Efficient calculation of expected shortfall contributions in large credit portfolios

Transcription

1 Effcent calculaton of expected shortfall contrbutons n large credt portfolos Mchael Kalkbrener 1, Anna Kennedy 1, Monka Popp 2 November 26, 2007 Abstract In the framework of a standard structural credt portfolo model, we nvestgate the Monte Carlo based estmaton of captal allocaton accordng to expected shortfall. We develop and analyze several varance reducton technques based on mportance samplng, analytc approxmatons of portfolo loss dstrbutons and a sem-analytcal allocaton technque. The focus of the paper s on the applcaton of these technques to large credt portfolos used n economc captal calculatons. Our results show that the nherent numercal problems of expected shortfall allocaton can be overcome and, as a consequence, economc captal allocaton accordng to expected shortfall s a vable opton for fnancal nsttutons. Key words: Monte Carlo smulaton, varance reducton, mportance samplng, portfolo credt rsk, expected shortfall allocaton 1 Introducton In a typcal bank, rsk captal for credt rsk far outweghs captal requrements for any other rsk class. Key drvers of credt rsk are concentratons n a bank s credt portfolo. These rsk concentratons may be caused by materal concentratons of exposure to ndvdual names as well as large exposures to sngle sectors (geographc regons or ndustres) or to several hghly correlated sectors. The most common approach to ntroduce sector concentraton nto a credt portfolo model s through systematc factors affectng multple borrowers. Condtonal on the systematc factors, the resdual default rsks of ndvdual borrowers are consdered ndependent and modeled by specfc (or dosyncratc) rsk factors. 3 In ths model, the credt worthness of each borrower s defned by a so-called Ablty-to-Pay varable that s 1 Deutsche Bank AG, Rsk Analytcs & Instruments. The vews expressed n ths artcle are the authors personal opnons and should not be construed as beng endorsed by Deutsche Bank. 2 Unversty of Lepzg, Department of Mathematcs. 3 In ths paper, we assume that the systematc and specfc factors follow a mult-varate normal dstrbuton as proposed by Gupton et al. (1997) n CredtMetrcs. We wll refer to ths model class as Gaussan mult-factor models. See Crouhy et al. (2000) and Bluhm et al. (2002) for a survey on credt portfolo modelng.

2 completely specfed by the systematc rsk factors and the specfc rsk factor of the borrower. In partcular, default and ratng of a borrower at the end of the plannng perod are determned by the value of ts Ablty-to-Pay varable. 4 Ths credt portfolo model captures credt losses due to default and due to ratng mgraton and provdes an approprate framework for assessng credt rsk at dfferent levels of the bank. Top level: quantfcaton of the rsk n the bank s credt portfolo, whch s usually expressed as the bank s economc captal for credt rsk. Lower levels: economc captal allocaton to subportfolos and ndvdual transactons. The standard approach n the fnance ndustry s to defne the economc captal n terms of a quantle of the portfolo loss dstrbuton. The captal charge of an ndvdual transacton s usually based on a covarance technque and called volatlty contrbuton. However, there s theoretcal and practcal evdence that the combnaton of quantles and covarances s not a satsfactory approach to rsk measurement and captal allocaton n credt portfolos (Kalkbrener et al., 2004). An alternatve defnton of economc captal s based on the expected shortfall, whch can ntutvely be nterpreted as the average of all losses above a gven quantle of the loss dstrbuton. It s well known that expected shortfall satsfes the axoms of coherent rsk measures (Acerb and Tasche, 2002) proposed n Artzner et al. (1999). Moreover, there s a natural way to allocate the expected shortfall of the portfolo: the expected shortfall contrbuton of a transacton s ts average contrbuton to the portfolo losses above the specfed quantle. It has been demonstrated n Kalkbrener et al. (2004) that expected shortfall allocaton detects concentraton rsk more accurately than covarance technques. Despte ts theoretcal and practcal advantages, there s a major obstacle to the applcaton of expected shortfall allocaton n Gaussan mult-factor models. Snce the loss dstrbutons of the portfolo and sngle transactons are not tractable n analytcal form, Monte Carlo technques are the standard approach to the actual calculaton of expected shortfall contrbutons. It s easy to see that due to statstcal fluctuatons the smulaton-based estmaton of ths condtonal expectaton s a demandng computatonal problem, 5 n partcular for large portfolos. The objectve of ths paper s the development of varance reducton technques for expected shortfall that make t practcally feasble to allocate economc captal to ndvdual transactons. The economc captal of a bank s derved from the loss dstrbuton of ts entre credt portfolo. Even after the applcaton of segmentaton 4 The mult-state model dstngushes between dfferent ratngs of a borrower whereas the twostate model only dentfes default and non-default events. 5 We refer to Kurth and Tasche (2003) for a computatonal approach to expected shortfall n the analytc framework of CredtRsk+. 2

3 technques, ths portfolo may consst of more than dfferent borrowers. As a consequence, 100 systematc and specfc factors are a realstc set-up for the economc captal calculaton n a bank. The focus of ths paper s therefore on the development of effcent numercal technques for expected shortfall allocaton that are talored to portfolos of that sze. A Gaussan mult-factor model wth ratng mgraton serves as quanttatve framework for the development of the algorthms. An mportant feature of ths model s the large number of ndependent specfc factors. Ths property can be utlzed by splttng the calculaton of expected shortfall contrbutons nto two steps (compare to Glasserman and L (2005) or McNel et al. (2005) n the more general context of Mxture Models): 1. Smulaton of systematc factors. Typcally, these factors are the man drvers for large portfolo losses. Effcent varance reducton technques are therefore partcularly mportant for the smulaton of the systematc factors. 2. Calculaton of expected shortfall contrbutons n each systematc scenaro. Condtonal on a systematc scenaro, loss varables of ndvdual borrowers are ndependent. There exst several optons how to explot condtonal ndependence for stablzng expected shortfall contrbutons. We use an mportance samplng technque to mprove the Monte Carlo smulaton of the systematc factors. In the credt rsk lterature, mportance samplng has been recently suggested n a number of papers, see Avrants and Gregory (2001), Glasserman and L (2005), Kalkbrener et al. (2004), Merno and Nyfelder (2004), Morokoff (2004), Glasserman (2005), Egloff et al. (2005) and Glasserman et al. (2007). In contrast to straghtforward Monte Carlo smulaton, mportance samplng puts more weght on the sample range of nterest, thereby makng the smulaton more effcent. However, t s generally far from obvous how such a change of measure should be obtaned n a practcal manner. In a Gaussan mult-factor model, a natural mportance samplng measure s a negatve shft of the systematc factors: a negatve shft enforces a hgher number of defaults and therefore ncreases the stablty of the MC estmate of expected shortfall. For calculatng the shft, Glasserman and L (2005) mnmze an upper bound on the second moment of the mportance samplng estmator of the tal probablty. Furthermore, they show that the correspondng mportance samplng scheme s asymptotcally optmal. Glasserman et al. (2007) use large devaton analyss to calculate multple mean shfts. Egloff et al. (2005) suggest an adaptve mportance samplng technque that uses the Robbns-Monro stochastc approxmaton method. Our approach s based on the nfnte granularty approxmaton of the portfolo loss dstrbuton (compare to Vascek (2002) and Gordy (2003)). More precsely, we approxmate the orgnal portfolo P by a homogeneous and nfntely granular portfolo P. The loss dstrbuton of P can be 3

4 specfed by a Gaussan sngle-factor model. The calculaton of the shft of the systematc factors s now done n two steps: n the frst step, we calculate the optmal mean n ths sngle-factor settng and then lft the scalar mean to a mean vector for the systematc factors n the orgnal mult-factor model. The effcency of the proposed mportance samplng scheme clearly depends on the qualty of the nfnte granularty approxmaton. By defnton, the analytc loss dstrbuton of the nfntely granular portfolo provdes an excellent ft to portfolo loss dstrbutons of large and well-dversfed portfolos. Snce these portfolo characterstcs are typcal for the credt portfolo of a large nternatonal bank, we experenced sgnfcant mprovements n the stablty of the economc captal calculatons: appled to a realstc test portfolo of loans t reduces the varance of the Monte Carlo estmate of expected shortfall - and therefore the number of requred smulatons - by a factor of 400. The average varance reducton experenced for expected shortfall contrbutons of ndvdual loans s of the order of 150. The second class of varance reducton technques presented n ths paper utlzes the ndependence of specfc rsk factors. We have combned dfferent approaches wth mportance samplng of systematc factors: 1. mportance samplng of specfc factors based on exponental twstng of default probabltes (Glasserman and L, 2005; Merno and Nyfeler, 2004), 2. analytc approxmatons of condtonal loss dstrbutons motvated by the applcaton of the central lmt theorem and 3. determnstc calculaton of the expected shortfall contrbuton of the -th borrower n scenaros where values of all systematc factors and all but the -th specfc factor have been smulated (compare to Merno and Nyfeler (2004)). The loan portfolo and a smaller portfolo of 1000 loans have been used for comparng the varance reductons for MC estmates of expected shortfall contrbutons obtaned by these technques. Our results ndcate that the last approach,.e. mportance samplng of systematc factors together wth condtonal allocaton, s partcularly well suted for large portfolos: on average, the varance of expected shortfall contrbutons of ndvdual loans s reduced by a factor of The paper has the followng structure. For the sake of smplcty we ntally develop all varance reducton technques n a Gaussan mult-factor model that does not dstngush between dfferent ratng states but only between default and non-default. Ths basc two-state model s presented n Secton 2.1. Secton 2.2 revews analytc technques for approxmatng the portfolo loss dstrbuton n the two-state model. Expected shortfall allocaton s formally ntroduced n Secton 3. Secton 4 s devoted to the development of the mportance samplng technque for systematc rsk factors. Varance reducton technques are appled to specfc rsk factors n Secton 5: mportance samplng, condtonal expected shortfall allocaton 4

5 and Gaussan approxmaton technques. Numercal results are presented n Secton 6. Secton 7 ntroduces the ratng mgraton model and generalzes the proposed technques to ths mult-state framework. 2 Loss dstrbutons of Gaussan mult-factor models 2.1 The two-state credt portfolo model For the sake of smplcty we develop and analyze varance reducton technques n the framework of a credt portfolo model n default-only mode. We refer to Secton 7 for generalzatons to models whch ncorporate ratng mgraton. The credt portfolo P conssts of n loans. Wth each loan we assocate an Abltyto-Pay varable A : R m+1 R, whch s a lnear combnaton of the m systematc varables x 1,..., x m and a specfc varable z : A (x 1,..., x m, z ) := m j=1 φ j x j + 1 R 2z (1) wth 0 R 2 1 and weght vector (φ 1,..., φ m ). The loan loss L : R m+1 R and the portfolo loss functon L : R m+n R are defned by L := l 1 {A D }, L := L, (2) =1 where 0 < l and D R are the (determnstc) loss-at-default and the default threshold respectvely. As probablty measure P on R m+n we use the product measure n P := N 0,C N 0,1, where N 0,1 s the standardzed one-dmensonal normal dstrbuton and N 0,C the m-dmensonal normal dstrbuton wth mean 0 = (0,..., 0) R m and non-sngular covarance matrx C R m m. Note that each x, z and A s a centered and normally dstrbuted random varable under P. We assume that the weght vector (φ 1,..., φ m ) has been normalzed n such a way that the varance of A s 1. Hence, the default probablty p of the -th loan equals =1 p := P(A D ) = N(D ), where N denotes the standardzed one-dmensonal normal dstrbuton functon. Ths relaton s used to determne the default threshold from emprcal default probabltes. 5

6 2.2 Analytc approxmatons The portfolo loss dstrbuton L defned n (2) can be consdered as a dscrete dstrbuton on a hgh dmensonal state space wth 2 n default/non-default states. It does not have an analytc form. Monte Carlo smulaton s the standard technque for the actual calculaton of rsk captal at portfolo and transacton level. However, Monte Carlo estmates of rsk measures derved from the tal of the dstrbuton - lke value-at-rsk or expected shortfall - tend to be numercally unstable n ths credt portfolo model. Analytc approxmatons have therefore been proposed 1. to calculate portfolo rsk and rsk contrbutons n a purely analytcal way or 2. for the development of mportance samplng technques n order to mprove Monte Carlo stablty. The mportance samplng technque proposed n ths paper utlzes the nfnte granularty approxmaton of loss dstrbutons of homogeneous portfolos Infnte granularty approxmaton for homogeneous portfolos Let χ = (χ 1,..., χ m ) R m be values of the m systematc varables. The specfc varables z are ndependent and therefore the L are ndependent on {x = χ}. Hence, f n s suffcently large the portfolo loss functon L = n =1 L can be approxmated on {x = χ} by applyng a lmt theorem to L/s, where s s an approprate scalng factor. The most straghtforward approxmaton s based on applyng the law of large numbers to (1/n) n =1 L,.e. the portfolo loss functon L s approxmated by ts condtonal mean on {x = χ}. Consder now a homogeneous portfolo P,.e. each loan has the same loss-atdefault l, default probablty p, R 2 and set of factor weghts (ρ 1,..., ρ m ) R m. The applcaton of the above strategy leads to the followng result. 6 Theorem 1 Let the loss functon L of the -th loan n the homogeneous portfolo P be defned by L := l 1 { Ā N 1 (p)}, where Ā denotes the -th homogeneous Ablty-to-Pay varable Then Ā (x 1,..., x m, z ) := lm (1/n) L n =1 = l N m ρ j x j + 1 R 2 z. j=1 ( N 1 (p) m j=1 ρ ) jx j 1 R 2 6 We refer to Vascek (1991) for a proof. Generalzatons are gven n Bluhm et al. (2002) and McNel et al. (2005). 6

7 holds almost surely on Ω. Note that f the lnear sum m j=1 ρ jx j of the systematc varables n the homogeneous m-factor model s consdered as one systematc factor then the m-factor model s transformed nto a one-factor model wth Ablty-to-Pay varables R 2 x + 1 R 2 z. In order to utlze ths analytc dstrbuton n our nhomogeneous mult-factor settng, the orgnal portfolo P has to be approxmated by a homogeneous and nfntely granular portfolo P. However, there s no unque procedure to establsh the homogeneous portfolo, whch s closest to a gven portfolo. We propose the followng technque for determnng the parameters of the homogeneous portfolo P,.e. loss-at-default l, default probablty p, R 2 and factor weghts ρ j, j = 1,..., m: Loss and default probablty. The homogeneous loss l s the average of the ndvdual losses l and the homogeneous default probablty p s the loss-at-default weghted default probablty of all loans n the portfolo: l := n =1 l, p := n n =1 p l n =1 l. (3) Weght vector. The homogeneous weght vector s the normalzed, weghted sum of the weght vectors of the ndvdual loans. In ths paper, the postve weghts g 1,..., g n R are gven by g := E(L ) = p l,.e. the -th weght equals the -th expected loss, and the homogeneous weght vector ρ = (ρ 1,..., ρ m ) s defned by ρ := ψ/s wth ψ = (ψ 1,..., ψ m ) := g (φ 1,..., φ m ). (4) =1 The scalng factor s R s chosen such that R 2 = holds, where R 2 s defned n (6). m ρ ρ j Cov(x, x j ) (5),j=1 R 2. The specfcaton of the homogeneous R 2 s based on the condton that the weghted sum of Ablty-to-Pay covarances s dentcal n the orgnal and the homogeneous portfolo. More precsely, defne R 2 := m k,l=1 ψ kψ l Cov(x k, x l ) n =1 g2 R2 ( n =1 g ) 2 n =1 g2 7 (6)

8 and the -th homogeneous Ablty-to-Pay varable by Ā (x 1,..., x m, z ) := m ρ j x j + 1 R 2 z. The specfcaton of the homogeneous R 2 s motvated by the followng result. j=1 Proposton 1 Equalty (7) holds for the weghted sum of Ablty-to-Pay covarances of the orgnal and the homogeneous portfolo: g g j Cov(A, A j ) =,j=1 g g j Cov(Ā, Āj). (7),j=1 Proof: We have g g j Cov(A, A j ) =,j=1 = m g φ k g j φ jl Cov(x k, x l ) + g 2 (1 R 2 ),j=1 k,l=1 m ψ k ψ l Cov(x k, x l ) + k,l=1 =1 =1 g 2 (1 R 2 ) (8) and, by (5), g g j Cov(Ā, Āj) =,j=1 = g g j,j=1 m k,l=1 g g j R 2 +,j=1 =1 ρ k ρ l Cov(x k, x l ) + g 2 (1 R 2 ) =1 g 2 (1 R 2 ). (9) If R 2 s defned by (6) then (8) equals (9) and the proposton s proved. Alternatve approaches to portfolo homogenzaton are proposed n Glasserman (2004). He presents two homogeneous sngle-factor approxmatons based on 1. matchng mean and varance of the loss dstrbutons or 2. approxmatons of the decay rate of the dstrbuton tal P(L > c). Compared to Glasserman s technques, the heurstc (6) has the advantage that t s extremely fast, even for large portfolos. We refer to Glasserman (2004) for a comparson of the three technques. 8

9 2.2.2 Moment generatng functons and saddle-ponts Saddle-pont approxmatons are another frequently used analytcal technque. Followng Glasserman and L (2005), we have mplemented an mportance samplng technque for specfc factors that uses saddle-ponts (see Secton 5.1). Here we brefly revew the defnton of saddle-ponts and ther calculaton n the portfolo model ntroduced n Secton 2.1. For a random varable X, the cumulant generatng functon (KGF) s defned as ψ X (θ) = log(e(e θx )) for complex θ. The tal probablty of X can be recovered from the KGF by a contour ntegral P(X > c) = 1 + e ψ X(θ) θ c dθ, (10) 2π θ n whch the path of ntegraton s up the magnary axs and runs to the rght of the orgn to avod the pole there. The KGF s a useful constructon because when ndependent random varables are added, ther KGFs are added. Ths feature s mportant for calculatng the KGF ψ L (θ, χ) := ψ L x=χ (θ) of the portfolo loss dstrbuton L condtonal on gven values χ = (χ 1,..., χ m ) R m of the systematc factors: on {x = χ}, the Ablty-to-Pay varables A 1,..., A n become ndependent wth condtonal default probabltes p (χ) := P(A D x = χ) = N N 1 (p ) m j=1 φ jχ j (11) 1 R 2 and therefore ψ L (θ, χ) = ψ L (θ, χ) = =1 log(1 + p (χ)(e θ l 1)). =1 On the real axs, ψ L (θ, χ) θc has a unque mnmum θ c (χ), the saddle-pont, that can be wrtten as { unque θ such that θ c (χ) = θ ψ L(θ, χ) = c f c > E(L x = χ), 0 f c E(L x = χ). Referrng to equaton (10), Martn et al. (2001a, 2001b) approxmate P(L > c) by a Taylor seres expanson of ψ L (θ, χ) θc around the saddle-pont θ c (χ). They show that ths technque works very well for partcular classes of credt portfolo 9

10 models. However, for Gaussan mult-factor models, ths approach would requre the calculaton of a multdmensonal ntegral whose dmenson corresponds to the number of systematc factors. For our purposes, ths s a hopeless task (Martn et al., 2001a). Applcaton of saddle-pont technques n the one-factor Vascek model can be found n Huang et al. (2006). Glasserman and L (2005) use saddle-ponts n ther mportance samplng for systematc as well as specfc factors. More detals are gven n Secton Coherent rsk measurement and captal allocaton The objectve of ths secton s the formal defnton of rsk measures and allocaton schemes, n partcular expected shortfall allocaton. After JP Morgan made ts RskMetrcs system publc n 1994 value-at-rsk became the domnant concept for rsk measurement. The value-at-rsk VaR α (L) of L at level α (0, 1) s defned as an α-quantle of L. More precsely, n ths paper VaR α (L) := nf{x R P(L x) α} s the smallest α-quantle. Whle the VaR methodology encourages dversfcaton for the specal case of an ellptcally dstrbuted random vector (X, Y ),.e. VaR(X + Y ) VaR(X) + VaR(Y ) (12) (McNel et al. 2005), n general subaddtvty (12) does not hold for value-at-rsk. Snce for typcal credt portfolos the assumpton of an ellptcal dstrbuton cannot be mantaned, dversfcaton, whch s commonly consdered as a way to reduce rsk, may ncrease value-at-rsk. Another dsadvantage of value-at-rsk s that the allocaton of portfolo VaR to subportfolos and ndvdual transactons s dffcult n credt portfolo models wth dscrete loss dstrbutons (Kalkbrener, 2005). The standard soluton s to allocate portfolo VaR proportonal to the covarances Cov(L 1, L),..., Cov(L n, L). (13) Ths allocaton technque, called volatlty allocaton, s the natural choce n classcal portfolo theory where portfolo rsk s measured by standard devaton (or volatlty). In general, combnng volatlty allocaton wth value-at-rsk works well as long as all loss dstrbutons are close to normal. However, for credt portfolos t does not: the captal allocated to a subportfolo P of P mght be greater than the rsk captal of P consdered as a stand-alone portfolo, the captal charge of a loan mght even be hgher than ts exposure (Kalkbrener et al., 2004). 10

11 An alternatve rsk measure s expected shortfall (see, for nstance, Rockafellar and Uryasev, 2000; Acerb and Tasche, 2002): the expected shortfall of L at level α s defned by 1 ES α (L) := (1 α) 1 VaR u (L)du. An equvalent defnton of expected shortfall s ES α (L) = (1 α) 1 (E(L 1 {L>VaRα(L)}) + VaR α (L) (P(L VaR α (L)) α)). (14) It s easy to see that for most loss dstrbutons the expected shortfall ES α s domnated by the frst term α E(L L > VaR α (L)) = (1 α) 1 E(L 1 {L>VaRα(L)}). (15) Intutvely, expected shortfall can therefore be nterpreted as the average of all losses above a gven quantle of the loss dstrbuton. The second term n (14) takes care of jumps of the loss dstrbuton at ts quantle and ensures coherence as defned n Artzner et al. (1997, 1999). In partcular, the subaddtvty property (12) holds for expected shortfall. Another mportant advantage of expected shortfall s the smple allocaton of rsk captal to subportfolos or ndvdual transactons: n accordance wth (14) the expected shortfall contrbuton of the -th loan s defned as wth ESC α (L, L) := (1 α) 1 (E(L 1 {L>VaRα(L)}) + β L E(L 1 {L=VaRα(L)})) (16) β L := P(L VaR α(l)) α. P(L = VaR α (L)) Agan the defnton (16) s usually domnated by ts frst term E(L L > VaR α (L)) = (1 α) 1 E(L 1 {L>VaRα(L)}). (17) Hence, the expected shortfall contrbuton of a loan can be consdered as ts average contrbuton to portfolo losses above quantle VaR α (L). 4 Importance samplng appled to systematc factors Monte Carlo smulaton s the standard technque for the actual calculaton of expected shortfall at portfolo and transacton level n the Gaussan mult-factor model presented n Secton 2.1. The man practcal problem n applyng expected shortfall to realstc credt portfolos s the computaton of numercally stable MC estmates. In the rest of the paper, we present technques to reduce the varance of Monte Carlo smulaton: 11

12 1. n ths secton, mportance samplng s appled to the Monte Carlo smulaton of systematc factors, 2. n Secton 5, varance reducton technques are developed that utlze the ndependence of loss varables of ndvdual borrowers condtonal on a systematc scenaro. 4.1 Straghtforward Monte Carlo smulaton The effcent computaton of expected shortfall (15) and expected shortfall contrbutons (17) s a challengng task for realstc portfolos and hgh confdence levels α. Straghtforward Monte Carlo smulaton does not work well due to the hgh varance of L 1 {L>VaRα(L)} and L 1 {L>VaRα(L)} respectvely (see (21) and Sectons 4.5 and 6). As an example, assume that we want to compute expected shortfall wth respect to the α = 99.9% quantle and compute ν = MC samples s 1 s 2... s ν of the portfolo loss L. Then ES α (L) becomes (1 α) 1 E(L 1 {L>c} ) = (1 α) L 1 {L>c} dp = s /100, (18) where c := VaR α (L). Snce the computaton of ES α (L) s only based on 100 samples t s subject to large statstcal fluctuatons and numercally unstable. Ths s even more true for expected shortfall contrbutons of ndvdual loans. A sgnfcantly hgher number of samples has to be computed whch makes straghtforward MC smulaton mpractcable for large credt portfolos. 4.2 Monte Carlo smulaton based on mportance samplng Importance samplng s a technque for reducng the varance of MC smulatons and - as a consequence - the number of samples requred for stable results. In our settng, the ntegral n (18) s replaced by the equvalent ntegral on the rght-hand sde of the equaton L 1 {L>c} dp = L 1 {L>c} f d P, (19) where P s contnuous wth respect to the probablty measure P and has densty f. The objectve s to choose P n such a way that the varance of the Monte Carlo estmate for the ntegral (19) s mnmal under P. Ths MC estmate s =1 ES α (L) ν, P := 1 ν ν L P() 1 {L P()>c} f(), (20) =1 where L P() s a realzaton of the portfolo loss L under the probablty measure P and f() s the correspondng value of the densty functon. 12

13 By the strong law of large numbers and the central lmt theorem, ES α (L) ν, P converges to (19) almost surely as ν and the samplng error converges as d ν (ESα (L) ν, P L 1 {L>c} dp) N(0, σ ESα(L)( P)), (21) where σ 2 ES α(l) ( P) s the varance of L 1 {L>c} f under P, that s: (L σes 2 ( P) α(l) = 1{L>c} f ) ( 2 d P L 1 {L>c} dp) 2. (22) In the followng we restrct the set of probablty measures P, whch we consder to determne a mnmum of (22): for every M = (M 1,..., M m ) R m defne the probablty measure P M by 7 P M := N M,C n N 0,1, (23) where N M,C s the m-dmensonal normal dstrbuton wth mean M and covarance matrx C. In other words, those probablty measures are consdered whch only change the mean of the systematc components x 1,..., x m n the defnton of the Ablty-to-Pay varables A 1,..., A n. Ths choce s motvated by the nature of the problem. The MC estmate (20) can be mproved by ncreasng the number of scenaros that lead to hgh portfolo losses,.e. portfolo losses above threshold c. Ths can be realzed by generatng a suffcently large number of defaults n each sample. Snce defaults occur when Ablty-to-Pay varables fall below default thresholds we can enforce a hgh number of defaults by addng a negatve mean to the systematc components. Havng thus restrcted mportance samplng to measures of the form (23) we consder σ 2 ES α(l) as a functon from Rm to R and rephrase The Varance Reducton Problem: compute a mnmum M = (M 1,..., M m ) of the varance ( ) 2 ( 2 σes 2 (M) = α(l) L 1 {L>c} n0,c dp M L 1 n {L>c} dp) (24) M,C n R m, where n 0,C and n M,C denote the probablty densty functons of N 0,C and N M,C respectvely. =1 We can formulate the mnmzaton condton as M σes 2 α(l) (M) = 0, = 1,..., m. (25) However, for realstc portfolos wth thousands of loans ths system s analytcally and numercally ntractable. 7 Note that the ntal measure P equals P 0. 13

14 4.3 Approxmaton by a homogeneous portfolo To progress we therefore approxmate the orgnal portfolo P by a homogeneous and nfntely granular portfolo P as descrbed n Secton 2.2. Based on Theorem 1 we defne the functon L : R R by ( N L 1 ) (p) x (x) := n l N (26) 1 R 2 and approxmate the portfolo loss functon L(x 1,..., x m, z 1,..., z n ) of the orgnal portfolo P by the loss functon m L m (x 1,..., x m ) := L ρ j x j (27) of the homogeneous and nfntely granular portfolo. The threshold c := VaR α (L m ) s defned as the α-quantle of L m wth respect to the m-dmensonal Gaussan measure N 0,C and σes 2 α(l m )(M) denotes the varance of j=1 L m 1 {L m >c } n0,c n M,C wth respect to N M,C. By approxmatng the fnte nhomogeneous portfolo P by an nfnte homogeneous portfolo we have transformed the varance reducton problem (24) to The Varance Reducton Problem for Infnte Homogeneous Portfolos: compute a mnmum M = (M 1,..., M m ) of the varance ( ) 2 σes 2 α(l m ) (M) = L m 1 {L m >c } n0,c dn M,C ( n M,C n R m. L m 1 {L m >c } dn 0,C ) 2 Note that we have acheved a sgnfcant reducton of complexty: the dmenson of the underlyng probablty space has been reduced from m + n to m and the loss functon L m s not a large sum but has a concse analytc form. We emphasze, however, that ths approxmaton technque s only used for determnng a mean vector M for mportance samplng. The actual calculatons of expected shortfall and expected shortfall contrbutons are based on Monte Carlo smulaton of the full portfolo model as specfed n Secton 2.1. In the next subsecton we wll present a smple and effcent algorthm whch solves the varance reducton problem for nfnte homogeneous portfolos wth arbtrary precson. (28) 14

15 4.4 Optmal mean for nfnte homogeneous portfolos The computaton of the mnmum of (28) s done n two steps: One-factor model: Instead of m systematc factors x 1,..., x m we consder the correspondng one-factor model and compute the mnmum µ (1) R of (28) n the case m = 1. We wll show that µ (1) s the mnmum of N 1 (1 α) (L 1 n 0,1) 2 dx. n M,1 Mult-factor model: The one-dmensonal mnmum µ (1) can be lfted to the m- dmensonal mnmum µ (m) = (µ (m) 1,..., µ (m) m ) of (28) by The one-factor model µ (m) := µ(1) m j=1 Cov(x, x j ) ρ j. (29) R 2 If the lnear sum m j=1 ρ jx j of the systematc varables n the homogeneous m-factor model s consdered as one systematc factor then the m-factor model s transformed nto a one-factor model wth Ablty-to-Pay varables R 2 x + 1 R 2 z and analytc approxmaton of the portfolo loss functon L 1 (x) := L ( R 2 x). Equaton (5) mples that the threshold c = VaR α (L m ) equals VaR α (L 1 ), the α-quantle of L 1 wth respect to the one-dmensonal Gaussan dstrbuton N 0,1. Snce L 1 (x) s monotonous, {x R L 1 (x) > c } = {x R x < N 1 (1 α)}. Hence, the varance σ 2 ES α(l 1 )(M) of under N M,1 can be wrtten as L 1 1 {L 1 >c } n0,1 n M,1 σ 2 ES α(l 1 )(M) = (L 1 1 {L 1 >c } n 0,1 ) 2 = N 1 (1 α) n M,1 ( dx L 1 ) 2 1 {L 1 >c } n 0,1 dx ( (L 1 n 0,1) 2 N 1 2 (1 α) dx L 1 n 0,1 dx). (30) n M,1 15

16 Snce the second ntegral n (30) does not depend on M, t suffces to compute a mnmum µ (1) of N 1 (1 α) (L 1 n 0,1) 2 dx. n M,1 Ths can be easly done by applyng numercal technques. The mult-factor model In order to solve the mnmzaton problem (28), µ (1) has to be transformed nto an m-dmensonal vector. The followng theorem shows that for nfnte homogeneous portfolos the vector µ (m) = (µ (m) 1,..., µ (m) m ) computed n (29) s optmal. Theorem 2 Let µ (1) R be a mnmum of σ 2 ES α(l 1 ) and µ(m) = (µ (m) 1,..., µ (m) m ) be defned by (29). Then σ 2 ES α(l m )(µ(m) ) = mn{σ 2 ES α(l m )(M) M Rm }. The proof of ths theorem s based on the followng proposton. It provdes a general technque for reducng the mnmzaton problem for a specfc class of multvarate ntegrals to the mnmzaton of one-dmensonal ntegrals. In ths proposton the varance and standard devaton of a random varable U : R m R wth respect to N 0,C are denoted by σ 2 (U) and σ(u) respectvely. C 1 and C T are the nverse and transpose of the matrx C. Proposton 2 Let A : R R + be a real-valued, non-negatve functon and M = (M 1,..., M m ) R m. Defne µ R by µ := Cov(U, V ) σ(u) = Cov(U, V ), (31) R 2 where U : R m R and V : R m R are the random varables m m U(x 1,..., x m ) := ρ j x j, V (x 1,..., x m ) := x M j Cj 1. j=1,j=1 Then m A ρ j x j n0,c(x 1,..., x m ) n M,C (x 1,..., x m ) dn 0,C j=1 ( ) A R2 x n0,1(x) n µ,1 (x) dn 0,1. (32) Equalty holds n (32) f and only f A = 0 a.s. or M and µ satsfy the addtonal equaton σ 2 (V ) = µ 2. (33) 16

17 Proof: Snce C 1 s symmetrc, V 2 can be wrtten n the form V 2 = (M T C 1 x)(x T C 1 M) and therefore Hence, σ 2 (V ) = M T C 1 CC 1 M = m,j=1 M M j C 1 j. n 0,C (x 1,..., x m ) n M,C (x 1,..., x m ) = e (1/2)( m,j=1 (x M )(x j M j )C 1 j m,j=1 x x j C 1 j ) = e (1/2)σ2 (V ) V. (34) Note that (U(x 1,..., x m ), V (x 1,..., x m )) and ( R 2 x 1, µ x 1 + σ 2 (V ) µ 2 x 2 ) are both 2-dmensonal Gaussan varables wth the same jont dstrbuton f consdered as random varables on (R m, N 0,C ) and (R 2, N 0,I ) respectvely, where I denotes the dentty matrx n R 2 2. It follows from (34) and the ndependence of x 1 and x 2 on (R 2, N 0,I ) that where B := m A( ρ j x j ) n0,c(x 1,..., x m ) n M,C (x 1,..., x m ) dn 0,C = j=1 A(U(x 1,..., x m )) e (1/2)σ2 (V ) V (x 1,...,x m) dn 0,C = A( R 2 x 1 ) e (1/2)σ2 (V ) (µ x 1 + σ 2 (V ) µ 2 x 2 ) dn 0,I = B B A( R 2 x) e (1/2)µ2 µ x dn 0,1 = A( R 2 x) n0,1(x) n µ,1 (x) dn 0,1, (35) e (1/2)σ2 (V ) σ 2 (V ) µ 2 x (1/2)µ 2 dn 0,1 = e σ2 (V ) µ 2. It follows from the Cauchy-Schwarz nequalty Cov(U, V ) 2 σ 2 (U) σ 2 (V ) 17

18 that µ 2 σ 2 (V ) and therefore B 1. Hence, m A( ρ j x j ) n0,c(x 1,..., x m ) n M,C (x 1,..., x m ) dn 0,C j=1 A( R 2 x) n0,1(x) n µ,1 (x) dn 0,1. Equalty holds f and only f A = 0 a.s. or B = 1 whch s equvalent to σ 2 (V ) = µ 2. Proof of Theorem 2: Defne the real-valued, non-negatve functon A : R R + by A(x) := (L (x) 1 {L (x)>c }) 2. Note that t follows from (5) and the defnton (29) of µ (m) = (µ (m) 1,..., µ (m) m ) that µ (m) and µ (1) satsfy equatons (31) and (33). Let M = (M 1,..., M m ) R m and defne µ R such that (31) s satsfed. By Proposton 2 and the defnton of µ (1), ( ) 2 L n 0,C m m 1 {L m >c } dn = n 0,C A( ρ µ(m),c j x j ) dn 0,C n µ (m),c = = j=1 n µ (m),c A( R 2 n 0,1 (x) x) n µ (1),1 (x) dn 0,1 A( R 2 x) n0,1(x) n µ,1 (x) dn 0,1 m A( ρ j x j ) n0,c dn 0,C n j=1 M,C ( ) 2 L m 1 {L m >c } n0,c dn M,C. n M,C Together wth the representaton (28) of σes 2 α(l m )(M), ths proves the theorem. 4.5 Numercal analyss Importance samplng based on the shft vector (µ (m) 1,..., µ (m) m ) n (29) mnmzes the Monte Carlo samplng fluctuaton for nfntely homogeneous portfolos. We do not know yet, however, whether ths technque leads to a sgnfcant error reducton n the Monte Carlo based estmaton of expected shortfall ES α (L) and expected shortfall contrbutons ESC α (L, L) for realstc portfolos. In order to assess ts mpact on the portfolo rsk measure expected shortfall ES α (L) we apply mportance samplng to a large loan portfolo and calculate the standard devaton of the Monte Carlo estmator for ES α (L) wth α = 99.9%. The test portfolo conssts of loans wth an nhomogeneous exposure and default probablty dstrbuton. The average exposure sze s 0.004% of the total 18

19 exposure and the standard devaton of the exposure sze s 0.026%. Default probabltes vary between 0.02% and 27%. The portfolo expected loss s 0.72% and the unexpected loss,.e. the standard devaton, s 0.87%. Default correlatons are specfed by the KMV factor model (see Kealhofer and Bohn (2001) for a descrpton of the model), comprsng 96 systematc country and ndustry factors. Although the portfolo s relatvely well dversfed there are concentratons caused by exposures to a sngle sector (geographc regon or ndustry) or to several hghly correlated sectors. Name concentratons do not play a domnant role. The test portfolo s a typcal example of a large credt portfolo n an nternatonal bank. We expect that the varance reductons reported n ths paper can be reproduced wth any portfolo of smlar characterstcs. In Fgure 1 we plot the standard devaton of the Monte Carlo estmator for ES (L) as a functon of the norm of the vector (µ (m) 1,..., µ (m) m ). A scalng factor of 0 corresponds to no mportance samplng, whereas a scalng factor of 1 corresponds to (µ (m) 1,..., µ (m) m ). The other ponts represent vectors wth dentcal drecton but a lnearly nterpolated/extrapolated norm,.e. 0.5 corresponds to the vector 0.5 (µ (m) 1,..., µ (m) m ). These results were obtaned from a sample of 40 ndependent Monte Carlo runs of smulaton trals for each scalng factor. On the rght-hand axs we plot the average result from the 40 runs wth the standard devaton n error bars. 8.00% 7.00% 6.00% Standard Error Average ESF 10.20% 10.00% 9.80% Standard Error 5.00% 4.00% 3.00% 9.60% 9.40% 9.20% Average ESF 2.00% 9.00% 1.00% 8.80% 0.00% Importance Samplng Shft 8.60% Fgure 1: Monte Carlo samplng error as a functon of the mportance samplng shft. From these results we draw two conclusons. Frst we have demonstrated that 19

20 mportance samplng can sgnfcantly mprove the qualty of the Monte Carlo estmate of the expected shortfall measure. The varance rato between the optmal pont n the graph and the no-shft case s 400,.e. the same precson wthout any mportance samplng would requre 400 tmes more smulatons. Improvements of a comparable magntude were found for the Monte Carlo estmate of the quantle of the loss dstrbuton,.e. the value-at-rsk. Secondly, we observe that our theoretcal optmal shft sze slghtly overestmates the emprcal optmal shft. Our explanaton for ths s that n our determnaton of the equvalent homogeneous portfolo we have overestmated the average correlaton R 2. Ths has been confrmed by the observaton that the R 2 determned from fttng a Vascek dstrbuton to our best Monte Carlo estmate for the 99.9% quantle s approxmately 10% smaller than the one calculated from the approxmaton procedure n Secton 2.2. Compared to the rsk measure expected shortfall, the calculaton of numercally stable expected shortfall contrbutons of ndvdual loans s an even more challengng task. Importance samplng on systematc factors also leads to a sgnfcant reducton n the volatlty of the ESC α (L, L): n the test calculatons n Secton 6, the varance s reduced by a factor of more than 100 (see Table 2), for 75% of the loans the standard devaton s below 5% f smulatons are calculated (see Fgure 3). However, for a number of transactons n the test portfolo the statstcal fluctuatons of ther expected shortfall contrbutons are stll unacceptably hgh. In the followng secton we obtan further mprovements by utlzng the ndependence of the loan loss varables L 1,..., L n condtonal on gven values χ = (χ 1,..., χ m ) of the systematc varables x = (x 1,..., x m ). 5 Varance reducton based on condtonal ndependence of specfc factors There exst several optons how to explot condtonal ndependence for stablzng expected shortfall contrbutons. In ths secton, we deal wth three dfferent technques: 1. mportance samplng of specfc factors based on exponental twstng of default probabltes, 2. determnstc calculaton of the expected shortfall contrbuton of the -th loan n scenaros where values of all systematc factors and all but the -th specfc factor have been smulated and 3. analytc approxmatons of condtonal loss dstrbutons motvated by the applcaton of the central lmt theorem. Numercal results and comparsons are presented n Secton 6. 20

21 5.1 Importance samplng on specfc factors Condtonal on {x = χ}, Glasserman and L (2005) and Merno and Nyfeler (2004) suggest mportance samplng based on exponental twsts of default probabltes to stablze expected shortfall. More precsely, they consder the related problem of mprovng the MC estmate of the tal probablty P (L > c). It s ntutvely clear that better estmates of P (L > c) can be obtaned by ncreasng the condtonal default probabltes of the ndvdual loans,.e by replacng each condtonal default probablty p (χ) = N N 1 (p ) m j=1 φ jχ j, = 1,..., n, (36) 1 R 2 by a hgher default probablty p (χ). Glasserman and L (2005) prove asymptotc optmalty for the exponental twst p (χ) := p (χ)e θc(χ) l 1 + p (χ)(e θc(χ) l 1), (37) where θ c (χ) denotes the saddle-pont defned n Secton mportance samplng dentty becomes ( n P (L > c) = E p 1 {L>c} =1 ( p (χ) p (χ) Hence, the basc ) A (χ,z ) ( ) ) 1 p (χ) 1 A (χ,z ), 1 p (χ) where E p denotes the expectaton usng the new default probabltes p 1 (χ),..., p n (χ) and A (χ, z ) s the -th Ablty-to-Pay varable restrcted to {x = χ}. 8 It s straghtforward to combne the mportance samplng technques on systematc and specfc factors: 1. Apply mportance samplng to the systematc factors x 1,..., x m and compute samples χ 1 = (χ 11,..., χ 1m ),..., χ k = (χ k1,..., χ km ). 2. For each of the k systematc samples χ j = (χ j1,..., χ jm ): calculate l IS samples of the specfc factors z 1,..., z n usng the default probabltes p (χ j ) n (37). The relatve mportance of both IS technques depends on the characterstcs of the portfolo, n partcular on the degree of correlaton (Glasserman and L, 2005). In our settng, the varance reducton on the systematc factors clearly domnates for the loan portfolo defned n Secton 4.5. Numercal results are presented n Secton 6. 8 In Glasserman (2005), exponental twstng s used to derve an asymptotc approxmaton to condtonal default probabltes. 21

22 5.2 Condtonal expected shortfall allocaton The varance reducton technque presented n ths subsecton can be combned wth mportance samplng on systematc and specfc factors. It utlzes the smple form of E(L L > c), = 1,..., n, condtonal on gven values of the systematc varables x 1,..., x m and the remanng specfc varables z 1,..., z 1, z +1..., z n (compare to Merno and Nyfeler (2004)). Let (χ, σ) = (χ 1,..., χ m, σ 1,..., σ n ) R m+n, {1,..., n}. Our objectve s to calculate E(L L > c) on Ω (χ, σ), where Ω (χ, σ) denotes the subset {x 1 = χ 1,..., x m = χ m, z 1 = σ 1,..., z 1 = σ 1, z +1 = σ +1,..., z n = σ n } of R m+n. Snce j L j s determnstc on Ω (χ, σ) we dstngush three cases: L j > c : E(L 1 {L>c} Ω (χ, σ)) = p (χ)l, P(L > c Ω (χ, σ)) = 1, j c j L j > c l : E(L 1 {L>c} Ω (χ, σ)) = p (χ)l, P(L > c Ω (χ, σ)) = p (χ), c l j L j : E(L 1 {L>c} Ω (χ, σ)) = 0, P(L > c Ω (χ, σ)) = 0, where the condtonal default probablty p (χ) of the -th loan on Ω (χ, σ) s specfed n (36). These smple formulae can be combned wth mportance samplng n order to mprove the stablty of expected shortfall allocaton. Let (χ j, σ j ) = (χ j1,..., χ jm, σ j1,..., σ jn ), j = 1,..., k be k samples of the systematc factors x 1,..., x m and the specfc factors z 1,..., z n and denote the probabltes of the samples by q(χ 1, σ 1 ),..., q(χ k, σ k ). The expected shortfall contrbuton of the -th loan equals E(L L > c) = E(L 1 {L>c} ) P(L > c) k j=1 q(χ j, σ j ) E(L 1 {L>c} Ω (χ j, σ j )) k j=1 q(χ, (38) j, σ j ) P(L > c Ω (χ j, σ j )) where (38) can be easly obtaned from the above formulae. The varance reducton n (38) s due to the fact that the smulaton of the -th specfc factor has been replaced by a determnstc calculaton of E(L L > c) n each scenaro Ω (χ j, σ j ). Ths smple technque s easy to mplement and does not requre much addtonal computng tme. The varance reducton obtaned n our test portfolo s sgnfcant, partcularly n combnaton wth mportance samplng on the systematc factors (see Secton 6). Furthermore, ths technque can be generalzed to mgraton mode n a straghtforward way (Secton 7). 22

23 5.3 Normal approxmatons Calculaton of expected shortfall for Gaussan dstrbutons Condtonal on a systematc scenaro {x = χ}, the volatlty of the expected shortfall estmates E(L L > c) can be completely elmnated f the condtonal portfolo loss s not smulated but approxmated by an analytc dstrbuton. The analytc approxmaton of the portfolo loss L n Theorem 1 has been obtaned by applyng the law of large numbers to the sum of the ndependent loss varables condtonal on x = χ. A more precse approxmaton of L s based on the central lmt theorem: approxmate L = n =1 L on {x = χ} by a normal dstrbuton L(χ) wth mean and varance µ(χ) := l p (χ), σ 2 (χ) := =1 l 2 p (χ) (1 p (χ)). (39) We wll now apply ths technque to obtan an approxmaton of E(L 1 {L>c} ) n a systematc scenaro {x = χ}. Frstly, {x = χ} s splt nto two components: {x = χ} {A D } and {x = χ} {A > D }. By the central lmt theorem, L can be approxmated by a normal dstrbuton L (χ) on {x = χ} {A D }, where the mean and varance of L (χ) are adjusted to µ (χ) := j=1, j Hence, on {x = χ} {A D }, l j p j (χ) + l, σ 2 (χ) = =1 j=1, j l 2 j p j (χ) (1 p j (χ)). P(L > c {x = χ} {A D }) P(L (χ) > c {x = χ} {A D }) = 1 N µ (χ),σ 2 (χ)(c). (40) Snce L = 0 on {A > D } and L = l on {A D }, the followng approxmaton of expected shortfall contrbutons on {x = χ} s derved from (40): E(L 1 {L>c} {x = χ}) p (χ) E(L 1 {L (χ)>c} {x = χ} {A D }) ) = p (χ) l (1 N µ (χ),σ 2 (χ)(c). (41) Combnng mportance samplng and analytc approxmatons The followng algorthm calculates expected shortfall contrbutons for a gven confdence level α (0, 1). It uses mportance samplng for the smulaton of the systematc factors and apples normal approxmatons to the condtonal loss dstrbutons. Algorthm: 23

24 1. Importance samplng on systematc factors: Compute k samples χ 1 = (χ 11,..., χ 1m ),..., χ k = (χ k1,..., χ km ) of the systematc factors x 1,..., x m usng the mportance samplng technque presented n Secton 4. Denote the probabltes of the samples by q(χ 1 ),..., q(χ k ). 2. Normal approxmatons of condtonal dstrbutons: For each j {1,..., k}: compute the mean µ(χ j ) and the varance σ 2 (χ j ) of the normal approxmaton L(χ j ) on {x = χ j } (accordng to defnton (39)). These normal dstrbutons defne a loss dstrbuton on the space k {x = χ j } j=1 whch approxmates the portfolo loss dstrbuton L. 3. Approxmaton of c = VaR α (L): An estmate of c = VaR α (L) can be backed out of the estmaton α k q(χ j ) N µ(χj ),σ 2 (χ j )(c). j=1 4. Calculaton of the expected shortfall contrbutons: For each {1,..., n}, the expected shortfall contrbuton ESC α (L, L) s now approxmated by k j=1 ESC α (L, L) q(χ j) E(L 1 {L>c} {x = χ j }), 1 α where E(L 1 {L>c} {x = χ j }) s calculated by (41),.e. t s derved from the normal approxmaton L (χ j ) on {x = χ j } {A D }. Analytc approxmatons nevtably ntroduce errors nto the calculaton of the captal estmates. The accuracy of the portfolo approxmaton and the estmates for ESC α (L, L) clearly depends on the characterstcs of the portfolo, n partcular the homogenety of the exposures. Snce the typcal credt portfolo of a large nternatonal bank s rather well-dversfed we experenced relatvely small errors n the calculaton of ESC α (L, L) (see Tables 1 and 2 n Secton 6). However, t s certanly a worthwhle topc for future research to develop analytc approxmaton technques that are applcable to large credt portfolos and provde a better ft to the condtonal dstrbuton of L n a systematc scenaro as well as effcent formulae for the calculaton of ES contrbutons. 24

25 6 Numercal results We now apply the dfferent varance reducton technques presented n ths paper to the test portfolo specfed n Secton 4.5, as well as to a smaller portfolo of 1000 loans. The objectve s to compare these technques n terms of the numercal stablty and accuracy of the results that they produce, and hence assess ther sutablty for allocatng economc captal to ndvdual transactons. A common feature of the analyzed algorthms s the splt of the calculaton of expected shortfall contrbutons nto two steps: 1. smulaton of systematc factors, 2. calculaton of expected shortfall contrbutons by utlzng the ndependence of loss varables n each systematc scenaro. The smulaton of the systematc factors s ether based on straghtforward Monte Carlo smulaton (MC) or on the mportance samplng technque (IS) developed n Secton 4. In each systematc scenaro, the followng technques, presented n Secton 5, are appled to the specfc factors: 1. straghtforward Monte Carlo smulaton (MC), 2. Monte Carlo smulaton wth mportance samplng based on exponental twstng (IS), 3. Monte Carlo smulaton wth the condtonal allocaton (CA), 4. approxmaton of the condtonal loss dstrbuton by a normal dstrbuton,.e. applcaton of the central lmt theorem (NA). The calculatons are based on 20 runs wth MC samples,.e smulatons of the systematc factors and one smulaton of all specfc factors (or a normal approxmaton of the condtonal loss dstrbuton) n each systematc scenaro. In order to assess the accuracy of the dfferent technques we compare the average ES contrbuton (calculated as the mean of 20 estmates) of a loan to a benchmark. The benchmark value s based on the average of 28 runs, where each run uses smulatons wth mportance samplng on the systematc factors and straghtforward MC smulaton on the specfc factors. Frstly we analyze the dfferent methods as appled to the loan portfolo used prevously. For each loan, the dfference between ts average ES contrbuton and the benchmark s calculated and expressed n % of benchmark value. In Table 1, the mean of these relatve dfferences s exhbted for the 8 calculaton methods. Next, we calculate the standard devaton of the smulated ES contrbutons. More precsely, for each loan the standard devaton of the 20 estmates of ts ES 25

26 % Dff Specfc Factors MC IS CA NA Systematc Factors MC IS Table 1: Average relatve error n loan portfolo. contrbutons s calculated. The second table shows the average standard devaton agan expressed n % of benchmark value. % StDev Specfc Factors MC IS CA NA Systematc Factors MC IS Table 2: Average standard devaton n loan portfolo. A comparson of the dfferent varance reducton technques shows that the mpact of IS on systematc factors s most sgnfcant. Ths s due to the characterstcs of the portfolo: our test portfolo s large and granular,.e. t s not domnated by ndvdual names. As a consequence, the homogeneous portfolo approxmaton used n Secton 4.3 as bass for IS on systematc factors, provdes a good representaton. Although the portfolo s relatvely well dversfed there are concentratons caused by exposures to a sngle sector (geographc regon or ndustry) or to several hghly correlated sectors. These concentratons are exploted by mportance samplng on systematc factors. Because of the portfolo characterstcs t s not surprsng that combnng IS on systematc factors wth IS on specfc factors only provdes a small addtonal reducton of the varance: the portfolo loss depends on concentraton rsks rather than on the behavour of ndvdual loans, and the exponental twst does not have the granularty to closely ft the requrements of such a large number of dverse loans. We have observed a better performance of mportance samplng on specfc factors for smaller portfolos wth low dependence on systematc factors (compare to Glasserman and L (2005)). As an example, we analyse the stablty and accuracy of the results on a smaller portfolo, made up of 1000 loans of our orgnal portfolo. Then mportance samplng appled to the specfc factors mproves the stablty and accuracy of results more than that appled to the systematc factors. Indeed, as we move to the smaller 1000 loan portfolo, the nfluence of the systematc factors on portfolo loss dmnshes whlst that of the specfc factors ncreases. Furthermore, the homogenous portfolo gves a less accurate representaton of the smaller portfolo, 26