Calculate the mean, variance, and skewness of the following two random variables:

w˜ 1 =

2.45 with probability 0.5141 , 7.49 with probability 0.4859 , w˜ 2 =

⎧

⎪⎨

⎪⎩

0 with probability 0.12096 , 4.947 with probability 0.750085 , 10 with probability 0.128955 .

You should see that w˜ 2 has a higher mean, lower variance, and higher skewness than w˜ 1. Show that, nevertheless, w˜ 1 is preferred to w˜ 2 by a CARA investor with absolute risk aversion equal to 1, by a CRRA investor with relative risk aversion equal to 1/2, and by an investor with shifted log utility log(1+w).