Brief synopsis concerning how to calculate risk premiums

The way to calculate a risk premium is to 1) compute expected utility, 2) set this equal to the utility of the certainty equivalent of wealth, 3) solve for the certainty equivalent of wealth, and 4) subtract the certainty equivalent of wealth from the expected value of wealth. For example, consider a simple coin toss where we assume a fair coin, a payoff of $200 if heads comes up and $100 otherwise. The expected payoff of this gamble is $150. For a person with U=W^.5, expected utility is E(U(W)) = 12.07 =U(W_CE). Thus W_CE=E(U(W))²= 12.07²= $145.68 and the risk premium is $4.32.

Last week, we also showed an alternative way to compute the risk premium involving an approximation based upon first and second order Taylor series expansions of U(W_CE) and E(U(W)) respectively. This analysis provides an important insight ? that the risk premium is equal to the product of ? of the variance of the gamble multiplied by the investor?s Arrow-Pratt absolute risk aversion coefficient. In other words, risk premiums depend upon two factors: 1) the risk itself as measured by variance, and 2) the investor?s degree of risk aversion, as measured by the risk aversion coefficient.

The risk aversion coefficient is equal to -1 multiplied by the ratio of the second derivative of utility divided by marginal utility. Taking the numerical example back up again, since the variance is $2,500 and the expected value is $150, this means that we can compute the risk premium via the alternative method by computing the following product: .5 x variance x risk aversion coefficient (evaluated at expected wealth). Since the risk aversion coefficient is .5/E(W) = 0.0033, we obtain (.5)2,500(0.0033) = $4.17 for our risk premium. Since W_CE=E(W) ? risk premium, W_CE= $150 ? $4.17 = $145.83.

Note that there is a 15 cent difference between the two estimates of the risk premium (the ?correct? risk premium is actually $4.32, whereas the Arrow-Pratt risk premium is $4.17). The reason why the second approach is slightly off is due to the manner in which we derived the Arrow-Pratt absolute risk aversion coefficient. The derivation itself creates some ?error? in the sense that we approximate U(W_CE) using a first order Taylor series expansion, whereas we approximate E(U(W)) using a second order Taylor series expansion (cf. pp. 17?19 of Decision Making under Risk and Uncertainty, part 2).

Source: http://risk.garven.com/2012/09/30/calculate-risk-premiums/

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