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Research Series
Factor Contributions and Hedging of Systemic Risk inMulti-Factor Credit Portfolio Models
Dan Rosen1 and David Saunders2
Multi-factor credit portfolio models are used widely today for measuringand managing economic capital as well as for pricing credit portfolio instru-ments such as collateralized debt obligations (CDOs). Commonly, practitionersallocate capital to the portfolio components, such as individual sub-portfolios,counterparties, or transactions. The hedging of credit risk is generally also fo-cused on the ”deltas” of the underlying names in the portfolio. Understandingthe contribution to economic capital or pricing of the various systemic factors(or credit drivers), which are at the heart of a multi-factor credit model, canlead to better methodologies for managing concentration risk and hedging creditportfolios effectively. This requires first decomposing the credit risk in a portfo-lio into its systemic and idiosyncratic components. The underlying multi-factormodel drives the correlations of obligor credit events and determines entirelythe systemic risk of the portfolio. In addition, the standard theory of marginalcapital contributions does not work well since the total capital is not a homoge-neous function of these factors. Finally, the most interesting cases, in practice,might require simulation of the multi-factor models.
This series of research papers presents several methodologies to explore thecontributions of systemic credit factors to economic capital and hedging of sys-temic risk in credit portfolios:
• The first paper of the series presents analytical results for hedging portfoliocredit risk with linear portfolios of the systemic factors. We focus on statichedges for the variance of credit losses of homogeneous, inhomogeneousportfolios and CDOs, as well as dynamic hedging in continuous time.
• The second paper presents extensions of this work to measure contribu-tions of systemic factors (analytically and numerically). We also showthe relationship between risk contributions and the hedging problem, andsuggest alternative functions of the risk factors for hedging the portfolio.
• The third paper obtains numerical methods for hedging credit portfoliosin other measures (such as expected shortfall and regret) as well as morecomplex portfolios and combination of instruments.
• The fourth paper focuses on discrete dynamic hedges using stochasticprogramming techniques.
1Fields Institute for Research in Mathematical Sciences, Toronto, [email protected]
2University of Waterloo, Waterloo, Canada. [email protected]
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Analytical Methods for Hedging Systemic Credit
Risk with Linear Factor Portfolios
Dan Rosen David Saunders
March 5, 2006
Abstract
This series of research papers present a series of methodologies to ex-plore the contributions of systemic factors to economic capital as well asthe hedging of systemic risk in credit portfolios. Multi-factor credit port-folio models are used widely today for measuring and managing economiccapital as well as for pricing credit portfolio instruments such as collater-alized debt obligations (CDOs). Commonly, practitioners allocate capitalto the portfolio components, such as individual sub-portfolios, counterpar-ties, or transactions. The hedging of credit risk is generally also focusedon the ”deltas” of the underlying names in the portfolio. In this paper,we present analytical results for hedging portfolio credit risk with lin-ear portfolios of the systemic credit factors. Formally, we minimize thesystemic variance of portfolio losses by using a linear combination of thesystemic risk factors. We develop the mathematical tools to solve these op-timization problems within a multi-factor Merton-type (or probit) creditportfolio model, and then apply it to various cases. First, we focus onstatic hedges of homogeneous and inhomogeneous credit portfolios. Wealso apply the methodology to hedge the systemic credit default losses ofCDOs. Finally we show the application of the methodology to dynamichedging strategies. In each case, we discuss the hedging portfolios, theeffectiveness of the hedges and provide numerical examples.
2
1 Introduction
Multi-factor credit portfolio models are used widely for measuring and managingeconomic capital as well as pricing credit portfolio instruments such as collater-alized debt obligations (CDOs). In particular, these models provide a naturalframework for analyzing diversification (or alternatively concentration risk) incredit portfolios, one of the key tools for managing credit risk and optimallyallocating credit capital. Thus, many institutions today have in productioneither internally developed or commercial multi-factor credit portfolio modelsto manage their credit risk (e.g. Gupton et al. 1997, Credit Suisse FinancialProducts 1997, Crosbie 1999). Generally, these models share an underlyingmathematical framework, which is referred to as the conditional independenceframework. Multi-factor credit portfolio models entail the use of Monte-Carlo(MC) simulation, although several analytical and semi-analytical methods havebeen developed in recent years (see Pykhtin 2004, and Gracia et al. 2005).
In addition to pricing instruments and measuring capital, multi-factor creditportfolio models can provide useful guidance regarding the allocation of capitaland the hedging of credit portfolios. Commonly, practitioners allocate capital tothe portfolio components, such as individual sub-portfolios, counterparties, ortransactions, using marginal risk contributions. The theory behind this proce-dure is well developed (Kalkbrenner et al. 2004, Gourieroux et al., 2000; Tasche,2000, 2002; see also Mausser and Rosen 2006 for a general presentation). Thehedging of credit risk is generally also focused on each underlying name in theportfolio and its ”deltas”, or sensitivities.
Understanding the contribution to economic capital, or pricing, of the vari-ous systemic factors (or credit drivers), which are at the heart of a multi-factorcredit model, can lead to better methodologies for managing concentration riskand hedging credit portfolios effectively. This requires first decomposing thecredit risk in a portfolio into its systemic and idiosyncratic components. Theunderlying multi-factor model drives the correlations of obligor credit eventsand determines entirely the systemic risk of the portfolio. In addition, the stan-dard theory of marginal capital contributions does not work well in this contextsince the total capital is not a homogeneous function of the factors. Finally, themost interesting cases, in practice, might require simulation of the multi-factormodels.
This series of research papers presents a set of methodologies to explore thecontributions of systemic factors to economic capital as well as the hedging ofsystemic risk in credit portfolios. In this first paper of the series, we presentanalytical results for hedging portfolio credit risk with linear portfolios of thesystemic factors. We can think of the hedging portfolio as a portfolio of spotpositions in the underlying indices. Formally, we seek to minimize the systemicvariance of portfolio losses with linear combinations of the systemic risk factors.We focus on default credit losses only in a multi-factor Merton-type (or probit)
3
credit portfolio model. We develop some of the mathematical tools to solvethese optimization problems and then apply it to various cases. First we focuson static hedges of homogeneous and inhomogeneous credit portfolios. We alsoapply the methodology to hedge the systemic credit default losses of CDOs. Fi-nally we show the application of the methodology to a dynamic hedging strategy.In each case, we discuss the effectiveness of the hedges and provide numericalexamples.
The rest of the papers in this series present extensions of this work to mea-sure contributions of systemic factors (analytically and numerically), as wellas to obtain numerical methods for hedging credit portfolios in other measures(such as expected shortfall), for more complex portfolios and combinations ofinstruments and for discrete dynamic hedges.
The rest of the paper is organized as follows. The second section reviewsthe multi-factor Merton model underlying our results, and sets the notationto be used in the remainder of the paper. The third section presents resultson optimal static hedging of portfolio losses with linear combinations of thesystemic factors. The fourth section presents analogous results for the systemiccredit default losses of CDOs. The fifth section disucsses hedging of systemiclosses in the dynamic case, where the weights in the linear portfolio of systemicfactors can be rebalanced continuously in time. The sixth section presentsconclusions, and discusses future work in this series. The proofs of the resultscited 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 atthe end of the horizon (say, one year) are described by a multi-factor Mer-ton model. Obligor i defaults when a continuous random variable Yi, whichdescribes its creditworthiness, falls below a given threshold. If we denote byPDi the obligor’s (unconditional) default probability and assume that thecreditworthiness is standard normal, we can express the default threshold asTi = Φ−1(PDi), where Φ is the standard cumulative normal distributionfunction. The creditworthiness of obligor i is driven by K factors through alinear model of the form:
Yi =K∑
k=1
βikZk + σiεi σ2i = 1 −
K∑
k=1
β2ik (1)
where the Zk are standard normal variables representing the systemic factorsdriving credit events, and the εi are independent standard normal variablesrepresenting the idiosyncratic movement of an obligor’s creditworthiness. We
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further assume, without loss of generality, that the systemic factors are uncor-related.1
For ease of notation, assume that obligor i has a single loan with (percentage of)
exposure at default wi,∑N
i=1 wi = 1. In order to simplify the presentation,we assume that loss given default is 100%. It is trivial to extend the results inthe paper to any other independent (possibly stochastic) loss given default.Total portfolio losses are thus given by the random variable:
Λ =
N∑
i=1
wi1{Yi≤Ti)} (2)
We refer to the case where βik ≡ βk, PDi ≡ PD, Ti ≡ T , σi ≡ σas the case of a “homogeneous portfolio”. The systemic component of theportfolio losses corresponds to replacing each individual loan by an infinitelygranular, homogeneous portfolio with the same probability of default and factordependencies (see, e.g. Gordy (2003)). It is thus given by:
L = E[Λ|Z1, . . . , ZK ] =N∑
i=1
Φ
(
Ti −∑K
k=1 βikZk
σi
)
(3)
It is well known, and easy to derive from Lemma (1) in the appendix, that
E[L] = E[Λ] =∑N
i=1 wiPDi. For future reference, we will also need thevariance of L, which is well known (the reader can calculate it easily using theresults in the appendix) to be:
var(L) =
N∑
i=1
N∑
j=1
wiwjΦ2
(
Ti, Tj;
K∑
k=1
βikβjk
)
−(
N∑
i=1
wiPDi
)2
(4)
In the case of a homogeneous portfolio, this reduces to:
var(L) = Φ2(T, T, 1 − σ2) − PD2 (5)
where Φ2(·, ·; ρ) is the bivariate cumulative normal distribution function withcorrelation ρ:
Φ2(z1, z2; ρ) =
∫ z1
−∞
∫ z2
−∞exp
(
−(w21 − 2ρw1w2 + w2
2)
2(1 − ρ2)
)
dw1 dw2
2π√
1 − ρ2
(6)
1This assumption is made to simplify the notation only, and we can think of these factors,for example, as the principal components resulting from some correlated factors.
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3 Static Hedging of Factor Risk for Credit Port-
folios
In this section, we study the problem of hedging the systemic risk of a portfoliousing a linear portfolio of the systemic 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 hedgesfor homogeneous and inhomogeneous credit portfolios, studying the quality ofthe 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 whichmay represent the loss of the entire portfolio, or some derivative on it (e.g. aCDO tranche). Thus, we are interested in the problem:
minα,c
‖Y − α −K∑
k=1
βkZk‖p (7)
The natural norm to take is the two norm, which leads to the analyticallytractable least-squares problem (the only other p norm that appears to produceanalytically tractable results is p = ∞, which is trivial, and not very useful, formost choices of Y ). The main tool that we will use is the following theorem.2
Theorem 1. Let Y be a random variable with finite variance andlet Z1, . . . , ZK be uncorrelated random variables with mean 0 andvariance 1. Then the optimal value of the minimization problem 7with p = 2 is:
minα,c
E[(Y − α −K∑
k=1
ckZk)2] = E[Y 2] − (α∗)2 −
K∑
k=1
(c∗k)2 (8)
= var(Y ) −K∑
k=1
cov(Zk, Y )2 (9)
where the optimal solution is:
α∗ = E[Y ] (10)
c∗k = E[ZkY ] k = 1, . . . , K (11)
A proof is provided in the appendix.
The coefficient of variation (R2) or the regression is defined to be one minusthe ratio of the optimal value to the variance of L, and may be taken as a simple
2Although we do not have a precise reference, we believe this to be a well known result.
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measure of the quality of the linear approximation of Y by α∗ +∑K
k=1 c∗kZk.It is easily calculated to be:
R2 =
∑Kk=1 cov(Zk, Y )2
var(Y )(12)
3.1 Homogeneous Portfolio
In this section, we consider the case of a homogeneous portfolio with independentfactors. This means that the probabilities of default and factor loadings are thesame for all obligors, i.e. PDi ≡ PD, Ti ≡ T, βik ≡ βk for i = 1, . . . , M .In this case the percentage portfolio systemic loss is simply given by the randomvariable:
L = Φ
(
T −∑K
k=1 βkZk
σ
)
(13)
where σ =√
1 −∑K
k=1 β2k .
We wish to find the weights that yield the best linear approximation (in theleast squares sense to the portfolio loss variable). The solution to this problemin the following proposition. Here φ is the standard normal probability densityfunction.
Proposition 1. The least squares hedging problem for the homoge-neous portfolio loss L has the optimal value:
minα,c
E
(
L − α −K∑
k=1
ckZk
)2
= Φ2(T, T, 1−σ2)−PD2−ϕ(T )2(1−σ2)
(14)with the optimal value being attained by
α∗ = PD (15)
c∗k = −βkϕ(T ) k = 1, . . . , K (16)
A proof of the proposition is given in the appendix. As noted in the generaltheorem in the previous section, the constant term α∗ (corresponding to a hedg-ing position in cash or a risk-free bond) is simply the expectation of the randomvariable being approximated, in this case the portfolio loss with E[L] = PD.The positions in the factors (c∗k) hedge the variance of the systemic losses. Theoptimal position in each factor is its weight in the creditworthiness index Yi (seeequation (1)), scaled by the density of the standard normal probability density
at the default threshold T = Φ−1(PD). The optimal weights underscore the
7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Probability of Default
Opt
imal
F
acto
r C
oeffi
cien
t
Figure 1: Optimal Coefficients for Least Squares Hedge of a Homogeneous Port-folio
factor that the multi-factor model with a homogeneous portfolio is really just asingle-factor model with factor:
Z =
∑Kk=1 βkZk√1 − σ2
(17)
The optimal hedge maintains the same relative weights on each factor as appear
in the aggregrate factor Z .
To see how the optimal weight in a given factor varies with the probability ofdefault, figure 1 shows the plot of c∗k against PD for a fixed value of βk = −0.5.
A simple calculation using equation (12) yields that for the least squares hedging
8
00.2
0.40.6
0.81
0
0.5
10
0.2
0.4
0.6
0.8
1
SigmaProbability of Default
RS
quar
e
Figure 2: Values of the R-Squared Coefficient for the Homogeneous Portfoliofor Different Probabilities of Default and Factor Betas
problem for the homogeneous portfolio is:
R2 =ϕ(T )2 · (1 − σ2)
Φ2(PD, PD, 1− σ2) − PD2(18)
A surface plot of the R2 against different values of PD and σ is given in figure2.
Notice that for the low probabilities of default often of interest in applica-tions, the value of R2 can be quite low, and the linear approximation does notprovide a good fit to the portfolio loss random variable. Observe that the valueof R2 is increasing in σ. At first this may seem counterintuitive, as σ = 1represents the case of purely idiosyncratic risk. However, one must rememberthat we are considering only the systemic component of portfolio losses, and inthis case that component is constant (indeed, one may observe that as σ → 1,
9
all the βk → 0 and therefore the optimal hedging coefficients c∗k → 0, butα∗ = PD). Furthermore, when σ = 1, the loss is independent of the factorvalues, and therefore the systemic component of the loss (= L = E[Λ|Z]) issimply the expected loss, equal to PD.
3.2 Inhomogeneous Portfolio
In this case the systemic portfolio loss is given by:
L =
N∑
i=1
wiΦ
(
Ti −∑K
k=1 βikZk
σi
)
(19)
where wi is the weight of the portfolio invested in counterparty i, i = 1, . . . , N .In the case of hedging the entire portfolio loss, linearity in equation (19) allowsus to obtain the optimal portfolio weights from the homogeneous case. Thefollowing proposition gives the optimal hedge for an inhomogeneous portfolio.
Proposition 2. The least squares hedging problem for the inhomo-
geneous portfolio loss L has the optimal value:
minα,c
E
(
L − α −K∑
k=1
ckZk
)2
=
N∑
i=1
w2i ri+2
∑
i<j
wiwjrij−(α∗)2−K∑
k=1
(c∗k)2
(20)where
ri = Φ2(Ti, Ti, 1 − σ2i ) (21)
rij = Φ2(Ti, Φ−1(PDj),
K∑
k=1
βikβjk) (22)
the optimal value being attained by
α∗ =
N∑
i=1
wiPDi (23)
c∗k = −N∑
i=1
wiβikϕ(Ti) k = 1, . . . , K (24)
A proof of the proposition is given in the appendix. Once again, the valueof the constant term (position in cash or a risk-free bond) is given by the
expected portfolio loss E[L] =∑N
i=1 wiPDi. The optimal hedging coeffi-
cients have a very simple form. If we define αi, cik to be the optimal hedging
10
weights for a homogeneous portfolio containing only loan i, then we have that
α∗ =∑N
i=1 wiαi, c∗k =∑N
i=1 cik. That is, the optimal weights for the in-
homogeneous portfolio are simply linear combinations of the optimal weightsfor each loan, weighted according to their contribution to the portfolio. Inparticular, the contribution of each loan to the hedging portfolio is portfolioinvariant in the sense of Gordy (2003), in that they only depend own thecharacteristics of the loan itself and not the other instruments in the portfolio.
A simple application of the general formula for R2, equation (12), together
with the known value of the portfolio variance (4) gives the R2 for the regressionfor an inhomogeneous portfolio:
R2 =
∑Kk=1
(
∑Ni=1 wiβikϕ(Ti)
)2
∑Ni=1
∑Nj=1 wiwjΦ2
(
Ti, Tj ;∑K
k=1 βikβjk
)
−(
∑Ni=1 wiPDi
)2
(25)
We also analyzed the impact of portfolio homogeneity on the quality of theregression fit. In particular, we consider a two-factor model with two loans ofequal credit quality in the portfolio, and the following factor weights.
β11 =√
1 − σ2 β12 = 0 (26)
β21 =√
(1 − σ2)λ β22 =√
(1 − σ2)(1 − λ), λ ∈ [0, 1] (27)
Thus the first loan is entirely dependent on the first factor, while the impact ofthe second factor on the second loan depends on the value of the parameter λ.In particular, with λ = 1 the portfolio is homogeneous, and depends entirelyon the first factor. While with λ = 0, the second loan is independent of thefirst loan. In this case, the general formula for the r-square of the fit reduces to:
R2 =12ϕ2(T )(1 − σ2)(1 +
√λ)
12(Φ2(T, T, 1 − σ2) + Φ2(T, T,
√λ(1 − σ2))) − PD2
(28)
Figure 3 gives the value of the r-square coefficient for different values oflambda, with a fixed default probability of PD = 0.01, and idiosyncraticweight σ2 = 0.7. The shape of the curve is quite interesting. First, we observethat the value of the r-square is relatively insensitive to the value of the param-eter λ. Second, we note that the best fit is achieved neither at a completelyhomogeneous portfolio, nor at a portfolio with the maximum granularity, butrather near the intermediate value of λ = 0.2.
4 Static Hedging of a CDO Tranche
In the homogeneous case, we can also obtain the best linear portfolio of factorsto approximate a CDO tranche. Denote the loss, over a fixed horizon, of the
11
0 0.2 0.4 0.6 0.8 10.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
Lambda
RS
quar
ed
Figure 3: Values of the R-Squared Coefficient for Different Values of the Homo-geneity Parameter λ
12
equity tranche3 with upper attachment point R by
LR = min(R, L) = min
(
R, Φ
(
T −∑K
k=1 βkZk
σ
))
(29)
Then we have the following result, whose proof is given in the appendix.
Proposition 3. The problem of least squares hedging problem forthe tranche LR has the optimal value:
minα,c
E
(
LR − α −K∑
k=1
ckZk
)2
= q∗ − (α∗)2 −K∑
k=1
(c∗k)2 (30)
where:
q∗ = P
[
X1 ≤σΦ−1(R) − T√
1 − σ2, X2 ≤ T, X3 ≤ T
]
+ R2Φ
(
T − σΦ−1(R)√1 − σ2
)
(31)
with (X1, X2, X3) jointly normally distributed variables with meanzero and variance-covariance matrix:
Σ =
1 −√
1 − σ2 −√
1 − σ2
−√
1 − σ2 1 1 − σ2
−√
1 − σ2 1 − σ2 1
(32)
In this case, the optimal value is attained by
α∗ = Φ2
(
σΦ−1(R) − T√1 − σ2
, T ;−√
1 − σ2
)
+ RΦ
(
T − σΦ−1(R)√1 − σ2
)
(33)
c∗k = −βkϕ(T ) · Φ(
Φ−1(R) − σT√1 − σ2
)
k = 1, . . . , K (34)
The formula for c∗ is very similar to the one for the optimal hedge of theentire portfolio, differing only by a multiplicative factor. At first glance, this
3We note that the above equation gives a tranche on only the systemic losses of theportfolio. The true portfolio tranche loss is given by ΛR = min(R, Λ). The systemic trancheloss LR will be a good approximation for the true CDO tranche loss only for large homogeneousportfolios (e.g. retail loan portfolios).
13
formula bears a striking resemblance to the probability that the systemic losswill be less than the attachment point:
P
[
Φ
(
T −∑K
k=1 βkZk
σ
)
≤ R
]
= Φ
(
σΦ−1(R) − T√1 − σ2
)
(35)
= Φ
(
σΦ−1(R) − Φ−1(PD)√1 − σ2
)
(36)
The multiplicative factor in the expression for c∗ is the same expression, with theroles of R and PD reversed. That is, it is the probability that a homogeneousportfolio with probability of default R will have systemic losses less than PD.
The R2 of the fit can be calculated based on the general equation (12).Values for various default probabilities and attachment points (with a fixed
σ2 = 0.7 are given in figure 4. 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 ofthe equity tranches LR−LR. The case of a CDO tranche on an inhomogeneousportfolio does not appear to be analytically tractable. We study this problemusing 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 factorsmight not provide sufficient hedges of portfolio losses. In this section, we brieflydiscuss how the above single step model can be embedded in a simple, continuoustime model that makes perfect hedging using a portfolio of the factors and therisk-free bond possible. For simplicity, we assume that interest rates are zero.Let the factor processes satisfy:
Zkt = Zk
0 +W k
t√τ
k = 1, . . . , K (37)
where W is a standard K dimensional Brownian motion, and τ is the timehorizon. Note that the (discounted) factor processes are martingales, and so weobtain the portfolio value in the homogeneous case to be:
f(t, z) = E
[
Φ
(
T −∑K
k=1 βkZkτ
σ
)
|Zt = z
]
(38)
14
00.1
0.20.3
0.4
0
0.1
0.2
0.3
0.40
0.2
0.4
0.6
0.8
1
Attachment PointDefault Probability
Figure 4: Values of the R-Squared Coefficient for Hedging a CDO Tranche
15
Now using the fact that:
Zkτ = ZK
t +
√
1 − t
τ· W k
τ − W kt√
τ − t(39)
= ZKt +
√
1 − t
τ· Zk
t,τ (40)
where Zkt,τ , k = 1, . . . , K are i.i.d. standard normal random variables, we get:
f(t, z) = E
[
Φ
(
T −∑Kk=1 βkZ
kτ
σ
)
|Zt = z
]
(41)
= E
Φ
T −∑Kk=1 βkzk
σ+
√
1 − tτ
√1 − σ2
σ·(−∑K
k=1 βkZkt,τ )√
1 − σ2
(42)
Using Lemma 1 from the appendix, we then obtain:
f(t, z) = Φ
T −∑K
k=1 βkzk
σ√
1 + τ−tσ2t
(43)
In order to obtain the hedging portfolio, we compute the partial derivative
fzk(t, z) =
−βk
σ√
1 + τ−tσ2t
ϕ
T −∑K
k=1 βkzk
σ√
1 + τ−tσ2t
(44)
The hedging portfolio then consists of holding fzk(t, Zt) in the kth factor at
time t, and f(t, Zt) −∑K
k=1 fzk(t, Zt) in the risk-free bond. The case of an
inhomogeneous portfolio is a straightforward extension using linearity.
It is also straightforward to derive the hedging weights for the factors for aCDO tranche on a homogeneous portfolio in the continuous time model. Thederivations follow exactly the same lines as those above, applying Lemma 5 tocompute the price and then differentiating in order to determine the optimalhedging weights.
6 Conclusion and Future Work
We present analytical results for hedging portfolio credit risk with linear port-folios of the systemic factors. Formally, for a default-only multi-factor Merton-type (or probit) credit portfolio model, we minimize the systemic variance of
16
portfolio losses with linear combinations of the systemic risk factors. The mathe-matical tools and results can be extended to other multi-factor portfolio models,within a conditional independence framework (such as the logit model). We de-velop the mathematical tools to solve these optimization problems and then ap-ply them to various cases: static hedges of homogeneous, inhomogeneous creditportfolios and CDOs, as well as dynamic hedging in continuous time. In eachcase, we discuss the effectiveness of the hedges and provide numerical examples.We observe that, in practice, the hedging effectiveness of purely homogeneousportfolios with simple linear factor portfolios is low. In this case, we essentiallyhave a one-factor problem and hence adding factors does not really improve theexplanation of credit volatility. We study the increase in effectiveness as port-folios are more inhomogeneous and the multiple factors start playing a biggerrole. If one is allowed to trade in continuous time, the portfolio systemic lossesover the horizon can be replicated fully.
There are various practical extensions of this work which are explored inother papers in this series:
• First, from a capital management perspective, the allocation of risk contri-butions to the systemic factors can provide useful tools for understandingthe structure of the portfolio and managing concentration risk.
• Second, we have explored in this paper only linear combinations of thefactors. Hedge effectiveness may be improved through the use of a linearcombination of non-linear functions of individual factors (for example, wecould use a portfolio CDS indices or defaultable bonds, each dependingon a single factor).
• Third, this paper uses variance (or standard deviation) as a measure ofrisk, given its widespread use and analytical tractability. However, creditloss distributions are far from normal and the work can be readily extendedto other measures which capture the risk in the tail of the loss distribution(such as expected shortfall). This generally requires some mathematicalprogramming techniques.
• Fourth, we can develop dynamic trading strategies in discrete time usingthe tools developed in stochastic programming.
6.1 Sensitivity Coefficients
An alternative to using the coefficients c∗k to measure sensitivity to the factorsis to use the following sensitivity indices (see Saltelli et al. (2004))
Sk =var(E[L|Zk])
var(L)(45)
=Φ2(T, T ; β2
k) − PD2
Φ2(T, T ; 1 − σ2) − PD2(46)
17
This calculation can be easily accomplished using lemma (2) in the appendix.Similarly, one gets
S∼k =var(E[L|Z∼k])
var(L)(47)
=Φ2(T, T ; 1 − σ2
k) − PD2
Φ2(T, T ; 1 − σ2) − PD2(48)
where σk =√
1 −∑j 6=k β2j .
A Proofs of Results
PROOF OF THEOREM 1: Define the function:
f(α, c) = E[(Y − α −K∑
k=1
ckZk)] (49)
then the minimization problem is clearly equivalent to
minα,c
f(α, c) (50)
Differentiating yields:
fα = −2E[Y ] + 2α + 2
K∑
k=1
ckE[Zk] (51)
= −2E[Y ] + 2α (52)
fck= −2E[ZkY ] + 2αE[Zk] + 2
K∑
j=1
cjE[ZkZj] (53)
= −2E[ZkY ] + 2ck k = 1, . . . , K (54)
and setting ∇f = 0 yields the candidate for optimality:
α∗ = E[Y ] (55)
c∗k = E[ZkY ] (56)
A second differentiation yields that the Hessian is constant and strictly positivedefinite Hf = 2IK+1, and thus α∗, c∗ is a global minimum. To compute the
18
optimal value:
E[(Y − α∗ −K∑
k=1
c∗kZk)2] (57)
= E[Y 2] − 2α∗E[Y ] − 2
K∑
k=1
c∗kE[ZkY ] + E[(α∗ +
K∑
k=1
c∗kZk)2] (58)
= E[Y 2] − 2(α∗)2 − 2K∑
k=1
(c∗k)2 + (α∗)2 + 2α∗
K∑
k=1
c∗kE[Zk] + E[(K∑
k=1
c∗kZk)2]
(59)
= E[Y 2] − (α∗)2 −K∑
k=1
(c∗k)2 (60)
This completes the proof of Theorem 1.
To prove Proposition 1 we begin with a series of lemmas. The first was takenfrom Kreinin and Nagy (2005).
Lemma 1. Let Z be a standard normal random variable, a, b ∈ R.Then
E [Φ (a + bZ)] = Φ
(
a√1 + b2
)
(61)
Proof. Letting X be a standard normal random variable indepen-dent of Z we have:
E[Φ(a + bZ)] = E[E[1{X≤a+bZ}|Z]] (62)
= E[1{X−bZ≤a}] = Φ
(
a√1 + b2
)
(63)
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 momentof the random variable.
Lemma 2. Let Z be a standard normal random variable, a, b ∈ R.Then
E[
(Φ(a + bZ))2] = Φ2
(
a√1 + b2
,a√
1 + b2;
b2
1 + b2
)
(64)
where Φ2(·, ·; ρ) is the cumulative bivariate normal distribution withcorrelation ρ.
19
Proof. Let X1, X2 be standard normal random variables such thatX1, X2, Z are independent.
E[Φ(a + bZ) · Φ(a + bZ)] = E[E[1{X1≤a+bZ} · 1{X2≤a+bZ}|Z]] (65)
= E[1{X1−bZ√1+b2
≤ a√1+b2
} · 1{X1−bZ√1+b2
≤ 1√1+b2
}]
(66)
= Φ2
(
a√1 + b2
,a√
1 + b2, ρ
)
(67)
where ρ is the correlation between (X1 − bZ)/√
1 + b2 and (X2 −bZ)/
√1 + b2 which is easily seen to be b2/(1 + b2).
The following lemma serves to simplify many calculations, particularly inthe 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 dif-
ferentiable in its jth variable, and that Z1, . . . , ZK are i.i.d. standardnormal random variables. Then:
E[Zjf(Z1, . . . , ZK)] = E[fj(Z1, . . . , ZK)] (68)
where the subscript denotes partial differentiation.
We need one final lemma before we can prove the proposition for the lossdistribution of the homogeneous portfolio.
Lemma 4. Suppose Z is a standard normal random variable, u, v ∈R. Then:
E[exp(−(u + vZ)2)] =exp
(
− u2
1+2v2
)
√1 + 2v2
(69)
Proof. The result follows simply by completing the square in the
20
expectation.
E[exp(−(u + vZ)2)] =
∫ ∞
−∞exp(−(u + vz)2) exp(−z2/2)
dz√2π
(70)
= exp
( −u2
1 + 2v2
)∫ ∞
−∞exp
(
−1
2
(√1 + 2v2 · z +
2uv√1 + 2v2
)2)
dz√2π
(71)
=exp
(
− u2
1+2v2
)
√1 + 2v2
·∫ ∞
−∞e−w2/2 dw√
2π(72)
=exp
(
− u2
1+2v2
)
√1 + 2v2
(73)
After the change of variables w =√
1 + 2v2z + 2uv√1+2v2 .
We are now ready to give the following:PROOF OF PROPOSITION 1: We begin by deriving the formulasfor α∗, c∗k. From Theorem 1 we have:
α∗ = E
[
Φ
(
T −∑Kk=1 βkZk
σ
)]
(74)
= E
[
Φ
(
T
σ−
√1 − σ2
σ·∑K
k=1 βkZk√1 − σ2
)]
(75)
= PD (76)
where the last line follows by applying Lemma (1) with a = Φ−1
σ, b =
√1−σ2
σ
and Z = −PK
k=1 βkZk√1−σ2 ∼ N(0, 1), and then an elementary simplification.
Next, we consider the optimal factor coefficients c∗j . From Theorem 1 we have:
c∗j = E
[
ZjΦ
(
T −∑Kk=1 βkZk
σ
)]
(77)
= E[Zjf(Z1, . . . , ZK)] (78)
21
Where
f(z1, . . . , zK) = Φ(
(
T −∑K
k=1 βkzk
σ
)
(79)
fj(z1, . . . , zK) = −β
σϕ
(
T −∑Kk=1 βkzk
σ
)
(80)
with the subscript denoting partial differentiation. Thus using Lemma 3:
c∗k = − β
σ√
2πE
exp
−1
2
(
T
σ−
√1 − σ2
σ·∑K
k=1 βkZk√1 − σ2
)2
(81)
and the result follows by applying Lemma 4 with u = Tσ√
2, v =
√1−σ2
σ√
2
and Z = −PK
k=1 βkZk√1−σ2 ∼ N(0, 1), and then performing a straightforward
simplification.
The expression for the optimal value follows immediately from Theorem 1 andLemma 2. The proof of proposition 1 is complete.
In order to prove proposition 3, we again need some preliminary lemmas.
Lemma 5. Let Z be a standard normal random variable, a ∈ R,b > 0, R ∈ (0, 1). Then:
E[min(R, Φ(a+bZ))] = Φ2
(
Φ−1(R) − a
b,
a√1 + b2
;−b√1 + b2
)
+RΦ
(
a − Φ−1(R)
b
)
(82)
Proof. Again, the result is achieved by considering a standard nor-mal 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]
(83)
= E[E[1Z≤Φ−1(R)−a
b
· 1X≤a+bZ |Z]] + RE[1Z≥Φ−1(R)−a
b
]
(84)
= E[1Z≤Φ−1(R)−a
b
· 1 X−bZ√1+b2
≤ a√1+b2
] + RΦ
(
a − Φ−1(R)
b
)
(85)
= Φ2
(
Φ−1(R) − a
b,
a√1 + b2
;−b√1 + b2
)
+ RΦ
(
a − Φ−1(R)
b
)
(86)
22
since corr(Z, (X − bZ)/√
1 + b2) = −b/√
1 + b2.
Lemma 6. Let Z be a standard normal random variable, a ∈ R,b > 0, R ∈ (0, 1). Then:
E[(min(R, Φ(a+bZ)))2] = P
[
X1 ≤Φ−1(R) − a
b, X2 ≤
a√1 + b2
, X3 ≤a√
1 + b2
]
+ R2Φ
(
a − Φ−1(R)
b
)
(87)
where (X1, X2, X3) are jointly normally distributed with mean zeroand variance-covariance matrix:
Σ =
1 − b√1+b2
− b√1+b2
− b√1+b2
1 b2
1+b2
− b√1+b2
b2
1+b21
(88)
Proof. Let X and Y be standard normal random variables indepen-dent 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]
(89)
= E[1Φ(a+bZ)≤RΦ(a + bZ)2] + R2Φ
(
a − Φ−1(R)
b
)
(90)
But
E[1Φ(a+bZ)≤RΦ(a + bZ)2] = E[E[1Z≤Φ−1(R)−a
b
· 1X≤a+bZ · 1Y ≤a+bZ |Z]]
(91)
= E
[
1Z≤Φ−1(R)−a
b
1 X−bZ√1+b2
≤ a√1+b2
· 1 Y −bZ√1+b2
≤ a√1+b2
]
(92)
and the result follows by computing the appropriate correlations.
Lemma 7. Let Z be a standard normal random variable, a ∈ R,b > 0, R ∈ (0, 1). Then:
E[
1Φ(a−bZ)≤R · ϕ(a − bz)]
=exp
(
−a2
2(1+b2)
)
√
2π(1 + b2)·Φ(
1
b
(
Φ−1(R)√
1 + b2 − a√1 + b2
))
(93)
23
Proof. The proof is accomplished by completing the square in theintegral.
E[
1Φ(a−bZ)≤R · ϕ(a − bz)]
(94)
=
∫ ∞
a−Φ−1(R)b
ϕ(a − bz)ϕ(z) dz (95)
=1
2π
∫ ∞
a−Φ−1(R)b
exp
(
−1
2
(
(1 + b2)z2 − 2abz + a2)
)
dz (96)
=1
2πexp
( −a2
2(1 + b2)
)∫ ∞
a−Φ−1(R)b
exp
(
−1
2
(√1 + b2 · z − ab√
1 + b2
)2)
dz
(97)
=exp
(
−a2
2(1+b2)
)
√
2π(1 + b2)
∫ ∞
√1+b2
“
a−Φ−1(R)b
”
− ab√1+b2
e−w2/2 dw√2π
(98)
=exp
(
−a2
2(1+b2)
)
√
2π(1 + b2)· Φ(
1
b
(
Φ−1(R)√
1 + b2 − a√1 + b2
))
(99)
after the change of variable w =√
1 + b2 · z − ab√1+b2
.
We are now ready to prove proposition 3.PROOF OF PROPOSITION 3:
We begin by computing α∗. By Theorem 1:
α∗ = E[LR] (100)
= E
[
min
(
R, Φ
(
T −∑K
k=1 βkZk
σ
))]
(101)
= E
[
min
(
R, Φ
(
T
σ−
√1 − σ2
σ·∑K
k=1 βkZk√1 − σ2
))]
(102)
and the result follows by applying Lemma 5 with a = Tσ
, b =√
1−σ2
σand
Z = −PK
k=1 βkZk√1−σ2 .
24
We now compute the optimal factor coefficients c∗j . By Theorem 1 and Lemma 3:
c∗j = E
[
Zj min
(
R, Φ
(
T −∑Kk=1 βkZk
σ
))]
(103)
= −βj
σE
[
1Φ
„
T−
PKk=1
βkZk
σ
«
≤R· ϕ(
T −∑Kk=1 βkZk
σ
)]
(104)
= −βj
σE
[
1Φ
„
Tσ−√
1−σ2
σ·
PKk=1
βkZk√1−σ2
«
≤R· ϕ(
T
σ−
√1 − σ2
σ·∑K
k=1 βkZk√1 − σ2
)]
(105)
and the result now follows by applying Lemma 7 with a = Φ−1
σ, b =
√1−σ2
σ
and Z =PK
k=1 βkZk√1−σ2 , and simplifying.
Finally, the formula for the optimal value follows from Theorem 1 and (to com-pute the first term) applying Lemma 6 with the obvious choices of a, b. Theproof of the proposition is complete.
For the proof of proposition 2, we need the following lemma.
Lemma 8. Let Z and Z be two standard normal random variableswith correlation ρ, a1, a2, b1, b2 ∈ R. Then:
E[Φ(a1+b1Z)·Φ(a2+b2Z)] = Φ2
(
a1√
1 + b21
,a2
√
1 + b22
;b1b2ρ
√
(1 + b21)(1 + b2
2)
)
(106)
Proof. As before, let X, Y be standard normal random variables,independent of each other and of Z, Z. Then:
E[Φ(a1 + b1Z) · Φ(a2 + b2Z)] = E[E[1X≤a1+b1Z · 1Y ≤a2+b2Z |Z1, Z2]]
(107)
= E[1 X−b1Z√1+b21
≤ a1√1+b21
· 1 Y −b2Z√1+b22
≤ a2√1+b22
]
(108)
= Φ2
(
a1√
1 + b21
,a2
√
1 + b22
; ρ
)
(109)
25
where
ρ =E[(X − b1Z)(Y − b2Z)]√
(1 + b21)(1 + b2
2)(110)
=b1b2ρ
√
(1 + b21)(1 + b2
2)(111)
PROOF OF PROPOSITION 2: The formulas for the optimal coef-ficients follow immediately from Theorem 1, linearity and the proof of Propo-
sition 1. Only the formula for the optimal value, and in particular for E[L2]requires any further work. We have:
E[L2] = E
(
N∑
i=1
wiΦ
(
Ti −∑K
k=1 βikZk
σi
))2
(112)
=N∑
i=1
w2i ri + 2
∑
i<j
wiwjrij (113)
where
ri = E
Φ
(
Ti −∑K
k=1 βikZk
σi
)2
(114)
= Φ2(Ti, Ti; 1 − σ2i ) (115)
rij = E
[
Φ
(
Ti −∑K
k=1 βikZk
σi
)
· Φ(
Φ−1(PDj) −∑K
k=1 βjkZk
σj
)]
(116)
= Φ2(Ti, Φ−1(PDj),
K∑
k=1
βikβjk) (117)
The result for ri follows by applying Lemma 2 with a = Ti
σi, b =
√1−σ2
i
σiand
Z =−
PKk=1 βikZk√1−σ2
i
∼ N(0, 1). The result for rij follows similarly by applying
Lemma 8 with a1 = Ti/σi, b1 =√
1 − σ2i /σi, a2 = Φ−1(PDj)/σj , b2 =
√
1 − σ2j /σj , Z = −
∑Kk=1 βikZk/
√
1 − σ2i , Z = −
∑Kk=1 βjkZk/
√
1 − σ2j .
26
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