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It is commonly believed that a large portfolio of independent insurance policies is a nec- essary and sufficient condition for an insurance company to shed its risk. The fact is that a multitude of independent insurance policies is neither necessary nor sufficient for a sound insurance portfolio. Actually, an individual insurer who would not insure a single policy also would be unwilling to insure a large portfolio of independent policies. Consider Paul Samuelsons (1963) story. He once offered a colleague 2-to-1 odds on a $1,000 bet on the toss of a coin. His colleague refused, saying, "I wont bet because I would feel the $1,000 loss more than the $2,000 gain. But Ill take you on if you promise to let me make a hundred such bets." Samuelsons colleague, like many others, might have explained his position, not quite correctly, as: "One toss is not enough to make it reasonably sure that the law of averages will turn out in my favor. But with a hundred tosses of a coin, the law of averages will make it a darn good bet." Another way to rationalize this argument is to think in terms of rates of return. In each bet you put up $1,000 and then get back $3,000 with a probability of one-half, or zero with a probability of one-half. The probability distribution of the rate of return is 200% with p 1⁄2 and 100% with p 1⁄2. The bets are all independent and identical and therefore the expected return is E(r) 1⁄2(200) 1⁄2 ( 100) 50%, regardless of the number of bets. The standard deviation of the rate of return on the portfolio of independent bets is15 (n) / n where is the standard deviation of a single bet: [1⁄2(200 50)2 1⁄2 ( 100 50)2]1/2 150% The average rate of return on a sequence of bets, in other words, has a smaller standard deviation than that of a single bet. By increasing the number of bets we can reduce the stan- dard deviation of the rate of return to any desired level. It seems at first glance that Samuel- sons colleague was correct. But he was not. The fallacy of the argument lies in the use of a rate of return criterion to choose from portfolios that are not equal in size. Although the portfolio is equally weighted across bets, each extra bet increases the scale of the investment by $1,000. Recall from your corporate finance class that when choosing among mutually exclusive projects you cannot use the in- ternal rate of return (IRR) as your decision criterion when the projects are of different sizes. You have to use the net present value (NPV) rule. Consider the dollar profit (as opposed to rate of return) distribution of a single bet: E(R) 1⁄2 2,000 1⁄2 ( 1,000) $500 R [1⁄2 (2,000 500)2 1⁄2 ( 1,000 500)2]1/2 $1,500     15 This follows from equation 8.10, setting wi l/n and all covariances equal to zero because of the independence of the bets. II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001

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