Proceedigs of the 2009 Witer Simulatio Coferece M. D. Rossetti, R. R. Hill, B. Johasso, A. Duki, ad R. G. Igalls, eds. A GENERAL FRAMEWORK OF IMPORTANCE SAMPLING FOR VALUE-AT-RISK AND CONDITIONAL VALUE-AT-RISK ABSTRACT Lihua Su L. Jeff Hog Departmet of Idustrial Egieerig ad Logistics Maagemet The Hog Kog Uiversity of Sciece ad Techology Clear Water Bay, Hog Kog, Chia Value-at-risk VaR ad coditioal value-at-risk CVaR are importat risk measures. Importace samplig IS is ofte used to estimate them. We derive the asymptotic represetatios for IS estimators of VaR ad CVaR. Based o these represetatios, we are able to give simple coditios uder which the IS estimators have smaller asymptotic variaces tha the ordial estimators. We show that the expoetial twistig ca yield a IS distributio that satisfies the coditios for both the IS estimators of VaR ad CVaR. Therefore, we may be able to estimate VaR ad CVaR accurately at the same time. INTRODUCTION Value-at-risk VaR ad coditioal value-at-risk CVaR are two widely used risk measures. They play importat roles i risk measuremet, portfolio maagemet ad regulatory cotrol of fiacial istitutios. The -VaR, defied as the quatile of a portfolio value L. The Basel Accord II has icorporated the cocept of VaR ad ecourages baks to use VaR for daily risk maagemet. It has the advatage of beig flexible ad coceptually simple. However, VaR also has some shortcomigs. It provides o iformatio about the amout of loss that ivestors might suffer beyod the VaR. Moreover, it has also bee criticized for ot beig a coheret risk measure because it is ot subadditive Artzer et al. 999. The -CVaR, defied as the average of VaR of L for 0 < <, has a log history of beig used i isurace idustry. It provides iformatio o the potetial large losses that a ivestor ca suffer. Rockafellar ad Uryasev 2000 showed that CVaR always satisfies the subadditivity ad is therefore a coheret risk measure. Heyde et. al 2007, however, argued that requirig subadditivity may lead to risk measures that are ot robust to the uderlyig models ad data. They proposed replacig subadditivity axiom by the comootoic subadditivity axiom ad foud that both VaR ad CVaR satisfy the ew axiom ad VaR is more robust. Because VaR ad CVaR both have advatages ad disadvatages, it is ofte difficult to decide which oe is better. Risk maagers may cosider both of them at the same time to complemet each other. There are typically three approaches to estimate VaR ad CVaR: the variace-covariace approach, the historical simulatio approach ad the Mote Carlo simulatio approach. Amog the three, the Mote Carlo simulatio approach is frequetly used, sice it is very geeral ad ca be applied to a wide rage of risk models. However, the Mote Carlo simulatio approach is ofte time cosumig. I risk maagemet, is typically close to 0. A large umber of replicatios are eeded to obtai accurate estimatio of the tail behavior. Therefore, variace reductio techiques are ofte used to icrease the efficiecy of the estimatio. Amog all variace reductio techiques, importace samplig IS is a atural choice, because it ca allocate more samples to the tail of the distributio that is most relevat to the estimatio of VaR ad CVaR. I this paper, we study the asymptotic properties of the IS estimators of VaR ad CVaR ad propose simple schemes for choosig IS distributios. Because IS estimators of both VaR ad CVaR are rather complicated compared to the typical sample meas, to aalyze their asymptotic properties, we use the method of asymptotic represetatios. Bahadur 966 used this method to aalyze the asymptotic properties of the ordiary estimator of VaR quatile by showig that the estimator ca be approximated by a sample mea except for a high-order term. The, the cosistecy ad asymptotic ormality of the estimator ca be derived easily. I this paper, we derive the asymptotic represetatios for the IS estimators of both VaR ad CVaR, ad use them to prove the cosistecy ad asymptotic ormality of both estimators. From the asymptotic ormality, we give simple coditios o the IS distributios uder which the IS scheme is guarateed to work asymptotically. A good feature of the coditios is that they are the same for both VaR ad CVaR. Therefore, oe ca estimate the VaR ad CVaR simultaeously 978--4244-577-7/09/$26.00 2009 IEEE 45
Su ad Hog usig the same IS distributio. This feature will greatly help those risk maagers who cosider both risk measures ad use them to complemet each others. We the cosider how to choose a good IS distributio for estimatig both VaR ad CVaR. We cosider this problem uder a special family of chage of measure that is kow as expoetial twistig ad show how to choose good IS distributios from the family. Iterestigly, the distributios that we choose for VaR ad CVaR are agai the same. We show that the chose IS distributio guaratees to reduce the asymptotic variaces for both VaR ad CVaR uder mild coditios. The literature o usig IS to estimate VaR is growig rapidly. For example, Glasserma et al. 2000, 2002 used IS to estimate the VaR of a portfolio loss for both light-tail ad heavy-tail situatios; Glasserma ad Li 2005 applied IS to estimate the VaR of portfolio credit risk; Glasserma ad Jueja 2008 used IS to estimate the VaR of a sum of i.i.d radom variables. To the best of our kowledge, however, there is o published work o usig IS to estimate CVaR. Our paper differs from others for several reasos. We cosider the geeral theory of usig IS to estimate VaR ad CVaR, without focusig o specific applicatios. We show that the variaces of VaR ad CVaR ca be reduced by usig the same IS distributio. Therefore, risk maagers who use both VaR ad CVaR ca obtai accurate estimates of both without ruig two separate simulatio experimets. The rest of the paper is orgaized as follows. Sectio 2 reviews the IS estimators of VaR ad CVaR. I sectio 3, we develop asymptotic represetatios for the IS estimators for both VaR ad CVaR. From these represetatios, we ca easily prove the cosistecy ad asymptotic ormality of the IS estimators. I sectio 4, we cosider the expoetial twistig to choose a IS distributio, followed by the coclusios i sectio 5. 2 IMPORTANCE SAMPLING FOR VAR AND CVAR Let L be a real-valued radom variable with a cumulative distributio fuctio c.d.f. F, adletv ad c deote the -VaR ad -CVaR of L, respectively, for ay 0 < <. The, v = F = if{x : Fx }, c = v Ev L] +, 2 where x + = max{x,0}. Note that v is also the -quatile of L, adc is the average of L coditioig that L v if L has a positive desity at v. Uder the defiitios, we are iterested i the left tail of the distributio of L. Therefore, is ofte close to 0. Sometimes, v ad c are defied for the right tail of a radom variable e.g., Hog ad Liu 2009. We may covert the right tail to the left tail by addig a egative sig to the radom variable. Covetioal Mote Carlo estimatio of v ad v ivolves geeratig idepedet ad idetically distributed i.i.d radom observatios of L, deoted as L,...,L, ad estimatig them by respectively, where ṽ = F = if{x : F x }, 3 c = ṽ ṽ L i ] +, 4 F x= I{L i x} 5 is the empirical distributio of L costructed from L,...,L ad I{ } is the idicator fuctio. Note that F x is a ubiased ad cosistet estimator of Fx. BySerflig 980, F =L :,wherel i: is the ith order statistic of the L. Serflig 980 showed that ṽ ad c are cosistet estimators of v ad c, respectively, as. As is typically chose close to 0, e.g., = 0.0 or 0.05, to estimate v ad c accurately usig the covetioal Mote Carlo method, the sample size ofte eeds to be very large. Therefore, it is importat to apply some variace reductio techiques to improve the estimatio efficiecy. Importace samplig IS is ituitively appealig i estimatig v ad c especially whe is close to 0, because it may allocate more samples to the left tail of the distributio that are importat i estimatig v ad c. Now we itroduce the IS estimators of v ad c. Suppose we choose a IS distributio fuctio G for which the probability measure associated to F is absolutely cotiuous with respect to that associated with G, i.e., Fdx =0if 46
Su ad Hog Gdx=0forayx. LetL x= Fdx Gdx,theL is called the likelihood ratio LR fuctio. The for ay x, we may estimate Fx by ˆF x= I{L i x}l L i. 6 Note that ˆF x is a ubiased ad cosistet estimator of Fx as F x that is defied i Equatio 5. Let ˆv ad ĉ deote the IS estimators of v ad c. Similar to Equatios 3 ad4, we defie ˆv = ˆF = if{x : ˆF x }, ĉ = ˆv ˆv L i] + L L i. I the rest of this paper, we aalyze the asymptotic properties of ˆv ad ĉ, ad discuss how to select G to reduce the variaces of ˆv ad ĉ. Throughout the rest of the paper, we make the followig assumptios.. Assumptio. There exists a > 0 such that whe x v,v + L has a desity f x > 0 ad f x is differetiable Assumptio requires that L has a desity i a eighborhood of v. It also implies that Fv = ad c = EL L v ] Hog ad Liu 2009. Assumptio 2. There exists a costat C > 0 such that L x C for all x < v +. To estimate v ad c, the samples i the set {L < v + } are most useful. The, ay reasoable IS distributio should allocate more samples to the regio, i.e., L x forallx < v +. Therefore, Assumptio 2 is a very weak assumptio o the IS distributio. 3 ASYMPTOTIC REPRESENTATIONS OF THE IS ESTIMATORS A complicated estimator ca ofte be represeted as the sum of several terms, where the asymptotic behaviors of all the terms are clear. The, the represetatio is called a asymptotic represetatio of the estimator. Based o the asymptotic represetatio of a estimator, may asymptotic properties of the estimator, e.g., cosistecy ad asymptotic ormality, ca be studied easily. A famous example is the asymptotic represetatio of the VaR estimator ṽ also kow as Bahadur represetatio of the quatile estimator. By Bahadur 966, uder Assumptio, ṽ = v + f v I{L i v } + R, 7 where R = O 3/4 log 3/4 with probability w.p.. The statemet Y = Og w.p. meas that there exist a set 0 with P 0 = such that for ay 0 we ca fid out a costat C satisfies Y Cg for all large eough. Give the represetatio, may asymptotic properties of ṽ ca be aalyzed easily. For istace, we may use it to prove the strog cosistecy ad asymptotic ormality of ṽ. By the strog law of large umbers Durrett 2005, I{L i v } Fv w.p. as. Furthermore, because Fv = by Assumptio, it is clear that ṽ v w.p. as. Thus, ṽ is a strogly cosistet estimator of v. Similarly, by the cetral limit theorem Durrett 2005, I{L i v } N0, 47
Su ad Hog as, where meas covergece i distributio ad N0, is stadard ormal radom variable. The, ṽ v N0, 8 f v as. Therefore, ṽ is asymptotically ormally distributed. I the rest of this sectio, we develop asymptotic represetatios of the IS estimators ˆv ad ĉ, ad use them to aalyze the cosistecy ad asymptotic ormality of ˆv ad ĉ. 3. Asymptotic Represetatio of ˆv We first cosider the asymptotic represetatio of ˆv. Note that by a secod order Taylor expasio, whe ˆv v <, where A, = O ˆv v 2. The, we have F ˆv Fv = f v ˆv v A,, ˆv = v + F ˆv Fv + f v f v A,. 9 Let A 2, = F ˆv + ˆF v ˆF ˆv Fv ad A 3, = ˆF ˆv Fv. It is easy to see that The by Equatio 9, we have F ˆv Fv =Fv ˆF v +A 2, + A 3,. ˆv = v + Fv ˆF v f v I the followig lemma, we provide the orders of A,, A 2, ad A 3,. + f v A, + A 2, + A 3,. 0 Lemma 3.. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, A, = O log,a 2, = O 4 3 log 3 4 ad A 3, = O w.p.. Let L i deote L L i for all i =,...,. By Lemma 3., we ca prove the followig theorem o the asymptotic expasio of ˆv. Theorem 3.. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, where A = O 4 3 log 3 4 w.p.. ˆv = v + f v Proof. By Assumptio, Fv =. The by Equatio 6, Fv ˆF v = I{L i v }L i + A, I{L i v }L i. Let A = f v A, + A 2, + A 3,.Sicef v > 0 by Assumptio, ad by Lemma 3., wehavea = O 4 3 log 4 3 w.p.. Therefore, the coclusio of the theorem follows directly from Equatio 0. 48
Su ad Hog Because ṽ is a special case of ˆv where L x=forallx, the Bahadur represetatio of Equatio 7 maybe viewed as a special case of Theorem 3.. Let E G ad Var G deote the expectatio ad variace uder the IS distributio G. From Theorem 3., it is also straight-forward to prove the followig corollary o the strog cosistecy ad asymptotic ormality of ˆv. Corollary 3.. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, ˆv v w.p. ad as. ˆv VarG I{L v }L L] v N0, f v Remark 3.. The coclusios of Corollary 3. have also bee proved by Gly 996 uder Assumptio ad the assumptio that E G L 3 L] <. Note that Var G I{L v }L L] = E G I{L v }L 2 L ] E 2 G I{L v }L L] = EI{L v }L L] 2. If the IS distributio allocates more samples to the left tail of the distributio of L, i.e., L x < forallx v,theby Equatio, Var G I{L v }L L] <. Compared to Equatio 8, the IS estimator ˆv has a smaller asymptotic variace tha the covetioal estimator ṽ. This explais why IS ca improve the efficiecy of VaR estimatio whe the IS distributio is selected appropriately. Theorem 3. ot oly ca be applied to obtai the strog cosistecy ad asymptotic ormality of ˆv, it ca also be applied to facilitate aother asymptotic aalysis related to ˆv. For istace, it helps i developig the asymptotic represetatio of ĉ as show i Sectio 3.2. 3.2 Asymptotic Represetatio of ĉ We ow cosider the asymptotic represetatio of ĉ. Note that Furthermore, ote that ĉ = ˆv = v = ˆv L i + L i v L i + L i +ˆv v ˆv L i + v L i ] L i = ˆv v I{L i ˆv }]L i + ˆv L i + v L i ] L i. 2 ˆv L ii{l i ˆv } v L i I{L i v }]L i v L i I{L i ˆv } I{L i v }]L i. 3 Sice ad ˆv v I{L i ˆv }]L i = ˆv v ˆF ˆv v L i I{L i ˆv } I{L i v }]L i ˆv v ˆF ˆv ˆF v, 49
Su ad Hog the by Equatios 2 ad3, where ĉ = v v L i + L i + B, 4 B ˆv v ˆF ˆv + ˆF ˆv ˆF v ˆv v 2 ˆF ˆv Fv + ˆF v Fv. I the followig lemma, we prove the order of B. Lemma 3.2. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, B = O log w.p.. The, we have the followig theorem o the asymptotic expasio of ĉ. Note that the coclusio of the theorem follows directly from Equatio 4 ad Lemma 3.2. Therefore, we omit the proof. Theorem 3.2. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, where B = O log w.p.. ĉ = c + v ] v L i + L i c + B, Note that E G v v L + L ] = c. The by the strog law of large umbers ad the cetral limit theorem, it is easy to prove the followig corollary o the strog cosistecy ad asymptotic ormality of ĉ. Corollary 3.2. Suppose that Assumptios ad 2 are satisfied. The for ay 0,, ĉ c w.p. ad as. ĉ VarG v L c + L L] N0, Furthermore, we may set L x=forallx. The, the coclusios of Theorem 3.2 ad Corollary 3.2 apply to c, the covetioal Mote Carlo estimator of c. We summarize the results i the followig corollary. Corollary 3.3. Suppose that Assumptio is satisfied. The for ay 0,, c = c + where C = O log w.p., c c w.p., ad v ] v L i + c +C as. ĉ Varv L c + ] N0, Remark 3.2. The strog cosistecy ad asymptotic ormality of c have also bee studied by a umber of papers i the literature, icludig Tridade et al. 2007, usig differet methods. If the IS distributio allocates more samples to the left tail of the distributio of L, i.e., L x < forallx v,the it is easy to show that Var G v L + L L] < Varv L + ]. Therefore, by Corollaries 3.2 ad 3.3, the IS estimator ĉ has a smaller asymptotic variace tha the covetioal estimator c. This explais why IS ca improve the efficiecy of CVaR estimatio whe the IS distributio is selected appropriately. 420
Su ad Hog 4 SELECTING IS DISTRIBUTIONS By Theorems 3. ad 3.2, it is clear that the variaces of ˆv ad ĉ are deomiated by respectively. Note that Var G Var G f v f v I{L i v }L i ] I{L i v }L i ] ad Var G v v L i + L i ] ], = f 2 v Var G I{L v }L ]= { EG I{L v f 2 }L 2] 2}, v Var G v ] v L i + L i = 2 Var G v L + L ] = 2 { EG v L 2 I{L v }L 2] E 2 v L +]}. Therefore, we oly eed to fid IS distributios that reduce E G I{L v }L 2] ad E G v L 2 I{L v }L 2]. To fid a good IS distributio fuctio G, we restrict our cosideratios to the techique of expoetial twistig, where we set L = e L+ ad =logee L, ad our goal is to choose a parameter that ca reduce E G I{L v }L 2] ad E G v L 2 I{L v }L 2]. We cosider < 0. The, L < e v ++ for L,v +. Therefore, Assumptio 2 is satisfied. Note that E G I{L v }L 2] = EI{L v }L ] EI{L v }]e v + 5 E G v L 2 I{L v }L 2] = E v L 2 I{L v }L ] E v L 2 I{L v } ] e v+. 6 Therefore, we propose to choose to miimize the upper bouds i Equatios 5 ad6, i.e., miimizig v +. Note that is called the cumulat-geeratig fuctio ad it is covex. Therefore, the optimal, deoted as, satisfies =v. The, we have the followig theorem. Theorem 4.. Suppose that Suppose that Assumptio is satisfied ad v < EL]. If the IS distributio G satisfies L = FL/GL=e L+,the Var G I{L v }L L] < VarI{L v }], Var G v L + L L ] < Var v L +]. Note that this applies to both the IS estimators of VaR ad CVaR. Therefore, this allows risk maagers to apply the IS distributio ad estimate these two risk measures at the same time. By Theorem 4., we show that the expoetial twistig with guaratees to reduce the asymptotic variaces of both ˆv ad ĉ. 5 CONCLUSIONS I this paper, we develop the asymptotic represetatios for the IS estimators of both VaR ad CVaR, ad derive the cosistecy ad asymptotic ormality for both estimators. We show that there are simple coditios for choosig IS distributios that guaratee to reduce the asymptotic variaces of the estimators. Furthermore, we show that the techique of expoetial twistig guaratees to fid a IS distributio that reduces the asymptotic variaces of both estimators at the same time. REFERENCES Artzer, P., F. Delbae, J. -M. Eber ad D. Heath. 999. Coheret measures of risk. Mathematical Fiace 93: 203-228. 42
Su ad Hog Bahadur, R. R. 966. A ote o quatiles i large samples. Aals of Mathematical Statistics 37: 577-580. Delbae, F. 200. Coheret risk measures. Lecture Notes, Scuola Normale Superiore di Pisa. Durret, R. 2005. Probability: Theory ad Examples, 3rd ed. Duxury Press, Belmot, MA. Glasserma, P. 2004. Mote Carlo Methods i Fiacial Egieerig. Spriger, New York. Glasserma, P., H. Heidelberger ad P. Shahabuddi. 2000. Variace reductio techiques for estimatig value-at-risk. Maagemet Sciece 46: 349-364. Glasserma, P. ad S. Jueja. 2008. Uiformly efficiet importace samplig for the tail distributio of sums of radom variables. Mathematics of Operatios Research 33: 35-50. Glasserma, P. ad J. Li. 2005. Importace samplig or portfolio credit risk. Maagemet Sciece 5:643-656. Gly, P. W. 996. Importace samplig for Mote Carlo estimatio of quatiles. Proceedigs of Secod Iteratioal Workshop o Mathematical Methods i Stochastic Simulatio ad Experimetal Desig. St. Petersburg, FL, 80-85. Heyde, C. C., S. G. Kou ad X. H. Peg. 2007. What is a good exteral risk measure: Bridgig the gaps betwee robustess, subadditivity, ad isurace risk measures. Workig paper, Departmet of Idustrial Egieerig ad Operatios research, Columbia Uiversity, New York. Hog, L. J. ad G. Liu. 2009. Simulatig sesitivities of coditioal value at risk. Maagemet Sciece 55: 28-293. Rockafellar, R. T. ad S. Uryasev. 2000. Optimizatio of coditioal value-at-risk. Joural of Risk 272-28. Serflig, R. J. 980. Approximatio Theorems of Mathematical Statistics. Wiley, New York. Tridade, A. A., S. Uryasev, A. Shapiro ad G. Zrazhevsky. 2007. Fiacial predictio with costraied tail risk. Joural of Bakig ad Fiace 3: 3524-3538. AUTHOR BIOGRAPHIES LIHUA SUN is a Ph.D. studet i the Departmet of Idustrial Egieerig ad Logistics Maagemet at The Hog Kog Uiversity of Sciece ad Techology. Her research iterests iclude simulatio methodologies ad fiacial egieerig. Her e-mail address is <sulihua@ust.hk>. L. JEFF HONG is a associate professor i the Departmet of Idustrial Egieerig ad Logistics Maagemet at The Hog Kog Uiversity of Sciece ad Techology. His research iterests iclude Mote Carlo method, fiacial egieerig, stochastic optimizatio. He is curretly a associate editor of Operatios Research, Naval Research Logistics ad ACM Trasactios o Modelig ad Computer Simulatio.His email address is <hogl@ust.hk>. 422