Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and

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1 MANAGMNT SCINC Vol. 5, No., Noveber 005, pp issn eissn infors doi 0.87/nsc INFORMS Iportance Sapling for Portfolio Credit Risk Paul Glasseran, Jingyi Li Colubia Business School, Colubia University, New York, New York 007 pg0@colubia.edu, jl976@colubia.edu} Monte Carlo siulation is widely used to easure the credit risk in portfolios of loans, corporate bonds, and other instruents subject to possible default. The accurate easureent of credit risk is often a rare-event siulation proble because default probabilities are low for highly rated obligors and because risk anageent is particularly concerned with rare but significant losses resulting fro a large nuber of defaults. This akes iportance sapling (IS potentially attractive. But the application of IS is coplicated by the echaniss used to odel dependence between obligors, and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used noral copula odel of portfolio credit risk. The procedure has two parts: One applies IS conditional on a set of coon factors affecting ultiple obligors, the other applies IS to the factors theselves. The relative iportance of the two parts of the procedure is deterined by the strength of the dependence between obligors. We provide both theoretical and nuerical support for the ethod. Key words: Monte Carlo siulation; variance reduction; iportance sapling; portfolio credit risk History: Accepted by Wallace J. Hopp, stochastic odels and siulation; received January 9, 004. This paper was with the authors onth for 3 revisions.. Introduction Developents in bank supervision and in arkets for transferring and trading credit risk have led the financial industry to develop new tools to easure and anage this risk. An iportant feature of odern credit risk anageent is that it takes a portfolio view, eaning that it tries to capture the effect of dependence across sources of credit risk to which a bank or other financial institution is exposed. Capturing dependence adds coplexity both to the odels used and to the coputational ethods required to calculate outputs of a odel. Monte Carlo siulation is aong the ost widely used coputational tools in risk anageent. As in other application areas, it has the advantage of being very general and the disadvantage of being rather slow. This otivates research on ethods to accelerate siulation through variance reduction. Two features of the credit risk setting pose a particular challenge: (i it requires accurate estiation of low-probability events of large losses; (ii the dependence echaniss coonly used in odeling portfolio credit risk do not iediately lend theselves to rare-event siulation techniques used in other settings. This paper develops iportance sapling (IS procedures for rare-event siulation for credit risk easureent. We focus on the noral copula odel originally associated with J. P. Morgan s CreditMetrics syste (Gupton et al. 997 and now widely used. In this fraework, dependence between obligors (e.g., corporations to which a bank has extended credit is captured through a ultivariate noral vector of latent variables; a particular obligor defaults if its associated latent variable crosses soe threshold. The Creditrisk+ syste developed by Credit Suisse First Boston (Wilde 997 has a siilar structure but uses a ixed Poisson echanis to capture dependence. We present an IS ethod for that odel in Glasseran and Li (003 together with soe preinary results on the odel considered here. In the noral copula fraework, dependence is typically introduced through a set of underlying factors affecting ultiple obligors. These soeties have a tangible interpretation for exaple, as industry or geographic factors but they can also be by-products of an estiation procedure. Conditional on the factors, the obligors becoe independent, and this feature divides our IS procedure into two steps: We apply a change of distribution to the conditional default probabilities, given the factors, and we apply a shift in ean to the factors theselves. The relative iportance of the two steps depends on the strength of the dependence across obligors, with greater dependence putting greater iportance on the shift in factor ean. The idea of shifting the factor ean to generate ore scenarios with large losses has also been suggested in Avranitis and Gregory (00, Finger (00, and Kalkbrener et al. (003, though with little theoretical support and priarily in single-factor odels. The single-factor case is also discussed in 643

2 644 Manageent Science 5(, pp , 005 INFORMS Glasseran (004, The ain contributions of this paper lie in providing a systeatic way of selecting a shift in ean for ultifactor odels, in integrating the shift in ean with IS for the conditional default probabilities, and in providing a rigorous analysis of the effectiveness of IS in siple odels. This analysis akes precise the role played by the strength of the dependence between obligors in deterining the ipact of the two steps in the IS procedures. The rest of the paper is organied as follows. Section describes the noral copula odel. Section 3 presents background on an IS procedure for the case of independent obligors, which we extend in 4 to the case of conditionally independent obligors. Section 5 cobines this with a shift in factor ean. We present nuerical exaples in 6 and collect ost proofs in the appendix.. Portfolio Credit Risk in the Noral Copula Model A key eleent of any odel of portfolio credit risk is a echanis for capturing dependence aong obligors. In this section, we describe the widely used noral copula odel associated with CreditMetrics (Gupton et al. 997 and related settings (Li 000. Our interest centers on the distribution of losses fro default over a fixed horion. To specify this distribution, we introduce the following notation. = nuber of obligors to which portfolio is exposed Y k = default indicator for kth obligor = if kth obligor defaults, 0 otherwise p k = arginal probability that kth obligor defaults c k = loss resulting fro default of kth obligor L = c Y + +c Y = total loss fro defaults The individual default probabilities p k are assued known, either fro credit ratings or fro the arket prices of corporate bonds or credit default swaps. We take the c k to be known constants for siplicity, though it would suffice to know the distribution of c k Y k. Our goal is to estiate tail probabilities PL>x, especially at large values of x. To odel dependence aong obligors we need to introduce dependence aong the default indicators Y Y. In the noral copula odel, dependence is introduced through a ultivariate noral vector X X of latent variables. ach default indicator is represented as Y k = X k >x k k = with x k chosen to atch the arginal default probability p k. The threshold x k is soeties interpreted as a default boundary of the type arising in the foundational work of Merton (974. Without loss of generality, we take each X k to have a standard noral distribution and set x k = p k, with the cuulative noral distribution. Thus, PY k ==PX k > p k = p k =p k Through this construction, the correlations aong the X k deterine the dependence aong the Y k. The underlying correlations are often specified through a factor odel of the for X k = a k Z + +a kd Z d + b k k ( in which Z Z d are systeatic risk factors, each having an N0 (standard noral distribution; k is an idiosyncratic risk associated with the kth obligor, also N0 distributed; a k a kd are the factor loadings for the kth obligor, a k + +a kd. b k = a k + +a kd so that X k is N0. The underlying factors Z j are soeties given econoic interpretations (as industry or regional risk factors, for exaple. We assue that the factor loadings a kj are nonnegative. Though not essential, this condition siplifies our discussion by ensuring that larger values of the factors Z i lead to a larger nuber of defaults. Nonnegativity of the a kj is often iposed in practice as a conservative assuption ensuring that all default indicators are positively correlated. Write a k for the row vector a k a kd of factor loadings for the kth obligor. The correlation between X k and X j, j k, is given by a k a j. The conditional default probability for the kth obligor given the factor loadings Z = Z Z d is p k Z = PY k = Z = PX k >x k Z = Pa k Z + b k k > p k Z ( ak Z + p = k ( b k 3. Iportance Sapling: Independent Obligors Before discussing IS in the noral copula odel, it is useful to consider the sipler proble of estiating loss probabilities when the obligors are independent. So, in this section, we take the default indicators Y Y to be independent; equivalently, we take all a kj = 0. In this setting, the proble of efficient estiation of PL>x reduces to one of applying IS to a su of independent (but not identically distributed rando variables. For this type of proble there is a fairly well-established approach, which we now describe.

3 Manageent Science 5(, pp , 005 INFORMS 645 It is intuitively clear that to iprove our estiate of a tail probability PL>x we want to increase the default probabilities. If we were to replace each default probability p k by soe other default probability q k, the basic IS identity would be [ PL>x= L>x k= ( pk q k Yk ( pk q k Yk ] (3 where denotes the indicator of the event in braces, indicates that the expectation is taken using the new default probabilities q q, and the product inside the expectation is the likelihood ratio relating the original distribution of Y Y to the new one. Thus, the expression inside the expectation is an unbiased estiate of PL>x if the default indicators are sapled using the new default probabilities. 3.. xponential Twisting Rather than increase the default probabilities arbitrarily, we apply an exponential twist: We choose a paraeter and set p p k = k e c k (4 + p k e c k If >0, then this does indeed increase the default probabilities; a larger exposure c k results in a greater increase in the default probability. The original probabilities correspond to = 0. With this choice of probabilities, straightforward calculation shows that the likelihood ratio siplifies to ( Yk ( pk Yk pk = exp L + (5 k= where p k p k = log e L = log + p k e c k k= is the cuulant generating function (CGF of L. For any, the estiator L > xe L+ is unbiased for PL>x if L is generated using the probabilities p k. quation (5 shows that exponentially twisting the probabilities as in (4 is equivalent to applying an exponential twist to L itself. It reains to choose the paraeter. We would like to choose to iniie variance or, equivalently, the second oent of the estiator. The second oent is given by M x = M x = [ L > xe L+ ] e x+ (6 where denotes expectation using the -twisted probabilities and the upper bound holds for all 0. Miniiing M x is difficult, but iniiing the upper bound is easy: we need to axiie x over 0. The function is strictly convex and passes through the origin, so the axiu is attained at unique solution to = x x > 0 x = (7 0 x 0 To estiate PL>x, we twist by x. A standard property of exponential twisting (e.g., as in Glasseran 004, p. 6, easily verified by direct calculation, iplies that L = and this facilitates the interpretation of x. The two cases in (7 correspond to x>l and x L. In the first case, our choice of twisting paraeter iplies x L = x = x thus, we have shifted the distribution of L so that x is now its ean. In the second case, the event L > x is not rare, so we do not change the probabilities. 3.. Asyptotic Optiality A standard way of establishing the effectiveness of a siulation estiator in a rare-event setting is to show that it is asyptotically optial as the event of interest becoes increasingly rare, eaning that the second oent decreases at the fastest possible rate aong all unbiased estiators (see, e.g., Heidelberger 995 for background. Asyptotic optiality results are often based on aking precise approxiations of the for PL>x e x M x e x (8 for soe >0. The key point here is that the second oent decays at twice the rate of the probability itself. By Jensen s inequality M x PL > x,so this is the fastest possible rate of decay. To forulate a precise result, we let the nuber of obligors increase together with the threshold x. This is practically eaningful bank portfolios can easily be exposed to thousands or even tens of thousands of obligors as well as theoretically convenient. We therefore need an infinite sequence p k c k, k = of obligor paraeters. Write for the CGF of the loss in the th portfolio. We require that = log + p k e c k (9 k=

4 646 Manageent Science 5(, pp , 005 INFORMS for all, for soe strictly convex. This holds, for exaple, if the p k have a it in 0 and the c k have a it in 0. Write L for the loss in the th portfolio, write for the value of x in (7 for the th portfolio, and write M x for the second oent of the IS estiator of PL >x using. Theore. Suppose (9 holds for all and there is a x > 0 at which x = x. Then log PL >x= log M x = where = supx Thus, the IS estiator is asyptotically optial. Proof. The result would follow fro a ore general result in Sadowsky and Bucklew (990 but for a difference in the choice of twisting paraeter. (They use x. Under (9, the Gärtner-llis theore (as in Debo and Zeitouni 998,.3 gives Using (6, we find that sup log PL >x (0 log M x sup x / sup x x x / = x x x = ( By Jensen s inequality, M x PL >x so the in (0 and the sup in ( hold as its. This use of Jensen s inequality in the proof of the it applies in the proofs of all other theores in this paper. So, it will suffice to prove the of the probability and the sup of the second oent as in Theore in subsequent theores. The its in Theore ake precise the approxiations in (8. This result (and its extension in the next section continues to hold if Y k c k k is replaced with a ore general sequence of independent light-tailed rando variables under odest regularity conditions. 4. Dependent Obligors: Conditional Iportance Sapling As a first step in extending the ethod of 3 to the noral copula odel of, we apply iportance sapling conditional on the coon factors Z. Conditional on Z =, the default indicators are independent, the kth obligor having conditional default probability p k as defined in (. We ay therefore proceed exactly as in 3. In ore detail, this entails the following steps for each replication:. Generate Z N0I,ad-vector of independent noral rando variables.. Calculate the conditional default probabilities p k Z, k =, in (. If L Z p k Zc k x k= set x Z = 0; otherwise, set x Z equal to the unique solution of Z = x with = log + p k e c k ( k= 3. Generate default indicators Y Y fro the twisted conditional default probabilities p p kx ZZ = k Ze xzc k k= + p k Ze xzc k 4. Copute the loss L = c Y + +c Y and return the estiator L > x exp x ZL + x Z Z (3 Although here we have tailored the paraeter to a particular x, we will see in the nuerical results that the sae value of can be used to estiate PL>x over a wide range of loss levels. The effectiveness of this algorith depends on the strength of the dependence aong obligors. When the dependence is weak, increasing the conditional default probabilities is sufficient to achieve substantial variance reduction. But this ethod is uch less effective at high correlations, because in that case large losses occur priarily because of large outcoes of Z, and we have not yet applied IS to the distribution of Z. To illustrate these points ore precisely, we consider the special case of a single-factor, hoogeneous portfolio. This eans that Z is scalar d =, all p k are equal to soe p, all c k are equal to a fixed value (which we take to be, and all obligors have the sae loading on the single factor Z. Thus, the latent variables have the for X k = Z + k k= (4 and the conditional default probabilities are p = PY k = Z = = PX k > p Z = ( + p = (5

5 Manageent Science 5(, pp , 005 INFORMS 647 As in Theore, we consider the it as both the nuber of obligors and the loss level x increase. We now also allow to depend on through a specification of the for = a/ for soe a, >0. Thus, the strength of the correlation between obligors is easured by the rate at which decreases, with sall values of corresponding to stronger correlations. If we kept fixed, then by the law of large nubers, L / would converge to pz and, with 0 <q<, PL >q PpZ>q ( q = p > 0 (6 Thus, with fixed, the event L >q does not becoe vanishingly rare as increases. By letting decrease with we will, however, get a probability that decays exponentially fast. Define ( p q ( p q log p<q Gp = q q (7 0 p q Gp 0, and expgp is the likelihood ratio at L = q for the independent case with arginal individual default probability p. Also define (with = a/ F a = Gp = G ( a + p (8 and F = q q + q = Gp (9 using the expression for p in (5; F a 0 and F 0 because Gp 0. The following lea is verified by differentiation. Lea. In the single-factor, hoogeneous portfolio specified by p k p, c k, and (4, the function F is increasing and concave. Write M q q for the second oent of the IS estiator (3 with x = q. Theore. Consider the single-factor, hoogeneous portfolio specified by p k p 0 /, c k, and (4. For constant q>p,if = a/, a>0, then (a For >/, log PL >q= F0 0 log M q q = F0 0 (b For = /, log PL >q= axf a / log M q q = axf a / (c For 0 <</, log PL >q = log M q q = a / with a = q p/a. This result shows that we achieve asyptotic optiality only in the case >/ (in which the correlations vanish quite quickly, because only in this case does the second oent vanish at twice the rate of the first oent. At = /, the second oent decreases faster than the first oent, but not twice as fast, so this is an interediate case. With </, the two decrease at the sae rate, which iplies that conditional iportance sapling is (asyptotically no ore effective than ordinary siulation in this case. The failure of asyptotic optiality in (b and (c results fro the ipact of the coon risk factor Z in the occurrence of a large nuber of defaults. When the obligors are highly correlated, large losses occur priarily because of large oves in Z, but this is not (yet reflected in our IS distribution. This is also suggested by the for of the its in the theore. The its in all three cases can be put in the sae for as (b (with (a corresponding to the iting case a 0 and (c corresponding to a once we note that F0 is independent of in (a and F a a = 0 in (c. In each case, the two ters in the expression F a / result fro two sources of randoness: the second ter results fro the tail of the coon factor Z and the first ter results fro the tail of the conditional loss distribution given Z. Because our conditional IS procedure is asyptotically optial for the conditional loss probability given Z, a factor of ultiplies F a in each second oent. To achieve asyptotic optiality for the unconditional probability, we need to apply IS to Z as well. 5. Dependent Obligors: Two-Step Iportance Sapling 5.. Shiftingthe Factors We proceed now to apply IS to the distribution of the factors Z = Z Z d as well as to the conditional default probabilities. To otivate the approach we take, observe that for any estiator p x of PL>x we have the decoposition Varp x = Varp x Z + Varp x Z Conditional on Z, the obligors are independent, so we know fro 3 how to apply asyptotically optial IS. This akes Varp x Z sall and suggests that

6 648 Manageent Science 5(, pp , 005 INFORMS in applying IS to Z we should focus on the second ter in the variance decoposition. When p x is the estiator of 4, p x Z = PL>xZ, so this eans that we should choose an IS distribution for Z that would reduce variance in estiating the integral of PL>x Z against the density of Z. The ero-variance IS distribution for this proble would saple Z fro the density proportional to the function PL>x Z = e / But sapling fro this density is generally infeasible the noraliation constant required to ake it a density is the value PL>x we seek. Faced with a siilar proble in an option-pricing context, Glasseran et al. (999 suggest using a noral density with the sae ode as the optial density. This ode occurs at the solution to the optiiation proble ax PL>x Z = e / (0 which is then also the ean of the approxiating noral distribution. Once we have selected a new ean for Z, the IS algorith proceeds as follows:. Saple Z fro NI.. Apply the procedure in 4 to copute x Z and the twisted conditional default probabilities p kx ZZ, k =, and to generate the loss under the twisted conditional distribution. 3. Return the estiator L > xe xzl+ x Z Z e Z+ / ( The last factor in the estiator (the only new one is the likelihood ratio relating the density of the N0I distribution to that of the NI distribution. Thus, this two-step IS procedure is no ore difficult than the one-step ethod of 4, once has been deterined. The key, then, is finding. xact solution of (0 is usually difficult, but there are several ways of siplifying the proble through further approxiation: Constant Approxiation. Replace L with L Z = and PL>xZ = with L Z = >x. The optiiation proble then becoes one of iniiing subject to L Z = > x. Noral Approxiation. Note that L Z = = k p k c k and VarL Z = = k c k p k p k and use the noral approxiation ( x L Z = PL>x Z = VarL Z = in (0. Saddlepoint Approxiation. Given the conditional cuulant generating function Z in (, one can approxiate PL>x Z = using any of the various saddlepoint ethods in Jensen (995, as in, for exaple, Martin et al. (00. These are approxiate ethods for inverting the conditional characteristic function of L given Z. (In cases for which precise inversion of the characteristic function is practical, one ay use the conditional Monte Carlo estiator PL>x Z and dispense with siulation of the default indicators. Tail Bound Approxiation. Define F x = x x + x ( this is the logarith of the likelihood ratio in (3 evaluated at L = x. The sallest axiier of F x is the solution using the constant approxiation described above because F attains its axiu value 0 only when L Z = x. The inequality y > x expy x, 0, gives PL>x Z = [ e xzl x Z = ] = e F x By treating this bound as an approxiation in (0 and taking logariths, we arrive at the optiiation proble Jx= ax Fx } (3 Other approxiations are possible. For exaple, in a related proble, Kalkbrener et al. (003 calculate the optial ean for a single-factor approxiation and then lift this scalar ean to a ean vector for Z. We focus on (3 because it provides the ost convenient way to blend IS for Z with IS conditional on Z. The expression in ( is also the exponent of the usual (conditional saddlepoint approxiation, which lends further support to the idea of using this upper bound as an approxiation. The various ethods discussed above will generally produce different values of. Our asyptotic optiality results are based on (3, but this does not preclude the possibility that the other ethods are asyptotically optial as well. 5.. Asyptotic Optiality: Large Loss Threshold We now turn to the question of asyptotic optiality of our cobined IS procedure. We establish asyptotic optiality in the case of the single-factor, hoogeneous portfolio used in Theore and then coent on the general case. As in 3 and 4, we consider a it in which the loss threshold increases together with the sie of the portfolio. In contrast to Theore, here we do not need to assue that the correlation paraeter decreases; this is the ain iplication of applying IS to the factors Z. As noted in (6, when the sie of the hoogenous portfolio increases, the noralied loss L / converges to a nonrando it taking values in the unit interval. In order that PL >x vanish as

7 Manageent Science 5(, pp , 005 INFORMS 649 increases, we therefore let x / approach fro below. The following specification turns out to be appropriate: ( x = q q = c log 0 <c< (4 Here q approaches fro below and q is O c /. Write for the axiier in (3 with x = x and write Jx for the axiu value. That the axiier is unique is ensured by Lea. It follows that the objective in (3 is strictly concave and that the axiu is attained at just one point. Write M x for the second oent of the cobined IS estiator of PL >x using. Theore 3. Consider the single-factor, hoogeneous portfolio specified by p k p, c k, and (4. With x as in (4 let = c then log log PL >x = log log M x = The cobined IS estiator is thus asyptotically optial. This result indicates that our cobined IS procedure should be effective in estiating probabilities of large losses in large portfolios. Because we have noralied the its by log, this result iplies that the decay rates are polynoial rather than exponential in this case: PL >x and M x. Although Theore 3 is specific to the single-factor, hoogeneous portfolio, we expect the sae IS procedure to be effective ore generally. We illustrate this through nuerical exaples in 6, but first we offer soe observations. The key to establishing the effectiveness of the IS procedure is obtaining an upper bound for the second oent. As a direct consequence of the concavity of F x, the arguent in the appendix shows that ( M x exp ax Fx }} (5 without further conditions on the portfolio, and this bound already contains uch of the significance of Theore 3. In fact, the assuption of concavity can be relaxed to the requireent that the tangent to soe optiier be a doinating hyperplane in the sense that F x F x + F x (6 with F x (a row vector the gradient of F x. In the proof of Theore 3, we show that the axiier of (3 coincides, in the it, with the sallest axiier of F x (the solution using the constant approxiation described in 5. in the sense that their ratio tends to. Property (6 is satisfied by any axiier of F x, so we expect it to be satisfied by a axiier of (3 as increases. This suggests that (5 should hold for large in greater generality than that provided by Theore 3. We view the iting regie in Theore 3 as ore natural than that of Theore because we are interested in the probability of large losses in large portfolios. In Theore, we are forced to let the correlation paraeter decrease with the sie of the portfolio in order to get a eaningful it for the second oent; this indicates that twisting the (conditional default probabilities without shifting the factor ean is effective only when the correlation paraeter is sall. In practice, we face fixed loss levels for fixed portfolios, not iting regies; we therefore suppleent our asyptotic optiality results with nuerical experients Asyptotic Optiality: Sall Default Probability We next show that our two-step IS procedure is also effective when large losses are rare because individual default probabilities are sall. This setting is relevant to portfolios of highly rated obligors, for which oneyear default probabilities are extreely sall. (Historical one-year default probabilities are less than 0.% for A-rated firs and even saller for firs rated AA or AAA. It is also relevant to easuring risk over short tie horions. In securities lending, for exaple, banks need to easure credit risk over periods as short as three days, for which default probabilities are indeed sall. For this iting regie, it is convenient to take x = q 0 <q< p = c c > 0 (7 Here p decreases toward ero and is Oexp c / We again consider a single-factor, hoogeneous portfolio in which all obligors have the sae default probability p = p and all have a loss given default of. Theore 4. For a single-factor, hoogeneous portfolio with default probability p and loss threshold x as in (7, let = c then log PL >x = log M x = The cobined IS estiator is thus asyptotically optial.

8 650 Manageent Science 5(, pp , 005 INFORMS 6. Nuerical xaples We now illustrate the perforance of our two-step IS procedure in ultifactor odels through nuerical experients. For these results, we shift the ean of the factors Z to the solution of (3 and then twist the conditional default probabilities by x Z, as in (. Our first exaple is a portfolio of = 000 obligors in a 0-factor odel. The arginal default probabilities and exposures have the following for: p k = 00 + sin6k/ k = (8 c k = 5k/ k= (9 Thus, the arginal default probabilities vary between 0% and % with a ean of %, and the possible exposures are, 4, 9, 6, and 5, with 00 obligors at each level. These paraeters represent a significant departure fro a hoogeneous odel with constant p k and c k. For the factor loading, we generate the a kj independently and uniforly fro the interval 0 / d, d = 0; the upper it of this interval ensures that the su of squared entries a k + +a kd for each obligor k does not exceed. Our IS procedure is designed to estiate a tail probability PL>x, so we also need to select a loss threshold x. The choice of x affects the paraeter x and the optial solution to (3 used as the ean of the factors Z under the IS distribution. For this exaple, we use x = 000. The coponents of the factor ean vector (the solution we found to (3 are all about 0.8, reflecting the equal iportance of the 0 factors in this odel. Using a odified Newton ethod fro the Nuerical Algoriths Group library (function e04lbc running on a SunFire V880 workstation, the optiiation takes 0. seconds. Although the IS distribution depends on our choice of x, we can use the sae saples to estiate PL>y at values of y different fro x. The IS estiator of PL>y is (see ( L > ye xz+ x Z Z e Z+ / (30 In practice, we find that this works well for values of y larger than x, even uch larger than x. Figure copares the perforance of iportance sapling (labeled IS and ordinary Monte Carlo siulation (labeled Plain MC in estiating the tail of the loss distribution, PL>y. The Plain MC results are based on 0,000 replications whereas the IS results use just,000. In each case, the three curves show the saple ean (the center line and a 95% confidence interval (the two outer lines coputed separately at each point. The IS estiates of PL>y are all coputed fro the sae saples and the sae IS distribution, Figure Coparison of Plain Monte Carlo Siulation (Using 0,000 Replications with IS (Using,000 Replications in stiating Loss Probabilities in a 0-Factor Model Tail probability Plain MC 0,000,000 3,000 4,000 5,000 Loss level Note. In each case, the three curves show the ean and a 95% confidence interval for each point. as in (30, with x fixed at,000. The figure indicates that the IS procedure gives excellent results even for values of y uch larger than x. With the loss distribution of L centered near,000 under IS, sall losses becoe rare events, so the IS procedure is not useful at sall loss levels. To estiate the entire loss distribution, one could therefore use ordinary siulation for sall loss levels and IS for the tail of the distribution. The IS ethods take approxiately twice as long per replication as ordinary siulation, but this is ore than offset by the draatic increase in precision in the tail. Our next exaple is a -factor odel, again with = 000 obligors. The arginal default probabilities fluctuate as in (8, and the exposures c k increase linearly fro to 00 as k increases fro to,000. The atrix of factor loadings, A = a kj k= 000 j =, has the following block structure: F G g A = R G= F G g (3 with R a colun vector of,000 entries, all equal to 0.8; F, a colun vector of 00 entries, all equal to 0.4; G a 00 0 atrix; and g, a colun vector of 0 entries, all equal to 0.4. This structure is suggestive of the following setting: The first factor is a arketwide factor affecting all obligors, the next 0 factors (the F block are industry factors, and the last 0 (the G block are geographical factors. ach obligor then has a loading of 0.4 on one industry factor and on one geographical factor, as well as a loading of 0.8 IS

9 Manageent Science 5(, pp , 005 INFORMS 65 Figure Coparison of Plain Monte Carlo Siulation (Using 0,000 Replications with IS (Using,000 Replications in stiating Loss Probabilities in a -Factor Model Tail probability Plain MC Loss level 0 4 Note. In each case, the three curves show the ean and a 95% confidence interval for each point. on the arketwide factor. There are 00 cobinations of industries and regions and exactly 0 obligors fall in each cobination. This highly structured dependence on the factors differentiates this exaple fro the previous one. The optial ean vector (selected through (3 for this exaple has.46 as its first coponent and uch saller values (around 0.0 for the other coponents, reflecting the greater iportance of the first factor. Results for this odel are shown in Figure, which should be read in the sae way as Figure. The paraeters of the IS distribution are chosen with x = 0000, but we again see that IS reains extreely effective even at uch larger loss levels. The effectiveness of IS for this exaple is quantified in Table. The table shows estiated variance reduction factors in estiating PL>y at various levels of y, all using the IS paraeters corresponding to x = The variance reduction factor is the variance per replication using ordinary siulation divided by the variance per replication using IS. As expected, the variance reduction factor grows as we ove farther into the tail of the distribution. The IS ethod takes Table Variance Reduction Factors Using IS at Various Loss Levels y PL>y V.R. factor 0, , , , , , IS approxiately twice as long per replication as ordinary siulation, so the (large saple efficiency ratios are about half as large as the variance ratios. This coparison does not include the fixed cost involved in finding the optial. In our experients, the tie required for the optiiation is equal to the tie to run 00,000 replications. There is no guarantee that a local optiu in (3 is a unique global optiu. If PL>x Z = is ultiodal as a function of, the optial IS distribution for Z ay also be ultiodal. A natural extension of the procedure we have described would cobine IS estiates using several local optia for (3, as in, e.g., Avraidis (00. However, we have not found this to be necessary in the exaples we have tested. 7. ConcludingRearks We have developed and analyed an IS procedure for estiating loss probabilities in a standard odel of portfolio credit risk. Our IS procedure is specifically tailored to address the coplex dependence between defaults of ultiple obligors that results fro a noral copula. The procedure shifts the ean of underlying factors and then applies exponential twisting to default probabilities conditional on the factors. The first step is essential if the obligors are strongly dependent; the second step is essential (and sufficient if the obligors are weakly dependent. We have established the asyptotic optiality of our procedure in singlefactor, hoogeneous odels and illustrated its effectiveness in ore coplex odels through nuerical results. Acknowledgents The authors thank the referees for their careful reading of the paper and detailed coents. This work is supported in part by NSF grants DMS and DMI Appendix. Proofs Proof of Theore. We first prove (b and then use siilar arguents for (a and (c. In all three cases, we prove an upper bound for the second oent and a lower bound for the probability, as these are the ost iportant steps. We oit the proof of the lower bound for the second oent and the upper bound for the probability because they follow by very siilar arguents. (b First we show that sup log M q q ax Define P as Binyqd PL=yZ d= Binypd F a } (3 p q p>q (33 whereas under the original distribution, PL= yz d = Biny pd. Here Bin p denotes the probability ass function of the binoial distribution with

10 65 Manageent Science 5(, pp , 005 INFORMS paraeters and p. As in (5, it follows that ( dpl Z pz L ( pz L d PLZ = pz q q q pz>q = exp q ZL + q Z Z with [ ] pzq + q Z = log qpz Writing M q q as an expectation under P, weget M q q = L>qexp q ZL+ q ZZ exp q Zq + q ZZ = expf Z (34 Here F is as defined in (9. The function F is concave fro Lea and F attains its axiu value 0 for all where = q a / p and p a = q This is obtained fro the fact that Gp attains its axiu value 0 for all p q and p is an increasing function in. Because q>p, we have > 0 for all sufficiently large. For any 0, the concavity of F iplies F F 0 + F 0 0. Then expf Z expf 0 + F 0 Z 0 = expf 0 F F 0 The ter F 0 appears because expaz = expa / for any constant a. Choose 0 to be the point at which F is axiied. Then F 0 = 0 and expf Z exp (F 0 0 ( = exp ax F } ( F = exp ax } (35 Because p 0 /, p < 0, coparison of (8 and (9 shows that F ( ( a + p = G ( ( G ( = F a a / a a / + p a / and therefore F ax } ( ax F a } a / = ax F a } a / a / ax F a } (36 Cobining this with (34 and (35 proves (3. Next we show that log PL>q ax F a } (37 For any >0, define P as in (33 but with q replaced by q +. It follows that with dpl Z d P L Z = exp ZL + Z Z [ Z = log ] pzq + + q pz Writing PL>q as an expectation under P,weget PL>q = L>qexp ZL + Z Z q < L q + exp Zq + + Z Z = q < L q + expg pz q < L q + expg pzpz q + (38 with ( p q ( p q+ log p q + G p = q q + 0 p>q+ Under P, L and Z are independent given pz<q+. So q < L q + expg pzpz q + = q < L q + pz<q+ expg pzpz q + (39 Also, given pz q +, the loss L has a binoial distribution with paraeters and q + under P. By the central it theore, for large enough, P q < L q + pz q + = P ( q + q < 0 pz q + L q + q + q (40 4 To bound the second factor in (39, we separate the cases q< and q.ifq<, we can choose sall enough so that q + <. On the event pz q +, we have a/ Z + p < 0 fro the expression for pz in (5. Then for arbitrarily sall >0 and >a /, ( a/ Z + p pz Define = q + p/a, cobining the fact that pz q + Z/ with (38 (40, PL>q ( ( }] 4 a/ Z + (G p Z = ( 4 (F a Z Z }] (4

11 Manageent Science 5(, pp , 005 INFORMS 653 Here ( a + F a = G ( p Applying an extension of Varadhan s Lea (as on p. 40 of Debo and Zeitouni 998, we get log PL>q log [ ( 4 exp (F a Z }] Z F a } F a } sup = ax (4 The equality is obtained fro the continuity of the function F a. Define F a = G a + p and let be the solution to the optiiation proble ax F a. As decreases to 0, the function F a increases to F a. With (4, this gives log PL>q ax F a } = ax F a } (43 The equality coes fro the fact that axiies F a and > 0 because q>pand sall enough. Siilarly, define to be the solution to the optiiation proble ax F a As decreases to 0, F a increases to F a. Cobining with (43, we obtain the in (37 for q<. If q, for >a /, ( q + a / p /a> Instead of (4, we get PL>q ( ( 4 a/ Z + (G p a / p Z }] a ( 4 (F a p a Z }] Applying the extension of Varadhan s Lea, we get log PL>q ax F a } p/a Because and are arbitrary, the arguent used for q< now shows that inf logpl>q ax F a } (44 p/a Also, for >a / and Z p/a, we have ( a/ Z + p pz for pz in (5. Because G is an increasing function, PL>q ( ( 4 a/ Z + (G p }] Z p a = ( 4 (F a }] Z p a Again applying the extension of Varadhan s Lea, we get inf logpl>q ax F a } (45 p/a Because is arbitrary, (45 holds with F replaced by F. Cobining this with (44 and using the fact that is arbitrary, we obtain the in (37 for q. (a Define a = a/ /, then = a /. First we show that sup log M q q F0 0 (46 By following the sae steps as in (34, (35, and (36, we get M q q exp ( a / ax Fa } (47 For arbitrarily sall > 0, we can find soe = / / so that a for. Define a to be the solution to the optiiation proble ax Fa and to be the solution at a =. It is obvious that a > 0 and > 0 because for any a>0, the function F a is increasing in 0. We have Fa F for any >0 and > since the function F a is also increasing in a for any >0 Then, with a > 0 and > 0, ax Fa } ax F } (48 Cobining (47 and (48, we get sup log M q q ax F } (49 >0 is arbitrary, so (49 holds at = 0. The function F0 has axiu F0 0. Next we show that log PL>q F0 0 (50 We also need to separate the cases q< and q as in (b. We give out only the proof for q<. Siilar proof follows for q. The arguent used to show (4 now yields PL>q ( 4 (F a Z }] Z q + p (5 a For >, we have a, so [ ( 4 exp (F a Z }] Z q+ p a ( 4 (F a Z }] Z q + p ( 4 (F 0 Z 0 Z }] q + p (5 The second inequality coes fro the fact that for any >0, the function F a is increasing in a. Cobining (5

12 654 Manageent Science 5(, pp , 005 INFORMS and (5, and applying the extension of Varadhan s Lea, we get log PL>q ax 0 q+ p/ F 0 } (53 Because and are arbitrary, the arguent used in (b now shows that log PL>q ax F0 } 0 q p/ = F0 0 (54 (c Recall a = a/ /, = a /. First we show that sup log M q q a (55 We use the upper bound on M q q in (47, which continues to apply. Define a = q p a then a = 0. The function Fa is concave (as can be verified by differentiation and takes its axiu value 0 for a. Let a be the point at which Fa is axiied. We will show that a / a ; i.e., we will show that for all sall >0, we can find large enough that a a a for all >. For this, it suffices to show that F a a a > 0 F a a a < 0 and where F denotes the derivative of F with respect to its second arguent. The second inequality holds because F a a = 0 and a > 0 when q>p. For the first inequality, the ean value theore yields, F a a = F a a F a a a (56 with a a point in a a. For < a, differentiation yields F a = a Ha (57 with Ha = and a + p a + pa + p H a H a = a + pa + p q +q q a + pa + p a + pa + p q Here is the probability density function of the standard noral distribution. If a + p 0, then H a p a a + p q + q q a + p + p a a + p q = a + p p a q > 0 a + p The inequality coes fro the fact that x/x x for x>0 (Feller 968, p. 75. Siilarly, if a + p > 0, then H a > 0. Also, Ha > 0 because H a > 0. Siple algebra also shows that Ha is bounded for a a. Thus, (57 iplies that F a a a C for soe constant C>0. We can now choose = /a C /, so that F a a < / for >. Cobining this with (56, we have F a a a = F a a F ( a a a a > 0 Therefore we have shown that for sall >0, we can find large enough that for >, we have a a a, i.e., a / a. The definitions of a and a give a = Fa a a Fa a a a so we also have ax Fa / a, and then, since a = a/ / and a = / /, ax Fa } x a (58 Applying this to (47 proves (55. Next we show that log PL>q a (59 Siilarly we also need to separate the cases q< and q as in (b. We give out only the proof for q<. Siilar proof follows for q. The arguent leading to (4 gives PL>q 4 (F (a Z }] Z (60 Recall that F a = 0, F a = 0, and arguing as in the proof of the upper bound, F a = c<0 for c soe positive constant. Thus we have F a = c<0 0 and for sufficiently sall, F a > c +.So if we choose = /c + /, then for sufficiently large, F a >. Therefore we have [ 4 exp (F (a Z }] Z ( 4 exp P < Z 4 exp exp ( = ( 4 exp exp c + / (6 The second inequality coes fro the fact that Px Z<x x if x>. Cobining (60 and (6, we have log PL>q ( + log c + / = and because and are arbitrary, the in (59 follows.

13 Manageent Science 5(, pp , 005 INFORMS 655 Proof of Theore 3. First we show that sup log log M x (6 Let denote expectation under the IS distribution. Then M x = [ L >x exp x ZL+ x ZZ exp ( ] Z + [ exp x Zx + x Z Z exp ( ] Z + = [ expf Z exp ( Z + ] (63 with F as in (9 but with q replaced by q. For any, the concavity of F iplies F F + F = F + (64 The equality coes fro the fact that F = because is the unique axiier in (3 with x = x. Substituting (64 into the upper bound in (63, we have [ expf Z exp ( ] Z + [ exp ( F + Z ] Z + = exp ( F (65 Define c log p = We have p = q and F takes its axiu value 0 for all. We show that for all sall >0, we can find large enough that for all >.It suffices to show that F > 0 and F < 0 (66 The second inequality holds because F = 0 and > 0 when p<q. For the first inequality in (66, we have ( F q = p q p ( + p Using the property that x x/x as x (Feller 968, p. 75, we conclude that and q p = O q p = o ( + p = O ( c / Therefore we have F = O ( c / and as 0 <c<, ( = O c log = o ( c / Then for large enough, we obtain (66. We have shown that for all >0, we have / for sufficiently large. It follows that for all > 0 and all sufficiently large we have <F <. Cobining this with (63 and (65, we get M x <exp, and as is arbitrary the sup in (6 follows. Next we show that For any > 0, log log PL >x (67 PL >x PL >x pz = q + PpZ q + (68 Given pz = q + >q, fro the definition of our change of easure, L is binoially distributed with paraeters and q +. Applying the bound (3.6 of Johnson et al. (993, we have PL >x pz = q + ( q + q It therefore suffices to consider the second factor in (68. If we take ( ( = c log + c log for soe >0, then q + = c log +. Thus, PpZ q + ( q = P Z + p ( q P + p Z q + ( exp q + p = ( exp c log + p and log log PpZ q + c log + = log (69 which proves (67. Proof of Theore 4. The proof is siilar to that of Theore 3. First we show that sup For this, set F = Gp where Define log M x (70 ( + p p = q + c =

14 656 Manageent Science 5(, pp , 005 INFORMS Here p = q and F takes its axiu value 0 for all. We show that for sall >0, we can find large enough that for all >. It suffices to show (66. The second inequality in (66 holds because F = 0 and > 0 when p <q. For the first inequality, we have ( F q = p q p ( c (7 Since ( q / ( p = O c and q p = O by applying the property that x/ x x as x to (7, we conclude that ( c F 3/ = O whereas = Oc. We thus obtain (66 for all sufficiently large. It follows that for >, /, and correspondingly <F <. Cobining this with (63 and (65, we have M x < exp. Since is arbitrary, we obtain the sup in (70. Next we show that For arbitrary >0, log PL >x (7 PL >x PL >x p Z=q +Pp Z q + (73 Given p Z = q + >q, fro the definition of our change of easure, L is binoial distributed with paraeter and q +. Applying the bound (3.6 of Johnson et al. (993, we have PL >x p Z = q + ( q + q (74 Also notice that for any constant c 0 > 0, Pp Z q + ( q + p = P Z ( q + p P Z q + p + c 0 ( exp ( q + p + c 0 c 0 = ( exp ( q + + c + c 0 c 0 (75 Cobining (73, (74, and (75, we obtain the in (7. By Jensen s inequality, M x PL >x, so the sup in (70 and the in (7 hold as its. References Avraidis, A. 00. Iportance sapling for ultiodal functions and application to pricing exotic options. Proc. Winter Siulation Conf., I Press, Piscataway, NJ, Avranitis, A., J. Gregory. 00. Credit: The Coplete Guide to Pricing, Hedging and Risk Manageent. Risk Books, London, UK. Debo, A., O. Zeitouni Large Deviations Techniques and Applications, nd ed. Springer-Verlag, New York. Feller, W An Introduction to Probability Theory and Its Applications, 3rd ed. John Wiley and Sons, New York. Finger, C. 00. nhancing Monte Carlo for credit portfolio odels. Presentation at The International Centre for Business Inforation Global Derivatives Conference, Juan-les-Pins, France. Glasseran, P Monte Carlo Methods in Financial ngineering. Springer, New York. Glasseran, P., J. Li Iportance sapling for a ixed Poisson odel of portfolio credit risk. Proc. Winter Siulation Conf., I Press, Piscataway, NJ. Glasseran, P., P. Heidelberger, P. Shahabuddin Asyptotically optial iportance sapling and stratification for pricing path-dependent options. Math. Finance Gupton, G., C. Finger, M. Bhatia CreditMetrics technical docuent. J. P. Morgan & Co., New York. Heidelberger, P Fast siulation of rare events in queueing and reliability odels. ACM Trans. Model. Coput. Siulation Jensen, J. L Saddlepoint Approxiations. Oxford University Press, Oxford, UK. Johnson, N. L., S. Kot, A. W. Kep Univariate Discrete Distributions, nd ed. John Wiley and Sons, New York. Kalkbrener, M., H. Lotter, L. Overbeck Sensible and efficient capital allocation for credit portfolios. Deutsche Bank, AG. Li, D On default correlation: A copula function approach. J. Fixed Incoe Martin, R., K. Thopson, C. Browne. 00. Taking to the saddle. Risk 4( Merton, R. C On the pricing of corporate debt: The risk structure of interest rates. J. Finance Sadowsky, J. S., J. A. Bucklew On large deviations theory and asyptotically efficient Monte Carlo estiation. I Trans. Infor. Theory Wilde, T CreditRisk + : A CreditRisk Manageent Fraework. Credit Suisse First Boston, London, UK.