Derivatives of Value at Risk and Expected Shortfall

Transcription

1 Dervatves of Value at Rsk and Exected Shortfall November 2003 Hans Rau-Bredow Dr. Hans Rau-Bredow Unversty of Cologne; Unversty of Würzburg Leo Wesmantel Str. 4 D Würzburg Germany hone.: +49(

2 2 Dervatves of Value at Rsk and Exected Shortfall Abstract Ths aer analyses dervatves of Value at Rsk (VaR and Exected Shortfall (ES. Frst, an elementary result s stated for contnuous robablty dstrbutons by whch dervatves of VaR and ES of arbtrary order can be derved through recursve alcaton. The case of dscrete dstrbutons wth only a fnte number of ossble values s also consdered. In ths case, exressons for the frst dervatves of VaR and ES can be develoed f an exceton s made for certan dscontnuty onts. JEL classfcaton: D 8, G Keywords : Rsk Management, Value at Rsk, Exected Shortfall, Margnal Rsk Contrbutons, Decomosng VaR and ES

3 3. Introducton Value at Rsk (VaR and Exected Shortfall (ES are two closely related and wdely used rsk measures. An mortant ssue are the dervatves of these rsk measures: If a new oston s added to the ortfolo, how does the rsk of the entre ortfolo change? In ractce, t s often assumed that margnal rsk contrbutons are roortonal to the contrbutons to the ortfolo standard devaton, but ths s only justfed under certan restrctons on the robablty dstrbuton: For examle, f returns are normal dstrbuted, VaR s always a multle of the standard devaton (e.g.,var 99% s 2.33 standard devatons. But t s wellknown that the assumton of a normal dstrbuton does not ft wth realty, n artcular because fnancal markets returns are fat taled and because credt rsk s asymmetrc and skewed. The queston arses f general exressons for the dervatves of VaR and ES exst whch do not rely on secfc assumtons about the robablty dstrbuton. A general and surrsngly smle result for the dervatve of VaR s due to Tasche (999, (2000, Lemus (999 and Goureroux/Laurent/Scallet (2000. These authors have shown that under certan contnuty assumtons, a frst order aroxmaton of VaR-contrbutons s gven by the condtonal exectaton of the margnal rsk, on condton that the ortfolo value s exactly equal to VaR. Smlarly, Tasche (999 has shown that the frst dervatve of ES s also gven as a condtonal exectaton, now on condton that losses are equal to or greater than VaR. A result for the second dervatve of VaR was gven by Goureroux/Laurent/Scallet (2000. For a smle ntuton of these results, consder a Monte Carlo smulaton wth e.g. 000 random scenaros, whch are ranked from the worst loss to the hghest gan. VaR 99% s then the outcome n the 0 th worst scenaro (losses do not exceed ths amount n 99% of all cases and ES s the average loss n the 0 worst scenaros. The smle fact that the value of the entre ortfolo s always gven as the sum of ts consttuents then suggests a decomoston of VaR and ES where the result of a artcular oston n the 0 th worst scenaro s consdered as ts contrbuton to VaR and the average loss of ths oston n the 0 worst scenaros s consdered as ts contrbuton to ES. Because of that, t s lausble that margnal contrbutons to VaR

4 4 or ES are both gven as condtonal exectatons, n case of VaR on condton that losses are exactly equal to VaR 2 and n case of ES on condton that losses are equal to or greater than VaR. However, t s not easy to comlete such smle consderatons to a rgorous roof. An llustraton of the roblems that wll arse can be gven wth reference to the work of Hallerbach (2002 (notatons translated: Assume, through varyng the exosure ε to Y, a small change n the allocaton of a ortfolo Z, where and Y are arbtrary random varables. Then, n wrtng VaR ( Z E( + ε Y Z VaR( Z ( Hallerbach (2002 wants to exlot the lnearty of the exectaton and concludes that VaR( Z E(Y Z VaR( Z (2 However, ths neglects that, because of Z, the ortfolo allocaton has also an nfluence on the condton Z VaR( Z. For (2 to be a correct result t must also be the case that E( + sy VaR( + sy s 0 s ε (3 Of course, because t has been roven n the lterature that the dervatve of VaR s gven as the condtonal exectaton and (2 s ndeed correct, (3 must also be true. But (3 n tself s not a trval result whch s obvous wthout a mathematcal roof. Therefore, although Hallerbach (2002 arves at a correct result, hs reasonng s not comletely convncng and cannot relace a rgorous roof. A dfferent dervaton for the frst and secnond dervatve of VaR, whch uses the Lalacwe transform, was gven by Martn/Wlde (2002.

5 5 Ths aer has two ams. Frst, an elementary result about the exectaton of a so called ndcator functon s derved, whch follows from the law of terated exectatons. Dervatves of VaR and ES of arbtrary order can then be derved through recursve alcaton of ths result. Ths wll be exlctly shown for the frst and second dervatves. The second am of the aer s to nvestgate the dervatves of VaR and ES n case of a dscrete robablty dstrbuton. Untl now, the case of a dscrete drtrbuton has not been studed n the lterature. It wll be shown by a counter-examle that the dervatve of VaR s not always gven by the condtonal mean n the dscrete case. However, t turns out that such counter-examles are related to certan dscontnuty onts of the condtonal mean. If an exceton s made for such dscontnutes, the results for the dervatves of VaR and ES can be generalzed to the dscrete case. The aer s organzed as follows: The next chater gves the formal defnton of VaR and ES for contnuous and dscrete dstrbutons. Thereafter, a basc result concerng the exectatons of the ndcator functon s develoed and aled n order to calculate dervatves of VaR and ES n the contnuous case. Subsequently, the case of a dscrete robablty dstrbutons s dscussed. A few concludng remarks follow. 2 But note that for a contnuous dstrbuton, the robablty that the ortfolo value wll be exactly equal to a gven VaR-number s always zero. The condtonal exectaton must then be nterretated n a non-elementary sense.

6 6 2. Defnton of Value at Rsk and Exected Shortfall Followng Tasche (2002, the formal defnton of VaR wth confdence level s as follows: ( VaR { t R Pr ob( t } nf (4 If the random varable descrbes gans (ostve values and losses (negatve values of a bank ortfolo, VaR accordng to ths defnton would be the mnmal amount of economc catal requred n order to reserve solvency ( + economc catal 0 wth a robablty of at least. If the nverse smlfes to: F of the cumulatve dstrbuton functon of exsts, the defnton VaR ( F (. (5 VaR does not dfferentate between small and very large volaton of the VaRthreshold. Indeed, VaR s not a so-called coherent rsk measure n the sense of Artzner et al. (999. For ths reason, Exected shortfall (ES has been roosed as an alternatve to VaR. ES s defned as the average loss on condton that losses are greater or equal than VaR 3. Ths defnton can be motvated by the fact that not only the robablty that the bank wll collase s of nterest to the deostors of the bank, but also f they wll loss everythng or only a small amount of money n case that the bank actually collases. ES then exresses the exected loss n case of a bank collase 4. Agan followng Tasche (2002, the formal defnton of ES s as follows: 3 The defnton of ES s smlar to that of other downsde rsk measures lke lower artal moments LPM. LPM are defned through LPM k E(max(t-,0 k. The dfference between LPM and ES s that n case of ES the target t VaR VaR( s not fxed at the outset but also deends on the ortfolo allocaton. Lower artal moments were dscussed n Fshburn (977.

7 7 ES ( E( { VaR ( } VaR ( ( Pr ob( > VaR ( (6 Here, A s the ndcator functon whch has value f A s true and zero otherwse. Frst consder a contnuous dstrbuton. Then t s always the case that Pr ob( > VaR ( and (6 reduces to: ES ( E( { VaR ( } E( VaR ( (7 The condtonal exectaton n (7 s sometmes also called Tal Value-at-Rsk (TalVaR 5. ES and TalVaR are therefore the same for a contnuous robablty dstrbuton. From the substtuton F ( t s t VaRs( one sees that ES or TalVaR can then also be wrtten as the average VaR for all confdence levels greater than : ES ( VaR ( tdf ( t VaR ( ds s (8 Conversely, VaR ( s the negatve dervatve wth resect to the confdence level of ES ( tmes -. Because of (8, ES ( s gven n fgure a by the hatched area tmes /(- ( 00 f 99%. Now consder a dscrete dstrbuton. If there are only a fnte number of ossble realzatons, then the robablty that losses are less than VaR could dffer from the confdence level. In ths case, a correcton has to be made n order to ensure that ES s the average loss n the worst (-% cases and not the average loss n all cases where losses are not less than VaR. In (6, ths correcton s gven by the second term of the numerator. In artcular f 4 However, ths mlctly assumes that deostors are rsk-neutral or that a seudo rsk-neutral robablty dstrbuton for the valuaton of state-deendent ay-offs has been used. 5 See for examle Artzner et al. (

8 8 < < (9 2 <... n and ( n, where the celng functon x denotes the least nteger greater than or equal to x, then VaR( and ES ( + ( [ n + ] (0 wth the notaton Pr ob(. Obvously, ES ( s the weghted sum of,...,, and the weght of s such that the weghts sum u to. The dfference between the contnuous and the dscrete case s also tred to llustrate by fgure b.

9 9 Pr ob( x Pr ob( VaR ( P VaR ( 0 x Fgure a: ES for a contnuous dstrbuton (hatched area tmes /(- Pr ob( x Pr ob( VaR ( VaR ( 0 x Fgure b: ES for a dscrete dstrbuton (hatched area tmes /(-

10 0 3. Dervatves of Value at Rsk and Exected Shortfall 3. Contnuous robablty dstrbutons In ths secton t wll be shown that the dervatves of VaR and ES can be derved by a recursve alcaton of an elementary result about the exectaton of an ndcator functon (theorem. For ths urose, the ndcator functon { t} s consdered whch s defned such that ts value s one f < s true and zero otherwse. The relatonssh to the condtonal exectatons s gven by the followng two equatons: E( Z t t t zf ( x,z dzdx f ( x,z dzdx E( Z{ E( { } t ( E( Z t zf ( t,z dz f ( t,z dz E( Z{ t E( { t } t (2 Also note that E({ t} Pr ob( t f ( t (3 t t where f (t s the densty of. The followng theorem analyses the mact f a small erturbaton ε Y s added to. It states that the dervatve wth resect to the exosure ε to Y can be relaced, after multlyng Z{ +ε Y by Y, by the dervatve wth resect to the uer threshold t:

11 Theorem : If, Y and Z are contnuously dstrbuted random varables, then: E( Z{ E( YZ{ t Proof: Frst note that for any fxed values Y y,z z : E( z{ + εy E( z{ t εy} ( t εy E( z{ s} E( z{ t εy} s t εy y s t Then, assumng that dervatve and exectatons can be nterchanged, by the law of terated exectatons 6 : E( Z{ E( E( Z{ Y,Z E( E( Z{ Y,Z E( E( YZ{ Y,Z E( YZ{ t t Results for hgher order dervatves can be get by a recursve alcaton of theorem 7 : n n n n E( Z{ E(( Y Z n { t (4 6 Defne g ( y, z E( Y y,z z. Consder the random varable g (Y,Z gven as a functon of the random varables Y and Z. The law of terated exectatons then states: E ( g(y,z E( E( Y,Z E(. 7 For a roof, nterchange the dfferentaton wth resect to ε and t and relace Z by (-Y n Z n each ste.

12 2 As an examle, the frst dervatve of VaR follows from theorem n the followng way: Frst note that VaR VaR( s mlctly defned as a functon of ε by: Pr ob( ε Y VaR E({ +εy VaR} const. (5 + Dfferentatng ths wth resect to ε and takng nto account that VaR deends on ε yelds E({ E({ 0 t VaR + t ε t VaR E( Y{ E({ t VaR + t t VaR t VaR VaR (6 and fnally because of (2: VaR E( Y{ t E({ t t t VaR VaR E( Y VaR (7 Other dervatves of VaR and ES can be calculated more or less n the same way. Theorem 2 collects the results for the frst and second dervatves of VaR and ES. Theorem 2 : Frst and second dervatves of VaR ( VaR and ES ( ES are gven as follows: VaR ( E( Y VaR 2 VaR ( 2 f ( σ ( VaR 2 (Y t t f ( t t VaR

13 3 ES ( E( Y VaR (v 2 2 (Y VaR f ES σ 2 ( VaR To roof (-(v, t s useful to frst show the followng generalzaton of (v: (v E( Z VaR cov(y,z VaR f +εy ( VaR Proof: ad(v: From equatons (, (2, (3 and (7 t follows that: E( Z VaR E( Z{ VaR} ( E( YZ{ E( Z{ VaR t t t VaR {[ E( YZ{ t E({ t E( Z{ t E( Y E({ t E({ t ] } t t VaR [ E(YZ t E(Y t E( Z t ] f ( t t VaR cov(y,z VaR f ( VaR

14 4 ad (: ES E( εy VaR E( sy s VaR s ε E( + εy + sy s VaR s ε E( Y + εy cov(y, VaR + εy + εy VaR f ( VaR E( Y + εy VaR + 0 where the last but one stes follows from (v wth Z εy ad (v: (v s a secal case of (v wth Z Y. ad(: It follows from (8 that VaR s the negatve dervatve wth resect to the confdence level of ES tmes -. Interchangng dervatves and alyng (v then yelds: 2 2 VaR 2 2 ( ( ES 2 [ σ (Y VaR f ( VaR ] f ( VaR 2 [ σ (Y t t f ( t ] t VaR The last equaton follows because of (5 and ( VaR ( F ( f ( VaR

15 5 Dervatves of VaR and ES of a degree hgher than 2 can be get by further recursve alcatons of theorem. Ths requres that the results of theorem 2, n artcular the condtonal exectatons and the densty f ( t, can be exressed wth the use of the ndcator functon, whch s ossble because of (, (2 and ( Dscrete robablty dstrbutons In ths secton t wll be studed whether the revous results are also vald for a dscrete robablty dstrbuton. Frst, a counter-examle wll be develoed whch shows that the dervatve of VaR not always concdes wth the condtonal exectaton. Consder the followng case: Y 2,2% 97,8% 50% 50% (8 Assume that, Y are stochastcally ndeendent. Then: 00 ε,% 00 + ε,% (9 00 ε 48,9% 00 + ε 48,9% It follows from defnton (4 that VaR of suffcently small ε s gven by: wth confdence level 99% for VaR 99 % ( + ε Y 00 + ε (20 The mnmal economc catal s VaR 00 + ε, because wth less economc catal the bank would collase wth robablty,% > %. Then for the dervatve of VaR:

16 6 VaR 99 % ( (2 On the other sde, because and Y are assumed to be stochastcally ndeendent, one gets for the condtonal exectaton on condton that the loss s equal to VaR at ε 0 : E( Y + ε Y VaR 0 E( Y 0 E( Y 0 (22 ε Obvously, ths condtonal exectaton dffers from the dervatve of VaR. Ths s a contradcton to theorem 2, result (. However, f ε dfferent from, but very close to zero, and wth the dstrbutons of and Y as resumed n the examle above, the only ossble soluton of VaR 00 ε s 00 and Y. Therefore, for a suffcently small ε 0, t s always E ( Y VaR. It seems that t s only the dscontnuty of the condtonal exectaton at ε 0 that leads to a contradcton to theorem 2. In the neghbourhood of ε 0, also n our dscrete examle the dervatve of VaR always concdes wth the condtonal exectaton. Theorem 3 shows that ths observaton can be generalzed: Theorem 3: Consder two random varables and Y wth only a fnte number (,... n and ( m Y,... Y of ossble realzatons. Denote VaR ( + ε Y VaR and ES ( + ε Y ES. Then for all suffcently small ε 0 n a certan neghborhood of ε 0 : ( VaR E( Y + ε Y VaR

17 7 ( ES E( VaR Y{ εy VaR} ( Pr ob( + + εy > VaR Proof: Frst note that f the two random varables and Y have a fnte number of n res. m ossble realzatons, then there s a strct rankng order of the realzatons of n m ossble and ths rankng order wll be the same for all suffcently small ε 0 n a certan neghborhood of ε 0 : k( + ε Y < <... < l( k( 2 l( 2 k( nm l( nm Necessary condtons for ths are k( k( + for all and Y l( < Yl( + f l( l( +. Wth ths notaton, the roof can be stated as follows: ad( VaR wth confdence level s the mnmal realzaton of on condton that less than ( n m realzatons of are below VaR. If the celng functon ( n m denotes the least nteger greater than or equal to ( n m, then: VaR k( ε Y l( VaR Y l( On the other sde, f the erturbaton ε Y of s suffcently small, and on condton that ε 0, there s only one soluton of VaR <> + ε Y, namely and k( l( k( Y Y l(. Therefore: E( Y VaR E( Y k( ; Y Y l( Y l( VaR

18 8 ad(: Wth notaton Pr ob( + ε Y k( l(, and recordng that small changes of ε have no effect on the rankng order, t can be stated that E(( εy { VaR ( Y } + ε [( Y k( ε l( ] ( Y l( E( Y { } VaR ( It s then also the case that ( Pr ob( εy < VaR ( εy 0 Alyng ths to the defnton of ES (equaton (6 mmedately yelds: ES E(( εy { VaR ( Y } ε ( ( VaR( ( Pr ob( > VaR( εy VaR E( Y{ VaR} ( Pr ob( > VaR It s clear that the that the above result for the frst dervatve of ES s a generalzaton of the result of theorem 2 for the contnuous case. Note that n the dscrete case, hgher order dervatves of VaR and ES are always zero, because VaR and ES are then locally lnear functons of ε. Ths can also be seen as n accordance wth the results n the contnuous case, because f the condton VaR determnes that Y Y k(, then Y becomes non-stochastc and the condtonal varance σ 2 (Y VaR, whch aears n theorem 2 results ( and (v, wll be zero.

19 9 4. Concludng remarks It has been shown n the exstng lterature that general exressons for margnal contrbutons to VaR and ES are both gven by condtonal exectatons, n case of VaR on condton that losses are exactly equal to VaR and n case of ES on condton that losses are equal to or greater than VaR. These results do not rely on any secfc assumtons about the robablty dstrbuton. In ths aer, a more elementary and general relatonsh (theorem has been derved from whch dervatves of arbtrary order can be get by recurrent alcaton. Ths relatonsh may also serve for a deeer understandng of the mathematcs behnd the result that margnal contrbutons to VaR and ES are n fact gven by condtonal exectatons. It has also been studed n ths aer whether these results are vald for random varables wth dscrete robablty dstrbutons. Although counter-examles n artcular for the frst dervatve of VaR can then be constructed, t s obvous that such counter-examles are caused by certan dscontnuty onts whch could emerge f the condtonal exectaton s consdered as a functon of the ortfolo allocaton. It has been shown that t s ossble to calculate frst order dervatves of VaR and ES also n the dscrete case rovded an exceton s made for such dscontnuty onts. However, hgher order dervatves are always zero n the dscrete case, because VaR and ES are then locally lnear functons of the ortfolo weghts 8. References Artzner, P., Delbaen F., Eber J.-M. and Heath, D. (999: Coherent Measures of Rsk, n: Mathematcal Fnance 9 3, Fshburn, P.C. (977: Mean Rsk Analyss wth Rsk Assocated wth below Target Returns, Amercan Economc Revew, 67,.6-26 Goureroux C., Laurent J.P., Scallet O. (2000: Senstvty Analyss of Values at Rsk, n: Journal of Emrcal Fnance Vol. 7 ( It could be added that racttoners are not only nterested n theoretcal results for the dervatves of VaR and ES, but also how these dervatves can be estmated n ractse. Ths aer has not focused on ths estmaton roblem, but see e.g. Goureroux (2000, Lemus (999, Hallerbach (2002, Tasche/Tblett (200.

20 20 Hallerbach W. (2002: Decomosng Portfolo Value-at-Rsk, A General Analyss, n: The Journal of Rsk 5/2, vol. 5/2, Wnter 2002,.-8 Lemus-Rodrguez, G. (999: Portfolo otmzaton wth quantle-based rsk measures, PhD thess, Massachusetts Insttute of Technology. Martn, R., Wlde, T. (2002: Unsystematc credt rsk, n: Rsk, Tasche, D. (999: Rsk Contrbutons and Performance Measurement, workng aer. Tasche, D. (2000: Condtonal exectaton as quantle dervatve, workng aer. Tasche, D., Tblett, L. (200: Aroxmatons for the Value-at-Rsk aroach to rsk-return analyss, workng aer.