Value-at-Risk and Stress Testing for Cash Flow Portfolios

Transcription

1 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Valu-at-Risk an Strss Tsting for Cash Flow Portfolios Carol Alxanr AIMS Summr School Cap Town, South Africa Fbruary /40

2 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Outlin 1 Introuction 2 VaR for Cash Flows 3 Principal Componnt Analysis 4 Cas Stuy 5 Volatility Ajustmnt of Historical VaR 2/40

3 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Introuction Rgulatory risk capital cannot b us for risky invstmnt Basl rcommnations 1996 Minimum rgulatory capital (MRC) for markt risk is a multipl of th (avrag) 1% Valu-at-Risk (VaR) of 10-ay profit & loss (P&L) aggrgat ovr all activitis in th traing book α% VaR is minus th α-quantil of a P&L istribution masur aily En-of-ay positions ar mapp to major risk factors whos volution ovr th risk horizon is simulat using historical or Mont Carlo (MC) mthos Simulation multivariat istribution for portfolio risk factor P&Ls portfolio 1% 10-ay VaR aggrgation MRC 3/40

4 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Portfolio VaR an Capital Allocation Simulation Multivariat istribution for portfolios risk factors Portfolio mapping Portfolio P&L istribution Propr aggrgation Aggrgat P&L istribution Quantil Portfolio VaR Simpl ruls Quantil Aggrgat VaR Intrnal ruls Economic capital allocation Extrnal ruls Minimum rgulatory capital 4/40

5 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Strss Tsting Nw Rgulations Basl rcommnations 2009 Augmnt MRC by aing capital to covr strss VaR (svar) svar = VaR bas on strss conitions for markt risk factors changs in volatilitis, corrlations, highr momnts Givn tim sris VaR t i an svar t i of 1% aily VaR an svar stimat on ay t i, th aily MRC at tim t is comput as max { VaR t 1, m c i=1 VaR t i } +max { svart 1, m s i=1 svar t i } Multiplicativ constants 3 m c, m s 4 ar st by rgulators aftr backtsting th VaR an svar stimats of th mol 5/40

6 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Supplmntary Raing An analytic formula is riv in MRA IV.2 for VaR unr th assumption that risk factors hav a multivariat normal (MVN) istribution whn th portfolio is a linar function of its risk factors This lctur follows a cash flow cas stuy similar to that in MRA IV.2.4 Also s PV01 in [MRA III.1.8] an cash flow mapping in [MRA III.5.2.1] an [MRA III.5.3] PCA is scrib in [MRA II.2] Strss tsts ar covr in MRA IV.7 aninparticularw follow MRA IV in this lctur Simulation mthos for VaR an strss tsting ar covr in MRA IV.3 an MRA IV.4 6/40

7 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Cash Flow Mapping Cash flows on bt portfolios ar mapp to th rlvant yil curv, i.. th curv corrsponing to th crit rating of th bt Intrst rats at fix vrtics.g. w shall us 50 vrtics at 0.5, 1, 1.5,..., 25 yrs Mapping must b to at last 3 narby vrtics to prsrv PV Duration Volatility Snsitivity of mapp cash flow C T at maturity T to its intrst rat rat R T is givn by th prsnt valu of a basis point mov or PV01, fin as [ ] PV01 T = C T (1 + R T +0.01%) T (1 + R T ) T 7/40

8 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Cash Flow Valu-at-Risk Portfolio risk factors ar basis point changs in intrst rats Δr =(ΔR T1,..., ΔR Tn ) In our cas, n =50 Ignoring trms of orr Δr 2 in Taylor xpansion ΔPV 50 n=1 PV01 Ti ΔR Ti = p Δr PV01 vctor p =(PV01 T1,..., PV01 Tn ) = constant Portfolio aily P&L has linar rlation to risk factors analytic VaR unr MVN assumption 8/40

9 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt VaR Mols [MRA IV.1.9] W consir two common mthos for gnrating scnarios on risk factors MVN assumption normal linar VaR Historical sampl historical VaR VaR mols iffr in th way thy gnrat scnarios on Δr Givn scnarios on risk factors Δr ovr risk horizon portfolio mapping portfolio P&L istribution lowr quantil 1=VaR Also common to us aily changs for Δr thn compoun VaR to a 10-ay horizon using th squar-root-of-tim rul 9/40

10 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Normal VaR Empricially, changs in intrst rat ar not normally istribut Nvrthlss, normal VaR oftn us in practic Consistnt with intrst rat option pricing mols Easy to comput in larg firm-wi risk systms Normal linar VaR is irctly proportional to volatility 10.g. 1% 10-ay normal VaR is σ whn σ nots th annual volatiity of th portfolio, if thr ar 250 traing ays pr yar i.. 1% 10-ay normal VaR is a littl lss than half (46.5%) of its annual volatility 10 / 40

11 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Historical VaR Historical VaR is favour by about 75% of banks Lgacy systms No paramtric assumption Naturally capturs co-pnncis btwn risk factors But it has substantial isavantags Not sufficintly risk-snsitiv Data-intnsiv A simpl EWMA volatility ajustmnt to historical VaR incrass its risk snsitivity [MRA IV.3.3.3] 11 / 40

12 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Qustion Mont Carlo (MC) VaR Anothr common mtho for scnario gnration is to simulat from an assum paramtric multivariat istribution for th risk factors, using Mont Carlo mthos With th stanar mapping for cash flow portfolios, which of th following is tru? 1 If w on t assum normality w must us MC VaR 2 If w o assum normality w might still us MC VaR 3 W woul nvr us MC VaR with this mapping 4 Non of th abov 12 / 40

13 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt What is an Eignvalu/Eignvctor? An n n matrix V is a linar transformation from R n to R n That is, it translats a vctor x R n to anothr vctor y R n such that Vx = y For most vctors x, th vctor y os not li on th sam lin through th origin as x, i..y λx for som λ R Howvr thr ar som spcial vctors, which w not w, that o hav th proprty Vw = λw for som λ R W call such a spcial vctor w an ignvctor of V an th scaling factor λ th ignvalu blonging to w 13 / 40

14 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt How Many Eignvalus/Eignvctors Ar Thr? Two vctors ar orthogonal if thy hav zro ot prouct (i.. th sum of th proucts of th lmnts is zro) In R 2 two orthogonal vctors will b at right angls If vctors ar tim sris, orthogonality zro corrlation An n n matrix V has n orthogonal ignvctors an ach has an ignvalu V has xactly n ignvalus (but thy ar not ncssarily all iffrnt) Howvr, ignvctors o not hav th sam uniqunss proprty as ignvalus in if w is an ignvctor with ignvalu λ thn so is kw for any k 0 W usually stanariz ignvctors to hav unit lngth but still, ths orthonomal ignvctors ar only uniqu up to sign 14 / 40

15 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Spctral Dcomposition Lt λ = {λ 1,..., λ n } b th ignvalus of V An suppos ths hav ignvctors w 1,..., w n so that Vw i = λ i w i for i =1,..., n Us th notation Λ = iag{λ 1,..., λ n } for th iagonal matrix of ignvalus, an W =(w 1,..., w n ) for th orthogonal matrix of ignvctors Thn th n quations abov may b writtn all togthr as VW = ΛW Sinc W is an orthogonal matrix W 1 = W,sothabov may b writtn in a form known as th spctral composition of V i.. V = W ΛW 15 / 40

16 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt What ar Principal Componnts? Now suppos th matrix V is th covarianc (or corrlation) matrix of a st of tim sris on n asst rturns or (as in our cas) basis point changs in intrst rats If thr ar T obsrvations on ach asst/intrst rat, thn ths ata may b summariz in a T n matrix X W multiply th ata matrix X by th matrix of ignvctors of V, that is, w st P = XW This tranforms th original (corrlat) ata X into an orthogonal rprsntation of th sam ata, P Th uncorrlat tim sris in th columns of P ar call th principal componnts of X 16 / 40

17 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Principal Componnt Rprsntation Writing P = XW in th form X = PW an splling out all th quations, w hav x i = w i1 p w in p n, i =1,..., n whr x i is th T 1 vctor of input ata on th ith tim sris (.g. th aily basis boint chang in th i-month intrst rat), p j is th jth principal componnt an w ij is th ijth lmnt of th ignvctor matrix W Ifthoriginalataarhighly corrlat w gt a goo approximation to ach x i using only th first fw principal componnts,.g. x i w i1 p 1 + w i2 p 2 + w i3 p 3, i =1,..., n Hr w orr th principal componnts by crasing siz of ignvalu an th accuracy from using th first k componnts is k 1 λ i/ n 1 λ i 17 / 40

18 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Financial Applications of PCA PCA is a common statistical tool for xtracting ky risk factors from corrlat asst rturns (or, in our cas, th ky risk factors riving changs in intrst rats along th yil curv) It can b appli to orthogonaliz any st of ata, but imnsion ruction is gratst whn th input ata ar vry highly corrlat Trm structurs typically rquir only th first thr componnts for mor than 95% accuracy! First ignvctor almost constant first (most important) componnt capturs an almost paralll shift Scon ignvctor almost linar scon componnt capturs tilt along th trm structur Scon ignvctor almost quaratic thir componnt capturs chang in convxity of trm structur 18 / 40

19 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Cas Stuy Using PCA for Cash Flow VaR an Strss VaR Portfolio UK govrnmnt bons with maturitis up to 25 yars PV01 vctor givn for 31 July 2008 Portfolio lost 2.5 million ovrnight, on 6 March 2009 Qustions How much accuracy is lost whn using PCA for VaR analysis? Di strss VaR prict th portfolio s lossss uring th banking crisis? Coul a longr history of ata hav hlp? If so, how far back o w n to go? 19 / 40

20 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Charactrising th Portfolio on 31 July 2008 Nt xposur 25 million GBP Total xposur 100 million GBP PV01 of cash flows in 000 GBP maturitis 6 months to 25 yars 20 / 40

21 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Qustion What s th bt on th yil curv? 1 Paralll shift up 2 Paralll shift own 3 Upwar tilt (short up, long own) 4 Downwar tilt (short own, long up) Yil Curv on 31 July 08 Maturity 21 / 40

22 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Historical Data UK Govrnmnt Liability Curv Daily, / 40

23 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt PCA (Daily, ) Eignvalus Prcntag Varianc Explain 87.7% 8.8% 1.7% Cumulativ Varianc Explain 87.7% 96.5% 98.1% Eignvctors w1 w2 w3 1yr yr yr yr yr yr yr yr yr / 40

24 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt First Thr Eignvctors Trn, tilt an curvatur componnts l l l l l l l l l l l l t t t 24 / 40

25 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Accuracy of PC VaR Combin th PC rprsntation of Δr with th PV01 vctor in th portfolio mapping PC rprsntation for th portfolio aily P&L Also ΔPV t =2, 635 p 1t 15, 436 p 2t +6, 795 p 3t Varianc ΔPV = 2, λ 1 +15, λ 2 +6, λ 3 = 2, , , % 1 ay normal PC VaR = 349, 544 1% 10 ay normal PC VaR = 1, 105, 354 Approx. rror (rlativ to normal VaR without PCA) 0.09% 25 / 40

26 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Strss Tsting a Tilt in th Yil Curv ArisinPC 2 corrspons to an upwar tilt in th yil curv i.. short rats up an long rats own th worst scnario A 6-sigma valu for PC 2 p 2 = = 50.7 Sinc ΔPV = 2, 635 p 1 15, 436 p 2 +6, 795 p 3 this 6-sigma tilt in th wrong irction woul inuc a larg aily loss for our portfolio ΔPV = 15, = 782, 898 So a 2.5 million loss appars to b an 18 sigma vnt! 26 / 40

27 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Strss Tsting Coul history hav prict th losss uring th crisis? 27 / 40

28 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Wkly Data Spanning 40 yars! UK Govrnmnt Liability Curv Monthly, / 40

29 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt PCA (Monthly, ) Eignvalus ,733 29,158 6,696 Prcntag Varianc Explain 71.0% 19.8% 4.5% Cumulativ Varianc Explain 71.0% 90.8% 95.3% Eignvctors w1 w2 w3 1yr yr yr yr yr yr yr yr yr / 40

30 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt First Thr Eignvctors Trn, tilt an curvatur componnts l l l l l l l l l l l l t t t 30 / 40

31 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt PC Rprsntation of Portfolio P&L (Monthly) 95.3% of th monthly variation in th yil curv is xplain by this 3-componnt rprsntation ΔPV t =8, 056 p 1t 15, 264 p 2t 167 p 3t Varianc ΔPV = 12 ( 8, λ 1 +15, λ ) λ whr λ 1 =10, 4733, λ 2 =29, 158, λ 3 =6, 696 1% 1 ay normal PC VaR = 1, 878, 897, 1% 10 ay normal PC VaR = 5, 941, 595 Approx. rror (rlativ to normal VaR without PCA) 0.04% 31 / 40

32 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Strss Tsting a Tilt in th Yil Curv Again, consir a 6-sigma valu for PC 2 an convrt th monthly stanar viation to a aily quivalnt using p 2 =6 29, = W hav ΔPV = 8, 056 p 1 15, 264 p p 3 So this 6-sigma tilt in th curv pricts an ovrnight loss of ΔPV = 15, = 3, 426, 139 A 3.5 million loss is a 6-sigma vnt. A 2.5 million loss is actually wll within strss tsts whn th right ata ar us 32 / 40

33 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt RiskMtrics Volatility of Portfolio Daily P&L Annualiz stanar viation bas on [MRA II.3.8] RiskMtrics Rgulatory mol, i.. qually wight avrag of squar P&L 1 yar RiskMtrics xponntially wight moving avrag (EWMA) with a smoothing constant of ,000,000 12,000,000 10,000,000 8,000,000 EWMA Volatility Rgulatory Volatility 6,000,000 4,000,000 2,000,000 0 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan / 40

34 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Daily P&L an Comparison of 1% VaR Graph assums PV01 vctor is hl constant Estimat aily 1% VaR Jan 04 Dc 2010 Only th EWMA normal VaR is sufficintly risk-snsitiv W> EsZ EsZ tdesz,sz 34 / 40

35 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt EWMA Volatility Ajustmnt of Historical VaR Stp 1 Ajust th P&L tim sris Divi th aily P&L tim sris by th EWMA volatility tim sris an thn multiply all points in th normaliz tim sris by th EWMA volatility at th tim th VaR is stimat Stp 2 Buil an ajust P&L istribution This is th istribution of th P&L aftr stp 1 ajustmnt Stp 3 Volatility-ajust VaR Th EWMA volatility ajust historical α% aily VaR is 1 th α quantil of th istribution built in stp 2 35 / 40

36 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Pr-Crisis 31 July 2008 Sampl 1 Aug Jul % aily VaR = 297,306 (stanar) 306,620 (ajust) s W> W> E & D E & D E & D E & D 36 / 40

37 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Start of Crisis 31 Octobr 2008 Sampl 1 Nov Oct % aily VaR = 308,490 (stanar) 652,926 (ajust) s W> W> E & D E & D E & D E & D 37 / 40

38 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Hight of Crisis 31 March 2009 Sampl 1 Apr Mar % aily VaR = 501,851 (stanar) 1,264,887 (ajust) K K K K s W> W> 38 / 40

39 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Post Crisis 30 Sptmbr 2010 Sampl 1 Oct Spt % aily VaR = 602,532 (stanar) 484,740 (ajust) W> s W> E & D E & D E & D E & D 39 / 40

40 Introuction VaR for Cash Flows PCA Cas Stuy EWMA Ajustmnt Summary an Conclusions 1 Sinc th banking crisis, simpl VaR mols hav bn substantially ovr-stimating risk 2 Volatility ajustmnt is a simpl rmy 3 Extrm losss uring th crisis coul hav bn prict but only using ata from th 1970 s 4 Basl II proposals for strss VaR a-on to usual VaR computation of rgulatory capital 5 Mor rcnt Basl proposals (May 2012) suggst xpct tail loss (ETL) shoul b us in plac of VaR 40 / 40