## Normal Returns

When all returns are normal random variables, the mean-variance criterion is also equivalent to the expected utility approach for any risk-averse utility function To deduce this, select a utility function U Consider a random wealth variable y that is a normal random variable with mean value M and standard deviation a, Since the probability distribution is completely defined by M and a, it follows that the expected utility is a function of M and a; that is,

(It may be impossible to determine the function / in closed form, but that does not matter ) If U is risk averse, then /(A/,cr) will be increasing with respect to M and decreasing with respect to a Now suppose that the returns of all assets are normal random variables, Then (and this is the key property) any linear combination of these assets is a normal random variable, with some mean and standard deviation. (See Appendix A.) Hence any portfolio of these assets will have a return that is a normal random variable The portfolio problem is therefore equivalent to the selection of that combination of assets that maximizes the function J(M, cr) with respect to all feasible combinations For a risk-averse utility this again implies that the variance should be minimized for any given value of the mean In other words, the solution must be mean-variance efficient. Therefore the mean-variance criterion is appropriate when all returns are normal random variables

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