n is the indicator of the event
n that obligor n defaults at the first common credit event and N nI ( H ) and N C ( H ) are the jump processes corresponding to . nI ( H ) and . C ( H ) ; using the above approximation a double-counting of defaults
that occurs from both common and individual credit events is ignored. Assumptions 1 and 2 imply the separation . . (H ) (s) . VC ( s )· . DC ( s ) and assumptions 3 and 4
VC allow for closed-form solutions of the characteristic function of the value component . and of the default component .
DC ( s) ( s ) respectively. Duffie and Pan (2001) apply this frame- work first to a portfolio of defaultable zero coupon bonds and, second, to a portfolio of options on equity indices with counterparty risk. Mainly, they assume a risk horizon of two weeks, which makes sense in the presence of the above assumptions, which get more and more rough with widening risk horizon.11 8 Reiß (2003) Reiß (2003) starts from the well-known two state credit portfolio model CreditRisk+TM, modifies it by using characteristic functions instead of probability generating functions12 and incorporates correlated sector variables and market risk into the model. In the following, a brief description of the approach of Reiß (2003) is given. The credit loss per unit exposure at default of a portfolio with N is:
L( H ) = 1{ n = H } ·Ln ,
n =1 N (2.9) where Ln denotes the loss given default per unit exposure, which is assumed to be deterministic13, and n is the default time of obligor n , which is assumed to be exponentially distributed with intensity . n . The credit quality of each obligor is considered to be influenced by K sectors and the affiliation to these sectors is given by the non-negative weights an , k ( n . {1,... , N } , k . {1,... , K } ), where an ,0 denotes the so-called idiosyncratic risk. Of course, for each obligor n the sector weights sum to one. The stochastic default intensity . n is defined by: . n ( R1 ,... , RK ) := pn · an,0 + an ,k ·Rk , k =1 K (2.10) where pn equals approximately, at least for small values of pn , the one year default probability of obligor n , and the stochastic vector R = ( R1 ,... , RK ) of sector variables consists of independent Gamma distributed random variables with expectation 1 and variance k2 > 0 . Conditional on the realization of R , all default times n ( n . {1,... , N } ) are assumed to be independent. Assuming small one year default probabilities pn , the characteristic function . 1{ n = H } R ( s ) of the individual default indicator functions, conditional on the realizations of the 9 sector variables R = ( R1 ,... , RK ) , can be approximated by the characteristic function of a Poisson distribution: . 1{ n = H } R ( ( s ) e. n ·H ·e i·s -1) , (2.11) where i = -1 again denotes the imaginary unit. Additionally exploiting the conditional independence (on R ) of the summands in (2.9), the independence of the sector variables R = ( R1 ,... , RK ) and the properties of the Gamma distribution, we get for the unconditional characteristic function .
L(H ) ( s ) of the loss variable L( H ) : . L ( H ) (s) = E . N ( s ) = E . . L( H ) R n =1 N ( s) = E . . n=1 1{ n = H } ·Ln R 1{ n = H } R ( Ln ·s ) K N N 1 s s = exp an ,0 · pn ·H · ei·Ln · - 1 - 2 ·ln 1 + k2 ·H · an , k · pn · 1 - ei·Ln · . k =1 k n =1 n =1 ( ) ( ) (2.12) Giving up the independence assumption for the sector variables R = ( R1 ,... , RK ) causes the problem that the expectation E .
L(H ) R ( s ) of the conditional characteristic function of
L(H ) L( H ) and hence the characteristic function . ( s ) can no longer be calculated in closed- form, but has to be computed by Monte Carlo methods.14 In order to incorporate market risk
' in the CreditRisk+TM-framework Reiß (2003) assumes the existence of K normally distributed ' market risk factors M l ( l . {1,... , K } ), which are correlated among themselves and with the sector variables.15 Reiß (2003, p. 117) claims that the credit portfolio value at the risk horizon
t = H can be written as: . (H ) = N
n =1 N n ( M , H ) ·qn ( R, H ) ·1{ n > H } , (2.13) where M = ( M 1 ,... , M K ) denotes the vector of market risk factors, N n ( M , H ) represents the ' nominal amount of the n th obligor discounted by the risk-free interest rate and
qn ( R, H ) = P ( n > Tn R = r . n > H ) (2.14) 10 is the real-world probability that obligor n survives up to the settlement date Tn of his liability, given he survives up to the risk horizon H and given the realization of the vector of sector variables R . Reiß (2003) proposes substituting N n ( M , H ) by a "delta" approximation around ( M 0 , 0) and evaluating the characteristic function . tions using the representation
. (H ) ( s ) by Monte Carlo simula- . . (H ) (s) = E . . ( H ) R ,1{1= H } ,...,1{ N = H } (s) . (2.15) Obviously, Reiß (2003) implicitly assumes the portfolio to be composed of defaultable zero coupon bonds with maturity Tn and a loss given default of 100% . But even with these implicit assumptions the structure of formula (2.13) for . ( H ) is not clear: Using a riskneutralized pricing approach, the t = H -price of a defaultable zero coupon bond with maturity
' Tn and a loss given default of 100% would be N n ( M , H ) ·P ( n >Tn R = r . n > H ) ·1{ n > H } , but only if M and n are independent, which is not the case in the model of Reiß as he assumes the market risk factors M and the sector variables R to be correlated. Furthermore, Reiß does not differentiate between the survival probability under the real-world probability
' measure P and that one under the risk-neutralized probability measure P . For short risk ho- rizons the difference might be negligible, but for longer time horizons, which are common practice in credit risk management, this assumption is rather questionable. III. CreditMetricsTM with integrated correlated interest rate and credit spread risk In this section the CreditMetricsTM-framework is extended by correlated interest rate and credit spread risk. It is assumed that the credit portfolio consists of N zero coupon bonds with identical face value F and maturity date T issued by N different corporates. Later, in section IV, the case that the portfolio is composed of European calls with counterparty risk on (default-) risk-free zero coupon bonds is considered. The risk horizon H of the credit portfo- 11 lio model is one year and P denotes the real-world probability measure. As additional risk factors interest rate risk and credit spread risk are introduced. It is further assumed that the return X n on firm n's assets can be described by a normally distributed random variable, which is ­ without loss of generality ­ standardized: X n = . V -.
2 r ,V ·Z + . r ,V ·X r + 1 - . V · n (. 2 r ,V =. V , n . {1,... , N } ),16 (3.1) where Z , X r , 1 ,... , N are mutually independent standard normally distributed stochastic variables. The stochastic variables Z and X r represent systematic credit risk, by which all firms are affected, whereas the n 's stand for idiosyncratic credit risk. The risk-free short rate is modeled for simplicity as a mean-reverting Ornstein-Uhlenbeck process introduced already by Vasicek (1977): ( dr (t ) = . ·. - r (t )) ·dt + r ·dWr (t ) , (3.2) where . . , . ! + are constants and Wr (t ) is a standard Brownian motion under P . The , process (r (t ))t. ! + always tends back to the mean level . ; the higher the value . the more unlikely are deviations from this level. The solution of the stochastic differential equation (3.2) is:
r r (t ) = . + (r (0) -. ) ·e-. ·t + ·(1 - e -2.· ·t ) ·X r , #$$ %$$$ $ & 2. P = E [ r ( t )] 2 (3.3) where X r N (0,1) enters the definition (3.1) of the firms' asset returns. As it can be easily seen, the definition (3.1) of the asset returns implies that all pairs of asset returns exhibit a correlation parameter of . V and that the asset returns X n and the interest rate factor X r (and hence the short rate r ( H ) ) are correlated with parameter .
r ,V . In this section, it is assumed 12 that the correlation . V between each pair of asset returns as well as the correlation . tween each asset return and the risk-free short rate are identical. r ,V be- As in the CreditMetricsTM methodology, the rating . n H of the N obligors at the risk horizon t = H is simulated by the N-variate normally distributed random vector X = ( X 1 ,... , X N ) , whose components exhibit means zero, variances one and equal pairwise correlations . V . An obligor n with current rating i is assumed to be in rating class k at the risk horizon if the realization of X n lies between two thresholds Rki +1 and Rki with Rki +1 < Rki . The numbers i and k identify the different rating classes with 1 representing the best rating and K being the default state. The thresholds Rki are derived from a one-year transition matrix Q = (qik )1=i ,k = K , whose elements qik specify the probability that an obligor migrates from the rating class i to the rating class k within one year (see table 1). The thresholds Rki ( 1 = i = K - 1 , 2 = k = K ) are computed by ensuring that the probability for the realization of a standardized normally distributed random variable X n to be in the interval [ Rki +1 , Rki ] coincides with the probability qik from the migration matrix:17 K Rki = -1 qil , l =k (3.4) where -1 ( ·) denotes the inverse of the cumulative density function of the standard normal
i i distribution. If X n = RK , the obligor has defaulted until the risk horizon, and if X n > R2 , the obligor is in the best rating class 1. - insert table 1 about here - The price of a zero coupon bond at the risk horizon H , whose issuer n has not already defaulted (nd) until H and exhibits the rating .
n H = k , is given by: 13
- ( R ( X r , H ,T ) + Sk ( H ,T ) )· T - H ) ( nd vn ( X r , k , H , T ) = F ·e , (3.5) where R( X r , H , T ) denotes the stochastic risk-free spot yield for the time interval [ H , T ] , and Sk ( H , T ) is the stochastic credit spread of rating class k for the time interval [ H , T ] . In the Vasicek model the stochastic risk-free spot yield R( X r , H , T ) can easily be calculated in closed-form18 and is a linear function of the risk factor X r appearing in (3.3). The rating specific credit spreads Sk ( H , T ) ( k . {1,... , K- 1} ) are assumed to be multivariate normally distributed random variables with expectation k and standard deviation k .19 The correlation parameters between the normally distributed credit spreads are denoted by .
i,k S . Furthermore, it is assumed that the random variable X r , which drives the term structure of risk-free interest rates, and the systematic credit risk factor Z respectively are both correlated with the credit spreads. For sake of simplicity, these correlation parameters are set equal to constants . and .
Z ,S X r ,S respectively, regardless of the rating grade. Besides, it is assumed that the idiosyn- cratic credit risk factors n ( n . {1,... , N } ) are independent of the credit spreads Sk ( H , T ) ( k . {1,... , K- 1} ). If the issuer n of a zero coupon bond has already defaulted (d) until the risk horizon H , the value of the bond is set equal to a beta-distributed fraction of the value p( X r , H , T ) of a risk-free but otherwise identical zero coupon bond:20
d vn ( X r , H , T ) = · p( X r , H , T ) , (3.6) where E [ ] = and Var ( ) = 2 . These first two moments of the distribution of the recovery rate can vary with the seniority of a claim and the value of individual collaterals. For simplicity, we use a uniform recovery rate distribution for all issuers, but for each defaulted issuer a beta-distributed recovery rate is drawn individually which ensures independence of the recovery rates across the different exposures. The recovery rate is assumed to be independent 14 from all other stochastic variables of the model, e.g. the systematic credit risk factors Z and X r , the idiosyncratic risk factors n and the credit spreads Sk ( H , T ) ( k . {1,... , K- 1} ). The value . ( H ) of the entire portfolio of defaultable zero coupon bonds at the risk horizon
H is: . (H ) =
N K -1 v
n =1 k =1 N K -1 nd , k n ( X r , k , H , T ) ·1{. n = k } + · p( X r , H , T ) · { 1.
H n H =K } = F ·e
n =1 k =1 - R ( X r , H ,T )· T - H ) ( ·e ( - S k ( H ,T )· T - H ) ( ·1{. n = k } + ·1{ .
H n H =K } ), (3.7) where the indicator function 1{. n = k } is one if obligor n is in the rating class k at t = H and
H zero otherwise. The probability of migrating from rating class i to k . {2,... , K- 1} until the risk horizon H , conditional on the realizations of the systematic credit risk factors Z and X r , is given by qi , k ( z , xr ) := P Rki +1 < X n = Rki Z = z , X r = xr ( )
(3.8) Ri - . -. 2 ·z - . ·x R i - . - . 2 ·z - . ·x V r ,V r ,V r V r ,V r ,V r k - k +1 . = 1- . V 1 -. V #$$$$$ %$$$$$ & #$$$$$%$$$$$& $ $
=: ti ,k ( z , xr ) =: ti ,k +1 ( z , xr ) The conditional default probability is
Ri - . -. 2 ·z - . ·x K V r ,V r ,V r i , qi , K ( z , xr ) := P X n = RK Z = z , X r = xr = 1- . V #$$$$$ %$$$$$ & ( ) (3.9) =: ti ,K ( z , xr ) and the conditional probability of being in the best rating class 1 equals
R i - . -. 2 ·z - . ·x 2 V r ,V r ,V r . qi ,1 ( z , xr ) := P X n > R Z = z , X r = xr = 1 - 1- . V #$$$$$ %$$$$$ & ( i 2 ) (3.10) =: ti ,2 ( z , xr ) 15 As (3.8), (3.9) and (3.10) show, the specification (3.1) of the multi-factor model for the individual asset returns implies that the transition process and the term structure of risk-free interest rates are correlated. The degree of correlation is determined by the value of the sensitivity . r ,V . As it is assumed that the random variable X r and the systematic credit risk factor Z re- spectively are correlated with the credit spreads, we even have a model, in which the transition process, the risk-free interest rates and the credit spreads are all pairwise correlated.21 Now, the methods of characteristic functions and inverse Fourier transformation are applied to this model in order to accelerate the computation of the probability distribution of . ( H ) .22 Conditional on the realizations of the stochastic variables Z , X r and S = ( S1 ( H , T ),... , S K -1 ( H , T )) all N summands of the outer sum in (3.7) are independent because the only remaining stochastic variables are the independent idiosyncratic risk factors n ( n . {1,... , N } ). Hence, at first the following conditional characteristic function is computed, where the initial rating of all obligors is assumed to be j . {1,... , K- 1} : . n n F ·e- R ( X r ,H ,T )·(T -H ) · e- Sk ( H ,T )·(T -H ) ·1{. H =k } + ·1{. H =K } X r , Z , S k =1 K -1 ( s) K- i·s · 1 F ·e- R ( xr ,H ,T )·(T -H ) · e- sk ( H ,T )·(T -H ) ·1 n + ·1 n . {. H =k } { H =K } e k =1 =E X r = xr , Z = z , S = ( s1 ( H , T ),... , sK -1 ( H , T ) ) = 1 8 0 -8 e K -1 e- R ( xr ,H ,T )·(T - H ) · e- sk ( H ,T )·(T - H ) · { >t } + e- sk ( H ,T )·(T - H ) · { t i ·s · F · 1 n j ,2 1 j ,k +1< n =t j ,k } + · { n =t j ,K } 1 k =2 · ( n ) · ( ) d n d ,23 where ( n ) and ( ) denote the density functions of the random variables n and respectively. Splitting up the integration path of n yields: 16 K -1 t j ,k t j ,K - R ( xr ,H ,T )· T - H ) - ( R ( xr , H ,T )+ sk ( H ,T ))· T - H ) ( ( · e i ·s ·F · · ( n ) d n + ei·s·F ·e · ( n ) d n -8 e k = 2 t j ,k +1 0 1 + t j ,2 8 F ei·s · ·e - ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( · ( n ) d n · ( ) d K -1 - ( R ( xr , H ,T )+ sk ( H ,T ))· T - H ) ( · - R ( xr ,H ,T )·(T -H ) = ei·s ·F ·e · (t j , K ) + ei·s·F ·e ·( (t j ,k ) - (t j , k +1 ) ) k =2 0 1 F + ei·s · ·e
- ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( ·(1 - (t j ,2 ) ) · ( ) d } =. ( s ·F ·e
i ·s · ·e F - R ( xr , H ,T )· T - H ) ( ) ·(t i ·s · · Fe j,K ) + e k =2 K -1 - ( R ( xr , H ,T )+·sk ( H ,T ))· T - H ) ( ·( (t j ,k ) - (t j ,k +1 ) ) +e - ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( ·(1 - (t j ,2 ) ) , (3.11) where . ( ·) denotes the characteristic function of the recovery rate .24 Note that as a consequence of the homogeneity assumption the term (3.11) does not depend on the identity of the obligor. Because of the conditional independence of the summands in the outer sum of (3.7) the conditional characteristic function of . ( H ) can be written as: . . (s) = . . ( H ) X r ,Z ,S
n =1 N n n F ·e- R ( X r ,H ,T )·(T -H ) · e- Sk ( H ,T )·(T -H ) ·1{. H =k } + ·1{. H =K } X r , Z , S k =1 K -1 ( s) = . Homog . ( F s ·F ·e- R ( xr , H ,T )·(T - H ) · (t j , K ) + ei·s · ·e
- ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( ) K -1 k =2 - ( R ( xr , H ,T )+ sk ( H ,T ))· T - H ) ( ·( (t j ,k ) - (t j ,k +1 ) ) (3.12) +e i ·s · · Fe ·(1 - (t j ,2 ) ) ) N . Finally, the unconditional characteristic function of . ( H ) is the expectation of the conditional characteristic function .
. ( H ) X r ,Z ,S (s) : (3.13) . . (H ) (s) = E . . ( H ) X r ,Z ,S (s) . Unfortunately, the above expectation can not be calculated in closed-form, but has to be computed by Monte Carlo simulations (e.g. using ,,quasi" random numbers). Of course, with one drawn sample of X r , Z and S = ( S1 ( H , T ),... , S K -1 ( H , T )) the conditional characteristic function .
. ( H ) X r ,Z ,S ( s ) can be computed for several values of s . Finally, having calculated 17 . . (H ) ( s ) , we get the distribution function of the credit portfolio value . ( H ) via formula (2.3) and numerical integration. Of course, it is not completely satisfying that we still have to employ simulation methods in order to compute the characteristic function (3.13), but, as the numerical example in section V will show, this method can be much faster than a full Monte Carlo simulation of the future credit portfolio value. However, the speed gain depends on the number of grid points s needed for the numerical integration in (2.3) because with a large number of grid points also a large number of expectations has to be calculated and hence a large number of function evaluations has to be done, which is a potential drawback of the method described above. Thus, the use of an numerical integration rule which only needs a moderate number of grid points for a sufficient accuracy is essential. IV. Extensions The purpose of this section is to discuss the computational consequences which result from changes of some of the assumptions made above. First, we want to analyze the consequences of giving up the homogeneity assumptions concerning the composition of the credit portfolio, i.e. dealing with inhomogeneous exposures, inhomogeneous initial ratings or inhomogeneous asset return correlations. Beginning with inhomogeneous exposures and assuming that there are E different exposure buckets with respective face values F1 ,... , FE , formula (3.12) would have to be altered as follows: .
E = . . e =1 . ( H ) X r ,Z ,S (s) ( F s ·Fe ·e - R ( xr , H ,T )·(T - H ) ·(t j , K ) + ei·s · e ·e
- ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( ) K -1 k =2 - ( R ( xr , H ,T )+ sk ( H ,T ))· T - H ) ( ·( (t j ,k ) - (t j ,k +1 ) ) (4.1) F + ei·s · e ·e ·(1 - (t j ,2 ) ) ) ne , 18 where ne ( e . {1,... , E} ) denotes the number of obligors whose zero coupon bond has a face value of Fe . Thus, within each simulation run of the conditional characteristic function . . ( H ) X r ,Z ,S ( s ) E conditional characteristic functions (instead of 1) have to be calculated for each grid point s so that the computational burden increases significantly with the number of exposure buckets rising. For inhomogeneous initial ratings the adapted formula (3.12) is: .
K -1 = . . j =1 . ( H ) X r ,Z ,S (s) ( s ·F ·e - R ( xr , H ,T )· T - H ) ( ) F · (t j , K ) + ei·s · ·e k =2 K -1 - ( R ( xr , H ,T )+ sk ( H ,T ))· T - H ) ( ·( (t j ,k ) - (t j ,k +1 ) ) F + ei·s · ·e - ( R ( xr , H ,T )+ s1 ( H ,T ))· T - H ) ( ·(1 - (t j ,2 ) ) ) (4.2) nj , where n j ( j . {1,... , K- 1} ) now denotes the number of obligors whose initial rating is j . Note that only the thresholds t j ,k depend on the initial rating index j , which implies that within each simulation run the exp-terms, which depend on s , have not to be recalculated for each rating grade j . Finally, assuming G different groups of obligors, in which each pair of asset returns exhibits a correlation parameter . Vg ( g . {1,... , G} ), yields an adapted formula (3.12) which resembles (4.2), but with K - 1 replaced by G in the upper index of the product. Again, only the thresholds t g,k depend on the asset return correlation index g . j Another kind of modification could concern the number of risk factors or their probability distribution. Increasing the number of risk factors, e.g. by employing more systematic credit risk factors which explain the obligors' asset returns or by introducing additional market risk factors, such as e.g. foreign exchange rates, would make it necessary to increase the number of simulations or to use more sophisticated variance reduction methods in order to obtain the same accuracy as before when calculating the unconditional characteristic function (3.13). Using a different probability distribution for the risk factors, e.g. a multivariate t ­distribution 19 for the asset returns, is less problematic. We only would have to use the inverse of the cumulative density function of the respective probability distribution when calculating the thresholds t j ,k and use its cumulative density function instead of ( ·) in (3.12). Furthermore, correlated recovery rates depending on the systematic credit risk factors as well as on the individual asset returns could easily be introduced, for example by the following representation:25 n ( Z , X r , n ,. n ) := min e
+ = min e
· + · ·Z + ·X r +. ·X n + 1- 2 - 2 -. 2 · n . ( ) ;1
;1 , (4.3) ( ·. . V -. 2 r ,V + ) ·Z + (. ·. ) r ,V + )·X r +. · 1- . V · n + 1- 2 - 2 - . 2 · n . where , . . ! + , . ! , 2 + 2 + . 2 = 1 and the . n N (0,1) ( n . {1,... , N } ) are independ- ent of each other as well as from all other random variables in the model, especially Z , X r and the n . As the recovery rates ( Z , X r , n ,. n ) are independent conditional on the realizations of Z and X r , the conditional characteristic function of the credit portfolio value equals (3.12) with .
+8 t j ,K ( s ·F ·e - R ( xr , H ,T )· T - H ) ( ) ·(t j,K ) replaced by the integral -8 -8 · ei·s ( z , xr , ,. )·F ·e - R ( xr , H ,T )· T - H ) ( · ( ) · (. ) d d. , (4.4) which has to be solved numerically. A third kind of modification could concern the type of credit-sensitive instrument the portfolio is composed of. Here, we want to consider the two examples of coupon bonds and European call options with counterparty risk on (default) risk-free zero coupon bonds. Let us first assume that the portfolio consists of N coupon bonds with identical face value F , maturity date T , coupon c and coupon dates H = t1 < ... < tM = T issued by N different corporates. The vector of stochastic credit spreads S = ( S1 ( H , t1 ),... , S K -1 ( H , t1 ), S1 ( H , t2 ), ... , S K -1 ( H , t M -1 ), S1 ( H , tM ),... , S K -1 ( H , t M ) ) 20 now consists of ( K - 1) ·M components where M denotes the number of coupon dates. Thus, now we also need to know the intertemporal correlations of credit spreads of different rating grades. Assuming a recovery payment of ·( F +c) in t = H in the case of a default until the risk horizon, the conditional characteristic function of the credit portfolio value is . . ( s) = . . ( H ) X r ,Z ,S
n =1 N n n c·e- ( R ( X r ,H ,tm )+Sk ( H ,tm ))·( tm -H ) +F ·e- ( R ( X r ,H ,T )+Sk ( H ,T ))·(T -H ) ·1{. H =k } + ·( c + F )·1{. H = K } X r , Z , S
k =1 m =1 K -1 M (s) = . ( s ·( F + c) ) ·(t j ,K ) + e
k =2 M e- ( R ( xr ,H ,tm )+ sk ( H ,tm ))·( tm - H ) + F · - ( R ( xr ,H ,T )+ sk ( H ,T ))·( T - H ) e K -1 i ·s · c· m=1 ·( (t j ,k ) - (t j ,k +1 ) ) +e M i ·s · c· - ( R ( xr ,H ,tm )+ s1 ( H ,tm ))·( tm - H ) + F · - ( R ( xr ,H ,T )+ s1 ( H ,T ))·( T - H ) e e m=1 ·(1 - (t j ,2 ) ) . N (4.5) As now each exp-term has M (instead of 1) exp-terms in its exponent, the computational burden increases with the number of coupon dates, but these additional exp-terms in the exponents have only to be calculated once (and not for each value of s ) within a simulation run. Next, let us assume that the portfolio consists of N European call options issued by N different corporates with identical expiration date T C , identical exercise price K and identical underlying (default) risk-free zero coupon bond p (r (t ), t , T ) with face value F and maturity date T = T C . Working within the term structure model of Vasicek (1977), the t = H -price C (r ( H ), H , K , T C , T ) of a European call option on a risk-free zero coupon bond without any counterparty risk is given by:26
-T r ( s ) ds ' P C C C H ·max { p (r (T ), T , T ) - K ,0} C (r ( H ), H , K , T , T ) = E e
C = p (r ( H ), H , T ) · (d1 ) - K · p(r ( H ), H , T C ) · (d 2 ) (4.6) with 21 d1 = p(r ( H ), H , T ) 1 ·ln + ·. , C . K · p(r ( H ), H , T ) 2 1 d 2 = d1 - . , . = C 1 2 · 3 · 1 - e -. ·(T -T ) 2 . ( ) - (e
2 - ·(T - H ) . - e -. ·(T C -H ) ) ,
2 ' where P denotes the risk-neutral probability measure relevant for pricing purposes. In order
to price a European call option on a (default) risk-free zero coupon bond with counterparty risk, we assume that a default is only possible at the maturity date T C of the option,27 and that in this case the recovery payment is a fixed exogenous fraction of the options regular pay off. Furthermore, for simplicity we assume for the pricing of the options independence between the movements of the risk-free interest rates and the credit quality changes of the counterparties.28 With these assumptions the price of a call written by counterparty n , whose rating at the risk horizon is .
C def (r ( H ),.
n H n H . {1,... , K- 1} , is given by: , H , K ,T C ,T ) ' = ·C (r ( H ), H , K , T C , T ) + (1 - ) ·C (r ( H ), H , K , T C , T ) ·P n > T C . ( n H ), (4.7) where n denotes the default time of counterparty n . Assuming that a default is an absorbing ' state under P , the event { n > T C } is equivalent to the event {. n TC . K } , whose probability can simply be calculated by summing up all individual risk-neutral probabilities for a rating change from .
n H to a non-default state within the time interval [ H , T C ] . Given the ­ for pric- ing purposes ­ assumed independence between the risk-free interest rates and the rating tran- ' sitions, the transition probabilities under P can easily calculated out of the prices of defaultable bonds of the respective counterparty.29 However, for the ease of exposition we do not differ between the real-world probability measure P and the risk-neutral probability measure ' P , but instead assume also for pricing purposes that the transition processes of all counterpar- 22 ties can be modeled by a time-homogeneous Markov chain with (real-world) one year transition matrix Q (see table 1), which is also used for modeling the rating transitions in the time interval [0, H ] . Of course, it has to be stressed that this approach does not reflect reality because the two measures will typically differ, especially over longer risk horizons used for credit risk management. The probabilities of rating changes within a T year horizon are then simply given by the matrix product
QT = Q ·... ·Q # %$ $ &
T times (4.8) of the one year transition matrix Q . The value . ( H ) of the entire portfolio of long positions in European call options with counterparty risk at the risk horizon H is:
. (H ) = C
n =1 k =1 N K -1 def (r ( H ), k , H , K , T C , T ) ·1{. n = k } + ·C (r ( H ), H , K , T C , T ) ·1{ .
H n H =K} , (4.9) where the second summand in the inner sum of (4.9) is the t = H -value of the recovery payment due at t = T C . Conditional on the realizations of the stochastic variables X r and Z all
N summands of the outer sum in (4.9) are independent because the only remaining stochastic variables are again the independent idiosyncratic risk factors n ( n . {1,... , N } ). Hence, we proceed as before. The initial rating of all obligors is assumed to be j . {1,... , K } :30 . n n C def ( X r , k , H , K ,T C ,T )·1{. H =k } + ·C ( X r , H , K ,T C ,T )·1{. H =K } X r , Z k =1 K -1 (s) K- i·s · 1C def ( xr ,k , H , K ,T C ,T )·1 n + ·C ( xr , H , K ,T C ,T )·1 n . {. H =k } { H =K } e k =1 =E X r = xr , Z = z = 8 -8 e K -1 i ·s · C def ( xr ,1, H , K ,T C ,T )· { n >t j ,2 } + C def ( xr , k , H , K ,T C ,T )· { t j ,k +1< n =t j ,k } + · ( xr , H , K ,T C ,T )· { n =t j ,K } 1 1 C 1 k =2 · ( n ) d n , where ( n ) again denotes the density function of a standard normal distribution. Splitting up the integration path of n yields: 23 t j ,K -8 e · C i ·s · ( xr , H , K ,T C ,T ) · ( n ) d n +
K -1 k =2 K -1 t j ,k k = 2 t j ,k +1
def e i ·s· def ( xr , k , H , K ,T C ,T ) C · ( n ) d n + 8 t j ,2 e i ·s · def ( xr ,1, H , K ,T C ,T ) C · ( n ) d n · = ei·s ·C ( xr , H , K ,T C + ei·s ·
def C ,T ) · (t j , K ) + ei·s·C ·(1 - (t j ,2 ) ) . ( xr , k , H , K ,T C ,T ) ·( (t j ,k ) - (t j ,k +1 ) ) ( xr ,1, H , K ,T C ,T ) Because of the conditional independence of the summands in the outer sum of (4.9) the conditional characteristic function of . ( H ) can be written as: . . ( s) = . . ( H ) X r ,Z
n =1 N n n C def ( X r ,k , H , K ,T C ,T )·1{. H =k } + ·C ( X r , H , K ,T C ,T )·1{. H =K } X r , Z k =1 K -1 (s) K -1 C def C · = ei·s ·C ( xr , H , K ,T ,T ) · (t j , K ) + ei·s·C ( xr , k , H , K ,T ,T ) ·( (t j ,k ) - (t j ,k +1 ) ) k =2 +e i ·s · def ( xr ,1, H , K ,T C ,T ) C ·(1 - (t j ,2 ) ) ) (4.10) N .
. (H ) Finally, the unconditional characteristic function . E P .
. ( H ) X r ,Z ( s ) of . ( H ) is again the expectation
. ( H ) X r ,Z ( s ) of the conditional characteristic function . ( s ) . Unfortunately, this expectation again can not be calculated in closed-form, but has to be computed by Monte Carlo simulations. Duffie and Pan (2001) propose to use a delta-gamma approximation for the option values at the risk horizon, which ­ together with additional assumptions ­ allows them to calculate the unconditional characteristic function of the credit portfolio value in closed-form. In the numerical example of section V we want to test whether the delta-gamma approximation approach is also appropriate for a risk horizon of one year and percentile calculations corresponding to high confidence levels as they are usual in credit risk management. For this purpose the t = H -price C (r ( H ), H , K , T C , T ) of the European call without counterparty risk and the t = H -price C def ( X r , k , H , K , T C , T ) of the corresponding option with counterparty risk respectively are approximated by a second order Taylor series expansion around the expected future value of the risk-free short rate r ( H ) at t = H :31 24 r ( H ) = E P [r ( H )] = . + (r (0) -. ) ·e-. ·H .
(3.3) (4.11) prices C (r ( H ), H , K , T C , T ) and In order
n H to shorten the notation the C def (r ( H ),. n , H , K , T C , T ) are abbreviated by C (r ( H ), H ) and C def (r ( H ),. H , H ) respec- tively. This yields:
, C (r ( H ), H ) C . (r ( H ), H )= C (r ( H ), H ) + . C (r ( H ), H ) ·( r ( H ) - r ( H ) ) . r(H ) r ( H )=r ( H ) 1 . 2C (r ( H ), H ) 2 + · ·( r ( H ) - r ( H ) ) , 2 2 (. r ( H )) r ( H )=r ( H ) and
n , C def (r ( H ),. H , H ) C def , . (r ( H ), . n H (4.12) ,H) , , ' = ·C . (r ( H ), H ) + (1 - ) ·C . (r ( H ), H ) ·P n > T C . ( n H ). (4.13) The value . ( H ) of the entire portfolio of long positions in European call options with counterparty risk at the risk horizon H is now approximated by:
, , . ( H ) . ( H ) . , = C def ,. (r ( H ), k , H ) ·1{. n = k } + ·C . (r ( H ), H ) ·1{ . n =1 k =1
H N K -1 n H =K} , (4.14) , and the conditional characteristic function of . ( H ) . is: K -1 , def ,. , · . ( xr , k , H ) ( s ) = ei·s ·C ( xr , H ) · (t j , K ) + ei·s·C ·( (t j ,k ) - (t j ,k +1 ) ) X r ,Z k =2 . . (H ) . , +e , i ·s · def ,. ( xr ,1, H ) C ·(1 - (t j ,2 ) ) ) (4.15) N . , Again, the unconditional characteristic function of . ( H ) . can not be calculated in closed- form. In a pure market risk context, one advantage of the delta-gamma approximation in the case of multivariate normally distributed risk factors is that the portfolio value can be expressed by a linear polynomial of independent chi-squared and normally distributed random variables. Based on this representation the characteristic function of the portfolio value at the risk horizon can be calculated in closed-form and the inversion theorem (2.3) can directly be applied. Unfortunately, this advantage is lost in the credit portfolio context (at least in the ex- 25 tended CreditMetricsTM-model described in section III) and the (unconditional) characteristic function of the credit portfolio value has to be computed again by Monte Carlo simulations. V. Numerical example Parameters In this section a numerical example is presented which demonstrates the differences between the percentile values when calculated on the one hand with a full Monte Carlo simulation and on the other hand by application of characteristic functions and the inversion theorem (2.3). For both methods we calculate the expectation of . ( H ) and the p % -percentiles p % (. ( H )) of the credit portfolio distribution with p . {0.1%,1%, 5%, 20%, 40%, 60%} . The percentiles corresponding to the probabilities 20%, 40% and 60% are only computed in order to check each method's accuracy for the body of the probability distribution. First, it is assumed that the portfolio consists of N = 500 defaultable zero coupon bonds, which are issued by N different obligors, but are otherwise identical. The face value is chosen to be F = 1 . The simulations are done for the homogeneous initial ratings . 0 . {Aa, Baa, B} . The parameters of the Ornstein-Uhlenbeck process (3.2) modeling the
risk-free short rate are from Lehrbass (1997), who estimated these parameters using the DEM-LIBOR overnight rates within the period July 31, 1991 to May 31, 1995. The market price of interest rate risk . , which is needed for calculating the price of a risk-free zero coupon bond, is the average of the values given by Lehrbass (1997). For simplicity, the recovery rate is set equal to a constant = 53.80% , which is Moody's mean recovery rate of senior unsecured bonds during 1970 to 1995.32 The employed transition matrix (see table 1) is also from Moody's. The time to maturity of the zero coupon bonds is chosen as T = 3 , implying a remaining time to maturity of two years at the risk horizon. The value of the correlation parameter . V of the asset returns is chosen as 10% , which is within the range of values pro- 26 posed by the Basle Committee on Banking Supervision for corporate exposures in the Internal Ratings-based approach33, and 40% respectively. The parameter .
r ,V , which determines the correlation between the firms' asset returns and the term structure of interest rates, is set equal to .
r ,V = -0.05 , implying a negative correlation between asset returns and interest rates. Tak- ing into consideration recent empirical studies of structural credit risk models34 this value seems reasonable. The means k and standard deviations k of the multivariate normally distributed rating grade specific credit spreads Sk ( H , T ) ( k . {1,... , K } ) as well as their correlation parameters .
j ,k S , which are used for simulating the credit spreads, can be seen in table 2 for T - H = 2 . These values are taken from Kiesel, Perraudin and Taylor (2003). The correlation coefficient .
X r ,S between the credit spreads and the risk-free interest rate factor is set
Z ,S equal to -0.1 . The correlation coefficient . between the systematic credit risk factor Z and the credit spreads, which is also independent of the rating grade, is assumed to be -0.1 , too. - insert table 2 about here - Afterwards, it is assumed that the portfolio consists of N = 500 European call options with counterparty risk on (default) risk-free zero coupon bonds, which are written by N different counterparties. The parameters of the short rate process (3.2), the recovery rate, the transition matrix, the asset return correlation parameter as well as the correlation parameter between the asset returns and the risk-free interest rates are chosen as above. Again