VaR and Stress Testing Debt Portfolios Before, During and After the Banking Crisis

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1 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion VaR an Strss Tsting Dbt Portfolios Bfor, During an Aftr th Banking Crisis Carol Alxanr ICMA Cntr, Hnly Businss School at Raing Wbinar Broacast on 10 Fbruary 2011 Profssional Risk Managr s Intrnational Association (PRMIA) Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

2 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Outlin Cas stuy portfolio UK govrnmnt bons from 6 months to 25 yars 31 July 2008 Part 1: Valu at Risk (VaR) Why mols fail to prict lossss uring th crisis How VaR mols can asily b improv Part 2: Strss tsting Coul history hav prict th losss ma? Introucing principal componnt analysis (PCA) as an ssntial intrst rat analysis tool Auinc participation through voting buttons plas Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

3 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion UK Govrnmnt Liability Curv: Daily,

4 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Cash Flow Mapping Cash flows on bt portfolios ar mapp to rlvant yil curv Intrst rats at fix vrtics in this cas thr ar 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 fin as: [ ] PV01 T = C T (1 + R T +0.01%) T (1 + R T ) T Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

5 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Typical 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 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

6 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Qustion 1: What s th bt on th yil curv? 1 Paralll shift up 2 Paralll shift own

7 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Part 1: Valu-at-Risk Portfolio Mapping 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 (31 July 2008) Portfolio aily P&L has linar rlation to risk factors Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

8 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion VaR Mols Scnarios on risk factors Δr ovr risk horizon portfolio mapping portfolio P&L istribution lowr quantil 1=VaR VaR mols iffr in th way thy gnrat scnarios on Δr Two common mthos for gnrating scnarios on risk factors: A multivariat normal assumption normal linar VaR A historical sampl historical VaR Also common to us aily changs for Δr thn compoun VaR using squar-root-of-tim rul Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

9 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Normal VaR Changs in intrst rat ar not normal Nvrthlss, normal VaR oftn us 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 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

10 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion RiskMtrics Volatility of Portfolio Daily P&L Annualiz stanar viation bas on RiskMtrics Rgulatory mol, i.. qually wight avrag of squar P&L 1 yar RiskMtrics xponntially wight moving avrag (EWMA) with a smoothing constant of 0.94 Portfolio lost 2.5 million ovrnight on 6 March 2009

11 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Daily P&L an Comparison of 1% VaR Graph assums th PV01 vctor is th sam at vry point in tim Estimat aily 1% VaR from Jan 04 to Dc 2010, rolling winow Only th EWMA normal VaR is sufficintly risk-snsitiv

12 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Qustion 2: 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 our portfolio mapping 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 MC is bst for computing strss VaR 5 Non of th abov Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

13 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Volatility Ajustmnt Historical VaR is oftn favour: 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 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

14 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion EWMA Volatility Ajustmnt of Historical VaR Stp1 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 Stp2 Buil an ajust P&L istribution: This is th istribution of th P&L aftr stp 1 ajustmnt Stp3 Volatility-ajust VaR: Th EWMA volatility ajust historical α% aily VaR is 1 th α quantil of th istribution built in stp 2 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

15 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Pr-Crisis: 31 July 2008 Sampl: 1 Aug Jul % aily VaR = 297,306 (stanar) 306,620 (ajust) GBP

16 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Start of Crisis: 31 Octobr 2008 Sampl: 1 Nov Oct % aily VaR = 308,490 (stanar) 652,926 (ajust) GBP

17 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Hight of Crisis: 31 March 2009 Sampl: 1 Apr Mar % aily VaR = 501,851 (stanar) 1,264,887 (ajust) GBP

18 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Post Crisis: 30 Sptmbr 2010 Sampl: 1 Oct Spt % aily VaR = 602,532 (stanar) 484,740 (ajust) GBP

19 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Part 2: Strss Tsting Coul history hav prict th losss uring th crisis?

20 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Strss Tsting via PCA Principal Componnt Analysis (PCA) is a statistical tool for xtracting th ky risk factors from a larg numbr of highly corrlat asst rturns (or changs in intrst rats) Principal componnt (PC) risk factors ar uncorrlat tim sris PCA has applications to all asst classs, but is most accurat whn rturns (changs) ar vry highly corrlat thn only a fw PCs ar n to approximat th ntir, larg systm Trm structurs oftn rquir only 3 PCs, which captur a paralll shift (PC1), tilt (PC2) an chang in convxity (PC3) Th principal componnt rprsntation is thn: ΔR Ti,t w i1 PC 1t + w i2 PC 2t + w i3 PC 3t, i =1,..., n, t =1,..., T Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

21 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion 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 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

22 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion First Thr Eignvctors Trn, tilt an curvatur componnts l l l l l l l l l l l l t t t Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

23 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion 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 PC 1t 15, 436 PC 2t +6, 795 PC 3t Varianc ΔPV = 2, λ 1 +15, λ 2 +6, λ 3 = 2, , , % 1 ay normal PC VaR = 349, 544 GBP 1% 10 ay normal PC VaR = 1, 105, 354 GBP Approximation rror (rlativ to normal VaR without PCA): -0.09% Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

24 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion 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 PC 2 = = 50.7 Sinc ΔPV = 2, 635 PC 1 15, 436 PC 2 +6, 795 PC 3 this 6-sigma tilt in th wrong irction woul inuc a larg aily loss for our portfolio: ΔPV = 15, = 782, 898 GBP So a 2.5 million loss appars to b an 18 sigma vnt! Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

25 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion UK Govrnmnt Liability Curv: Monthly,

26 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Qustion 3: Will UK Inflation Soon Incras? Which is NOT currntly an inflationary prssur in th UK? 1 Low intrst rats put ownwar prssur on GBP 2 Consumption will incras u to low borrowing costs 3 Inflation is th only way to ruc govrnmnt bt 4 Political unrst in mil ast will incras oil prics 5 All of th abov ar inflationary prssurs Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

27 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion 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 Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

28 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion First Thr Eignvctors Trn, tilt an curvatur componnts l l l l l l l l l l l l t t t Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

29 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion 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 PC 1t 15, 264 PC 2t 167 PC 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 GBP 1% 10 ay normal PC VaR = 5, 941, 595 GBP Approximation rror (rlativ to normal VaR without PCA): -0.04% Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

30 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion Strss Tsting a Tilt in th Yil Curv Again, consir a 6-sigma valu for PC 2, but now convrt its (monthly) stanar viation to a aily quivalnt using PC 2 =6 29, = : W hav ΔPV = 8, 056 PC 1 15, 264 PC PC 3 Hnc, this 6-sigma tilt in th curv pricts an ovrnight loss of: ΔPV = 15, = 3, 426, 139 GBP Sinc 3.5 million loss is a 6-sigma vnt, a 2.5 million loss is actually wll within strss tsts, but only if th right ata ar us Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

31 UK Bon Portfolio Portfolio VaR Strss Tsting an PCA Conclusion In brif... 1 Sinc th banking crisis, simpl VaR mols hav bn substantially ovr-stimating risk but thr is a simpl rmy 2 Extrm losss uring th crisis coul hav bn prict using ata from th 1970 s 3 Basl II proposals for strss VaR ar riiculous s forthcoming issu of Intllignt Risk Furthr information: Rsarch, books an othr publications Th MRA 4-volum sris vios an iscussion forums Carol Alxanr UK Bon Risk Analysis Fbruary 10, / 31

Holding period yield the distinction between interest rates and

Holding period yield the distinction between interest rates and LECTURE 3 Hamza Ali Malik Econ 315: Mony and anking Wintr 005 Holding priod yild th distinction btwn intrst rats and rturns. Th pric of most forms of dbt fluctuats ovr tim whil th yild to maturity can