Proof by Certainty Equivalents That Diversification-Across-Time Does Worse, Risk Corrected, Than Diversification-Throughout-Time

Abstract

For those with constant relative-risk-aversion, one can calculate an easy and exact measurement of their risk-corrected total return per period by use of an appropriate "power mean". For them, this approach can dramatize the inefficiency of being (say) half the time in each of two independent and identically distributed securities; 100% is then lost of the benefit from being all the time 50-50 in each; actually, being half the time in each is as bad as being all the time in either one, which is equivalent to being completely undiversified. More generally, there is proved here that, for any risk-averse U(W) and time-independent probabilities, optimal diversification within each time period outperforms generically any and all patterns of across-time diversification. The variety of proposed risk-corrected returns can give useful approximations for different classes of investors–widows and orphans, pension fiduciaries, high-flying plungers, and so forth–to replace or extend Markowitz, Sharpe, Treynor, or Modigliani-Modigliani measures of corrected performance.