Some Properties of Absolute Return, an Alternative Measure of Risk

Let a(theta) represent the absolute value of the daily return (after subtraction of mean return) raised to the power theta, so that a(two) is just the return squared. Duncan Luce (Theory and Decision, 1980) using an axiomatic approach suggested that a(theta) was a suitable class of measures of risk, with theta as the choice of the individual. The empirical properties of these measures are investigated using daily Samp;P 500 index data for the period January 3, 1928 to August 30, 1991. It is found that: (i) correlation (a(theta, day t), a(theta, day t-k)) remains positive and significant for many lags k (up to two thousand) for most positive values of theta. (ii) this quot;long-memoryquot; property is strongest for theta=1 compared to theta values both larger and smaller than one. (iii) if a few outliers are reduced, the marginal distribution of the absolute return has approximately an exponential distribution. Reasons for and implications of these results are explored