臺灣博碩士論文加值系統

近幾年，風險管理受到大家廣為重視，Value-at-Risk (VaR)則是最常用來衡量風險的工具。Basak and Shapiro (2001)是首位將涉險值（VaR）的限制式納入效用函數內，再極大化投資人之效用函數而求出最適資產配置。依據他們的方法，本文的第一部分（見第二章）探討當資產報酬分配呈左偏和肥尾時，對風險管理者資產配置之影響。許多實證研究顯示資產報酬分配呈左偏和肥尾。本文採用Gram-Charlier expansion近似資產報酬分配，探討當資產報酬分配在非常態分配下，其資產配置的變化。對風險管理者而言，最重要的工作就是準確預測損失與發生損失的機率。瞭解資產報酬的型態將有助於準確的預測損失，我們無法降低損失，但可以降低發生損失的機率，本文建議可以降低 值（期末財富損失大於VaR之機率）來達成，而降低 值會使期末財富在好的狀態與壞的狀態的財富稍減。利率是影響使用金融工具的主要因素，本文的第二部分（見第三章）探討VaR風險管理者當考慮利率風險時如何配置其資產，本文採用Vasicek-type模型描述隨機利率，探討在隨機利率的情況下，財富配置於現金、股票與債券之比例。本文將這些參數以數值代入，分析VaR風險管理者期末財富的分配情況以及期中現金、股票與債券之配置情形。本文的第三部分（見第四章）探討VaR風險管理者當考慮利率與通膨風險時如何配置其資產。本文採用correlated Ornstein-Uhlenbeck過程描述隨機實質利率與通膨率，探討當考慮利率與通膨風險的情況下，VaR風險管理者財富配置於現金、股票與債券之比例。對風險管理者而言，最重要的工作就是準確預測期末財富與損失。研究發現忽略通膨風險將使風險管理者嚴重低估期末財富與損失。

Risk management has received much attention in the last few years. Value-at-Risk (VaR) is widely used by corporate treasurers, fund managers and financial institution (Hull, 2000). A vast amount of literature considered a simple one-period asset allocation problem under VaR constraint. Furthermore, the aggregation of single-period optimal decisions across periods might not be optimal for multi-period as a whole. Basak and Shapiro (2001) were the first to address VaR-related issue in a dynamic general equilibrium setting. This dissertation builds upon the work of Basak and Shapiro (2001) to discuss three issues about dynamic asset allocation.
The first topic focuses on how deviations from normality affect asset choices made by risk managers. This study utilizes the Gram-Charlier expansion to approximate asset returns with negatively skewed and excess kurtosis. This work examines how negatively skewed and excess kurtosis affects asset allocations when investors manage market-risk exposure using Value-at-Risk-based risk management (VaR-RM). It is important for risk managers to precisely forecast the loss. The analytical results imply that the impact of leptokurtic asset returns is based on the shape of asset returns, and a correct measurement of leptokurtic asset returns is helpful to risk managers seeking to precisely forecast the loss. A risk manager cannot reduce the loss in bad states, but can reduce the value of , the probability that a loss exceeds VaR, and the agent will suffer from reduced terminal wealth in both the good and bad states.
The second topic solves an optimal investment problem involving a VaR risk manager who must allocate his wealth among cash, stocks and bonds. This study incorporates a stochastic interest rate process into the optimization problem. A Vasicek（1977）one-factor model governed the dynamics of the term structure of interest rates and risk premia are constant. Closed form formulate for the optimal investment strategy are obtained by assuming complete financial markets. Moreover, this study provides numerical examples to analyze the optimal terminal wealth and portfolio weights in stocks and bonds of the VaR risk manager. This work demonstrated the bond-stock allocation puzzle of Canner et al. (1997) that the bond-to-stock weighting ratio increases with risk aversion in popular investment advice in contradiction with standard two fund separation.
Finally, this work derives the optimal portfolio selection of the VaR manager by assuming complete financial markets and that the inflation and real interest rates follow correlated Ornstein-Uhlenbeck processes. This study provides numerical examples to analyze the optimal terminal real wealth and optimal portfolio in stocks and two nominal bonds with different maturities. Furthermore, this work studies the influence of the parameters of inflation on the solution. This work illustrated that the younger VaR agent who has a long investment horizon invests the fraction of wealth in stock varies with the state price. It is not consistent with the Samuelson puzzle.

摘要……………………………………… ……………………………………………..i
Abstract…………………………………………………………………………………ii
Chapter 1 Introduction………………………………………………………………1
1.1 Review of risk management……………………………………………………1
1.2 Motivations of this dissertation……………………………………………...…2
1.3 Purposes of this dissertation……………………………………………………3
1.4 Contents of this dissertation……………………………………………………4
Chapter 2 Optimal Asset Allocation with Extreme Returns and a VaR Constraint…………………………………………………………………7
2.1 Introduction………………..…..…………………………….…………………7  2.2 Economic setting…………………………………………………………......11
2.2.1 Asset returns with normal distribution………………………………....12
2.2.2 Asset returns with fat-tailed distribution…………………………….…14
2.3 Portfolio Optimization under VaR-RM………………………………………16
2.4 Numerical Illustrations……………………………………………………….21
2.5 Discussions……………………………………………………………….…..27
Chapter 3 Dynamic Asset Allocation with Stochastic Interest Rates and a VaR Constraint………………………………………………………………..31
3.1 Introduction……………………………………………...……………………31 3.2 Economic setting……………………………………………………….…….33
3.3 Portfolio Optimization under VaR-RM………………………………………37
3.4 Numerical Illustrations……………………………………………………….40
3.5 Discussions……………………………………………………………….…..51
Chapter 4 Dynamic Asset Allocation with Stochastic Inflation Rates and a VaR Constraint………………………………………………………………..55
4.1 Introduction…………………………………………………………………...55
4.2 Economic setting……………………………………………………………..57
4.3 Portfolio Optimization under VaR-RM………………………………………62
4.4 Numerical Illustrations……………………………………………………….65
4.5 Discussions…………………………………………………….……………..75
Chapter 5 Conclusions and Future Researches…………………………...………78
Appendix ………………………………………………………………………………80
A Proof of equation (2.8)………………...……..………………………………...80
B Proof of equation (2.11)…………………………..……………………………81
C Proof of equation (2.12)…………………………..……………………………87
D Proof of equation (3.10)…………………………..……………………………91
E Proof of equation (3.11)………………………….….…………………………93
F Proof of equation (3.12)………………………………………..………………95
G Proof of equation (4.5)…………………………………..…….……….………99
H Proof of equation (4.12)……………………………..………………….….…101
I Proof of equation (4.13)………………………………………………...……103
J Proof of equation (4.14) ………………………………………………..……106
References………………………………………………………………………….…111

Alexander, G. J. and Baptista A. M. (2002), “Economic Implications of Using a Mean- VaR Model for Portfolio Selection: A Comparison with Mean-Variance Analysis”, Journal of Economics, Dynamics and Control, Vol. 26, 1159-1193.
_____________________________(2003), “Portfolio Performance Evaluation Using Value at Risk”, Journal of Portfolio Management, 29, 93-102.
Alexander S., T. F. Coleman and Y. Li. (2006), Minimizing CVaR and VaR for a Portfolio of Derivatives, Journal of Banking and Finance, 30, pp.583–605.
Andersson, F., H. Mausser, D. Rosen and S. Uryasev (2001), Credit Risk Optimization with Conditional Value-at-Risk Criterion. Mathematical Programming, Series B 89, pp. 273-291.
Bajeux-Besnainou I., Jordan, J. V. and Portait R. (2001), “An Asset Allocation Puzzle: Comment”, American Economic Review, Vol. 91, 1170-1179.
Bajeux-Besnainou I. and Portait R. (1998), “Dynamic Asset Allocation in a Mean-Variance Framework”, Management Science, Vol. 44, 79-95.
Basak, S. (1995), “A General Equilibrium Model of Portfolio Insurance”, The Review Financial Studies, Vol.8, 1059-1090.
Basak, S., & Shapiro, A. 2001. Value-at-risk based risk management: Optimal policies and Asset prices. The Review of Financial Studies, 14 (2): 371-405.
Bidarkota, P.V., & McCulloch, J.H. (2003), Consumption asset pricing with stable shocks: exploring a solution and its implications for mean equity returns. Journal of Economic Dynamics and Control, 27 (3): 399-421.
Breeden, D. T. (1979), “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities”, Journal of Financial Economics, 265-296.
Brennan, M. J., E. S. Schwartz and Lagnado, R. (1997), “Strategic Asset Allocation”, Journal of Economics, Dynamics and Control, Vol. 21, 1377-1403.
Brennan, M. J. and Xia, Y. (2002), “Dynamic Asset Allocation under Inflation”, Journal of Finance, 57(3), 1201-1238.
Campbell, J. Y., Lo, A. W., & Mackinlay, A. C. 1997. The Econometrics of Financial Markets. Princeton: Princeton University Press.
Campbell, J. Y. and Viceira, L. M. (2001), “Who Should Buy Long-Term Bonds? ” , American Economic Review, 91(1), 99-127.
Campbell, R., R. Huismanet and K. Koedijk (2001), “Optimal Portfolio Selection in a Value at Risk Framework”, Journal of Banking and Finance, 25, 1789-1904.
Chang Shih-Chieh and Yi-Feng Li (2006), “Controlling the shortfall Risks in Dynamic Asset Allocation”, Review of Securities and Futures markets, , forthcoming.
Corrado, C. J. & Su, T. 1996. S&P 500 index option tests of Jarrow and Rudd’s approximate option valuation formula. Journal of Futures Markets, 16 (6): 611-629.
____________________. 1997. Implied volatility skews and stock index skewness and kurtosis implied by S&P 500 index option prices. Journal of Derivates, 4 (1): 8-19.
Cox, J., & Huang, C. 1989. Optimum consumption and portfolio policies when asset price follow a diffusion process. Journal of Economic Theory, 49 (1): 33-83.
________________ (1991), “A Variational Problem Arising in Financial Economics”, Journal of Mathematical Economics, 20 (5), pp. 465-487.
Deelstra, G., Grasselli, M. and Koehl, P. F. (2000), “Optimal Investment Strategies in a CIR Framework”, Journal of Applied Probability, Vol.37, 936-946.
de Vries, C. G. 1994. Stylized facts of nominal exchange rate return. The Handbook of International Macroeconomics, Cambridge: Blackwell.
Embrechts, P., Kluppelberg, C., & Mikosch T. 1997. Modelling Extremal Events for Insurance and Finance. Berlin: Spring Verlag.
Engle, R. F. 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50 (4): 987-1007.
Fama, E. 1965. The behavior of stock prices. Journal of Business, 38 (2): 244-280.
Hales, S.J. 1997. “Valuation of Foreign Currency Options with the Paretian Stable Option Pricing Model.” Unpublished doctoral dissertation, Ohio State University, U.S.A.
Gallant, A. R., Hansen, L. P., & Tauchen, G. 1990. Using conditional moments of asset payoffs to infer the volatility of intertemporal marginal rates of substitution. Journal of Econometrics, 45 (2): 141-179.
Gallant, A. R., & Nychka, D. W. 1987. Seminonparametric maximum likelihood estimation. Econometrica, 55 (2): 363-390.
Gallant, A. R., & Tauchen, G. 1989. Seminonparametric estimation of conditionally constrained heterogeneous processes: asset pricing applications. Econometrica, 57 (5): 1091-1120.
Gordon J. Alexander and Alexandre M. Baptista (2004), A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model, Management Science, Vol. 50, No. 9, pp. 1261-1273.
Grossman, S. and Zhou, Z. (1996), “Equilibrium Analysis of Portfolio Insurance”, Journal of Finance, Vol.51, 1379-1403.
Hull, J. C. (2000), Options, Futures and Other Derivates, Prentice-Hall, Upper Saddle River, NJ.
Jarrow, R., & Rudd, A. 1982. Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics, 10 (3): 347-369.
Jensen, B. A., and Sorensen, C. (2001), “Paying For Minimum Interest Rate Guarantee: Who Should Compensate Who?”, European Financial Management, Vol.7, 183-211.
Jorion, P. 1996. Risk2: Measuring the risk in value at risk. Financial Analysts Journal, 52 (6): 47-56.
Jondeau, E., & Rockinger, M. 2001. Gram-Charlier densities. Journal of Economic Dynamic & Control, 25 (10): 1457-1483.
Jondeau, E. and M. Rockinger (2006), Optimal Portfolio Allocation Under Higher Moments, European Financial Management, Vol. 12, No. 1, pp. 29-55.
Kim, T., and Omberg, E. (1996), “Dynamic Nonmyopic Portfolio Behavior”, Review of Financial Studies, Vol.9, 141-161.
Longstaff, F. 1995. Option pricing and the martingale restriction. Review of Financial Studies, 8 (4): 1091-1124.
Lucas, A., & Klaassen, P. 1999. Extreme returns, downside risk, and optimal asset allocation. Journal of Portfolio Management, 25 (1): 71-79.
Merton, R. C. (1971), “Optimum Consumption and Portfolio Rules in a Continuous Time Model”, Journal of Economic Theory, Vol.3, 373-413.
Munk, C., C. Sørensen, and T. N. Vinther (2004), “Dynamic Asset Allocation Under Mean-Reverting Returns, Stochastic Interest Rates and Inflation Uncertainty”, International Review of Economics and Finance, Vol.13(2),141–166.
Nelson, D. B. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 51 (2): 347-370.
Pirvu, T. A. (2007), Portfolio Optimization under Value-at-Risk Constraint, Quantitative Finance, Vol. 7, pp. 125-136.
Praetz, P. D. 1972. The distribution of share price changes. Journal of Business, 45 (1): 49-55.
Rachev, S. T., & Mittnik, S. 2000. Stable paretian models in finance. Wiley.
Rockafellar, R.T. and S. Uryasev (2000), Optimization of conditional value-at-risk. Journal of Risk, 2 (3), pp. 21–41.
Rockafellar, R.T. and S. Uryasev (2002), Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7), pp.1443–1471,2002.
Samorodnitsky, G., & Taqqu, M. S. 1994. Stable non-gaussian random processes: stochastic models with infinite variance. Chapman and Hall, New York -London.
Samuelson, P. A. (1991), Long-run Risk Tolerance When Equity Returns are Mean Regressing: Pseudoparadoxes and Vindication of “Businessmen’s Risk”, In W. C. Nordhaus, and H. W. Watts (eds.), Money, Macroeconomics and Economic Policy. Cambrige, Mass.: MIT Press.
Sharpe, W. F. (1991), “Capital Asset Prices with and without Negative Holdings,” Journal of Finance, Vol.64, 489-509.
Sorensen, C. (1999), “Dynamic Asset Allocation and Fixed Income Management”, Journal of Financial and Quantitative Analysis, Vol.34, 513-531.
Stulz, R. M. 2002. Risk management and derivatives. Southwestern Publishing Company.
Tokat, Y., Rachev, S.T., & Schwartz E.S. 2003. The stable non-Gaussian asset allocation: a comparison with the classical Gaussian approach. Journal of Economic Dynamics and Control, 27 (6):937-969.
Vasicek, O. (1977), “An Equilibrium Characterization of The Term Structure”, Journal of Financial Economics, Vol.5, 177-188.
Wachter, J. A. (2002), “Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets”, Journal of Financial and Quantitative Analysis, Vol.37, 63-91.
Zhao, Y., Haussann, U. and Ziemba, W. T. (2003), “A Dynamic Investment Model with Control on The Portfolio's Worst Case Outcome”, Mathematical Finance, Vol.13, 481-501.