Here are the solutions for the class problem today; I also added a column for log utility. Log utility is the most risk averse, with Arrow-Pratt coefficient of 1/W. Square root utility is the least risk averse, with Arrow-Pratt coefficient of .5/W, and U = W^.25 is less risk averse than log utility and more risk averse than square root utility, with Arrow-Pratt coefficient of .75/W. These differences in degree of risk aversion are all reflected in the spreadsheet below. Also, keep in mind that the Arrow-Pratt equation is an approximation; thus the (relatively minor) differences reflected here between “exact” and “approximate”.
It turns out that if initial wealth is higher, then this mutes the differences between the exact and approximate methods. For example, suppose initial wealth is tripled to $30 and the decision-maker is exposed to the same bet size (win $6 if heads, lose $6 if tails). Then this one change produces the following table:
See Arrow-Pratt Spreadsheet for a spreadsheet template that you can use (if you like) to replicate the results shown here.
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