Journal of Risk

Risk.net

Analytical portfolio value-at-risk

Guy Kaplanski

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.

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