邱顯比、林清珮、楊宗庭

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1 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 funds in Taiwan, and presents an application of VaR to asset allocation on mutual fund portfolio. The result shows that the best approach is NAV (Net Asset Value) method of Historical Simulation approach and the second best is EWMA (Exponential Weighted Moving Average) method of Variance-Covariance approach. Furthermore, from the outcome of sensitivity analysis of the portfolio compositions, in order to maintain accuracy of VaR, we suggest the risk manager must dynamically adjust the weight of each security in the portfolio. In the aspect of the application of VaR, we replace standard deviation with VaR to establish efficient frontier called MvaR (Mean-Value-at-Risk) efficient frontier. After depicting MV (Mean-Variance) and MvaR on the same picture, we find MvaR efficient frontier is better in terms of downside risk than MV efficient frontier. Keywords Mutual Funds, Value at Risk, Efficient Frontier * Tel (02) ext Fax (02) [email protected] ** *** Tel (02) ext. 612 Fax (02) [email protected] 1

2 1 Makowitz (Market Portfolio) (Maximum Possible Loss) (Value at Risk) 2 Jorion (1996) Lu and et al. (2000) PaR (Project at Risk) BOT BOT BOT Dowd (1999) IVaR (Incremental VaR) 1 : (Basle Supervisory Committee) standard model 2

3 Murray (1999) BRVaR (Benchmark-Relative Value at Risk) Chow and Kritzman (2001) Risk Budget 3 Duarte and Alcantara (1999) 15 4 (2000) MVaR 3 Risk Budget 3

4 PR (1) (2) ( )

5 (Covariance Matrix) ( weight matrix) (RP) ( Covariance Matrix) J.P Morgan % 99% EWMA VaR(t 1, (1- )% t 2 (1- )% 5 6 Kolmogorov-Smirnov 7 =0.94 5

6 1 - I. Variance-Covariance Approach I. Variance-Covariance A (1)Standard Deviation (2)EWMA (1)Standard Deviation VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) VaR(30,0.05) VaR(30,0.01) VaR(30,0 $1.65 $2.33 $1.96 $2.76 $9.03 $12.75 $ $2.53 $3.57 $2.99 $4.22 $13.85 $19.56 $ $2.10 $2.97 $2.48 $3.51 $11.52 $16.27 $ $1.73 $2.44 $1.92 $2.71 $9.46 $13.36 $ $2.12 $2.99 $2.46 $3.47 $11.60 $16.39 $ $1.38 $1.95 $1.57 $2.21 $7.58 $10.70 $2.66 $3.76 $3.20 $4.52 $14.57 $20.58 $ $2.82 $3.99 $3.49 $4.93 $15.47 $21.84 $ $2.29 $3.24 $3.05 $4.31 $12.56 $17.74 $ $2.47 $3.48 $3.19 $4.50 $13.51 $19.08 $ * - (Standard Deviation) (EWMA) *VaR(t, ) t (1- )% * $100 6

7 2 II. Historical Simulation II. Historical Simula (1)Portfolio (2)NAV (1)Portfolio VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) VaR(30,0.05) VaR(30,0.01) VaR(30,0 $1.51 $2.76 $2.73 $4.57 $8.25 $15.12 $ $2.42 $4.43 $3.14 $5.09 $13.27 $24.24 $ $2.01 $4.12 $3.42 $4.95 $10.99 $22.57 $ $1.77 $2.84 $2.78 $4.74 $9.67 $15.58 $ $2.18 $3.73 $3.33 $4.74 $11.94 $20.44 $ $1.18 $2.49 $3.60 $5.48 $6.46 $13.64 $ $2.67 $4.60 $3.03 $5.10 $14.63 $25.22 $ $2.73 $5.06 $3.28 $4.85 $14.94 $27.71 $ $2.18 $3.80 $3.13 $4.56 $11.91 $20.83 $ $2.48 $4.17 $3.37 $5.42 $13.60 $22.81 $ * (Portfolio) (NAV) *VaR(t, ) t (1- )% * $100 7

8 t 2 VaR( t2, α) = VaR( t1, α) t ( ) n n (Portfolio method) % (Net Asset Value; NAV) 10 (NAV method) (Portfolio Method) (NAV Method)

9 - ( ) (process) 10, III. Monte Carlo Simulation Normal Dist III. Monte Carlo Simulation Normal Dist VaR(1,0.05) VaR(1,0.01) VaR(30,0.05) VaR(30,0.01) $1.71 $2.34 $9.35 $12.82 $2.39 $3.53 $13.11 $19.33 $2.03 $2.82 $11.15 $15.47 $1.68 $2.39 $9.20 $13.12 $2.11 $2.69 $11.57 $14.74 $1.50 $2.09 $8.21 $11.47 $2.67 $4.00 $14.65 $21.93 $2.72 $4.26 $14.92 $23.35 $2.24 $3.03 $12.25 $16.60 $2.46 $3.40 $13.47 $18.64 *VaR(t, ) t (1- )% * $100 9

10 ( ) 4 - (EWMA) 4 I. Variance-Covariance Approach (1)Standard Deviation (2)EWMA VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) $ $ $ $ (1)Portfolio II. Historical Simulation (2)NAV VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) $ $ $ $ VaR(1,0.05) III. Monte Carlo Simulation VaR(1,0.01) $ $ *VaR(t, ) t (1- )% * $100-10

11 5 III. Mon I. Variance-Covariance Approach II. Historical Simulation Simu (1)Standard (2)EWMA (1)Portfolio (2)NAV Norm Deviation VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) * * 11

12 ( ) Basle Committee (1996) (Back Test) % 25 99% 5 Z score = X T α T T α ( 1 α) X T T (1- )% 95% 99% Z-score

13 7 95% 99% 99% ? 6 - I. Variance-Covariance Approach (1)Standard Deviation (2)EWMA VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) * *95% %

14 7 (1)Portfolio II. Historical Simulation (2)NAV VaR(1,0.05) VaR(1,0.01) VaR(1,0.05) VaR(1,0.01) * *95% % III. Monte Carlo Simulation VaR(1,0.05) VaR(1,0.01) * *95% %

15 ( ) % 9 20% * % 31.28% 17.54% 9 99% 15% % 9 III. Monte I. Variance-Covariance II. Historical Carlo Approach Simulation Simulation (1)Standard (2)EWMA Portfolio Normal Dist Deviation 20% $ $ $ $ % $ $ $ $ % $ $ $ $ % $ $ $ $ % $ $ $ $ % $ $ $ $ % $ $ $ $ * 10% 1.2 * * VaR(1,0.05)

16 10 10% I. Variance-Covariance Approach (1)Standard II. Historical Simulation III. Monte Carlo Simulation Deviation (2)EWMA Portfolio Normal Dist 20% % % % % % % * Z-score I. Variance-Covariance Approach (1)Standard Deviation II. Historical Simulation III. Monte Carlo Simulation (2)EWMA Portfolio Normal Dist 20% % % % % % % * *95% %

17 MV MVaR MV MVaR MV w = 1 i n w i i= 1 17

18 σ 2 p = [ w w Kw ] 1 2 n 2 σ 1σ 12σ 13Kσ 1n w1 2 σ w 21σ 2σ 23Lσ 2n 2 MO M 2 σ n n n L n w 1σ 2σ 3 σ n 13 σ p = w' Σ w W 95% 14 =-1.65 VaR p [ w w' ] W 1 VaR p = α 2 MV MVaR 1 MV MV MVaR MV MVaR MV MVaR 13 - w 14-18

19 1 ( ) *MV MV MVaR MVaR *MV MVaR 1 J. P. Morgan RiskMetrics TM 19

20 (Mean-Variance Efficient Frontier) MVaR (Mean-Vaule-at-Risk Efficient Frontier) MV (Mean-Variance Efficient Frontier) MVaR MV Alexander and Chibumba, 1998, Orthogonal Factor Garch, University of Sussex, Centre for Statistics and Stochastic Modeling. Basle Committee on Banking Supervision, 1996, Amendment to the capital accord to incorporate market risks. Basle:Bank for International Settlement. Bauer, Sep/Oct 2000, Value at Risk Using Hyperbolic Distributions, Journal of Economics and Business, pp Best, 1998, Implementing Value at Risk, John Wiley and Sons. Bollersev, 1987, A Conditional Heteroskedastic Model for Speculative Prices and Rates of Return Journal of Econometrics, pp Chow and Kritzman, 2001, Risk Budget, Journal of Portfolio Management, pp Culp, 1998, Value at Risk for asset managers, Derivatives Quarterly, pp Dowd, 1998, Beyond Value at Risk, John Wiley and Sons. 20

21 Dowd, Spring 2000, Assessing VaR Accuracy, Derivatives Quarterly, pp Dowd, Spring 2001, Estimating VaR with Order Statistics, Journal of Derivatives, pp Duarte and Alcantara, winter 1999, Mean-Value-at-Risk Optimal Portfolios with Derivatives, Derivatives Quarterly, pp Fong, summer 1999, A new analytical approach to Value at Risk, Journal of Portfolio Management, pp Jorion, Nov/Dec 1996, Risk 2 : Measuring the Risk in Value at Risk, Financial Analysts Journal, pp Jorion, 1996, Value at Risk: the new benchmark for controlling market risk, Chicago: Irwin. J. P. Morgan and Company, December 1996, RiskMetrics TM Technical Document, Morgan Guaranty Trust Company of New York. Kupiec, 1995, Technique for Verifying the Accuracy of Risk Management Models, Journal of Derivatives, pp Gujarati, 1995, Basic Econometrics, McGraw-Hill Gupta, Stubbs and Thambiah, summer 2000, U.S. Corporate Pension Plans, Journal of Portfolio Management, pp Hull and White, spring 1998, Value at Risk When Daily Changes in Market Variables are not Normal Distributed, Journal of Derivatives, pp9-20. Lu, Wu, Chen and Lin, winter 2000, BOT Projects in Taiwan: Financial Modeling Risk, Term Structure of Net Cash Flows, and Projects at Risk Analysis, Journal of Project Financial, pp Nathan, 1999, Performance Evaluation Using Performance Score, Derivatives Quarterly, pp Naftci, spring 2000, Value at Risk Calculations, Extreme Events, and Tail Estimation, Journal of Derivatives, pp

22 Wilson, 1993, Infinite wisdom, Risk 6, pp37-45 Scott Murray, 1999, Benchmark-Relative Value at Risk, Derivatives Quarterly, pp Zangari, 1995, Statistics of market moves. In RiskMetrics TM technical document, New York: Morgan Guaranty Trust Company Global Research. ; 1998, ; 2000, :, 2000, :,,,,,, 2000, : (VaR), 1999,, 1999,, 2000, 22

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