In recent years, particular stress has been laid on the substitution of variance as a risk measure in the standard Markowitz [11] mean-variance problem. Since it makes no distinction between positive and negative deviations from the mean, variance is a good measure of risk only for distributions that are (approximately) symmetric around the mean, such as the normal distribution or, more generally, elliptical distributions [12]. However, in most cases, such as in portfolios containing options, one deals with wealth distributions that are highly skewed. It is thus more reasonable to consider asymmetric risk measures since individuals are typically loss averse. In this regard, Value-at-Risk (VaR), a downside risk measure, has emerged as an industry standard with regulatory authorities, such as the Basle Committee on Banking Supervision which enforces its use [9].
Despite its widespread acceptance, VaR is known to possess unappealing features. Artzner et al. [3] proposed an axiomatic foundation for risk measures, by identifying four properties that a reasonable risk measure should satisfy. They also provided a characterization of the risk measures satisfying these properties, which they called coherent risk measures. By these axioms, VaR is not coherent. Tail Conditional Expectation (TCE), on the other hand, is a coherent risk measure for an underlying continuous distribution [14].
The focus in this paper is the dynamic portfolio choice of a trader subject to a risk limit specified in terms of TCE. We maximize the agent’s utility over wealth throughout the investment horizon. The TCE constraint is re-calculated and re-imposed at every short interval of time, throughout the investment horizon. The portfolio is assumed constant over each such short interval (re-evaluation horizon), as is the case in practice.
This art of dynamically re-evaluating the constraint has not yet received adequate attention in the existing literature. We show through numerical simulations, by applying an algorithm similar to that in Yiu [15], that the introduction of a TCE constraint causes investment in risky assets2 to be controlled. In our numerical experiments we use two risky assets, as opposed to just one, which is common in the existing literature.
Download
Optimal portfolio strategies under a shortfall constraint
