Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student earlier today asking for some clarification regarding the Arrow-Pratt method vis–à–vis (what I like to refer to as) the “exact” method for calculating risk premiums. As I showed in class, the Arrow-Pratt method is an alternative method for calculating the risk premium. Thus, either approach (the “exact” method or the Arrow-Pratt method) is perfectly acceptable for calculating risk premiums.

The value added of Arrow-Pratt is (as I point out in my two page Finance 4335 synopsis) that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, 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 the square root investor. Another important insight yielded by Arrow-Pratt (at least for the types of utility functions we have considered in Finance 4335; i.e., power and logarithmic utilities) is the notion of decreasing absolute risk aversion. Other things equal, investors become less (more) risk averse as wealth increases (decreases).

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