is a fair game in that the expected profit is zero. Suppose, however, that the curve in Figure 6B.1 represents the investors utility value of wealth, assuming a log utility func- tion. Figure 6B.2 shows this curve with numerical values marked. Figure 6B.2 shows that the loss in utility from losing $50,000 exceeds the gain from winning $50,000. Consider the gain first. With probability p .5, wealth goes from $100,000 to $150,000. Using the log utility function, utility goes from ln(100,000) 11.51 to ln(150,000) 11.92, the distance G on the graph. This gain is G 11.92 11.51 .41. In expected utility terms, then, the gain is pG .5 .41 .21. Now consider the possibility of coming up on the short end of the prospect. In that case, wealth goes from $100,000 to $50,000. The loss in utility, the distance L on the graph, is L ln(100,000) ln(50,000) 11.51 10.82 .69. Thus the loss in expected utility terms is (1 p)L .5 .69 .35, which exceeds the gain in expected utility from the possi- bility of winning the game. We compute the expected utility from the risky prospect:   E[U(W)] pU(W1) (1 p)U(W2) 1⁄2 ln(50,000) 1⁄2 ln(150,000) 11.37   If the prospect is rejected, the utility value of the (sure) $100,000 is ln(100,000) 11.51, greater than that of the fair game (11.37). Hence the risk-averse investor will reject the fair game. Using a specific investor utility function (such as the log utility function) allows us to compute the certainty equivalent value of the risky prospect to a given investor. This is the amount that, if received with certainty, she would consider equally attractive as the risky prospect. If log utility describes the investors preferences toward wealth outcomes, then Figure 6B.2 can also tell us what is, for her, the dollar value of the prospect. We ask, "What sure II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001