Increasing risk, decreasing absolute risk aversion and diversification

This paper defines conditions for 'Increasing Risk' when the utility functions of risk averse investors are characterized by decreasing absolute risk aversion (DARA). Rothschild and Stiglitz (Journal of Economic Theory 1970, 2, 225-243, and 1971, 3, 66-84) define cases when a random variable Y is 'more risky' (or 'more variable') than another variable X for the utility functions of risk averse investors. They conclude that Y is riskier than X if G, the cumulative distribution of Y, can be formed from F, the cumulative distribution of X, by adding a series of mean preserving spread (MPS) steps to F. This paper suggests considering a sequence of steps which are denoted by 'mean preserving spread and antispread' (MPSA). We define the condition under which a random variable (r.v.) Y is 'more risky' (or 'more variable') than another variable X for DARA utility functions. We prove that for DARA utility functions, Y is riskier than X if and only if G, the cumulative distribution function of Y, can be formed from F, the cumulative distribution function of X by adding a series of MPSA steps to F, under the restrictions stated in the paper. The economic intuition and impact of MPS and MPSA steps on the optimum diversification strategy are demonstrated.