Moving the Goalposts: Subjective Performance Benchmarks and the Aumann-Serrano Measure of Riskiness

This paper parameterizes the well-known Aumann and Serrano (2008) measure to explicitly allow the riskiness of financial instruments to be examined relative to benchmark levels of return other than zero. Using real data, we demonstrate that the ordering of two financial instruments in terms of their relative riskiness may change even with only small changes in the benchmark level of return. We then show that one asset may be (uniformly) riskier than another in the sense that every member of this generalized class ranks the two assets in the same way, and that this occurs only when the Laplace transforms of the assets can be ordered over a compact subset of the positive real line. A statistical test for this Laplace order is proposed, and a simulation study shows that this test is reliable in finite samples. Finally, the proposed methodology is illustrated using data on global stock index returns.