Weak Dependence of CRRA on Standard Deviation in the Case of Truncated Normal Distribution of Returns

Abstract

This paper analyzes the dependence of the Certainty Equivalent Return of a Constant Relative Risk Aversion, CER[CRRA], on the Standard Deviation of the Return with the hypothesis of a Truncated Normal distribution of returns and for some level of Relative Risk Aversion (RRA) parameter. The paper compares this dependence with those detected by an Annualized Geometrical Return (AGR) and by CER of the Quadratic Utility Function, CER[Q]. The behavior of CER[CRRA] is more similar to AGR than CER[Q] and only for a higher value of RRA is it possible to find substantial differences, even if in this case we find values of Standard Deviation that have discontinuity points for the concavity. Using a ranking criteria equal to the one introduced by Morningstar for a set of Funds, the paper shows that, in a wide range for monthly Standard Deviation and Mean of the Returns, the ranking done by CER[CRRA] is similar to the one induced by AGR, and that a CER[Q] has essentially different behavior. It will be shown that Morningstar ranking may be considered a particular case of the CER[CRRA] and thus all the considerations can be applied to the well-known Morningstar Rating methodology. An application is made to Italian Pension Funds.