![]() ![]() Its popularity stems from the fact that, under the assumption of quadratic utility, mean-variance analysis is optimal. I'll make a note that we look to revise this solution going forwards - it's not wrong - but could be simplified. Quadratic Utility In financial economics, the utility function most frequently used to describe investor behaviour is the quadratic utility function. However, Investor B is still risk averse (just to a lesser degree as wealth increases) whereas Investor A is not risk averse at all! This means, as Investor B becomes wealthier, he or she becomes less risk averse. However, I agree with your comment that Investor B exhibits decreasing absolute risk aversion. I'm not convinced that we need to consider A(w) and R(w) to answer this question. We can check that U'(w) = w^(-0.5) and U''(w) = - 0.5w^(-1.5) 0. The power utility function is with positive or negative, but non-zero, parameter a < 1. Investor B's utility curve is a graph of the square root of w, which has a concave shape. We can check that U'(w) = 1 and U''(w) = 0. Investor A's utility curve is a straight line. ![]() The graph of U(w) against w will be concave. For each additional $1 of wealth, the extra utility derived reduces. The assumption of quadratic utility function is convenient in portfolio theory because it is possible to demonstrate that if the portfolio returns are not normally distributed, the mean-variance approach is still best (best in the sense that any other distributional properties is amenable into mean and variance. Pages 10 to 12 in Chapter 2 show some illustrations of the shapes of the U(w) graphs for investors that are risk-averse, risk-neutral and risk-seeking.įor a risk-averse investor, we are looking for diminishing marginal utility of wealth. I wonder if an easier way of looking at it is to consider the utility function itself.Ĭan you visualise a plot of these two functions? ![]()
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