VaR and Low Interest Rates

Transcription

1 VaR and Low Ineres Raes Presened a he Sevenh Monreal Indusrial Problem Solving Workshop By Louis Doray (U de M) Frédéric Edoukou (U de M) Rim Labdi (HEC Monréal) Zichun Ye (UBC) 20 May 2016 P r e s e n e d a h e S e v e n h M o n re a l I n d u s ri a l P r o b l e m S o l v i n g W o r k s h o p

2 Plan of he alk 1. Inroducion + Financial Mahemaics The daa 3. The CDPQ model 4. Scenario generaion: he CEV model Esimaion and esing Weekly VaR 5. Annualizaion 6. Furher work 2 Plan of he alk

3 Inroducion Ideal eam composed of Financial engineer (or acuary) Probabilis Compuer exper Saisician 3 Inroducion

4 Financial Mahemaics 101 Le Y = value of asse a ime ; Reurn r = (Y +1 -Y )/Y ; Log-reurn ln r = ln(y +1 -Y ); If r is small, ln (1+r ) r (r > 0). Assuming ln r ~ N(0, σ 2 ), his implies r ~ Lognormal LN(0, σ 2 ) and ensures r is never negaive. The log-reurns over a period (week) is he sum of he log-reurns over sub-periods (days). Y 5 Y 5 Y 4 Y 3 Y 2 Y 1 Z ln ln Y Y 4 Y 3 Y 2 Y 1 Y ln( r r r r r ) 4 lnr k k 4 Inroducion

5 Financial Mahemaics 101 Assuming ln r are i.i.d. ~N(0, σ 2 ); This implies Z = lny +5 /Y ~N(0, 5 σ 2 ). Under his model, i can be shown ha correlaion beween (Monday o Monday) and (Tuesday o Tuesday) log-reurn is 0.8, because of he overlapping inervals. Z Z Z Z , Z, Z, Z, Z Z, Z k k 0, for 5. 5 Inroducion

6 Daa: Ineres Rae Sar : January End : April Mauriy : 20 Observaions by mauriy : 4260 nb. of days Mauriy 1 Day (d) 7 Week (w) 30 Monh (m) 90 Quarer(q) years (30y) 6 Daa

7 Daa A summary of numbers of iid mauriy daa. 30y 25y 20y 15y 12y 10y 9y 8y 7y 6y 5y 4y 3y 2y 1y s q m w d Daa

8 Daa Hisogram for mauriy w 8 Daa

9 Daa non saionary process: mean, variance and covariance change over ime.( firs difference is saionary) nonlinear process: he exisence of differen saes of he world (or regimes). volailiy clusering : "large changes end o be followed by large changes, of eiher sign, and small changes end o be followed by small changes. 9 Daa

10 The CDPQ model For CDPQ, simulaions of ineres rae are based on he assumpion ha variaions are independen of he level of ineres raes. In order o simulae he value of a bond, re-value he bond under he simulaed ineres raes and hen calculae corresponding VaR (weekly). In pracice, here are wo problems associaed wih his mehod. 10 The CDPQ model

11 The CDPQ model Scenario generaion: he way CDPQ simulaes ineres movemens is by simply adding he hisorical variaion o curren values. However, he disribuion of ineres rae variaions seems o depend on he level. Source: Slides from Yannis Papageorgiou, Caisse de dépô e placemen du Québec 11 The CDPQ model

12 The CDPQ model Annualizaion: in order o ransform VaR from he ime scale of he measuremen frequency o ha of he invesmen horizon, we need our observaions o be i.i.d. and normal. he following graphic illusraes a heavy-ail disribuion of weekly profis and losses. Source: Slides from Yannis Papageorgiou, Caisse de dépô e placemen du Québec 12 The CDPQ model

13 Scenario generaion: Model To evaluae he influence of ineres rae level o is disribuion, we apply CEV model (a special case of SABR model). Coninuous version of CEV model (W is he sandard BM) dr r dw Discree version (by proper scaling of sigma, Z ~ N(0,1)): r r Z Given he ineres rae level, he change in ineres rae follows a normal disribuion wih variance depending on curren level. r N r 2 2 ~ (0, ) 13 Scenario generaion

14 Scenario generaion: Model Based on above CEV model, he formula of he evoluion of ineres rae becomes r m, 0 m,, r 0 m, 0 rm, w rm, rm, w The change of ineres rae are re-scaled by he level of ineres rae. When β = 0, he addiive model (wihou scale). When β = 1, he muliplicaive model. r ( ) 14 Scenario generaion

15 Scenario generaion: MLE Recall ha given he ineres rae level, Likelihood funcion L(σ, β) MLE (σ, β) maximize L(σ, β) or equivalenly he log-likelihood funcion l (σ, β) = ln L(σ, β). MLE (σ, β) soluions of sysem of equaions Soluion: 1. Find σ numerically as he roo of an equaion; 2. Calculae β = g(σ). r N r 2 2 ~ (0, ) independen. 15 Scenario generaion

16 Scenario generaion: MLE The observed informaion marix can be obained from he inverse of minus he marix of he second parial derivaives of he log-likelihood funcion wih respec o he 2 parameers σ and β. Asympoic disribuion of MLE (σ, β) is normal; MLE: asympoically unbiased; efficien esimaor. The disribuion of MLE permis he calculaion of confidence inervals for σ and β and o do hypohesis esing. 16 Scenario generaion

17 Scenario generaion For example, o es he simple hypohesis H 0 : β =0 (CDPQ model), use es saisic z 0 = β /s.e.(β ). If z 0 >1.96, rejec H 0 a 5% level (CDPQ model). For example, wih daa of ineres rae wih 1 year mauriy (852 iid observaions from weekly non-overlapping inervals), esimaed value β= Conclusion: rejec H 0 : β =0 (CDPQ model). 17 Scenario generaion

18 Scenario generaion: VaR Wih above mehod of maximum likelihood esimaion and nonoverlapping daa, Here is an example of esimaions for wo mauriy. Mauriy 180 (Semi annual) 365 (Annual) β Comparison of empirical (black) and heoreical (red) densiy. level = 0.01 level = Scenario generaion

19 Scenario generaion: VaR As a es, we calculae he VaR of a shor posiion in following wo kinds of bonds: Zero coupon bond wih one year mauriy; 4% coupon bond wih one year mauriy and semi-annual coupon paymen. Time horizon: 1 week. Confidence level: 99%. For each case, we generae 500 scenarios. Bond 4% Coupon Zero Coupon VaR -0.18% -0.22% 19 Scenario generaion

20 Annualizaion: Self-Similar Process To fi he heavy ail disribuion while keeping he form of square-roo-of-ime rule, we generalize BM o self-similar process {X()} saisfying: d X ( k) k H X ( ) H: Hurs coefficien. Examples of self-similar process: Brownian moion: H = 0.5; α-sable process: H = 1/ α. Heavy ail when H < 0.5! If he underlying prices is a self-similar process wih Hurs coefficien H, hen h VaR( k) k VaR( ) 20 Annualizaion

21 Annualizaion: Esimaion The esimaion of Hurs coefficien H is based on he he Rescaled Range (R/S) Calculaion. Regression: log(range/s.d) ~ log(scale). Log(R/S) v.s. Log(Scale) Daa: ineres raes corresponding o one year mauriy, Annualizaion

22 Annualizaion: Esimaion As an example, we run he regression for he daa of ineres raes corresponding o one year mauriy, As he weekly VaR for one year zero coupon is -0.22%, hen he annualized VaR is -0.22%*52^0.056 = %. 22 Annualizaion

23 Furher work There are sill los of models o be sudied for ineres rae process 1. ARCH-GARCH models; 2. Comparison of VaR from independen vs dependen observaions; 3. GNL (Brownian-Laplace moion); 4. HMM; 23 Furher work

24 Furher work: GARCH GARCH (Generalized Auo-Regressive Condiional Heeroskedasiciy) models volailiy clusering. In his model :he volailiy process is ime varying and is modeled o be dependen upon boh he pas volailiy and pas innovaions. 24 Furher work

25 Furher work:ms-garch Markov-swiching GARCH model (MS-GARCH) is an exension of GARCH. The condiional mean and variance swich in ime from one GARCH process o anoher. The condiional variance may originae from srucural changes in he variance process which are no accouned for by sandard GARCH models. Esimae a model ha permis regime swiching in he parameers. S: saes or régimes. Transiion probabliliies: 25 Furher work

26 Quesion Time 26 Quesion Time

27 27 Sevenh Monreal Indusrial Problem Solving Workshop