CUTTIG EDGE. PORTFOLIO RISK AALYSIS Crackng VAR wth kernels Value-at-rsk analyss has become a key measure of portfolo rsk n recent years, but how can we calculate the contrbuton of some portfolo component? Eduardo Epperlen and Alan Smlle show how kernel estmators can be used to provde a fast, accurate and robust estmate of component VAR n a smulaton framework The noton of component value-at-rsk (CVAR) orgnated n the papers of Garman (1996, 1997) and Ltterman (1997a, 1997b), and has been used by banks as a practcal rsk analyss tool snce at least Epperlen & Sondh (1997). The goal s to calculate how much some component of a portfolo contrbutes to the total VAR of that portfolo. We denote the proft and loss (P&L) of the portfolo as PL and the P&L of the th component as PL, so that: PL = PL Then the CVAR of the th component s defned to be the expected value of PL gven that the portfolo P&L s equal to the VAR, that s: = E PL PL (1) For notatonal smplcty, VAR s treated as the approprate percentle of the P&L dstrbuton, so VAR and CVAR wll usually be negatve. Component VAR has the desrable property of addtvty: f we sum all the CVARs of a gven portfolo we recover the portfolo VAR: () Also note that s always defned wth respect to a parent portfolo PL and a chld portfolo PL. Our defnton (1) s dfferent from that often seen n the lterature, where s defned as the ncremental change n the portfolo VAR gven a small change n the sze of the th exposure w, tmes the sze of that exposure, that s: VAR = w (3) w Subect to some techncal condtons, the two defntons are equvalent (see Goureroux, Laurent & Scallet, ), but we consder (1) to be more ntutve, at least n the smulaton settng we shall employ. Estmators of component VAR When all the components of a portfolo are ellptcally dstrbuted 1, the CVAR can be calculated analytcally, as shown by Carroll et al (1): = E PL + cov PL, PL ( VAR E PL var PL ) (4) The assumpton of ellptcally dstrbuted P&L s of course very restrctve, and wll be volated by a market rsk portfolo contanng optons, for example. A more general approach to VAR estmaton s Monte Carlo smulaton. We generate random scenaros for the P&L of each component, and sum to fnd the portfolo P&L. We wrte the P&L of the th component n the th Monte Carlo scenaro as () PL, so that the smulated portfolo P&Ls are: PL = PL Equaton (1) then suggests an estmator of the form: S CVAR ( n) = PL (5) where n denotes the VAR scenaro, that s, the scenaro such that: PL ( n) In ths approach, whch we shall refer to as scenaro extracton, the estmator wll automatcally satsfy the addtvty property, that s: [ S ] We shall see later that ths estmator gves an unbased estmate for CVAR but suffers badly from nose. We can amelorate (5) by takng an average of the values of PL around the nth value n the estmator, and weghtng them accordng to ther dstance from n. Ad hoc smoothers are used by Ltterman (1997b) and Hallerbach (). We attempt to make the method more rgorous by usng a kernel estmator to measure the dstance from the nth scenaro. A kernel s a functon of the form: K x;h ( ) = K x h whch s symmetrc about zero, takes a maxmum at x = and s 1 ote that the multvarate normal dstrbuton s an example of an ellptcal dstrbuton 7 Rsk August 6
non-negatve for all x. A partcularly smple example s the trangle kernel: ( ) = max 1 x h, K x;h whch we have chosen for ease of mplementaton. A kernel estmator of CVAR can thus be constructed as: K PL ( VAR;h)PL K CVAR [ ] =1 = =1 K ( PL VAR;h) The numerator n the above expresson can be seen as the weghted average of PL, whle the denomnator s a normalsaton factor. A smlar approach s hnted at by Goureroux, Laurent & Scallet (), though these authors never make the use of kernels explct nor conduct any statstcal tests of the accuracy of the estmator n the fnte sample. Usng (6) we fnd that the sum of the CVARs s not n general equal to the VAR, volatng the addtvty property (). Ths s because the kernel estmator s based low 3 due to asymmetry n the P&L dstrbuton around the pont PL (there are more scenaros wth loss a lttle smaller than VAR than wth loss a lttle greater than VAR). We can correct for the bas by rescalng accordng to the bas n the kernel estmate of the VAR tself, whch leads to a rescaled kernel estmator for CVAR: K,R K PL ( VAR;h)PL [ ] =1 K ( PL VAR;h)PL =1 Usng estmator (7) guarantees that the addtvty property holds and, as we see n the next secton, that the estmate s unbased. The performance of the kernel estmator depends on the choce of the smoothng parameter h: n our applcaton, lettng h = corresponds to the scenaro extracton estmator (5), whle lettng h corresponds to takng an unweghted average of PL. It can be shown (see Slverman, 1986) that the optmal choce for h (n the sense of mnmsng the mean square error) for the trangular kernel s: h =.575σ 1 5 where σ s the standard devaton of PL. We examne the performance of the estmators (5) and (7) n what follows, but frst we menton two alternatve numercal approaches as a bass for comparson. The alternatve defnton of CVAR (3) suggests usng numercal dfferentaton, as proposed by Epperlen (1998). Lettng: VAR + ( PL + δpl ) and: VAR ( PL δpl ) be the estmates of VAR wth the th exposure perturbed upward and downward respectvely, a fnte dfference estmator for s: [ D ] VAR + = VAR δ (6) (7) The parameter δ controls the sze of the perturbaton, and plays a role analogous to h n the kernel estmator. Ths tme t s not clear how to choose δ a pror, but numercal expermentaton shows that a value of δ =.1 acheves a reasonable compromse between bas and varance. Due to Monte Carlo error, the estmated CVARs wll agan fal to sum to the VAR, but a rescalng method smlar to (7) can be used to correct for ths, yeldng the rescaled fnte dfference estmator: [ D ] [ D,R] CVAR [ D ] (8) Fnally, we menton a very smple method to calculate CVAR from a smulated sample. If we are wllng to assume that the P&Ls are approxmately ellptcally dstrbuted, equaton (4) suggests the sem-parametrc estmator proposed by Carroll et al (1): PL [ P ] = + PL PL PL PL ( n) PL The sem-parametrc estmator s addtve by constructon. Statstcal comparson of the estmators We shall now compare the estmators (5), (7), (8) and (9) for a selecton of sample portfolos. We use Monte Carlo smulaton wth = 1, and calculate VAR at a confdence level of 99%. The experment s run 1, tmes, and the mean and standard devaton of the estmated s recorded. For comparson, we have also ncluded results on the error n the Monte Carlo estmate of the VAR tself. Example 1: lnear portfolo. Frst, let us consder a very smple example where the portfolo rsk factors are normally dstrbuted wth zero mean, zero correlaton and unt varance, that s: RF 1 RF ~, 1 1 We assume lnear exposure to w 1 and w unts of RF 1 and RF respectvely, so we have payout functons: PL 1 = w 1 RF 1 PL = w RF Whle ths may seem lke a trval example, t s of nterest snce n ths case (4) can be used to calculate VAR, CVAR 1 and analytcally as: VAR = Φ 1 (.1) w 1 + w CVAR 1 = Φ 1 (.1) = Φ 1 (.1) w 1 w 1 + w w w 1 + w umercal experments (not shown) ndcate that our conclusons are unaffected by the partcular form of kernel functon used 3 A referee has ponted out that ths can be avoded by estmatng the VAR tself usng kernels. Ths wll work, but only at the expense of nducng a smlar bas n the VAR. Also note that ths bas s not the same as the boundary bas observed when kernels are used for non-parametrc densty estmaton rsk.net 71
CUTTIG EDGE. PORTFOLIO RISK AALYSIS A. Comparson of the estmators for lnear, uncorrelated components Mean Std dev Std dev/mean Scenaro extracton (5) CVAR 1 1.3.98 89.49% 4.17.98.6% (7) CVAR 1 1.4.7 6.71% 4.16.93.3% Fnte dfference (8) CVAR 1 1.5.17 16.4% 4.15.178 4.8% Sem-parametrc (9) CVAR 1 1.4.9.79% 4.16.7 1.73% VAR 5..85 1.63% B. Comparson of the estmators for non-lnear components Mean Std dev Std dev/mean Scenaro extracton (5) CVAR 1 1.89.81 4.86% 1.35.84 59.56% (7) CVAR 1 1.88.155 8.4% 1.36.11 1.% Fnte dfference (8) CVAR 1 1.9.1 1.47% 1.3.178 13.48% Sem-parametrc (9) CVAR 1.79.4.5%.45.46.38% VAR 3.4.73.7% 1 ose n the CVAR estmators as a functon of exposure sze 1, Scenaro extracton Fnte dfference 1, Sem-parametrc 1 CVAR varyng wth confdence level for a portfolo contanng non-lnear exposures CVAR 1 /VAR (%) 8 6 4 Fnte dfference Sem-parametrc.5 1. 1.5. Sze of w 1 where Φ 1 denotes the nverse standard normal dstrbuton functon. Monte Carlo smulatons usng (5), (7), (8) and (9) wth w 1 = 1 and w = yeld the results n table A. The mean estmates for CVAR 1 and are close to ther analytc values, ndcatng that the estmators are unbased, and thanks to the rescalng procedure addtvty holds exactly for each estmator. The man dfference s n the nose of the estmate the sem-parametrc method gves the best results, followed by the kernel. The fnte dfference method s somewhat worse, and the scenaro extracton approach leads to very sgnfcant nose even n ths smple example. otce how the sze of the nose relatve to the mean s greater for the smaller component ths behavour s typcal of numercal procedures to estmate CVAR. The relatve nose can become very large for components that make a mnmal contrbuton to the VAR, as llustrated n fgure 1, where we have plotted the relatve nose n CVAR 1 (that s, standard devaton/mean) usng each of the estmators for w 1 =.,.4,..., whle w =. The very hgh levels of nose for low values of w 1 may be of less concern than we mght ntally expect, however, snce for such small exposures the poston contrbutes a very small proporton of the total rsk. Example : non-lnear portfolo. ext, we consder a portfolo contanng an nstrument wth non-lnear payout functon. The underlyng rsk factors are agan standard normal and ndependent, but ths tme we set: PL 1 = max( RF 1 1, ) PL = RF 9 9 94 96 98 VAR confdence level (%) PL 1 represents an opton-lke payout, specfcally a short poston n out-of-the-money puts close to expry. The non-lnearty n PL 1 means that n ths case we are unable to calculate VAR or CVAR analytcally. Runnng the same test as before gves the results n table B. The results for the scenaro extracton, rescaled kernel and rescaled fnte dfference estmators are roughly n agreement, and exhbt smlar levels of nose to the prevous example. The results for the sem-parametrc method are very dfferent to the rest, suggestng that n ths example estmator (9) does not provde a good approxmaton to the CVAR. Ths s unsurprsng snce PL 1 and PL do not follow an ellptcal dstrbuton. Snce t s not clear how ths error could have been estmated ex ante and thus corrected, the presence of such a large bas must be consdered a serous flaw n the sem-parametrc estmator. We can use ths portfolo to hghlght another property of CVAR: ts dependence on the VAR confdence level. In fgure, we plot CVAR 1 as a proporton of the total VAR for confdence levels from 9 99.9% usng the kernel, fnte dfference and semparametrc estmators (we omt scenaro extracton snce the results are very close to the former two estmators). otce how the non-lnear component contrbutes a greater proporton of the VAR at hgher confdence levels, but the sem-parametrc estmator fals to capture ths. Example 3: correlated portfolo. Fnally, we examne a specal case where all the proposed estmators perform poorly. We return to the example of normally dstrbuted P&L, but ths tme we have a correlaton of ρ and set w 1 = w = 1: 7 Rsk August 6
Usng (3) we calculate: RF 1 RF ~, 1 ρ ρ 1 PL 1 = RF 1 PL = RF VAR = Φ 1.1 CVAR 1 = Φ 1 (.1) = Φ 1 (.1) ( ) + ρ 1 + ρ 1 + ρ In ths example, both of the CVARs are sgnfcant, n the sense that they are of the same order of magntude as the VAR (ndeed, they are each 5% of the VAR). Fgure 3 shows the relatve nose of each estmator for a range of values of ρ. Overall, the error s of an acceptable sze, but for strong negatve correlaton between the components t ncreases substantally. Intutvely, ths s because the nose n the CVAR estmator s affected by the magntude of PL, whle the magntude of the s proportonal to the magntude of the VAR. For strong negatve correlaton, the magntude of the VAR falls but the magntude of PL does not, causng the relatve nose to ncrease. Such an ncrease s observed n all the estmators, though the mpact on the fnte dfference estmator seems to be a lttle smaller than for the others. For very strong negatve correlatons (ρ <.95), the fnte dfference estmator actually gves a better performance than the kernel. It may seem that ths s of purely academc nterest snce market factors wll very rarely exhbt such strong negatve correlaton, but recall that components can comprse any subset of a portfolo. We could have, for example, two components of a tradng portfolo that are constructed to hedge each other, where we would expect to see strong negatve correlaton between the components. Implementaton We have shown how the kernel method yelds the most accurate and robust CVAR estmates for a general portfolo structure, but practoners wll also be nterested n practcal ssues such as computatonal tme. Computatonal tmes for CVAR usng a desktop computer are gven n table C, where CVARs refers to the computaton of the CVAR for components of the same parent portfolo. Clearly the queston of whch method s fastest s of somewhat academc nterest, snce for all the approaches the tme to compute s lkely to be much less than the tme to compute the VAR tself. Scenaro extracton s, unsurprsngly, the most effcent approach, snce n ths case the computaton amounts to lookng up the nth scenaro for PL. The kernel and sem-parametrc methods are also very effcent, partcularly where we wsh to calculate multple CVARs on the same parent portfolo. For the kernel method, ths s because the weghts K(PL () VAR; h) depend only on the PL, so only have to be computed once for each parent portfolo. To speed up the calculaton further, we have chosen to use a very smple form of kernel based on a trangle functon, whch means that most (typcally 95% for a VAR confdence level of 99%) kernel weghts are zero (n contrast to a Gaussan kernel, for example, where each scenaro receves some fnte postve weght). The fnte dfference estmator s a lttle less effcent, 3 ose n the CVAR estmators as a functon of correlaton largely because t requres the computaton of two new VARs for every CVAR that s requred. We mght expect that the nose n the CVAR estmators wll follow the usual Monte Carlo rule and fall as. We nvestgate ths by computng the standard devaton n CVAR 1 from example 1 usng ncreasng values of. The results are recorded n fgure 4. We see that the square-root rule does ndeed apply for the kernel, fnte dfference and sem-parametrc estmators, but not for scenaro extracton. The reason for ths s qute subtle: recall that s defned as the expectaton of the component P&L, gven that the parent P&L s equal to the VAR. Usng estmator (5), one computes: not: 1, 1 PL PL Scenaro extracton Fnte dfference Sem-parametrc 1..5.5 1. Correlaton C. Computng tme (n seconds) for the CVAR estmators = 1, = 5, 1 CVARs 1 CVARs Scenaro extracton (5).1..1. (7).69.113.119.455 Fnte dfference (8).8.159.413 4.9174 Sem-parametrc (9).58.356.186 1.335 4 ose n the CVAR estmators as the number of Monte Carlo scenaros s ncreased 8 6 4 Scenaro extracton Fnte dfference Sem-parametrc 1 1 3 1 4 1 5 1 6 umber of MC scenaros rsk.net 73
CUTTIG EDGE. PORTFOLIO RISK AALYSIS D. CVARs and errors for market rsk on the bond portfolo (% of portfolo VAR) E. CVARs and errors for credt rsk on the CDS portfolo (% of portfolo VAR) Rsk ratng Scenaro extracton Fnte dfference Sem-parametrc Rsk ratng Scenaro extracton Fnte dfference Sem-parametrc AAA 4.4 (1.) 4.7 (.1) 4.8 (.5) 4.7 (.1) AA 1. (1.8) 11.8 (.3) 1. (1.) 11.8 (.) A 7.8 (1.8) 8. (.3) 7.9 (.8) 8. (.1) BBB 44.5 (5.8) 46. (1.) 45.6 (.4) 46.4 (.7) BB 33.7 (5.7) 3.5 (.6) 3.7 (.) 3.6 (.6) B 1.5 (5.7) 13.1 (.5) 1.8 (.4) 13.3 (.) CCC 1.1 (3.4) 9.6 (.5) 9.7 (.) 9.7 (.) AAA.4 (7.7).1 (.). (.). (.) AA 1. (13.8) 1.4 (.9) 1.4 (1.) 3.1 (.7) A 5.4 (7.3) 5.4 (1.7) 5.5 (.3) 5.8 (.5) BBB 94.1 (57.5) 93.5 (4.) 94.6 (5.3) 4. (1.1) BB 5. (19.4) 5.4 (1.) 5. (.4) 1.3 (.3) B 9.5 (3.9) 1.7 (1.9) 1. (3.5) 3.5 (.7) CCC 15.8 (3.6) 16.5 (1.9) 16.9 (3.5) 9.9 (.6) E PL PL Thus, by usng scenaro extracton, we smply draw a sngle sample from a (condtonal) probablty dstrbuton. The dstrbuton wll have some varance, whch we cannot expect to reduce smply by ncreasng the number of Monte Carlo runs. The Monte Carlo error could also be reduced by applyng one of the numerous varance reducton technques avalable n the lterature, but we consder a dscusson of these methods to be beyond the scope of ths artcle. ote that the bas n the sem-parametrc estmator when a portfolo contans non-lnear nstruments (such as n example ) cannot be reduced by ncreasng the number of Monte Carlo scenaros or by applyng a varance reducton technque. Traded portfolos We conclude by presentng two examples of CVAR appled to realstc tradng portfolos, analysed usng nternal rsk models. We look at market rsk on a portfolo of 1,5 US corporate bonds and credt rsk on a portfolo of 7 credt default swaps (CDSs). In both cases we am to assess the VAR contrbuton by rsk ratng, and report the mean and standard devaton of CVAR as a percentage of the total portfolo VAR. The composton of the portfolos and the detals of the rsk models used are propretary, but the key pont s that snce the bond portfolo contans only lnear exposures the P&L wll be approxmately ellptcal, whle the credt loss dstrbuton on the CDS portfolo wll exhbt fat tals and non-gaussan dependence. The market rsk model uses 1, smulatons, whle the credt loss model uses 5,. CVARs (standard devaton n parenthess) for the bond portfolo are gven n table D. We can see that most of the rsk n the bond portfolo comes from BBB and BB rated ssuers, and that n ths case the sem-parametrc estmator yelds the best estmate of CVAR. Results for the CDS portfolo are shown n table E. Ths portfolo s manly comprsed of exposures to AA, A and BBB grade ssuers, but almost all the default rsk comes from the latter due to the small probablty of default on the AA and A rated ssuers. The loss dstrbuton of ths portfolo s fat-taled, whch means the CVAR estmates suffer more from nose than for an ellptcal portfolo, even gven the far hgher number of Monte Carlo smulatons used. The volaton of the ellptcal dstrbuton assumpton means that the sem-parametrc estmator ms-estmates the CVARs consderably, falng, for example, to detect the negatve CVAR that ndcates a hedge poston n CCC ssuers. Summary The rescaled kernel exhbts the best performance overall, snce t s robust n the presence of non-lnearty, s less nosy than the fnte dfference method for all portfolos wthout strong negatve correlaton, and s hghly effcent, partcularly where we wsh to compute many CVARs from a sngle portfolo. Where we can be sure that all the components follow an ellptcal dstrbuton (for example, when we have lnear exposures to multvarate normal rsk factors), the sem-parametrc estmator s more effectve, but n ths case both VAR and CVAR can be computed analytcally so there s lttle reason to use Monte Carlo smulaton at all. We remark that although we have focused on parametrc Monte Carlo smulaton, the proposed methods could also be used to decompose the rsks n a hstorcal smulaton-based rsk engne, though here the relatvely low number of scenaros may be problematc. -based estmators can also be appled to compute component expected shortfall (see Scallet, 4). Eduardo Epperlen s managng drector and head of market, model and counterparty rsk analytcs at Ctgroup. Alan Smlle s a quanttatve analyst n Ctgroup s rsk archtecture group. Emal: eduardo.epperlen@ctgroup. com, alan.smlle@ctgroup.com References Carroll R, T Perry, H Yang and A Ho, 1 A new approach to component VAR Journal of Rsk 3(3), pages 57 67 Epperlen E, 1998 Component VAR n large scale Monte Carlo smulatons and ts relatonshp to scenaro extracton ICBI Rsk Management Conference, December Epperlen E and P Sondh, 1997 Are factor senstvty lmts consstent wth value-at-rsk lmts? Workng paper, Ctbank Garman M, 1996 Improvng on VAR Rsk May, pages 61 63 Garman M, 1997 Takng VAR to peces Rsk October, pages 7 71 Goureroux C, J-P Laurent and O Scallet, Senstvty analyss of value-at-rsk Journal of Emprcal Fnance 7(3), pages 5 45 Hallerbach W, Decomposng portfolo value-at-rsk: a general analyss Journal of Rsk 5(), pages 1 18 Ltterman R, 1997a Hot spots and hedges (I) Rsk March, pages 4 45 Ltterman R, 1997b Hot spots and hedges (II) Rsk May, pages 38 4 Scallet O, 4 onparametrc estmaton and senstvty analyss of expected shortfall Mathematcal Fnance 14, pages 115 19 Slverman B, 1986 Densty estmaton for statstcs and data analyss Chapman & Hall, London 74 Rsk August 6