Analytical Portfolio Value-at-Risk

Kaplanski, Guy (2005): Analytical Portfolio Value-at-Risk. Published in: Journal of Risk , Vol. 7, No. 2 (2005): pp. 33-54.

Abstract

The paper develops analytical tools used to calculate the VaR of a portfolio composed of generally distributed assets. Accordingly, the VaR of a portfolio is analytically constructed from the conditional returns of the individual assets. This analytical VaR can then be used to construct optimal portfolios of generally distributed assets for the case in which the target function and/or constraints are expressed in terms of VaR. The proposed method is applicable in a wide range of practical problems such as utility maximization under a VaR constraint. The article demonstrates this method by developing a minimal VaR rule that identifies the proportions that minimize the portfolio VaR. This rule is used to compare the minimal VaR portfolio with the minimal standard deviation portfolio in the case of the lognormal distribution. This example illustrates the importance of downside risk in optimal asset allocation even under modest deviations from the normal distribution such as in the case of the lognormal distribution.

• Item Type: MPRA Paper
• Language: English
• Keywords: Value-at-Risk, Risk measurement, Portfolio Optimization, Downsize Risk
• Subjects: C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65 - Miscellaneous Mathematical Tools

References

Ahn, D.H., Boudoukh, J., Richardson, M., and R.F., Whitelaw, "Optimal Risk Management Using Options", J. of Finance, 54(1), February 1999, 359-375.

Alexander, J.G., and A.M., Baptista, "Economic Implications of Using a Mean-VaR Model for Portfolio Selection: A Comparison with Mean-Variance Analysis", Journal of Economic Dynamics and Control, 2002, 1159-1193.

Artzner, P., Delbaen, F., Eber, J.M., and D., Heath, "Coherent Measures of Risk", Mathematical Finance, 9, 1999, 203-228.

Basak, S., and A., Shapiro, "Value-at-Risk Based Risk Management: Optimal Policies and Asset Prices", Review of Financial Studies, 14(2), 2001, 371-405.

Basel Committee on Bankig Supervision, Amendment to the Capital Accord to Incorporate Market Risks, January 1996.

Campbell, R., Huisman, R., and K., Koedijk, "Optimal Portfolio Selection in a Value-at-Risk Framework", J. of Banking and Finance, 25, 2001, 1789-1804.

Crouhy, M., Galai, D., and R., Mark, Risk Management, (Eds.), McGraw-Hill, 2001.

Cuoco, D., He, H., and S., Issaenko, "Optimal Dynamic Trading Strategies with Risk Limits", University of Pennsylvania working paper, 2001.

Duffie, D., and J., Pan, "An Overview of Value at Risk", J. of Derivatives, 4, Spring 1997, 7-49.

Duffie, D., and J., Pan, "Analytical Value-at-Risk with Jumps and Credit Risk", Finance and Stochastic, 5, 2000, 155-180.

Emmer, S., Klüppelberg, C., and R., Korn, "Optimal Portfolios with Bounded Downside Risks", Mathematical Finance, 2001, 365-384.

Harvey, C.R., and A., Siddique, "Conditional Skewness in Asset Pricing Tests", J. of Finance, 55, June 2000, 1263-1295.

Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, December 2000.

Jorion, P., "Portfolio Optimization with Constraint on Tracking Error", Working paper, University of California at Irvine”, 2001.

Kaplanski, G., and Y., Kroll, "VaR Risk Measures versus Traditional Risk Measures: An Analysis and a Survey ", Journal of Risk, 4(3), 2002, 1-27.

Levy, H., and Y., Kroll, "Ordering Uncertain Options with Borrowing and Lending", J. of Finance, 33(2), May 1978, 552-573.

Rothschild, M., and J.E., Stiglitz, "Increasing Risk. I: A Definition", J. of Economic Theory, 2, 1970, 225-243.

Sentana, E., "Mean Variance Portfolio Allocation with a Value at Risk Constraint", C.E.P.R., Discussion Papers, 2997, 2001.

Yiu, K.F.C., "Optimal Portfolios under a Value-at-Risk Constraint", J. of Economic Dynamics and Control, 28(7), April 2004, 1317-1334.