Generalized Safety First and a New Twist on Portfolio Performance

Abstract

We propose a Generalization of Roy's (1952) Safety First (SF) principle and relate it to the IID versions of Stutzer's (Stutzer's 2000, 2003) Portfolio Performance Index and underperformance probability Decay-Rate Maximization criteria. Like the original SF, the Generalized Safety First (GSF) rule seeks to minimize an upper bound on the probability of ruin (or shortfall, more generally) in a single drawing from a return distribution. We show that this upper bound coincides with what Stutzer showed will maximize the rate at which the probability of shortfall in the long-run average return shrinks to zero in repeated drawings from the return distribution. Our setup is simple enough that we can illustrate via direct calculation a deep result from Large Deviations theory: in the IID case the GSF probability bound and the decay rate correspond to the Kullback–Leibler (KL) divergence between the one-shot portfolio distribution and the “closest” mean-shortfall distribution. This enables us to produce examples in which minimizing the upper bound on the underperformance probability does not lead to the same decision as minimizing the underperformance probability itself, and thus that the decay-rate maximizing strategy may require the investor to take positions that do not minimize the probability of shortfall in each successive period. It also makes clear that the relationship between the marginal distribution of the one-period portfolio return and the mean-shortfall distribution is the same as that between the source density and the target density in importance sampling. Thus Geweke's (1989) measure of Relative Numerical Efficiency can be used as a measure of the quality of the divergence measure. Our interpretation of the decay rate maximizing criterion in terms of a one-shot problem enables us to use the tools of importance sampling to develop a “performance index” (standard error) for the Portfolio Performance Index (PPI). It turns out that in a simple stock portfolio example, portfolios within one (divergence) standard error of one another can have very different weights on individual securities.