انتخاب سبد سهام بهینه در یک چارچوب ارزش در معرض خطر

In this paper, we develop a portfolio selection model which allocates financial assets by maximising expected return subject to the constraint that the expected maximum loss should meet the Value-at-Risk limits set by the risk manager. Similar to the mean–variance approach a performance index like the Sharpe index is constructed. Furthermore when expected returns are assumed to be normally distributed we show that the model provides almost identical results to the mean–variance approach. We provide an empirical analysis using two risky assets: US stocks and bonds. The results highlight the influence of both non-normal characteristics of the expected return distribution and the length of investment time horizon on the optimal portfolio selection.

Modern portfolio theory aims to allocate assets by maximising the expected risk premium per unit of risk. In a mean–variance framework risk is defined in terms of the possible variation of expected portfolio returns. The focus on standard deviation as the appropriate measure for risk implies that investors weigh the probability of negative returns equally against positive returns. However it is a stylised fact that the distribution of many financial return series are non-normal, with skewness and kurtosis pervasive.1 Furthermore there is ample evidence that agents often treat losses and gains asymmetrically. There is a wealth of experimental evidence for loss aversion (see, for example, Kahneman et al., 1990). The choice therefore of mean–variance efficient portfolios is likely to give rise to an inefficient strategy for optimising expected returns for financial assets whilst minimising risk. It would therefore be more desirable to focus on a measure for risk that is able to incorporate any non-normality in the return distributions of financial assets. Indeed risk measures such as semi-variance were originally constructed in order to measure the negative tail of the distribution separately. Typically mainstream finance rests on the assumption of normality, so that a move away from the assumption of normally distributed returns is not particularly favoured; one drawback often stated is the loss in the possibility of moving between discrete and continuous time frameworks. However it is precisely this simplifying approach, whereby any deviations from the square root of time rule are ignored, which needs to be incorporated into current finance theory. The ability to focus on additional moments in the return distribution with the possibility of allowing for skewed or leptokurtotic distributions enables additional risk factors (along with the use of standard deviation) to be included into the optimal portfolio selection problem.2 In this paper, we develop an optimal portfolio selection model which maximises expected return subject to a downside risk constraint rather than standard deviation alone. In our approach, downside risk is written in terms of portfolio Value-at-Risk (VaR), so that additional risk resulting from any non-normality may be used to estimate the portfolio VaR. This enables a much more generalised framework to be developed, with the distributional assumption most appropriate to the type of financial assets to be employed. We develop a performance index similar to the Sharpe ratio, and for the case that financial assets are assumed to be normally distributed, provide a model similar to the mean–variance approach. The plan of the paper is as follows: We introduce the framework in Section 2. Section 3 then provides empirical results of the optimal portfolio allocation for a US investor. In Section 4 we address the importance of the non-normal characteristics of expected return distributions in such a framework. Conclusions and practical implications are drawn in Section 5.

Focussing on downside risk as an alternative measure for risk in financial markets has enabled us to develop a framework for portfolio selection that moves away from the standard mean–variance approach. The measure for risk depends on a portfolio's potential loss function, itself a function of portfolio VaR. Introducing VaR into the measure for risk has the benefit of allowing the risk–return trade-off to be analysed for various associated confidence levels. Since the riskiness of an asset increases with the choice of the confidence level associated with the downside risk measure, risk becomes a function of the individual's risk aversion level. The portfolio selection problem is still to maximise expected return, however whilst minimising the downside risk as captured by VaR. This allows us to develop a very generalised framework for portfolio selection. Indeed the use of certain parametric distributions such as the normal or the student-t allows for a market equilibrium model to be derived, with the assumption of normality enabling the model to collapse to the CAPM. We illustrate just how great the impact is on the portfolio selection decision from non-normalities, alternative time horizons, and alternative risk specifications.