Financial Risk Measurement/Management

Transcription

1 Fnancal Rsk Measureent/Manageent Week of epteber 30, 03 Volatlty Where we are Last week: Interest Rate Rsk and an Introducton to Value at Rsk (VaR) (Chapter 8-9) Ths week Fnsh-up a few tes for VaR: Includng, Margnal VaR and back-testng; then ove to Volatlty (Chapter 0) Next week: Correlaton and the Copula odel (Chapter ) Assgnent For epteber 30 th (Ths Week) Read: Hull Chapters 0 - (Volatlty, Correlaton and the Copula odel) Probles (Due epteber 30 th ) Chapter 8: 6, 7, 0, ; 6 Chapter 9:, 3, 4, 5 (a) (d); (a) (d) Probles (Due Oct 7 th ) Chapter 9: 5 (e) ; (e) Chapter 0 (Oct 7):,, 5, 7, 8, 9,, 5, 6; 9, Assgnent For Oct 7 th (Next Week) Read: Hull Chapter (Correlaton and Copula odels) Probles (Due Oct 7 th ) Chapter 9: 5 (e) ; (e) Chapter 0 (Oct 7):,, 5, 7, 8, 9,, 5, 6; 9, Probles (Due Oct 5 th ) Chapter 0 (Oct 5 th ): Chapter ( Oct 5 th ):,, 5, 6, 8, 0, 4;

2 Assgnent Mdter: October 30, 03 Fnal Exa Wednesday, Deceber 8 th ; 9a - Noon haffer 0 Coherent Rsk Measure A nuber of propertes have been proposed as desrable for a rsk easure f t s to be used to set captal requreents A rsk easure that satsfes all these propertes s called a coherent rsk easure These propertes follow Coherent Rsk Measures Propertes of coherent rsk easures If one portfolo always produces a worse outcoe than another ts rsk easure should be greater (onotoncty) If we add an aount of cash K to a portfolo ts rsk easure should go down by K (translaton nvarance) buffer aganst loss Changng the sze of a portfolo by should result n the rsk easure beng ultpled by (hoogenety) x the PF; x loss The rsk easures for two portfolos after they have been erged should be no greater than the su of ther rsk easures before they were erged (subaddtvty) allows for dversfcaton to reduce rsk 4.7 VaR vs Expected hortfall VaR satsfes the frst three condtons but not always does t satsfy the fourth one Expected shortfall (C-VaR) satsfes all four condtons. 4.8

3 VaR vs Expected hortfall Exaple 9.6 : Two $0 llon one-year loans each of whch has a.5% chance of defaultng. All recoveres between 0 and 00% are equally lkely. If there s no default the loan leads to a proft of $0. llon. If one loan defaults t s certan that the other one wll not default. 4.9 VaR vs Expected hortfall Exaple 9.6 (Contnued): ngle Loan -yr 99% VaR s $.5% chance of loss; f loss, then 80% chance t s > $ (unfor) Uncondtonal prob of loss > $ s 80% of.5% = % (99% VaR) Two Loan PF -yr 99% VaR s $5.8 Default occurs.5% of te, but never together Pr a default occurs s.5% ( = probablty of a default) If there s a loss, then 40% chance t s > $6 Uncondtonal prob of loss > $6 s 40% of.5% = % (99% VaR) A proft of $. on other loan plus loss of $6 => $5.8 Two Loans eparately = + = $4 Two Loans Together = $5.8 > eparately => No ubadd. 4.0 VaR vs Expected hortfall Exaple 9.8 : Consder sae stuaton agan, but for C-VaR C-VaR (Expected hortfall) fro loan for -year and 99% confdence level s Expected Loss condtonal on loss > $ = 99% -yr VaR W/unfor loss [0,$0]; expected loss, condtoned on > $,s halfway along the nterval [$,$0] = $6 VaR for PF of loans = $5.8 C-VaR s expected loss on PF condtonal on loss > $5.8 When loan defaults, other doesn t; outcoes = unfor [+$.,-$9.8] Expected loss gven we are n the part of the dstrbuton [$5.8,$9.8] s $7.8 Note: $6 + $6 > $7.8, so C-VaR s subaddtve 4. pectral Rsk Measures Rsk easures are characterzed by the weghts assgned to quantles of the loss dstrbuton VaR assgns all weght to Xth quantle Expected shortfall assgns equal weght to all quantles greater than the Xth quantle and zero to all below the Xth quantle We now consder alternatves 4. 3

4 pectral Rsk Measures pectral Rsk Measures A spectral rsk easure ay be defned by akng other assuptons about weghts assgned to quantles of the loss dstrbuton A spectral rsk easure s coherent, that s, It satsfes the ubaddtvty condton, IF The weghts are a non-decreasng functon of the quantles such as wth Condtonal VaR Another possblty for a rsk easure s to ake the weght assgned to q-th quantle proportonal ( to q )/ e where γ s a constant Ths s called the exponental spectral rsk easure Noral Dstrbuton Assupton The splest assupton s that daly gans/losses are norally dstrbuted and ndependent It s then easy to calculate 99%, -Day VaR fro the standard devaton of the -day loss dstrbuton (-day 99% VaR =.33) nce N(.33) = 0.0 or N(.33) = 0.99 The N-day VaR equals N tes the one-day VaR Assung the daly changes are..d. noral, N ( 0, σ ) Regulators allow banks to calculate the 0 day VaR as 0 tes the one-day VaR 4.5 Independence Assupton n VaR Calculatons (Equaton 9.3, page 93) When daly changes n a portfolo are dentcally dstrbuted and ndependent the varance over N days s N tes the varance over one day N-Day VaR = N -Day VaR When there s frst-order autocorrelaton, correlaton n the changes equal to the ultpler of the varance,, s ncreased fro N to 3 N ( N ) ( N ) ( N 3) N Where the correlaton between P jand P j s and P s the change n the PF value on day 4.6 4

5 Ipact of Autocorrelaton: Rato of N-day VaR to -day VaR (Table 9., page 93) N= N= N=5 N=0 N=50 N=50 = = = = As correlaton s present & ncreases fro zero, the approxaton of ultplyng the varance by N can ore serously understate VaR 4.7 Choce of VaR Paraeters Te horzon should depend on how quckly portfolo can be unwound. Regulators n effect use -day for bank arket rsk and -year for credt/operatonal rsk. Fund anagers often use one onth Confdence level depends on objectves. Regulators use 99% for arket rsk and 99.9% for credt/operatonal rsk. A bank wantng to antan a AA credt ratng wll often use 99.97% for nternal calculatons. VaR for hgh confdence levels cannot be observed drectly fro data and ust be nferred n soe way One approach: nce VaR( X ) N ( X ) then for the sae loss dstrbuton * * VaR( X ) VaR( X ) * N ( X ) VaR( X ) VaR( X ) * N ( X) N ( X ) N ( X) Deterne one confdence level VaR fro another Better approach s to use EVT 4.8 VaR PF Measures: An Aount x s Invested n the th ub-pf Margnal VaR: enstvty of VaR to the sze of the th subportfolo, x : (VaR) x Margnal VaR s related to the CAPM and f a sub-pf s beta s hgh, the argnal VaR wll be large Increental VaR: Increental effect of th sub-pf on VaR The dfference n VaR wth/wthout the th sub-pf Can calculate w/brute force: Fnd VaR w/wo the th sub-pf 4.9 VaR PF Measures: An Aount x s Invested n the th ub-pf Coponent VaR: Margnal effect of th sub-pf on N VaR leads to the forula: VaR C Where the parttoned coponent s sub-pf Coponent VaR: (VaR) C x x Add-up to the total VaR Alternatvely, consstent VaR (sae te horzon and confdence) ay be aggregated VaR VaRVaR total j j j 4.0 5

6 Exaple of Coponent VaR Consder the $00llon PF of assets: Exp Corr % 95% Y Asset Ret % Vol % 3 Alloc VAR U tocks $5.3 U Bonds $0.9 Intl tocks $5.9 Portfolo $6.9 VaR =.645 x 0.3% x $00 x 00% = $6.9 As w ww j j j j Deonstrates the beneft of dversfcaton 4. Exaple of Coponent VaR Wth our defnton of argnal VaR: Increasng the sze of the Allocaton to U tocks fro 60% to 6% ncreases PF VaR fro $0.3 to $0.35, an ncrease of.44% (slarly for other assets) % Margnal Cop Asset Vol % Alloc Rsk Rsk, C U tocks U Bonds Intl tocks Portfolo Back-testng Back-testng a VaR calculaton ethodology nvolves lookng at how often exceptons (loss>var) occur If exceptons occur ore or less frequently than the odel VaR predcts s there a proble? We copare what we see wth what we expect and test f the exceptons are consstent wth the populaton to soe level of confdence 4.3 Back-testng uppose that the theoretcal probablty of an excepton s p (=-X/00). The probablty of or ore exceptons n n days s: n n! k nk p ( p k!( n k)! ) k An often used confdence level n statstcs s 5% If the probablty of VaR level beng exceeded or ore days s less than 5%, we reject the hypothess that the probablty of an excepton s p If the probablty of the VaR level beng exceeded on or ore days s greater than 5%, the hypothess s not rejected and that the probablty of an excepton s p the odel for VaR s a good one 4.4 6

7 Exaple of Backtestng uppose we back-test VaR w/600 days of data. VaR confdence s 99% and we observe 9 exceptons (the expected nuber of exceptons s 6) hould we reject the odel? Probablty of 9 or ore exceptons n EXCEL s BINOMDIT (8,600,0.0,TRUE) = 0.5 >.05 At 5% confdence level we should not reject the odel Probablty of or ore exceptons n EXCEL s BINOMDIT (0,600,0.0,TRUE) = 0.09 <.05 Probablty of 0 or ore exceptons n EXCEL s BINOMDIT (9,600,0.0,TRUE) >.05 The odel s stll a good one up thru 0 observatons w/95% confdence not too low an estate for rsk 4.5 Back-testng In the exaple, the odel was for a 99% VaR In 600 observatons that equated to 6 tes Wth 9 exceptons, we were able to conclude that the odel was stll good not too rsky On the other hand, f we observed exceptons should we reject the odel as beng too conservatve Here, f the probablty of an excepton s p (=-X/00), the probablty of or fewer exceptons s n! k nk p ( p) k 0 k!( n k)! Ths s copared to 5% as before 4.6 Exaple of Backtestng uppose agan we back-test VaR w/600 days of data. VaR confdence s 99% and we observe excepton) hould we reject the odel as too conservatve? Probablty of 0 or exceptons n EXCEL s BINOMDIT (,600,0.0,TRUE) = 0.07 <.05 At 5% confdence level we should reject the odel However, f the nuber of exceptons had been or ore (up to 6) we would not reject the odel wth 95% confdence 4.7 Back-testng Alternatvely, there s a relatvely powerful -sded test. If the probablty of an excepton under VaR s p and exceptons n n trals are observed, then n n ln ( p) p ln ( / n) ( / n) should be ch-squared wth -degree of freedo The value of the statstc s very hgh for ether low or hgh occurrences of exceptons There s a probablty of 5%that the ch-squared varable wth -degree of freedo s greater than 3.84 If the above s greater than 3.84 we should reject the odel ether at the low end or the hgh end 4.8 7

8 Exaple of Backtestng uppose we agan have a stuaton lke the prevous two exaples where we back-test 99% VaR w/600 days of data. The value of the statstc s greater than 3.84 when the nuber of exceptons s and less or and ore Therefore we accept the VaR odel when and reject otherwse The End for VaR For now we shall return and address ore about VaR later Background on Volatlty The volatlty of a varable s the standard devaton of ts return wth the return beng expressed wth contnuous copoundng The varance rate s the square of volatlty Ipled volatltes are the volatltes pled fro opton prces Norally days when arkets are closed are gnored n volatlty calculatons (5 days per year; see Busness napshot 0., page 07) Ipled Volatltes Of the varables needed to prce an opton the one that cannot be observed drectly s volatlty We can therefore ply a volatltes fro arket prces and vce versa

9 Prce as the Market Varable For the non-dvdend payng stock, we assued a odel for stock prce oveent where the expected percentage return s ndependent of stock prce If s the stock prce at te t, the expected drft rate n should be assued to be µ x for soe constant paraeter µ Ths eans that n a short nterval of te, Δt, the expected ncrease n s µ Δt. The paraeter µ s the expected annual rate of return on the stock so The ncreent Δt, s n decal years If volatlty of the stock prce s always zero, then ths odel d ples Δ = µ Δt and as t 0 d = µ dt or dt T Ths ples, ntegratng fro 0 to T, that T 0e When the varance rate s zero, the stock prce grows at a contnuously copounded rate of µ per unt te 4.33 Prce as the Market Varable In practce a stock prce does exhbt varance fro expectaton A reasonable assupton s that the varablty of the percentage return n a short perod, Δt, s the sae regardless of stock prce Ths suggests that the standard devaton n a short perod of te Δt should be proportonal, σ, to the stock prce and leads to the odel d dt dz or d dt dz The process dz s the Wener process wth drft 0 and varance rate The above process s known as geoetrc Brownan oton and σ s the volatlty of the arket varable per year The dscrete te verson s t t or t t Where Δ s the change n stock prce n a sall te nterval Δt and ε s standard noral (0,) 4.34 Prce as the Market Varable In suary, Δ/, s norally dstrbuted wth ean µ Δt and standard devaton t (varance t ) ~ ( t, t) Consder the process followed by ln, where follows geoetrc Brownan oton G G G Defne G = ln, then snce,, and 0 t t follows fro Ito s lea that the process followed by G s dg dt dz nce µ and σ are constant, G = ln follows a generalzed Wener process wth constant drft rate and constant varance rate σ Prce as the Market Varable Ths eans that the change n ln between 0 and a future te T s norally dstrbuted wth ean ( ) and varance T T ln T ln 0 ~ ( ) T, T or ln ~ ln 0 ( ), T T T where ϕ(,v) denotes a noral dstrbuton wth ean and varance v Reeber that a basc property of the Wener process dz s that [z(t)-z(0)] s norally dstrbuted wth ean 0 and standard devaton T The equaton above shows that ln T s norally dstrbuted Furtherore, a varable s sad to have a lognoral dstrbuton f the natural logarth of the varable s norally dstrbuted

10 Prce as the Market Varable For the non-dvdend payng stock, we assue a odel for stock prce oveent where the expected percentage return s ndependent of stock prce For ths process, as we have developed, ~ ( t, t) where we started wth d dt dz And we have shown that T ln ln T ln 0 ~ ( ) T, T and 0 ln T ~ ln 0 ( ) T, T In ether case, sae varance rate (volatlty) We use both odels nterchangeably n quantfyng volatlty 4.37 Prce as the Market Varable For the non-dvdend payng stock, we assue a odel for stock prce oveent where the expected percentage return s ndependent of stock prce For ths process, ~ ( t, t) Where we started wth Geoetrc Brownan Moton d dt dz 4.38 Prce as the Market Varable If we consder the process followed by ln, where follows geoetrc Brownan oton and G G G Defne G = ln, then snce,, and 0 t t follows fro Ito s lea that the process followed by G s dg dt dz We get T ln ln T ln 0 ~ ( ) T, T and 0 t ln ~ ( ) t, t and t In ether case, sae varance rate (volatlty) We use both odels nterchangeably n quantfyng volatlty 4.39 Prce as the Market Varable Deternng volatlty fro hstorcal prce data We usually observe prces at fxed ntervals (daly, weekly, or onthly) Defne n : Nuber of observatons : tock prce at the end of the th nterval, 0,,..., n : Length of the te nterval n years Let u ln for,,..., n o the usual estate, s, of the standard devaton of the u s gven by s u u n n ( ) or n n s u u n n( n) where u s the ean of the u: u u n n

11 Prce as the Market Varable T t Fro ln ~ ( ) T, T & ~ ( ) t, t 0 t we see that the varance of the u s σ τ/n (where τ/n s the te nterval assocated wth each u The varable s s therefore an estate of σ τ and t follows that σ tself can be estated as ˆ s ˆ n The standard error of ths estate s approxately More data adds to the accuracy of the estate, but the assupton that volatlty s a constant s only an expedence; t does change over te and old data ay not be relevant for predctng future volatlty Most analyses for volatlty derved VaR depends on the Ito process wth Gaussan randoness n the te seres Is that a good/reasonable assupton? 4.4 Are Daly Changes n Exchange Rates Norally Dstrbuted? 4.4 Are Daly Changes n Exchange Rates Norally Dstrbuted? Table 0., page 09 Alternatves to Noral Dstrbutons: The Power Law: Real World (%) Noral Model (%) > D >D >3D >4D >5D >6D Prob(v > x) = Kx - ees to ft the tal behavor of the returns for any arket varables better than the noral dstrbuton where x s the nuber of standard devatons and two paraeters defne the exponental dstrbuton o we have ln[prob(v > x)] = ln K α ln x 4.44

12 Alternatves to Noral Dstrbutons: The Power Law Alternatves to Noral Dstrbutons: The Power Law (x = # standard devatons) Let x be the nuber of standard devatons fro Table 0. for the Real World data Alternatves to Noral Dstrbutons: The Power Law Usng the data for x = 3, 4, 5, & 6 standard devatons: Pr ob( v x) ln x o.06 K e Hence 5.5 Pr ob x.88x For the specfed nuber of standard devatons, x Estatng Volatlty tandard Approach The alternatve to the noral dstrbuton the power law shows prose n characterzng the tal behavor (we wll coe back to ths dea later) In the ean te ost rsk anagers stll work wth, and prefer, volatlty-based VaR We look at standard technques and refneents that are popular

13 Estatng Volatlty tandard Approach Defne n as the volatlty per day between day n- and day n, as estated at end of day n- Defne as the value of arket varable at end of day Defne u = ln( / - ), the contnuously copounded return durng day, and where we use only the last days of data n ( u n u) u n u for an unbased estate of the varance rate per day, n 4.49 Estatng Volatlty Refnng the tandard Approach For purpose of ontorng daly volatlty, the prevous estate s changed n the followng ways Defne u as ( - - )/ - the percent day-change Assue that the ean value of u s zero Replace - by (nvokes a axu lkelhood estate vs. an unbased estate ore later) Ths gves n u n 4.50 Estatng Volatlty Refnng the tandard Approach Estatng Volatlty Varatons to the tandard Approach Instead of assgnng equal weghts to the observatons u we can alternatvely set n u n where Where, f we choose j for j, less weght s gven to older observatons An extenson to ths dea s to assue there s a long run average varance rate, V L, and that ths should be gven soe weght also, n V L u n where The ARCH() odel (AutoRegressve Condtonal Heteroscedacty) We can alternatvely denote V L In an exponentally weghted ovng average odel (EWMA), the weghts assgned to the u declne exponentally ovng back through te In n u n where we let Ths leads to a recursve relatonshp ) u ( n Whereby lttle data needs to be saved; only the last estate and the new update (estate for day n) A sall λ leads to heavy weght beng put on ost recent observaton; a large λ provdes an estate that changes ore slowly, wth less volatlty n the estate RskMetrcs has found the value λ=0.94 to be satsfactory n n 3

14 Exaple Estatng Volatlty Varatons to the tandard Approach 4.53 The next approach s a sple extenson to EWMA, whch adds the feature of a long-run average varance, V L It s called GARCH(,) Generalzed AutoRegressve Condtonal Heteroscedacty GARCH (,) assgns a weght to the long-run average varance rate and ncorporates an updatng procedure slar to EWMA: n VL un n nce weghts ust su to : Reduces to EWMA when 0,, 4.54 Estatng Volatlty Varatons to the tandard Approach Exaple GARCH (,) ndcates the estate s calculated fro the sngle ost recent u and the sngle ost recent estate of the varance rate The ore general GARCH (p,q) estates fro the p ost recent u and the q ost recent estates of the varance rate ettng V L, the GARCH (,) odel can also be wrtten as n u n where V n L For stable GARCH, so weght on V L s > 0 p q For GARCH (p,q) n un jn j j

15 Estatng Volatlty Paraeterzng the Key to akng the Volatlty Estaton approaches effectve s to have a way to paraeterze the odels that gves good results n reproducng the data In axu lkelhood ethods we choose paraeters that axze the lkelhood of the odel to predct the data observed Reeber axu lkelhood technques? Estatng Volatlty Paraeterzng the A sple exaple of the axu lkelhood (ML) technque n statstcs We observe that a certan event happens one te n ten trals. What s our estate of the proporton of the te, p, that t happens? The probablty of the event happenng on one 9 partcular tral and not on the others s p( p) We axze to obtan a ML estate 9 L( p) p( p), then ax L( p) s found by solvng L (p)=0 p Result: p= Estatng Volatlty Paraeterzng the A ore pertnent exaple Estate the varance of observatons, u, u,, u fro a noral dstrbuton wth ean zero The lkelhood of the observatons appearng as they were observed s u The ML estate of v exp v v u ax exp v v v Is equvalent to takng logarths and axzng u u ln( v) or ln( v) v v Whch results n: v u 4.59 Estatng Volatlty Paraeterzng the For applcaton of ML estaton to paraeters of GARCH(,), the proble s to fnd the best paraeters (ω, α, β)n the expresson for the estator u v Ths s u ax exp v (,, ) v v Or equvalently u ax ln( v ) v (,, ) v

16 Estatng Volatlty Paraeterzng the (Exaple: Table 0.4) Estatng Volatlty Paraeterzng the (Table 0.4) An exaple of ML estaton for the GARCH(,) odel Data for Yen/U$ exchange rate ee the Table 0.4 tart wth tral values of,, and Update varances v u v Calculate u ln( v ) v Use solver to search for values of,, and that axze ths objectve functon Iportant note: set up spreadsheet so that you are searchng for three nubers that are the sae order of agntude (?!) Estatng Volatlty Paraeterzng the One way of pleentng GARCH(,) that ncreases stablty s by usng Varance Targetng We set the long-run average volatlty equal to the saple varance Calculated fro the data or another reasonable value Only two other paraeters need be estated V L Indeed, snce When EWMA s used, And only λ need be estated so V L ( ) 0,, and Estatng Volatlty Paraeterzng the How dfferent/good are the varous estators For coparson n the exaple 3-paraeter G(,): Objectve functon =, paraeter w/varance targetng: =, paraeter EWMA: =,

17 Estatng Volatlty Paraeterzng the Estatng Volatlty Paraeterzng the Elnatng autocorrelaton: If the estate for s free of any of the autocorrelaton present n the u ; the odel for has succeeded n explanng that autocorrelaton and can be judged to be a good odel ee the data n Table 0.5 on the next page for yen-dollar fro before The table shows that the autocorrelaton are postve for u for all lags between and 5 On the other hand, for u /, soe are postve and soe are negatve and always of saller agntude suggestng a good result u Estatng Volatlty Paraeterzng the But, we can be ore scentfc usng the Ljung- Box statstc to confr ths observaton If a seres has observatons the Ljung-Box statstc s K where: k s the autocorrelaton of lag k w k k K s the nuber of lags, and k wk k For K = 5, zero autocorrelaton can be rejected wth 95% confdence when L-B statstc s greater than 5 For Table 0.5 where K s 5 : For the u, the L-B = 3 > 5 => autocorrelaton s present For the u /, the L-B = 8. < 5 => autocorrelaton reoved