Journal of Risk

Space–time diversification: which dimension is better?

Moshe Arye Milevsky


In this paper I provide a quantifiable link between the two dimensions by examining the impact of both space and time on the equity shortfall risk (ESR) of a portfolio. The ESR, originally introduced into finance by A. D. Roy in 1952 and adopted by many practitioners since, is defined as equal to the probability that a portfolio or individual security will underperform the return from the risk-free asset. This framework allows me to compute the marginal benefit of investing for one more year, versus investing in one more asset. To that end, I provide a variety of numerical examples and general theorems about the relative benefits of space versus time diversification when Roy’s criterion is used as the measure of risk and returns are lognormal. There is a general perception, especially in industry, that the equity shortfall risk goes to zero as time gets large. Whether or not this is relevant to asset allocation is secondary. The fact is that for highly volatile or concentrated portfolios this is mathematically untrue. Furthermore, even when the conditions are such that the statement itself is correct – and the probability does go to zero – it can take a surprisingly long time.