During today’s Finance 4335 class meeting, we compared and contrasted two methods for calculating risk premiums.
Under the so-called “exact” method, one 1) calculates expected utility, 2) sets expected utility equal to the utility of the certainty-equivalent of wealth, 3) solves for the certainty-equivalent of wealth, and 4) gets the risk premium by calculating the difference between expected wealth and the certainty-equivalent of wealth. The Arrow-Pratt method is an alternative method for calculating the risk premium based upon Taylor series approximations of expected utility of wealth and the utility of the certainty equivalent of wealth (the derivation for which appears on pp. 6-8 of http://fin4335.garven.com/fall2019/lecture6.pdf). Both approaches for calculating risk premiums are perfectly acceptable for Finance 4335.
The value added of Arrow-Pratt is that it analytically shows how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as showed by variance), and 2) the degree to which the decision-maker is risk averse. For example, we showed that the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the square root investor. Another important insight yielded by Arrow-Pratt (at least for the utility functions considered so far in Finance 4335) is decreasing absolute risk aversion (DARA). Other things equal, an investor with DARA preferences becomes less (more) risk averse as wealth increases (decreases). Such an investor increases (reduces) the dollar amount that she will put at risk as she becomes wealthier (poorer).