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1 Academy Papers * The Testing and Application of the VaR Model for Bank Portfolio - A Case Study Mei-YingLiu Soochow University Department of Business Administration ( )( )( ) * ** 10056TEL #3602 FAX meiying@scu.edu.tw

2 Review of Financial Risk Management Abstract Given their function both as internal risk management tools and as potential regulatory measures of risk exposure, it is important to quantify the accuracy of an institution's VaR estimates. The purpose of this study is to set up the market risk measurement models for a bank in Taiwan. First, three popular methods: the variance-covariance method, the historical simulation method and Monte Carlo simulation method are used to construct the VaR models. Next, we employ backtesting to verify the accuracy of VaR models for day to day risk management. Finally, we apply the various VaR measures to evaluate the investment performance of the bank portfolio. Our results suggest that financial asset returns are found to be nonnormal and leptokurtic with fat tails. The historical simulation method without normality assumption yields more conservative, accurate and efficient measures of tail probabilities than parametric and Monte Carlo simulation VaR model. The portfolio diversified effect between asset classes is larger than that within asset class which demonstrates the importance of asset allocation in bank risk management. During the economic recession period, the more conservative investment strategy of the bank outperforms the benchmark portfolios of market indexes. Unexpectedly, the VaRs calculated by the three internal VaR models are substantially less than the capital requirement under the standard method. This may discourage the banks to develop their own internal models. Key Words: VaR, backtesting, performance measure, capital requirement (Asset and Liability Management, ALM) ALM ALM (mismatch) (gap analysis) (duration analysis)

3 Academy Papers 3 ALM (risk-adjusted return) (credit derivatives) (Option-Adjusted Spread, OAS) (simulation analysis)(value-at-risk, VaR) VaR VaR (dayto-day risk management ) (mark to market) VaRVaR (Bank for International Settlement, BIS)(the Basel Committee on Banking Supervision) 1996 (Amendment to The Capital Accord to Incorporate Market Risks) ( ) ( internal model ) ( Dimson & Marsh, 1995) (Beder, 1995; Hendricks, 1996; Alexander & Leigh, 1997; Guermat & Harris, 2002 etc. ) (, 2003;, 2003;, 2003) (2001) (EWMA) Jackson, Maude and Perraudin (1997)

4 Review of Financial Risk Management 3 Engel & Gizycki (1999) (2000) (2001) (2003) (2003) (2003) (2000) (2003) Crouhy et al. (1998 ) Csaba SOCZO (2001) (, 2003) ( ) (variance-covariance method) ( ) (historical simulation method) ( ) (Monte Carlo simulation method)

5 Academy Papers 5 2. A 1, ( ) 4 A ( ) A 2.2 (Taiwan Economic Journal, TEJ ) ( ) (Grand Cathay Government Bond Index) TEJ ( trading book ) Basel II1996 BaseI IIBasel II ( IRB ) 1996 Basel II 2 ()

6 Review of Financial Risk Management A (outpurchase, OP ) (repurchase, RP ) ( resale, RS ) (1) OP RS RP(1) A19 A TEJ 6 MATLAB A ( variancecovariance method ) Delta-Normal Delta ( ) Jorion (1995)00 =0

7 Academy Papers 7 1 (2) p w / V V P 99% Z( )2.33 (3) (4) i t (decay factor), 0 1 t J. P. Morgan RiskMetrics (diagonal matrix) 8 J.P.Morgan RiskMetrics

8 Review of Financial Risk Management (historicalsimulation method) ( )t ( ) 2 A ( ) (Monte Carlo simulation method ) (Geometric Brownian (5) Motion, GBM) n i +1 ( ) P t ( )1 t (7) dz 0 dt t t t t t (6) dt t t

9 Academy Papers 9 T =T-1 VaR P t+1 n (8) (10) 1 A (11) (9) ( ) N(0,1) n x 1 An x 1 n ( )n ( )Cholesky Lower Triangular A (12) ( )

10 Review of Financial Risk Management 3.2 BIS (backtesting) ( 250 ) 9 ( ) BISEngel & Gizycki (1999) (, 2003; Liu, 2005) () Hendricks(1996) (Mean Relative Bias, MRB ) T N VaR i MRBVaR VaR BIS VaR VaR VaR VaR VaR (Engel & Gizycki, 1999; Christoffersen, 1998) (13) ( ) (Binary Loss Function, BLF ) 9 BIS (multiplication factor) ( multiplication factor )

11 Academy Papers 11 VaR Lopez (1998) (General Loss Function ) t VaR i (14) VaR Coverage, MOC ) VaR VaR VaR VaR VaR VaR (equivalent multiplier ) VaR (16) (15) VaR (- )(outlier) VaR (- ) VaR 95%BLF %BLF 0.01 ( ) (Multiple to Obtain VaR 1VaR 1 VaR 1 ( ) (Average Uncovered Losses to VaR Ratio, AUL ) VaRVaR AUL VaR

12 Review of Financial Risk Management VaR VaRVaR (Conditional VaR, CVaR) 10 VaR ( ) (Maximum Uncovered Loss to VaR Ratio, MUL) AUL VaR MUL (17) VaR MUL M VaR AUL ( stress testing ) VaR AUL VaR VaR 5%AUL (benchmark)1.251% AUL 1.14 (Engel & Gizycki, 1999)AUL AUL (18) MUL AUL MUL ( ) (LR Test of Unconditional Coverage, LRuc ) T N Kupiec (1995) ( likelihood ratio test ) 10 (VaR)(threshold) VaR Artzner et al. (1997)Rockafellar & Uryasev (2000) (Conditional VaR, CVaR)((Beyond VaR)) X VaR a (x)(1- ) VaR

13 Academy Papers 13 LR uc LR uc P :VaR :VaR LR uc (19) ( ) LR(LR Test of Independence, LRind) VaR Christoffersen (1998) LR ind VaR LR ind (20) T ij : i j 1 VaR () 0 VaR : LR ind VaR (VaR ) () VaR (Mean Relative Scaled 1 Bias, MRSB) MOC MRB MOC MRB

14 Review of Financial Risk Management (21) (22) i (VaR ) 3.3 Gaman(1996, 1997) ( risk profile ) (portfolio VaR) (individual VaR) (2) i (diversified effect) (hedge effect) (2) (23) 11

15 Academy Papers =1 (24) (24) =-1 (25) (25) ( ) ( ) (27) (long position) i (marginal VaR, MVaR ) i 12 =0 (26) (28) (26) (29) (29) (28) i MVaR (diversified benefit, DB) (30) 12 Jorion (2000) ( continuous ) ( discrete ) (incremental VaR )

16 Review of Financial Risk Management (30) (VaR adjust return on capital, VARC ) ( component Va R, C VaR) ( risk contribution ) 13 (33) R VaR VARC (31) W i i (31) ( )( absolute VARC, AVARC ) (34) ( ) (relative (32) VARC, RVARC ) 13 ( Taylor's expansion ) Garman (1996,1997 )

17 Academy Papers 17 (benchmark) (35) AVARC b A % 79.92% 83.84% 16.07% 2 A 276, DB b () 2. (4) = (1) + (2) +(3) 3. (6) = (4) - (5)

18 Review of Financial Risk Management 226,437 56, % Delta Normal Model 14 ( fat-tailed ) A 1 Delta Normal Model 3(Liu, 2005) 1%1 LR uc LR ind 16 LR ind VaR VaR Kolomgorov- Smirnovt Hill(1975) Huisman, Koedijk, Kool & Palm (2001) 16 LR ind T 11 ( ) VaR 0T 11 0 (3.5.8) L A 0 lnl A LR ind

19 Academy Papers VaR 1.Liu (2005) 7 2.NaN 3. * 5%LR ** 1%LR

20 Review of Financial Risk Management 1 /8.008*10-3 ( =10,753/1,342,817 ) 1.489*10-3 ( =11,231/7,543,617 ) *10-3 ( =63,807/46,099,787 ) Delta Normal Model A A % 42.15%

21 Academy Papers 21 4

22 Review of Financial Risk Management 4 ( ) 1.DB t (%) 4.4 A

23 Academy Papers A A A

24 Review of Financial Risk Management ( International Swap and Derivatives ISDA BIS ISDA Jackson, Maude & Perraudin (1997) (2003) 3 Association, ISDA ) ( ) (1) (2)

25 Academy Papers ( ) ( )( ) VaR A A 57.85% 42.15% A A

26 Review of Financial Risk Management ( 2000 ) ( 2003 )Power EWMA ( 2003 ) - ( 1998 ) ( 2001 ) ( 2003 ) ( 2003 ) ( 2003 )- Alexander, C. O. and C. T. Leigh ( 1997 ), On the Covariance Matrices Used in Value at Risk Models, The Journal of Derivatives, Vol.4, No.3, Artzner, P., F. Delbaen, J-M Eber and D. Health (1997), Thinking Coherently, Risk, Vol.10, Basel Committee on Banking Supervision (1996), Supervisory Framework for the Use of Backtesting in Conjunction with the Internal Models Approach to Market Risk Capital Requirements. Basel Committee on Banking Supervision (1996), Amendment to the Capital Accord to Incorporate Market Risks. Beder, T. S. (1995), VaR: Seductive but Dangerous, Financial Analysts Journal, Vol. 51, No.5, Christoffersen, P. (1998), Evaluating Interval Forecasts, International Economic Review, Vol.39, Crouhy, M., D. Galai, and R. Mark (1998), The New 1998 Regulatory Framework for Capital Adequacy, In: Alexander, C. (Ed.), Risk Management and Analysis, Vol. 1. Wiley, New York, Chapter 1, Csaba SOCZO (2001), Comparison of Capital Requirements Defined by Internal (VAR) Model and Standardized Method, Periodica Polytechnica SER. SOC. MAN. SCI., Vol.10, No.1, Dimson, E. and P. Marsh (1995), Capital Requirements for Securities Firms, Journal of Finance, Vol.50, No.3,

27 Academy Papers 27 Engel, J. and M. Gizycki (1999), Conservatism, Accuracy & Efficiency: Comparing Value-at- Risk Models, Working Paper 2, Australian Prudential Regulation Authority. Garman, M. (1996), Improving on VAR, Risk, Vol.9, Garman, M. (1997), Taking VAR to Pieces, Risk, Vol.10, Guermat, C. and R. D. F. Harris (2002), Robust Conditional Variance Estimation and Value at Risk, Journal of Risk, Vol.4, Hendricks, D. (1996), Evaluation of Value-at- Risk Models Using Historical Data, Economic Policy Review, Vol.2, No.1, Hill, B. (1975), A Simple General Approach to Inference about the Tail of a Distribution, Annals of Statistics, Vol.3, Huisman, R., K. G. Koedijk, C. J. M. Kool and F. Palm (2001), Tail-Index Estimates in Samll Samples, Journal of Business and Economic Statistics, Vol.19, Jackson, P., D. J. Maude and W. Perraudin (1997), Bank Capital and Value at Risk, The Journal of Derivatives, Vol.4, No.3, Jorion, P. (1995), Predicting Volatility in the Foreign Exchange Market, Journal of Finance, Vol.50, Jorion, P. (2000), Value at Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill Companies, Chicago. JP Morgan. (1996), RiskMetrices Technical Document, Fourth Edition, New York, J. P. Morgan. Kupiec, P. H. (1995), Techniques for Verifying the Accuracy of Risk Measurement Models, The Journal of Derivatives, Vol.3, No.2, Liu, M.Y. (2005), VaR Evaluation of Bank Portfolio Conservativeness, Accuracy and Efficiency, Journal of Financial Studies, Vol.13, No.2, Lopez, J. (1998), Methods for Evaluating Valueat-Risk Estimates, Federal Reserve Bank of New York Research Paper, No Rockafellar, R. and S. Uryasev (2000), Optimization of Conditional Value-at-Risk, The Journal of Risk, Vol.2, No.3,

邱顯比、林清珮、楊宗庭

邱顯比、林清珮、楊宗庭 The Estimation and Application of Value at Risk in Mutual Fund Shean-Bii Chiu * Ching-Pei Lin ** Tzung-Ting Yang *** Abstract This article uses several approaches to evaluate Value at Risk (VaR) of mutual