Comparison of back-testing results for various VaR estimation methods. Aleš Kresta, ICSP 2013, Bergamo 8 th July, 2013

Transcription

1 Comparison of back-esing resuls for various VaR esimaion mehods, ICSP 3, Bergamo 8 h July, 3

2 THE MOTIVATION AND GOAL In order o esimae he risk of financial invesmens, i is crucial for all he models o esimae he fuure variance correcly. I is obvious ha, if he modelling of volailiy is inaccurae for a single asse (one-dimensional problem), increasing he dimensionaliy of he problem (incorporaing he dependence o i) would no improve he resuls (accuracy of he join model). The goal of he paper is o back-es he differen volailiy models, concreely GARCH model and GJR model wih Gaussian and Suden innovaions, on he chosen ime series and invesigae he imporance of he lengh of he period uilized for parameers esimaion. ICSP 3, Bergamo 8 h July, 3

3 CONTENT Volailiy models VaR and is backesing procedure Empirical par ICSP 3, Bergamo 8 h July, 3

4 Volailiy models Alhough he fuure value of many financial ime series is unpredicable, here is a clusering in volailiy. (Engle, 98) We assume log-reurns x ~ ~ N R x N x ~ i i i, or ~ ~, o be modelled as follows, where is modelled sandard deviaion (volailiy) and is a random number from chosen probabiliy disribuion. ~ ICSP 3, Bergamo 8 h July, 3

5 Volailiy models GARCH-n and GARCH- models GARCH model was proposed as he exension of ARCH model in order o avoid problemaic esimaion of parameers, GJR-n and GJR- models I was shown ha here is usually differen impac of he posiive and negaive shocks on he volailiy. Glosen, Jagannahan and Runkle proposed he model, which akes his ino accoun, Q j j j P i i i Q j j j j Q j j j P i i i i ICSP 3, Bergamo 8 h July, 3

6 VaR and is backesing procedure Value a Risk (VaR) a a given confidence level α is he hreshold which will no be exceeded a α percen of he cases. VaR X infx R : F x i.e. only a -α percen of he cases he loss will be bigger. Mosly uilized values of -α are 5% (Solvency II), 5% (original mehodology proposed by JP Morgan), % (Basel II) and.5% (Solvency II). Based on our simple models we esimae fuure VaR as follows, VaR R, : VaR x ˆ ˆ x ˆ i X q i i ICSP 3, Bergamo 8 h July, 3

7 VaR and is backesing procedure Is he model correc? To check his we have o backes i on he pas daa. By means of backesing procedure he model is verified. This procedure is based on he comparison of he risk esimaed a ime for ime + wih he rue loss observed a ime +. I if if VaR VaR Based on VaR definiion we should experience losses bigger han VaR esimae in (-α) N cases. r r ICSP 3, Bergamo 8 h July, 3

8 VaR and is backesing procedure For he esing of model accuracy we can employ Kupiec s es (henceforh K-es), n LR ln n n ex n obs This es akes ino accoun only he number of excepions (and probabiliy of heirs occurring). The disribuion of excepions in ime should be also esed. ex obs ICSP 3, Bergamo 8 h July, 3

9 VaR and is backesing procedure The disribuion of excepions over ime can be esed by Chrisoffersen s es (C-es): LR ln n n n obs obs n n n The null hypohesis is ha he probabiliy of excepion occurring is independen on he informaion wheher he excepion has occurred also previous day. ICSP 3, Bergamo 8 h July, 3

10 Empirical par Four differen models for VaR esimaion were backesed. Backesing was applied on American sock marke index S&P 5. Daase covers he period January 3, 95 o December 3, (i.e. 5,85 daily logreurns). I will focus on: whole daase parameers esimaion; raios of observed and expeced quaniies of excepions; p-values of Kupiec s es; p-values of Chrisoffersen s es; differen seups of he GJR model. ICSP 3, Bergamo 8 h July, 3

11 Inpu daa S&P 5,% 8,% 6,% 4,%,%,% -,% -4,% -6,% -8,% -,% Basic descripive characerisics are: mean is.8%, median is.46% (average loss>average profi); skewness is -.3; kurosis is ICSP 3, Bergamo 8 h July, 3

12 Whole daase parameers esimaion Only firs wo auoregressive coefficiens and only firs order of coefficiens for volailiy equaion were found saisically significan. Thus, only -- model is furher assumed. Esimaed parameers AR()-GARCH(,)-n AR()-GARCH(,)- AR()-GJR(,)-n AR()-GJR(,)- LLF Akaike informaion crieria Bayesian informaion crieria e e e ICSP 3, Bergamo 8 h July, e

13 The raios - S&P 5,5 GARCH-n,5 GARCH-,5,5, ,5 3,5,5,5 7 7 GJR-n, ,5 3,5,5,5 7 7 GJR- 5% 5% %,5% ICSP 3, Bergamo 8 h July, 3

14 The raios - S&P 5 The raio S&P 5 (.3%) GARCH-n GARCH- GJR-n GJR- ICSP 3, Bergamo 8 h July, 3

15 Kupiec s es (p-values) GARCH,,8,6,4,,,8,6,4,,,8,6,4,,,8,6,4, 5% 5% %,5% 5% 5% %,5% For boh 5% and 5% levels he observed quaniies of excepions are close o he assumed quaniies (he model generally can be acceped as accurae) blue and yellow. For some period lenghs also % probabiliy levels esimaion can be acceped (bu, generally model should no be acceped) - red. Esimaion of VaR a.5% probabiliy level canno be acceped for any period lengh. ICSP 3, Bergamo 8 h July, 3

16 Kupiec s es (p-values) GJR,,8,6,4,,,8,6,4,,,8,6,4,,,8,6,4, 5% 5% %,5% 5% 5% %,5% GJR wih Gaussian innovaions is generally no accurae model for VaR esimaion. I can be acceped only for 5% probabiliy level and longer esimaion periods - blue. The graph of GJR- is ineresing. We can observe differen accuracy resuls in dependence on he period lengh and VaR probabiliy level. For model o be accurae, wih lowering probabiliy level he longer esimaion period should be uilized. Why? The longer esimaion period -> more exreme values we encouner -> esimaed Suden disribuion has faer ails. ICSP 3, Bergamo 8 h July, 3

17 Chrisoffersen s es (p-values) - GARCH,,8,6,4,,,8,6,4,,,5,,5 5% 5% %,5% 5% 5% %,5% The bunching of excepions can be generally rejeced. However, here are periods in which he bunching canno be rejeced. 5% and 5% probabiliy levels for Gaussian disribuion blue and yellow % and.5% probabiliy levels for Suden disribuion red and green. ICSP 3, Bergamo 8 h July, 3

18 Chrisoffersen s es (p-values) GJR,,,5,5,,,5,5 5% 5% %,5% 5% 5% %,5% For GJR-n he bunching of excepions is presen (for shorer esimaion periods and 5% probabiliy level) - blue. For GJR- he bunching of excepions is presen (for longer esimaion periods and % probabiliy level) - red. ICSP 3, Bergamo 8 h July, 3

19 Differen model seups Will he resuls change if we change he number of esimaed parameers? We assumed he model AR(R)-GJR(P,Q) wih differen srucures: R= 5; P= 5; Q= 5. The lengh of esimaion period was se o 4 days. ICSP 3, Bergamo 8 h July, 3

20 Differen model seups Will Rank he resuls change Model if we change he number of esimaed 5% 5% %.5% parameers? (he bes) AR()-GJR(3,) We assumed he model AR(R)-GJR(P,Q) wih differen srucures: R= 5; P= 5; AR()-GJR(,)- Q= The lengh of AR()-GJR(,)- esimaion period was se o.99 4 days AR(3)-GJR(,) AR()-GJR(,) AR(3)-GJR(5,4) AR()-GJR(5,5) AR(3)-GJR(5,5) AR()-GJR(5,5) (he wors) AR(5)-GJR(5,5) ICSP 3, Bergamo 8 h July, 3

21 Conclusion Swif changes in volailiy are challenging ask for any risk model. I was found ou ha for Value a Risk esimaion of invesmen ino S&P 5 index: assuming probabiliy levels 5% and 5% he bes model is GARCH model wih Suden innovaions, assuming probabiliy levels % and.5% he mos accurae model is GJR model wih Suden innovaions. By increasing he order of dependence in volailiy equaion he resuls do no improve (necessarily). ICSP 3, Bergamo 8 h July, 3

22 Lieraure Alexander, C., and Sheedy, E.: Developing a sress esing framework based on marke risk models. Journal of Banking and Finance 3 (8), -36. Black, F.: The pricing of Commodiy Conracs. Journal of Financial Economics 3 (976), Bollerslev, T.: Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics 3 (986), Engle, R. F.: Auo-regressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 5 (98), Glosen, L. R., Jagannahan, R., and Runkle, D. E.: On he relaion beween he expeced value and he volailiy on he nominal excess reurns on socks. Journal of Finance 48 (993), Huang, J. J., Lee, K. J., Liang, H., and Lin, W.F.: Esimaing value a risk of porfolio by condiional copula- GARCH mehod. Insurance: Mahemaics and Economics 45 (9), Hull, J.: Risk Managemen and Financial Insiuions. Prenice Hall, Upper Saddle River, 6. Chrisoffersen, P.F.: Evaluaing Inerval Forecass. Inernaional Economic Review 39 (998), Ignaieva, K., and Plaen, E.: Modelling Co-movemens and Tail Dependency in he Inernaional Sock Marke via Copulae. Asia-Pacific Financial Markes 7 (), 6 3. Jorion, P.: Value a Risk: The New Benchmark for Managing Financial Risk. 3 rd ed. McGraw-Hill, New York, 6. Kupiec, P.: Techniques for verifying he accuracy of risk measuremen models. Journal of Derivaive 3 (995), Resi, A., and Sironi, A.: Risk managemen and shareholders Value in banking: from risk measuremen models o capial allocaion policies. John Wiley & Sons Inc., New York, 7. ICSP 3, Bergamo 8 h July, 3