Analytical Methods for Hedging Systematic

Credit Risk with Linear Factor Portfolios∗

Dan Rosen† David Saunders‡

∗This version: November, 2006. First version: January, 2006. The authors are grateful to Roger Stein for many helpful comments.

†Fields Institute for Research in Mathematical Sciences, Toronto, Canada. drosen@fields.utoronto.ca

‡University of Waterloo, Waterloo, Canada. dsaunders@math.uwaterloo.ca

1

Abstract

This paper is part of a series explaining various methodologies for defining and measuring the contributions of systematic factors to eco­nomic capital as well as for hedging systematic risk in credit portfolios.

Multi-factor credit portfolio models are used widely today for mea­suring and managing economic capital as well as for pricing credit portfolio instruments such as collateralized debt obligations (CDOs). Commonly, practitioners allocate capital to the portfolio components, such as individual sub-portfolios, counterparties, or transactions. The hedging of credit risk is generally also focused on the ”deltas” of the underlying names in the portfolio. In this paper, we present analyt­ical results for hedging portfolio credit risk with linear portfolios of the systematic credit factors. Formally, we minimize the systematic variance of portfolio losses by using a linear combination of the sys­tematic risk factors. We review the mathematical tools to solve these optimization problems within a multi-factor Merton-type (or probit) credit portfolio model, and then apply them to various cases. First, we focus on static hedges of homogeneous and inhomogeneous credit portfolios. We also apply the methodology to hedge the systematic credit default losses of CDOs. Finally we show the application of the methodology to dynamic hedging strategies. In each case, we discuss the hedging portfolios, the effectiveness of the hedges and provide nu­merical examples.

2

1 Introduction

Multi-factor credit portfolio models are used widely for measuring and man­aging economic capital as well as for pricing credit portfolio instruments such as collateralized debt obligations (CDOs). In particular, these models provide a natural framework for analyzing diversification (or alternatively concentration risk) in credit portfolios, one of the key tools for managing credit risk and optimally allocating credit capital. Thus, many institutions today have implemented either internally developed or commercial multi­factor credit portfolio models to manage their credit risk (e.g. Gupton et al. (1997), Credit Suisse Financial Products (1997), Crosbie (1999)). Gen­erally, these models share an underlying mathematical framework, which is referred to as the conditional independence framework. Multi-factor credit portfolio models entail the use of Monte-Carlo (MC) simulation, although several analytical and semi-analytical methods have been developed in re­cent years (see e.g. Pykhtin (2004)).

In addition to pricing instruments and measuring capital, multi-factor credit portfolio models can provide useful guidance regarding the allocation of cap­ital and the hedging of credit portfolios. Once the economic capital has been computed with a credit portfolio model, practitioners commonly allocate capital to the portfolio components, such as individual sub-portfolios, coun­terparties, or transactions, using marginal risk contributions. The theory be­hind this procedure is well developed (Kalkbrenner et al. (2004), Gourieroux et al. (2000), Tasche (2000, 2002); see also Mausser and Rosen (2006) for a general presentation). The hedging of credit risk is generally also focused on each underlying name in the portfolio and its “deltas”, or sensitivities.

Understanding the contribution to economic capital and pricing of the various systematic factors (or credit drivers), which are at the heart of a multi-factor credit model, can lead to better methodologies for managing concentration risk and hedging credit portfolios effectively. This requires first decomposing the credit risk in a portfolio into its systematic and idiosyncratic compo­nents. The underlying multi-factor model drives the correlations of obligor credit events and determines entirely the systematic risk of the portfolio. However, the standard theory of marginal capital contributions cannot be applied directly in this context since the total capital and the price of credit portfolio instruments are not homogeneous functions of the factors. Finally,

3

the most interesting and realistic cases, in practice, may require simulation of the multi-factor models.

This series of research papers presents a set of methodologies for defining and measuring the contributions of systematic factors to economic capital as well as for hedging of systematic risk in credit portfolios. In this first paper of the series, we present analytical results for hedging portfolio credit risk with linear portfolios of the systematic factors. We can think of the hedging portfolio as a portfolio of spot positions in the underlying indices. 1 Formally, we seek to minimize the systematic variance of portfolio losses with linear combinations of the systematic risk factors. We focus on default credit losses only in a multi-factor Merton-type (or probit) credit portfolio model. We review the mathematical tools to solve these optimization problems and then apply them to various cases. First we focus on static hedges of homogeneous and inhomogeneous credit portfolios. We then apply the methodology to hedge the systematic credit default losses of CDOs. Finally, we show that the hedging problem can be solved exactly in continuous time, and derive the corresponding dynamic hedging strategy. In each case, we discuss the effectiveness and practicality of the hedges and provide numerical examples.

The other papers in this series present extensions of this work to measure contributions of systematic factors (analytically and numerically), as well as to obtain numerical methods for hedging credit portfolios in other measures (such as expected shortfall), for more complex portfolios and combinations of instruments and for discrete multi-period dynamic hedges.

The rest of the paper is organized as follows. The second section reviews the multi-factor Merton model underlying our results, and sets the notation to be used in the remainder of the paper. The third section presents results on optimal static hedging of portfolio losses with linear combinations of the systematic factors. The fourth section presents analogous results for the systematic credit default losses of CDOs. The fifth section discusses hedging

1The difficulty of actually implementing such a hedge will vary depending on the inter­pretation of the systematic factors in the multi-factor credit risk model. Two observations are pertinent in the event that the hedge does not correspond directly to a tradeable port­folio. First, it will often be possible to find a closely correlated tradeable portfolio that will produce a near optimal hedge. Second, the coefficients of the optimal linear hedge can still be used in assessing the contribution of each individual systematic factor to overall portfolio losses.

4

� �

of systematic losses in the dynamic case, where the weights in the linear portfolio of systematic factors can be rebalanced continuously in time. The sixth section presents conclusions, and discusses future work in this series. The proofs of the results presented in the main body of the paper are provided in an appendix.

2 Background

Consider a portfolio with N obligors. For each obligor, default events at the end of the horizon (say, one year) are described by a multi-factor Merton model. Obligor i defaults when a continuous random variable Yi, which describes its creditworthiness, falls below a given threshold. If we denote by PDi the obligor’s (unconditional) default probability and assume that the creditworthiness is standard normal, we can express the default threshold as Hi = Φ−1(PDi), where Φ is the standard cumulative normal distribution function. The creditworthiness of obligor i is driven by K factors through a linear model of the form:

K K

Yi = βikZk + σiεi σi 2 = 1 − β2 (1) ik

k=1 k=1

where the Zk are standard normal random variables representing the sys­tematic factors driving credit events, the βik are the “factor loadings” for the ith instrument (

�K βik < 1), and the εi are independent standard normal k=1

variables representing the idiosyncratic movement of an obligor’s creditwor­thiness. We further assume, without loss of generality, that the systematic factors are uncorrelated. 2

For ease of notation, assume that obligor i has a single loan with (percentage �Nof) exposure at default wi, = 1. Without loss of generality, we i=1 wi

assume that loss given default is 100%. The methodology and results apply in general to stochsatic, but independent losses given default (LGD). 3 Total

2This assumption is made to simplify the notation only. We can think of these factors, for example, as the principal components resulting from some correlated factors.

3There is a growing literature on the correlation of LGD and credit events (see, e.g. Altman et al. (2005)). The tools used in this paper can be generally applicable in this case. However, the analysis becomes substantially more complicated. Numerical methods can also be applied in this case, as will be discussed in future work.

5

�

� � �

� � � �

�

portfolio losses are given by the random variable:

N

Λ = wi1{Yi≤Hi)} (2) i=1

We refer to the case where βik ≡ βk, PDi ≡ PD, Hi ≡ H , σi ≡ σ as the case of a “homogeneous portfolio”. The systematic component of the portfolio losses corresponds to replacing each individual loan by an infinitely granular, homogeneous portfolio with the same probability of default and factor dependencies (see, e.g. Gordy (2003)). In this case, portfolio losses are:

N �K

σL = E[Λ|Z1, . . . , ZK ] = Φ

Hi − k=1 βikZk (3)

ii=1

It is well known (and easy to derive from Lemma (1) in the appendix) that E[L] = E[Λ] =

�N wiPDi. For future reference, we also need the variance i=1

of L, which is well known (the reader can calculate it easily using the results in the appendix) to be:

� � � �2N N K N

var(L) = wiwjΦ2 Hi, Hj; βikβjk − wiPDi (4) i=1 j=1 k=1 i=1

where Φ2(·, ·; ρ) is the bivariate cumulative normal distribution function with correlation ρ:

� z2 2 2

Φ2(z1, z2; ρ) = z1 �

exp

� (w1 − 2ρw1w2 + w2) dw1 dw2

� (5) − 2(1 − ρ2) 2π 1 − ρ2 −∞ −∞

In the case of a homogeneous portfolio, this reduces to:

var(L) = Φ2(H, H, 1 − σ2) − PD2 (6)

The idiosyncractic losses of the portfolio may be defined as LI = Λ − L. Notice that E[LI ] = 0, as in general idiosyncractic effects could either increase or decrease portfolio losses. The variance of the full portfolio (including both

6

�

systematic and idiosyncratic losses) can be calculated to be:

N �

var(Λ) = i=1

w 2 i PDi(1 − PDi)

� K

� � �

+ 2 wiwj Φ2(Hi, Hj; βikβjk) − PDiPDj

i<j k=1

The systematic factors Zk can only be employed in hedging the variance of the systematic losses, given by (4). Similarly, when measuring factor contributions, it is logical to measure constributions of factors to systematic losses. 4 The remainder of this paper focuses on factor contributions and hedging of this risk.

3 Static Hedging of Systematic Risk for Credit

Portfolios

We now study the problem of hedging the systematic risk of a portfolio using a linear portfolio of the systematic factors and a position in a risk-free bond. We begin by presenting a general theorem which gives the solution to the least-squares hedging problem. We then apply this result to derive optimal hedges for homogeneous and inhomogeneous credit portfolios, studying the quality of the hedge in each case.

We are interested in the problem of finding the “portfolio” of factor weights (together with a risk-free bond) that best hedges a random variable Y which may represent the loss of the entire portfolio, or some derivative on it (e.g. a CDO tranche). Thus, we are interested in the problem:

K

min bkZk�p (7) a,b

�Y − a − k=1

The natural norm to take is the two norm, which leads to the analytically

4These estimates of factor contributions to systematic losses could be combined with an assessment of the overall contributions of systematic and idiosyncratic risk, if the contributions of the factors to the total portfolio losses were required.

7

� �

�

tractable least-squares problem. 5 The main tool that we will use is the fol­lowing theorem.

Theorem 1. Let Y be a random variable with finite variance and let Z1, . . . , ZK

be uncorrelated random variables with mean 0 and variance 1. Then the op­timal value of the minimization problem (7) with p = 2 is:

K K

min E[(Y − a − bkZk)2] = E[Y 2] − (a∗)2

k)2 (8)

a,b − (b∗

k=1 k=1

K

= var(Y ) − cov(Zk, Y )2

k=1

where the optimal solution is:

a∗ = E[Y ]

b∗ = E[ZkY ] k = 1, . . . , K k

This is a direct consequence of the projection theorem for Hilbert spaces. See, for example, Rudin (1987) Theorem 4.14. A more detailed study of the Least-Squares problem (including a discussion of the case when the factors are not independent) is contained in Chapter 4 of Luenberger (1969).

The coefficient of variation (R2) of the regression is defined to be one minus the ratio of the optimal value to the variance of Y , and is given by:

�K k=1 cov(Zk, Y )2

R2 = (9) var(Y )

It may be taken as a simple measure of the quality of the linear approximation of Y by a∗ +

�K b∗Zk.k=1 k

To conclude this section, we observe that the coefficients of the best linear approximation to Y are the same as those of the best linear approximation to Y = E[Y |Z]. 6 This has important consequences for applications in credit

5The only other p norm that appears to produce analytically tractable results is p = ∞, which is trivial, and not very useful, for most choices of Y .

6This is a direct consequence of the tower law and the properties of conditional expec­tation E[Y Zk] = E[E[Y Zk Z]] = E[ZkE[Y Z]]. | |

8

�

� �

�

risk, as it implies that whether we approximate the total portfolio loss, or only its systematic component, the optimal weights on the systematic factors in the best linear approximation will stay the same. (The quality of the approximation, for example as indicated by R2 will be different in these cases, see below.)

3.1 Homogeneous Portfolio

Consider the simplest case of a homogeneous portfolio with independent fac-tors. This means that the probabilities of default and factor loadings are the same for all obligors, i.e. PDi ≡ PD, Hi ≡ H, βik ≡ βk for i = 1, . . . , M . In this case the percentage portfolio systematic loss is simply given by the random variable:

� � βkZkk=1 L = Φ

H − �K

(10) σ

1 − �Kwhere σ = k=1 βk2 .

The loss distribution and moments of L are well known (see, e.g. Va­sicek (2002)). In particular, we have:

FL(x) = P(L ≤ x) = Φ σΦ−1(x) − H √

1 − σ2

µL = E[L] = PD

σ2 = Φ2(PD, PD, 1− σ2) − PD2 L

We wish to find the weights that yield the best linear approximation (in the least squares sense) to the portfolio loss variable. The solution to this problem is given by by applying the general theorem, leading to the following proposition.

Proposition 1. The least squares hedging problem for the homogeneous port­folio loss L has the optimal value:

⎡ ⎤ � �2K

min E ⎣ L − a − bkZk ⎦

a,b k=1

= Φ2(H, H, 1 − σ2) − PD2 − ϕ(H)2(1 − σ2) (11)

9

with the optimal value being attained by

a∗ = PD

b∗ = −βkϕ(H) k = 1, . . . , K k

Here ϕ is the standard normal probability density function. A proof of the proposition is given in the appendix. As noted in the general theorem in the previous section, the constant term a∗ (corresponding to a hedging position in cash or a risk-free bond) is simply the expectation of the random variable being approximated, in this case the portfolio loss with E[L] = PD. The positions in the factors (b∗

Y

k) hedge the variance of the systematic losses. The optimal position in each factor is its weight in the creditworthiness index

i (see equation (1)), scaled by the standard normal probability density at the default threshold H = Φ−1(PD). The optimal weights underscore the fact that the multi-factor model with a homogeneous portfolio is really just a single-factor model with factor:

�K βkZkk=1 Z = (12) √1 − σ2

The optimal hedge maintains the same relative weights on each factor as they appear in the aggregrate factor Z.

To see how the optimal weight in a given factor varies with the probability of default, Figure 1 shows the plot of b∗

βk against PD for a fixed value of

k = −0.5.

A simple calculation using equation (9) yields the following R2 coefficient for the least squares hedging problem for the homogeneous portfolio:

R2 ϕ(H)2 · (1 − σ2) =

Φ2(H, H, 1 − σ2) − PD2 (13)

A surface plot of the R2 against different values of PD and the idiosyncratic risk σ is given in Figure 2.

Notice that for the low probabilities of default often of interest in applica­tions, the value of R2 may be quite good, and the linear approximation does provide a good fit to the portfolio loss random variable. For example, for a

10

Opt

imal

Fac

tor

Coe

ffici

ent

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Probability of Default

Figure 1: Optimal Coefficients for Least Squares Hedge of a Homogeneous Portfolio

11

1

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

SigmaProbability of Default

R S

qu

are

d

Figure 2: Values of the R Squared Coefficient for the Homogeneous Portfoliofor Different Probabilities of Default and Sigmas

12

0.8

0.85

0.9

0.95

1

0

0.05

0.10

0.2

0.4

0.6

0.8

1

SigmaProbability of Default

R S

qu

are

d

Figure 3: Values of the R Squared Coefficient for the Homogeneous Portfoliofor Different Probabilities of Default and Sigmas

large homogeneous portfolio of mortgages with asset correlations β = 0.15and default probability 2%, the value is R2 = 0.9539. Figure 3 gives a closeupof the surface plotted in Figure 2 over the range of parameter values thatis most financially relevant. Observe that the value of R2 is increasing inσ. At first this may seem counterintuitive, as σ = 1 represents the case ofpurely idiosyncratic risk. However, one must remember that we are con-sidering only the systematic component of portfolio losses, and in this casethat component is constant (indeed, one may observe that as σ → 1, all theβk → 0 and therefore the optimal hedging coefficients b∗k → 0, but a∗ = PD).Furthermore, when σ = 1, the loss is independent of the factor values, andtherefore the systematic component of the loss (= L = E[Λ|Z]) is simply theexpected loss, equal to PD.

13

As noted above, the optimal coefficients in the approximation are the same whether we consider total losses Λ =

�N i=1 wi1{Yi≤Hi)} or systematic

losses L = E[Λ|Z], however the quality of the fit R2 can change as a function of N .

It is easy to show that for the total loss variable Λ the R2 of a completely homogeneous portfolio is:

R2 ϕ(H)2 · (1 − σ2) = 1(Φ2(H, H, 1 − σ2) − PD2) +

N (PD − Φ2(H, H, 1 − σ2)

(14)

The relation to (13) is clear. We illustrate this using a homogeneous port­folio. All N loans are taken to have a default probability of 0.01 and a factor correlation β = 0.16. The R2 coefficients for the approximation of systematic and total risk as a function of the number of instruments in the homogeneous portfolio are compared in Figure 4. Observe that the fit is always worse for the total losses7, and that as the number of instruments in the portfolio increases, the total losses approach the systematic losses, and the R2 coefficients converge.

As mentioned earlier, in the case of a homogeneous portfolio, as the number of loans N increases, systematic risk dominates and Λ (which implicitly depends on N) converges to L. Furthermore, as the value of β increases (with N held fixed), systematic risk becomes more important. Thus, we would expect linear portfolios of systematic factors to be most effective in explaining total portfolio loss variance when N and β are both large. This fact is evidenced by Figure 5, which gives values of the R2 of the fit for a completely homogeneous portfolio with PD = 0.01, with different values for β and N .

Observe in particular from Figures 3 and 5 that when β is near zero, the R2

of the fit to the systematic losses L is very good (nearly one) but the R2 of the fit to the total losses Λ is nearly zero. In this case, the systematic factors are nearly irrelevant, and the high R2 for systematic risk stems from the con­stant term providing a good approximation to the systematic portfolio losses. However, in this case systematic losses are far from the total losses (which are almost entirely driven by idiosyncratic risk), and the contribution of the systematic factors to total portfolio loss variance is negligible, producing the low R2 when Λ is considered the loss variable to be hedged.

7This is a consequence of the factor that var(Λ) = var(E[Λ|Z])+E[var(Λ Z)] = var(L)+ |E[var(Λ Z)] ≥ var(L). |

14

0 2000 4000 6000 8000 10000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R S

quar

ed

Systemic Losses Total Losses

Number of Loans

Figure 4: Values of the R Squared Coefficient for Systematic and Total Risk of a Homogeneous Portfolio

15

00.05

0.10.15

0.2

0

5000

100000

0.2

0.4

0.6

0.8

1

BetaNumber of Loans

R S

quar

ed

Figure 5: Values of the R Squared Coefficient for Total Losses of a Homoge-neous Portfolio for Different Portfolio Sizes and Factor Loadings.

16

� � �

�

� � �

�

�

�

3.2 Inhomogeneous Portfolio

The systematic portfolio losses for an inhomogeneous portfolio are given by:

N

k=1 βikZkL = wi · Φ

σ˜ Hi −

�K

(15) ii=1

where wi is the weight of the portfolio invested in counterparty (or a ho­mogeneous sector) i, i = 1, . . . , N . In the case of a single systematic factor Z, Gordy (2003) presents analytical formula for portfolio losses L, extending the Vasicek (2002) results. However, there is no closed-form solution for the loss distribution in the multi-factor case.

Linearity in equation (15) allows us to obtain the optimal portfolio weights from the homogeneous case. The following proposition gives the optimal linear hedge.

Proposition 2. The least squares hedging problem for the inhomogeneous portfolio loss L in equation (15) has the optimal value:

⎡ ⎤ � �2K

min E ⎣ ˜ ⎦L − a − bkZk a,b

k=1

N K

= wi 2 ri + 2 wiwjrij − (a∗)2

k)2 (16) − (b∗

i=1 i<j k=1

where

ri = Φ2(Hi, Hi, 1 − σi 2)

K

rij = Φ2(Hi, Hj , βikβjk) k=1

the optimal value being attained by

N

a∗ = wiPDi

i=1

N

b∗ = wiβikϕ(Hi) k = 1, . . . , K k −i=1

17

�

�

� �

A proof of the proposition is given in the appendix.

Once again, the value of the constant term (position in cash or a risk-free ˜bond) is given by the expected portfolio loss E[L] =

�N wiPDi. The i=1

optimal hedging coefficients have a very simple form as in the previous case. If we define ai, b

i k to be the optimal hedging weights for a homogeneous

portfolio containing only loan i, then we have that

N

a∗ = wiai, (17) i=1

N

b∗ = wibi

k k

i=1

That is, the optimal weights for the inhomogeneous portfolio are simply linear combinations of the optimal weights for each loan, weighted according to their contribution to the portfolio. In particular, the contribution of each loan to the hedging portfolio is portfolio invariant in the sense of Gordy (2003), in that it only depends on the characteristics of the loan (or sector) itself, and not the other instruments in the portfolio.

A simple application of the general formula for R2, equation (9), together with the known value of the portfolio variance (4) gives the R2 for the re­gression:

� �2

R

�Kk=1

2 = �

�N i=1 wiβikϕ(Hi)

� � �2 (18) �N �N �N

i=1 j=1 wiwjΦ2 Hi, Hj ; �K βikβjk − wiPDik=1 i=1

To analyze the impact of portfolio homogeneity on the quality of the regres­sion fit, consider a two-factor model with two loans of equal credit quality in the portfolio, and the following factor weights.

β11 = √

1 − σ2 β12 = 0

β21 = (1 − σ2)λ β22 = (1 − σ2)(1 − λ), λ ∈ [0, 1]

Thus the first loan is entirely dependent on the first factor, while the impact of the first and second factors on the second loan depends on the value of the

18

� � ��

| � |

parameter λ. For example, λ = 1 implies a homogeneous portfolio, which depends entirely on the first factor; while λ = 0 denotes two loans which are independent.

In this case, the formula for the R2 of the fit reduces to:

1ϕ2(H)(1 − σ2)(1 + √

λ)2R2 =

1 2(Φ2(H, H, 1 − σ2) + Φ2(H, H,

√λ(1 − σ2))) − PD2

Figure 6 gives the value of the R2 coefficient for different values of λ, with a fixed default probability of PD = 0.01, and idiosyncratic weight σ2 = 0.7. The shape of the curve is quite interesting. First, we observe that the value of the R2 is relatively insensitive to the value of the parameter λ (ranging between 0.47 and 0.52). Second, we note that the best fit is achieved neither at a completely homogeneous portfolio, nor at a portfolio with the maximum granularity, but rather near the intermediate value of λ = 0.2.

4 Static Hedging of a CDO Tranche

In the homogeneous case, we can also obtain an analytical solution for the best linear portfolio of factors to approximate the losses of a CDO tranche. Denote the loss, over a fixed horizon, of the equity tranche8 with upper attachment point R by

�K βkZkk=1 LR = min(R, L) = min R, Φ H −

(19) σ

Then we have the following result, whose proof is given in the appendix.

8We note that the above equation gives a tranche on only the systematic losses of the portfolio. The true portfolio tranche loss is given by ΛR = min(R, Λ). The equal­ity of optimal solutions observed in the portfolio loss case does not hold here since E[min(R, Λ) Z] = min(R, E[Λ Z]). The systematic tranche loss LR will be a good ap­proximation for the true CDO tranche loss only for large homogeneous portfolios (e.g. retail loan portfolios).

19

0.46

0.47

0.48

0.49

0.51

0.52

0.53

R S

qu

are

d 0.5

0 0.2 0.4 0.6 0.8

Lambda

Figure 6: Values of the R Squared Coefficient for Different Values of the Homogeneity Parameter λ

20

1

� �

� �

� �

� � � �

� �

� � � �

� �

Proposition 3. The least squares hedging problem for the tranche LR has the optimal value:

⎡ ⎤ � �2K K

min E ⎣ LR − a − bkZk ⎦ = q∗ − (a∗)2

k)2 (20)

a,b − (b∗

k=1 k=1

where:

q∗ = P X1 ≤ σΦ−1(R) − H, X2 ≤ H, X3 ≤ H√

1 − σ2

+ R2Φ H − σΦ−1(R) √

1 − σ2

with (X1, X2, X3) jointly normally distributed variables with mean zero and variance-covariance matrix:

⎛ ⎞ 1

Σ = ⎝−√

1 − σ2

−√

1

1

− σ2 −√

1 − σ2

⎠

−√

1 − σ2 1 − σ2

1 −1

σ2

In this case, the optimal value is attained by

a∗ = Φ2 σΦ−1(R) − H

, H ;−√

1 − σ2 + RΦ √1 − σ2

√1 − σ2

H − σΦ−1(R)

b∗ = −βkϕ(H) Φ Φ−1(R) − σH

k = 1, . . . , K k · √1 − σ2

The formula for b∗ k is very similar to the one for the optimal hedge of the entire portfolio, differing only by a multiplicative factor. At first glance, this formula bears a striking resemblance to the probability that the systematic loss will be less than the attachment point:

�K � � H − k=1 βkZk ≤ R

σΦ−1(R) − H P Φ = Φ (21)

σ √

1 − σ2

σΦ−1(R) − Φ−1(PD) = Φ √

1 − σ2

21

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.40

0.2

0.4

0.6

0.8

1

Attachment PointDefault Probability

R S

qu

are

d

Figure 7: Values of the R Squared Coefficient for Hedging a CDO Tranche

The multiplicative factor in the expression for b∗ is the same expression,with the roles of R and PD reversed. That is, it is the probability thata homogeneous portfolio with probability of default R will have systematiclosses less than PD.

The R2 of the fit can be calculated based on the general equation (9). Valuesfor various default probabilities and attachment points (with a fixed σ2 = 0.7)are given in Figure 7. In particular, we find a good fit near the diagonalR = PD, and generally a better fit for lower probabilities and attachmentpoints.

The tranche with upper attachment point R and lower attachment point Rcan also be solved analytically since its loss is simply given by the difference

22

� � � �

�

�

� � � �

�

of the equity tranches L ¯ The case of a CDO tranche on an inho-R − LR. mogeneous portfolio does not appear to be analytically tractable. We study this problem using mathematical programming techniques in a later paper in this series.

5 The Dynamic Case

As shown in the previous sections, static linear portfolios involving the factors might not provide sufficient hedges of portfolio losses. In this section, we briefly discuss how the above single step model can be embedded in a simple, continuous time model that achieves perfect hedging (i.e. completes the market) using a portfolio of the factors and the risk-free bond. For simplicity, we assume that interest rates are zero. Let the factor processes satisfy:

W k tZk = Z0

k + k = 1, . . . , K (22) t √T

where W is a standard K dimensional Brownian motion, and T is the time horizon. Note that the (discounted) factor processes are martingales, and so we obtain the portfolio value in the homogeneous case to be:

H − �K βkZk

f(t, z) = E Φ k=1 T |Zt = z (23) σ

Now using the fact that:

Zk = Zk + 1 − t WTk − Wt

k

(24) T t T · √

T − t

Zk = Zk + 1 − t t t,T T ·

Zkwhere ˜t,T , k = 1, . . . , K are i.i.d. standard normal random variables, we get:

�K βkZk Tf(t, z) = E Φ

H − k=1 |Zt = z (25) σ

⎡ ⎛ ⎞⎤ �K t �K βkZ

k k=1 βkzk k=1 t,T )

= E ⎣Φ⎝ H − +

1 −T

√1 − σ2

(− ⎠⎦

σ σ · √

1 − σ2

23

Using Lemma 1 from the appendix, we then obtain:

⎛ ⎞ �K βkzkk=1 f(t, z) = Φ⎝ H −�

⎠ (26) T−tσ 1 + σ2t

In order to obtain the hedging portfolio, we compute the partial derivative

⎛ ⎞ �K βkzkk=1 fzk

(t, z) = ∂f(t, z)

= �−βk

ϕ⎝ H −�

⎠ (27) ∂zk T−t T−tσ 1 +

σ2t σ 1 +

σ2t

The hedging portfolio then consists of holding fzk(t, Zt) in the kth factor at

time t, and f(t, Zt) − �K fzk(t, Zt) in the risk-free bond. k=1

The case of an inhomogeneous portfolio is a straightforward extension using linearity. Furthermore, one can derive the hedging weights for the factors for a CDO tranche on a homogeneous portfolio in the continuous time model. The derivations follow exactly the same lines, applying Lemma 6 to compute the price and then differentiating in order to determine the optimal hedging weights.

6 Conclusion and Future Work

We present analytical results for hedging portfolio credit risk with linear portfolios of the systematic factors. Formally, for a default-only multi-factor Merton-type (or probit) credit portfolio model, we minimize the systematic variance of portfolio losses with linear combinations of the systematic risk factors. The mathematical tools and results can be extended to other multi­factor portfolio models, within a conditional independence framework (such as the logit model). We review the mathematical tools to solve these opti­mization problems and then apply them to various cases: static hedges of homogeneous and inhomogeneous credit portfolios and CDOs, as well as dy­namic hedging in continuous time. In each case, we discuss the effectiveness of the hedges and provide numerical examples. We observe that, in practice, the hedging effectiveness of purely homogeneous portfolios with simple lin­ear factor portfolios can be low. We study the increase in effectiveness as portfolios increase in size and factor loadings increase. We also note that

24

if one is allowed to trade in continuous time, the portfolio systematic losses over the horizon can be replicated fully, and derive the replicating portfolio in this case.

There are various practical extensions of this work which are explored in other papers in this series:

• First, from a capital management perspective, the allocation of risk contributions to the systematic factors can provide a useful tool for understanding the structure of the portfolio and managing concentra­tion risk.

• Second, we have explored in this paper only linear combinations of the factors. Hedge effectiveness may be improved through the use of a linear combination of non-linear functions of individual factors (for example, we could use a portfolio CDS indices or defaultable bonds, each depending on a single factor).

• Third, this paper uses variance (or standard deviation) as a measure of risk, mainly due to its widespread use and analytical tractability. However, credit loss distributions are far from normal and variance can be a misleading measure of risk in this context. At the cost of analytic tractability, the present work can be readily extended to other measures which capture the risk in the tail of the loss distribution (such as expected shortfall). This generally requires the application of techniques from mathematical programming.

• Fourth, we can develop dynamic trading strategies in discrete time using the tools developed in stochastic programming.

References

[1] E. Altman, A. Resti, and A. Sironi, eds., Recovery Risk: The Next Challenge in Credit Risk Management, Risk Books, 2005.

[2] Credite Suisse Financial Products, CreditRisk+: A Credit Risk Management Framework, 1997.

[3] P. Crosbie, Modeling default risk, tech. report, KMV Corporation, 1999.

25

[4] M. Gordy, A risk-factor model foundation for ratings-based bank cap­ital rules, Journal of Financial Intermediation, 12 (2003), pp. 199–232.

[5] C. Gourieroux, J. Laurent, and O. Scaillet, Sensitivity analysis of values at risk, Journal of Empirical Finance, 7 (2000), pp. 225–245.

[6] G. Gupton, C. Finger, and M. Bhatia, CreditMetrics technical document, tech. report, J.P. Morgan & Co., 1997.

[7] M. Kalkbrener, H. Lotter, and L. Overbeck, Sensible and ef­ficient capital allocation for credit portfolios, Risk, (2004), pp. 19–24.

[8] A. Kreinin and A. Nagy, Calibration of the default probability model. To appear, European Journal of Operational Research.

[9] D. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, 1969.

[10] H. Mausser and D. Rosen, Economic credit capital allocation and risk contributions, in Handbook of Financial Engineering, V. Linetsky and J. Birge, eds. Forthcoming.

[11] M. Pykhtin, Multi-factor adjustment risk, Risk, (2004), pp. 85–90.

[12] D. Rosen and D. Saunders, Measuring capital contributions of sys­temic factors in credit portfolios. Working Paper, Fields Institute of Mathematical Research, 2006.

[13] W. Rudin, Real and Complex Analysis, McGraw-Hill, third edition ed., 1987.

[14] D. Tasche, Conditional expectation as a qunatile derivative. Working Paper, Technische Universit¨ unchen, 2000. at M¨

[15] , Expected shortfall and beyond. Working Paper, Technische Uni-versit¨ unchen, 2002. at M¨

[16] O. Vasicek, Loan portfolio value, Risk, (2002), pp. 160–162.

26

� �

|� �

� �

� �

A Proofs of Results

To prove Proposition 1 we begin with a series of lemmas. The first was taken from Kreinin and Nagy (2005).

Lemma 1. Let Z be a standard normal random variable, a, b ∈ R. Then

a E [Φ (a + bZ)] = Φ (28) √

1 + b2

Proof. Letting X be a standard normal random variable independent of Z we have:

E[Φ(a + bZ)] = E[E[1{X≤a+bZ} Z]] (29)

a = E[1{X−bZ≤a}] = Φ √

1 + b2

using the tower law and the fact that X − bZ ∼ N(0, √

1 + b2).

The next lemma employs a similar technique to compute the second moment of the random variable.

Lemma 2. Let Z be a standard normal random variable, a, b ∈ R. Then

� � a E (Φ(a + bZ))2 = Φ2

a, ;

b2

(30) √1 + b2

√1 + b2 1 + b2

where Φ2(·, ·; ρ) is the cumulative bivariate normal distribution with correla­tion ρ.

Proof. Let X1, X2 be standard normal random variables such that X1, X2, Z are independent.

E[Φ(a + bZ) · Φ(a + bZ)] = E[E[1{X1≤a+bZ} 1{X2≤a+bZ} Z]] (31) · |= E[1 X1−bZ a 1 X1−bZ 1 { √

1+b2 ≤√

1+b2 } · {√

1+b2 ≤√

1+b2 }]

a a = Φ2 , , ρ √

1 + b2 √

1 + b2

where ρ is the correlation between (X1−bZ)/√

1 + b2 and (X2−bZ)/√

1 + b2

which is easily seen to be b2/(1 + b2).

27

�

� � � �

� �

The following lemma serves to simplify many calculations, particularly in the homogeneous portfolio case. Its proof is an elementary integration by parts, using that ϕ�(z) = −zϕ(z), and is therefore omitted.

Lemma 3. Suppose that f : RK R is piecewise continuously differentiable9 →

in its jth variable, and that Z1, . . . , ZK are i.i.d. standard normal random variables. Then:

E[Zjf(Z1, . . . , ZK)] = E[fj(Z1, . . . , ZK)] (32)

where the subscript denotes partial differentiation.

We need one final lemma before we can prove the proposition for the loss distribution of the homogeneous portfolio.

Lemma 4. Suppose Z is a standard normal random variable, u, v ∈ R. Then:

� � 2uexp

1+2v2

E[exp(−(u + vZ)2)] = −

(33) 2

√1 + 2v

Proof. The result follows simply by completing the square in the expectation.

dz 2E[exp(−(u + vZ)2)] =

∞ exp(−(u + vz)2) exp(−z /2) (34) √

2π−∞ � �

1 2uv dz = exp

−u2 ∞ exp

√1 + 2v2 z + √

1 + 2v

�2

1 + 2v2 −

2 ·

2 √

2π−∞ 2u

�exp 1+2v2 ∞

e−w2/2 dw −=

2√

1 + 2v· √

2π � � −∞

2uexp 1+2v2−

= 2

√1 + 2v

2uv After the change of variables w = √

1 + 2v2z + .2√

1+2v

9We note in passing that this condition is sufficient, but by no means necessary, for this result to hold.

28

� � ��

� � ��

� � ��

� �

� �

We are now ready to give the following:

PROOF OF PROPOSITION 1: We begin by deriving the formulas for a∗, b∗ k. From Theorem 1 we have:

H − �K βkZk

a∗ = E Φ k=1 (35) σ

�KH √

1 − σ2 βkZkk=1 = E Φ σ

− σ

· √1 − σ2

= PD

Hwhere the last line follows by applying Lemma (1) with a = σ , b =

√1−σ2

σ PK

k=1−and Z = βkZk ∼ N(0, 1), and then an elementary simplification. √

1−σ2

Next, we consider the optimal factor coefficients b∗ j . From Theorem 1 we have:

βkZkk=1 b∗ j = E ZjΦ H −

�K

(36) σ

= E[Zjf(Z1, . . . , ZK)]

Where

H − �K βkzkf(z1, . . . , zK) = Φ( k=1

σ

βj βkzkk=1 fj(z1, . . . , zK) = −σ

ϕ H − �

σ

K

with the subscript denoting partial differentiation. Thus using Lemma 3: ⎡ ⎛ ⎞⎤

� �2 �Kβj 1 H

√1 − σ2 βkZkk=1 b∗ j = E ⎣exp ⎝−

2 σ −

σ · √

1 − σ2 ⎠⎦ (37) −

σ√

2π

and the result follows by applying Lemma 4 with u = H = √

1−σ2 and

σ√

2, v

σ√

2 PK

k=1−Z = βkZk N(0, 1), and then performing a straightforward simplifi-√

1−σ2 ∼cation.

29

� �

� �

= �

� �

The expression for the optimal value follows immediately from Theorem 1 and Lemma 2. The proof of proposition 1 is complete.

For the proof of proposition 2, we need the following lemma.

ˆLemma 5. Let Z and Z be two standard normal random variables with correlation ρ, a1, a2, b1, b2 ∈ R. Then:

E[Φ(a1 + b1Z) · Φ(a2 + b2Z)]

a1 a2 b1b2ρ = Φ2 � ,� ; � (38)

1 + b2 1 + b2 (1 + b22)1)(1 + b2

1 2

Proof. As before, let X, Y be standard normal random variables, independent of each other and of Z, Z. Then:

ˆE[Φ(a1 + b1Z) · Φ(a2 + b2Z)] = E[E[1X≤a1+b1 ˜1Y ≤a2+b2Z Z1, Z2]] (39) Z · ˆ|

= E[1 X−b1Z ˆ a2 ]˜ a1

1 Y −b2Z·√1+b2

≤√1+b2

√1+b2

≤√1+b2 1 1 2 2

a1 a2 = Φ2 � ,� ; ρ

1 + b2 1 + b2 1 2

where

ˆE[(X − b1Z)(Y − b2Z)]ρ = �

(1 + b22)1)(1 + b2

b1b2ρ

(1 + b22)1)(1 + b2

PROOF OF PROPOSITION 2: The formulas for the optimal coefficients follow immediately from Theorem 1, linearity and the proof of Proposition 1.

˜Only the formula for the optimal value, and in particular for E[L2] requires any further work. We have:

⎡ ⎤ � � ��2

˜� N

k=1 βikZk

σE[L2] = E ⎣ wiΦ

Hi − �K

⎦ (40) ii=1

N

= wi 2 ri + 2 wiwjrij

i=1 i<j

30

�

� �

� �

� � � �

� �

� � � �

where⎡ ⎤ � �2

k=1 βikZkHi − �K

σri = E ⎣Φ ⎦

i

= Φ2(Hi, Hi; 1 − σi 2)

� � � � �� �K �K

σrij = E Φ

Hi − k=1 βikZk Φ

Hj − k=1 βjkZk

i

· σj

K

= Φ2(Hi, Hj), βikβjk) k=1

√1−σ2

iThe result for ri follows by applying Lemma 2 with a = Hi , b = σi

and σi

PK

Z = − k=1 βikZk N(0, 1). The result for rij follows similarly by applying √

1−σ2 ∼

i

Lemma 5 with a1 = Hi/σi, b1 = 1 − σi 2/σi, a2 = Hj/σj , b2 = 1 − σj

2/σj , �KZ = − k=1 βikZk/ 1 − σi

2 , Z = �Kˆ

k=1 βjkZk/ 1 − σ2 .j−

In order to prove proposition 3, we again need some preliminary lemmas.

Lemma 6. Let Z be a standard normal random variable, a ∈ R, b > 0, R ∈ (0, 1). Then:

E[min(R, Φ(a + bZ))]

Φ−1(R) − a a a − Φ−1(R) = Φ2 , ;

−b + RΦ (41)

b √

1 + b2 √

1 + b2 b

Proof. Again, the result is achieved by considering a standard normal random variable X independent of Z and using the tower law:

E[min(R, Φ(a + bZ))] = E[1Φ(a+bZ)≤RΦ(a + bZ)] + RE[1Φ(a+bZ)≥R] (42)

= E[E[1 1X≤a+bZ Z]] + RE[1 Φ−1(R)−a ]Z≤ Φ−1(

b

R)−a · |Z≥

b

a − Φ−1(R) = E[1

Z≤ Φ−1(b

R)−a 1√1+b2

≤√1+b2

] + RΦX−bZ a· b

Φ−1(R) − a a a − Φ−1(R) = Φ2 , ;

−b + RΦ

b √

1 + b2 √

1 + b2 b

since corr(Z, (X − bZ)/√

1 + b2) = −b/√

1 + b2 .

31

� �

� �

� �

� �

� �

� �

Lemma 7. Let Z be a standard normal random variable, a ∈ R, b > 0, R ∈ (0, 1). Then:

E[(min(R, Φ(a + bZ)))2]

a a = P X1 ≤ Φ

−1(R) − a, X2 ≤ , X3 ≤

b √

1 + b2 √

1 + b2

+ R2Φ a − Φ−1(R)

(43) b

where (X1, X2, X3) are jointly normally distributed with mean zero and variance­covariance matrix:

⎛ 1 b b

⎞ √

1+b2−√

1+b2 −

b2 ⎜ b ⎟

Σ = ⎝

−√1+b2

1 1+b2 ⎠ (44)

b b2 √1+b2 1+b2

1−

Proof. Let X and Y be standard normal random variables independent of Z and each other. Then:

E[(min(R, Φ(a + bZ)))2]

= E[1Φ(a+bZ)≤RΦ(a + bZ)2] + R2E[1Φ(a+bZ)≥R] (45)

= E[1Φ(a+bZ)≤RΦ(a + bZ)2] + R2Φ a − Φ−1(R)

b

But

E[1Φ(a+bZ)≤RΦ(a + bZ)2]

= E[E[1 1X≤a+bZ · 1Y ≤a+bZ Z]] (46)

Z≤ Φ−1(b

R)−a · |

= E 1 Φ−1(R)−a 1 X−bZ a 1 Y −bZ a Z≤

b √

1+b2 ≤√

1+b2 · √

1+b2 ≤√

1+b2

and the result follows by computing the appropriate correlations.

Lemma 8. Let Z be a standard normal random variable, a ∈ R, b > 0, R ∈ (0, 1). Then:

E 1Φ(a−bZ)≤R · ϕ(a − bz) 2−a � � �� exp

2(1+b2) 1 a = �

2π(1 + b2) · Φ Φ−1(R)

√1 + b2 − √

1 + b2 (47)

b

32

� �

�

� � �

� � �

= �

� �

� �

Proof. The proof is accomplished by completing the square in the integral.

E 1Φ(a−bZ)≤R · ϕ(a − bz) (48) ∞

= ϕ(a − bz)ϕ(z) dz a−Φ−1(R)

b

1 ∞ 1 � 2 2�

= exp −2

(1 + b2)z − 2abz + a dz a−Φ−1(R)

b2π

2−a� �

�exp 2(1+b2)

∞ 1 ab = exp

√1 + b2 · z − √

1 + b2

�2

dz a−Φ−1(R)2π

−2

b

2−a�exp

2(1+b2) ∞

e−w2/2 dw “ ”

a−Φ−1(R) ab 2π(1 + b2) √1+b2

b −√

1+b2

√2π

2−a � � �� exp 2(1+b2) 1 a

= � 2π(1 + b2)

· Φ Φ−1(R)√

1 + b2 − √1 + b2b

ab after the change of variable w = √

1 + b2 · z − √1+b2

.

We are now ready to prove proposition 3.

PROOF OF PROPOSITION 3:

a

We begin by computing a∗. By Theorem 1:

∗ = E[LR] (49)� � � ���

H − �K βkZk

= E min R, Φ k=1

σ � � � ���

�KH √

1 − σ2 βkZk = E min R, Φ k=1

σ −

σ · √

1 − σ2

Hand the result follows by applying Lemma 6 with a = σ , b =

√1−σ2

and σ

PKk=1−Z =

βkZk .√1−σ2

33

� � ��

We now compute the optimal factor coefficients b∗ j . By Theorem 1 and Lemma 3:

� � � ��� �K βkZkk=1 b∗ j = E Zj min R, Φ

H − (50)

σ �K

− βσ j

1 Φ

„

H−

PK « ϕ

T − k=1 βkZk = E βkZkk=1

σ ≤R

· σ

Hand the result now follows by applying Lemma 8 with a = σ , b =

√1−σ2

and σ

PKk=1Z =

βkZk , and simplifying. √1−σ2

Finally, the formula for the optimal value follows from Theorem 1 and (to compute the first term) applying Lemma 7 with the obvious choices of a, b. The proof of the proposition is complete.

34