White Pape Decomposing potolio isk using Monte Calo estimatos D. Mossessian, dmossessian@actset.com V. Vieli, vvieli@actset.com
Decomposing potolio isk using Monte Calo estimatos Contents Intoduction............................................................ 2 Risk measues, maginal isk contibutions and isk measue components......................... 2 Monte Calo estimates o VaR and maginal contibutions to VaR.............................. 3 Monte Calo estimatos o component isk in MAC model.................................. 5 Potolio VaR estimato................................................... 6 Secuity contibution to VaR................................................. 6 Facto Contibution to VaR................................................. 7 Secuity and Facto contibution.............................................. 7 Choice o ε and numeical examples.............................................. 8 Conclusion............................................................ 9 Bibliogaphy........................................................... 9 Appendix............................................................. 11 Maginal contibution to VaR as conditional expectation................................... 11 Secuity and acto coection paametes........................................... 11 Intoduction Investment pactitiones ely on vaious isk measues - Tacking Eo Volatility (TEV), Value-at-Risk (VaR), Expected Shotall (ES) - and thei decomposition into contibutions om secuities and isk actos to guide investment decisions. In this note we analyze decomposition o potolio isk into maginal contibutions (measuing individual component impact on total isk), o additive components (the components that will sum to the total potolio isk measue), each component being associated with a single secuity, an investment secto, an asset class, o a isk acto. The key popety undelying this decomposition is positive homogeneity o the isk measue, a popety that states that the isk o a potolio scales in popotion to the size o the potolio. Taking this popety into account, these isk measue decompositions ae easily computed unde the assumption o joint nomality o the component etuns. In most o actual cases, howeve, the assumption o nomally distibuted etuns does not apply. Fat tailed distibutions ae ule athe than exception o inancial maket actos and the inclusion o non-linea deivative instuments in the potolio gives ise to distibutional asymmeties. Wheneve these deviations om nomality ae expected to cause seious biases in VaR calculations, one has to esot to eithe altenative distibution speciications o simulation methods. In this note we pesent the methodology o estimating these metics in the context o Monte Calo simulations. The maginal isk contibutions and components associated with both Value at Risk and Expected Shotall can be epesented as conditional expectations o component etuns, conditioned on events in the tail o the loss distibution o the ull potolio. The aity o these tail events pesents an obstacle to pactical calculation o these conditional expectations. Each contibution depends on the pobability o a ae event (a lage loss o a paticula component) conditional on an even ae event (an exteme loss o the potolio as a whole). This note descibes methods and techniques we use to addess the pactical diiculties o calculating these expectations. 2 FactSet Reseach Systems Inc. www.actset.com
Risk measues, maginal isk contibutions and isk measue components The distibution o etuns in a potolio is typically summaized though a scala measue o isk. Two o the most commonly used isk measues ae Value-at-Risk (VaR) and Expected Shotall (ES). The VAR associated with pobability 1 α (eg, α = 1%) is the lowe bound o the loss incued by a potolio with pobability α: V ar α : P ( p V ar α ) = α (1) whee p is the potolio etun. The coesponding expected shotall is the conditional expectation ES α = E[ p p V AR α ] (2) and can be intuitively intepeted as the aveage o all losses above a given quantile o the loss distibution. Value-at-Risk is in moe widespead use, but expected shotall is coheent (in the sense o [1]) while VaR is not. VaR is not in geneal subadditive, which means that the sum o the VaRs o two potolios is not equal to the VaR o the combined potolio. In paticula this means that the VaR o a potolio cannot be decomposed as a sum o the standalone VaRs o its components. Fo the pupose o isk management it is not suicient to just estimate a single measue o the potolio isk as a whole. Fo capital allocation, measuement o isk-adjusted peomance, developing hedging stategies and in geneal undestanding the impact o dieent isk actos and component on potolio isk it is useul to allocate the isk to elements o the potolio based on thei maginal contibution to total isk. To see how the maginal contibution to VaR can be calculated let us conside the etun p o a potolio that consists o n secuities, each having etun i, with the weight o the i th secuity in the potolio denoted as w i : n p = w i i (3) i=1 We should note that om the point o view o isk acto model, the etun o the potolio can be similaly epesented as a weighted sum o the acto etuns. Thus, equation (3) can be used to look at the potolio isk decomposition by acto etuns, as well as by secuity etuns. The maginal contibution to potolio VaR om component i, MV ar α,i, is the change in potolio VaR esulting om a maginal change in the i th component position : MV ar α,i = V ar α w i (4) This metic allows potolio manages to ind the components that can be used to signiicantly evise the oveall isk o the potolio with the minimal change to capital allocation. We can us the maginal contibutions to VaR to deine the additive VaR components that will sum to the total potolio isk as V ar α = i CV ar α,i. To do that we should note that VaR deines a quantile o the potolio etun distibution and, as potolio etun, is a homogeneous unction o component weights (meaning that multiplying all weights by the same numbe leads to VaR scaling by that numbe). Thus, accoding to Eule homogeneous unction theoem [2], VaR can be decomposed as V ar α = n i w i V ar α w i (5) This means that we can deine an additive component o VaR in tems o the maginal contibution to VaR as CV ar α,i = w i MV ar α,i (6) It can be shown (see Appendix ) that the maginal contibution to potolio VaR is the conditional expectation o the component etun, conditioned on ae values o the potolio etun V ar α : MV ar α,i = V ar α w i = E[ i p = V ar α ] (7) 3 FactSet Reseach Systems Inc. www.actset.com
When VaR is estimated using a linea nomal model, calculating contibution to VaR is ast and easy - one just need to use equation (4) and dieentiate the paametic expession o VaR. But in case o Monte Calo simulation contibution to VaR has some sevee poblems - the sampling vaiability o the estimate is lage and will not go down as we incease the numbe o samples o the simulation. The poblem is that the contibution to VaR om a given component depends on the single etun sample that happens to be the α th etun obsevation o the potolio (the simulated V ar α ). The contibution to VaR depends on that single etun obsevation in such a way that the sampling vaiability does not change with the numbe o tials in the simulation. Next section descibes Monte Calo techniques used to compute VaR, explains in moe details the poblem with estimating VaR components and outlines the methodology used to ovecome the poblem (o detailed theoetical teatment o the poblem and analysis o solution methods see, o example, [3],[4], and [5]). Monte Calo estimates o VaR and maginal contibutions to VaR Estimation o the isk decomposition descibed by (4) and (6) by Monte Calo is a two steps pocedue. Fist, the Value-at-Risk (and Expected Shotall) is estimated, and then the isk contibutions ae computed using the value o VaR om the ist step in place o the tue VaR in the conditional expectations (7). To ceate Monte Calo estimato o potolio VaR we should ist wite the equation (11) o VaR though the conidence level α as α = P (x V ar α ) = V ar α (x)dx (8) and note that this equation can be ewitten as an expectation o the indicato unction deined as { 1 i x a I(x a) = 0 i x > a (9) as α = I(x V ar α ) (x)dx = E [I(x V ar α )] (10) The expectation epesentation can be used to compute VaR when we geneate, using Monte Calo method, a sample o independent and identically distibuted potolio etuns i, i = 1,..N. In this case the estimato o the expectation is: α = 1 N N I( i V ar α ) (11) i=1 and in this om it can be used to ind VaR om a soted list o sample etuns given a value o α. Suppose we pick a value o VaR equal to the n th etun in the sample V ar α = n. Then o evey sample with i n in the soted list the indicato unction is equal one, and o evey sample with i > n the indicato unction is zeo. Thus, the conidence level o that value o VaR is α V ar = n N (12) So we can just ind the value n in the list such that n N is closest to the given value o α and use the sample n as an estimate o VaR. 4 FactSet Reseach Systems Inc. www.actset.com
When estimating VaR o a potolio that consists o S secuities we ae geneating K i.i.d. vectos o secuity etuns i T = ( 1i, 2i,..., Si ) whee i is a single (i th ) sample o a joint distibution o individual secuity etuns ji. Fom these vectos we compute K i.i.d. potolio etuns ip = w T i, and use these samples to estimate potolio VaR. An estimato o a conditional expectation (7) o the k th secuity etun in this case will be K i=1 E[ k p = V ar α ] = kii( ip = V ar α ) N i=1 I( ip = V ar α ) (13) Unotunately, i we just geneate a single sample o K values o ip the sums in the estimato o conditional expectation will only have a single non-zeo tem. In othe wods, we will have a single MC sample in the egion o inteest. We can emedy this situation by geneating a numbe o samples o K potolio etuns, such that we will have multiple ealizations o ip = V ar α. This, howeve, is extemely ineicient. Instead, we can elax the condition in the expectation (13) om p = V ar α (14) to p + V ar α < ε (15) so that the estimato o the maginal contibution to VaR becomes: MV ar α,k = E[ k p + V ar α < ε] = K i=1 kii( ip + V ar α < ε) N i=1 I( ip + V ar α < ε) (16) (hee the symbol indicates that this is a biased estimato o maginal contibution). The size o the neighbohood ε will detemine the numbe o active points in the estimato (16). We need to have a easonable numbe o points in the neighbohood o VaR to bing the vaiance o the estimato down, but at the same time we have to estict the width o the egion to limit the vaiability o the potolio etun within the neighbohood. Because the aveaging egion o omula (16) p + V ar α < ε is located in the tail o the potolio etuns distibution, the median o the samples in the egion will be less negative than the mean, and the weighted aveage o the conditional mean etuns will be less negative than the potolio quantile etun. In othe wods, the weighted sum o maginal contibution estimatos is expected to be less than the estimated potolio VaR: n MV ar α,i < V ar α (17) To coect that we intoduce the nomalization acto ω deined as and deine the adjusted estimato o maginal contibution to VaR as i w i V ar α ω = n i w (18) imv ar α,i MV ar α,i = ωmv ar α,i (19) The coesponding estimate o the CVaR ollows om eq.(6). Due to the adjustment acto ω the sum o the CVaRs exactly equals the initially estimated oveall potolio VaR, as equied by CVaR deinition. Intoduction o the nomalization acto ω is vey simila in natue to the method o contol vaiates in Monte Calo estimations. The method elies on knowing the expectation o an auxiliay simulated andom vaiable, called a contol. The known expectation is compaed with the estimated expectation obtained by simulation. The obseved discepancy between the two is then used to adjust estimates o othe (unknown) quantities that ae the pimay ocus o the simulation. In ou case the potolio VaR is used as a contol vaiate o component VaR estimatos. The moe detailed analysis o the adjusted estimato and justiication o the nomalization pocedue can be ound in [4]. 5 FactSet Reseach Systems Inc. www.actset.com
Monte Calo estimatos o component isk in MAC model Fo a potolio o S secuities we geneate K Monte Calo samples (typically K = 5000) o each o the F actos. As a esult we have a 3D matix o etuns o each secuity, each acto, each sample. We can visualize the matix as having F vetical slices, each slice is a matix o S ows and K columns. Each ow o the slice matix is the set o etuns o one secuity/one acto, obtained on K Monte Calo samples. That is, each ow o each slice o the matix is a distibution o etuns om one secuity/one acto. I we add all vetical slices togethe, we will obtain the S K matix, whee each ow will be a distibution o etuns o individual secuity. I we slice the 3D matix hoizontally, instead o vetically, (imagine hoizontal 2D matices stacked on top o each othe), multiply each hoizontal slice by the appopiate secuity weight and add the slices togethe, we will have F K matix whee each ow is the distibution o etuns om individual acto. S Secuities 1,1,1.... 1,K,1 1,1,1.... 1,K,1 1,1,1.... 1,K,1 1,1,1.... 1,K,1 1,1,1.... 1,K,1. 1,1,1.... 1,K,1. 1,1,1.... 1,K,1. 1,1,1.... 1,K,1. 1,1,1..... 1,K,1. 1,1,1........... 1,K,1 S,1,1...... S,K,1 S,1,1...... S,K,1 S,1,1...... S,K,1 S,1,1...... S,K,1 S,1,1..... S,K,1 S,1,1..... S,K,1 S,1,1..... S,K,1 S,1,1..... S,K,1 S,1,1.... S,K,1 S,1,1......... S,K,1 F Factos K Samples Figue 1: Results o Monte Calo sampling with K samples o MAC on a potolio o S secuities Each element o the 3D sample matix sk is the etun o the s th secuity,om the th acto, o the k th Monte Calo sample. The potolio etun distibution sample p k can be computed as p k = S i w s sk (20) k We can wite this sum in two ways. Eithe as a weighted sum o etuns o the s th secuity om all the actos sk = M sk =1 S p k = w i sk (21) O as a sum o etuns om the th acto om all the secuities in potolio k = S w s sk s s=1 F p k = k (22) Potolio VaR estimato To compute Monte Calo estimato o the potolio VaR we ist use omula (20) to compute all K samples p k om potolio etun distibution. These samples ae soted and the estimato (11) is used to get the value o potolio VaR o a given pobability α ( V ar α ) and the index o the location o the estimato V ar α in the vecto o potolio etun samples. The soting ode o the potolio etuns samples is stoed - we will denote the soted sample index k to distinguish it om the oiginal index sample k. The obtained estimato o V ar α, and its location in the soted vecto o potolio etun samples kv ar is then used o computing maginal contibutions and components VaR in both secuity and acto spaces. 6 FactSet Reseach Systems Inc. www.actset.com
Secuity contibution to VaR To calculate secuity contibutions to VaR we will use the omula (21) o potolio distibution sample expessed though the etuns o individual secuities. S p k = w s sk s whee sk is the sum o etuns o the s th secuity om all the actos o Monte Calo sample k. Note the usage o the k index - each secuity distibution sample vecto s = { sk, k = 1,..., K} is soted in the same ode as the vecto o potolio etuns distibution. To compute the estimato o maginal contibution to VaR om secuity s using equation (16) we ist need to deine the aveaging egion bounday ε. We typically pick the width o the aveaging egion in omula (16) coesponding to cetain pecentage o the total Monte Calo samples. Fo example, i we use K = 5000 Monte Calo samples and deine the width o the aveaging egion as 5% that would coespond to 250 samples in the aveaging egion, o the value o ε = 125. Having deined ε we compute each secuity s estimato o maginal contibution to VaR MV ar s as MV ar α,s = E( s p = V ar) = k=k V ar ε sk whee is the total numbe o sample points in the aveaging egion. We can then compute the nomalization constant ω using equation (18) and obtain the adjusted estimato o maginal contibution to VaR om secuity s as Finally, the contibution to VaR om a secuity s is computed as (23) MV ar α,s = ω MV ar α,s (24) CV ar s = w s MV ar α,s (25) Facto Contibution to VaR Computation o the maginal contibution to VaR om a given isk model acto is simila to computation o secuity contibution. We stat om the epesentation o the potolio etun distibution sample as a sum o acto etuns (Eq.(22)) p k = k (note again the usage o odeed potolio sample index k ). The acto maginal contibution to VaR is then computed in the same way as secuity contibution, but stating om the 3D Monte Calo etun matix aggegated along the secuity diection (vetical aggegation in the igue (1)). Fo each acto the unadjusted estimato o the maginal contibution to VaR is computed as k=k V ar ε k MV ar α, = The nomalization coeicient ω is again computed in the same way as o secuity contibutions: (26) ω = M V ar α MV ar α, (27) And, inally, adjusted maginal acto contibution to VaR is computed as MV ar α, = ω MV ar α, 7 FactSet Reseach Systems Inc. www.actset.com
It is impotant to note that the nomalization coeicients ω computed using estimatos o maginal secuity contibutions, o maginal acto contibutions, ae exactly the same (see Appendix()). This allows us to compute ω only once, when ist set o maginal contibutions is estimated, and also use the same value o ω to adjust the individual secuity and acto contibution estimates (see below). Secuity and Facto contibution The most ganula decomposition has to be computed diectly om the 3D Monte Calo matix. Eectively, this decomposition is based on epesentation o the potolio VaR as the sum o conditional expectation o individual secuity etuns o each individual acto S F V ar = E( s p = V ar) (28) s w s Analogous to the maginal secuity o acto contibution, we can compute the adjusted maginal contibution om individual acto and individual secuity as MV ar s = ω k=k V ar ε sk This esults in S F matix o maginal contibutions MV ar 11... MV ar 1F..... (30) MV ar S1... MV ar SF (29) By constuction, the sum o all element in a ow o the maginal contibution matix is the estimato o the maginal secuity contibution to potolio VaR (om eq. (23)): MV ar s = ω =1 sk k=kv ar ε =1 = ω sk k=kv ar ε = MV ar s (31) At the same time (see eq. (26), the sum o all elements in a column o the contibution matix is the estimato o the maginal acto contibution to potolio VaR: MV ar s = MV ar (32) =1 Choice o ε and numeical examples This section analyzes numeical estimates o isk measues o a sample potolio (a Baclays Aggegate Index). The ocus is on the analysis o the eos in Monte Calo estimates o VaR and its components. The eos in Monte Calo estimatos aise due to statistical eos in numeical simulations o the andom distibutions. Monte Calo estimatos ae based on the weak law o lage numbes, that basically states that when the numbe o samples is α V ar α σ V ar σ V ar V ar α % ES α σ ES σ ES ES α % 90 4.79 0.13 2.66 6.55 0.11 1.60 95 6.16 0.14 2.23 7.69 0.10 1.26 97 7.06 0.12 1.63 8.44 0.10 1.16 99 8.61 0.13 1.55 9.92 0.13 1.29 Table 1: Estimatos and eos o VaR and ES o Baclays EUR Aggegate Index o dieent conidence levels α 8 FactSet Reseach Systems Inc. www.actset.com
inceased towads ininity, the estimatos tend towads the tue values o estimated quantities. Howeve, since in all pactical applications the estimatos ae based on a inite numbe o samples, they always have an uncetainty associated with them. This uncetainty can be educed by inceasing the numbe o samples. In most cases the vaiance o the estimato that elects that uncetainty is invesely popotional to the numbe o Monte Calo samples used to obtain the estimato. Monte Calo pocess itsel can also povide an estimate o the vaiance o the estimato. Fom the same computation one can obtain both estimated esult and an objective measue o the statistical uncetainty in the esult. In ou case, we use multiple Monte Calo simulations o a potolio VaR and its components at dieent conidence levels with K = 5000 samples as descibed above. We un each simulation ten times, and ecod the means and standad deviations o estimated VaR and MVaR values. We use this data to evaluate the adequacy o the numbe o samples o ou puposes and to establish an acceptable aveaging egion ε o computing maginal contibutions to VaR that povide easonable balance between bias and vaiance o the estimatos. Table (1) shows the estimated Value-at-Risk (V ar α ) and Expected Shotall (ES α ) values o dieent conidence levels α computed o Baclays EUR Aggegate Index. Also shown ae Monte Calo standad deviations o the estimatos σ V ar and σ ES and coesponding elative eo o each estimato, 250 200 150 100 50 0 0.0 0.1 0.2 Relative Eo σ/µ 1% ange 5% ange Figue 2: Distibutions o component VaR Monte Calo eos o two width o aveaging egions - 1% (coesponding to 50 aveaged points) and 5% (coesponding to 250 aveaged points) expessed as a pecentage o the estimato itsel. It is clea that with the employed numbe o samples (K = 5000) the eo o both isk measues neve exceeds 3%. The index we use o testing puposes contains aound 4000 secuities. We analyze peomance o the Monte Calo algoithm by compaing the distibutions o the eos o the components coesponding to each secuity obtained at dieent values o the aveaging egion width paamete ε (eq. (16)). Some o the secuities have negligible impact on potolio VaR, and the value o thei coesponding components ae vey small. This components, i consideed, will contibute lage elative eos to the distibution o eos even when Monte Calo vaiance is small in absolute measue. To avoid distoting the distibution o eos with this components we do not conside the components with the absolute value less then CV ar s < 10 5. Figue (2) shows two such distibution o V ar 95 obtained at values o ε coesponding to 1% and 5% o total numbe o samples. The distibution o eos obtained with ε = 1% (geen bas on the igue) has the mean o µ = 0.12 and standad deviation o σ = 0.08. In othe wods, most o the eos ae lying below the value o 16% (µ + σ 2 ). Inceasing the value o ε to 5% o the numbe o samples lead to signiicantly naowe distibution o eos. Now the mean o the distibution is at µ = 6%, and the standad deviation is σ = 4%, thus the majoity o the eos in this case ae less than 8%. Moeove, close examination o the distibution o absolute and elative eos shows that the elative eos 0.3 Mean σ α 1% 5% 1% 5% 90 0.15 0.07 0.11 0.05 95 0.12 0.06 0.08 0.04 97 0.1 0.04 0.08 0.03 99 0.08 0.05 0.06 0.04 0.4 Table 2: Paametes o eo distibutions lage than 20% ae obseved only on vey small CV ar values. In othe wods, only components with contibutions that ae not signiicant o the pupose o potolio isk analysis will have lage elative estimation eos. Finally, Table (2) shows the mean and standad deviation o the distibution o eos obtained with dieent values o aveaging egion width paamete ε (1% and 5%) o dieent VaR theshold pobabilities α. With the value o ε equal to 5% o the total numbe o samples, the cente o the eo distibution is located aound 6%, while its width neve exceed 9 FactSet Reseach Systems Inc. www.actset.com
the value o 5%. Thus, the value o ε = 5% is suicient to keep the eos o the VaR components within 10% ange. Conclusion Decomposition o potolio isk measues into components by secuity, asset class o acto, while elatively staightowad unde the assumption o nomal etun distibutions, becomes complicated when the nomality assumption does not hold. The decomposition o isk measues equies computations o conditional expectations o component etuns, conditioned on ae tail events in potolio etun distibution. The aity o the conditioning event is expessed in elatively lage eos o components estimates obtained using Monte Calo methods. In this note we have pesented the Monte Calo methodology o potolio isk decomposition that limits the eos o component estimates to acceptable levels. The eos can be uthe educed by applying moe sophisticated Monte Calo techniques (like, o example, impotance sampling) o estimating conditional expectations o tail events. These techniques, howeve, ae not diectly applicable to the case o high-dimensional isk acto model, such as MAC model, because they sue om sevee eduction o eiciency when dimensionality o the model inceases to seveal hundeds o actos o highe (see o example [6]). Application o such techniques to isk decomposition poblems in the amewok o the MAC model is the subject o ou cuent eseach. Reeences [1] Atzne, P., et. al. (1999). Coheent Measues o Risk. Mathematical Finance, 9(3), 203-228. [2] Hazewinkel, M. Encyclopedia o Mathematics, Vol.4. Spinge (1989) [3] Glasseman, P. (2005). Measuing Maginal Risk Contibutions in Cedit Potolios. FDIC Cente o Financial Reseach Woking Papes, 1-41. [4] Hallebach, W. G. (2002). Decomposing potolio value-at-isk: A geneal analysis. Jounal o Risk, 5, 1 18. [5] Gouieoux, C., Lauent, J. P., Scaillet, O. (2000). Sensitivity analysis o Values at Risk. Jounal o Empiical Finance, 7, 225-245. [6] Bengtsson, T., Bickel, P., Li, B. (2008). Cuse-o-dimensionality evisited: Collapse o the paticle ilte in vey lage scale systems. Pobability and Statistics, 2, 316-334 10 FactSet Reseach Systems Inc. www.actset.com
Maginal contibution to VaR as conditional expectation Suppose we have a bivaiate continuous andom vaiable (X, Y ) and its quantile q(α) deined as We want to compute the deivative P (X + ϵy < q(α)) = α dq(α) dϵ Let s wite the pobability as integal ove the distibution unction Dieentiating with espect to ϵ gives leading to P (X + ϵy < q(α)) = x+ϵy<q (x, y)dxdy = [ ] dq dϵ y (q ϵy, y)dy = 0 q ϵy (x, y)dx dy dq dϵ = y(q ϵy, y)dy (q ϵy, y)dy = E [ Y X + ϵy = q ] Secuity and acto coection paametes Let us compae the expessions o the secuity and acto contibutions coection paametes: ω = V ar α (33) M w s MV ar α,s s and ω = M V ar α MV ar α, (34) We can ewite the equation o maginal secuity contibutions (23) as MV ar α,s = k=k V ar ε sk = sk k=kv ar ε =1 11 FactSet Reseach Systems Inc. www.actset.com
and the equation o acto contibutions Eq (26) as MV ar α, = k=k V ar ε k = k=kv ar ε s=1 S w s sk Using these two expessions we can ewite the sums o the contibutions ove all secuities and actos that ae used in the denominatos o the omulas (33,34) o coection paametes: S w s MV ar α,s = s S s k=kv ar ε =1 w s sk and MV ar α, = k=kv ar ε s=1 S w s sk which shows that the sums ae equal, and, theeoe, the acto and secuity coection paametes ae equal. 12 FactSet Reseach Systems Inc. www.actset.com