the total profit from n bets is the sum of the profits from the single bets. Therefore, with n bets   E[R(n)] $500n n R Variance ( Ri) n 2 i 1 R(n) n 2 R n   so that the standard deviation of the dollar return increases by a factor equal to the square root of the number of bets, n, in contrast to the standard deviation of the rate of return, which decreases by a factor of the square root of n. As another analogy, consider the standard coin-tossing game. Whether one flips a fair coin 10 times or 1,000 times, the expected percentage of heads flipped is 50%. One expects the actual proportion of heads in a typical running of the 1,000-toss experiment to be closer to 50% than in the 10-toss experiment. This is the law of averages. But the actual number of heads will typically depart from its expected value by a greater amount in the 1,000-toss experiment. For example, 504 heads is close to 50% and is 4 more than the expected number. To exceed the expected number of heads by 4 in the 10-toss game would require 9 out of 10 heads, which is a much more extreme departure from the mean. In the many-toss case, there is more volatility of the number of heads and less volatility of the percentage of heads. This is the same when an insurance company takes on more policies: The dollar variance of its portfolio increases while the rate of return variance falls. The lesson is this: Rate of return analysis is appropriate when considering mutually exclusive portfolios of equal size, which is the usual case in portfolio analysis, where we consider a fixed investment budget and investigate only the consequences of varying investment proportions in various assets. But if an insurance company takes on more and more insurance policies, it is increasing the size of the portfolio. The analysis called for in that case must be cast in terms of dollar profits, in much the same way that NPV is called for instead of IRR when we compare different-sized projects. This is why risk-pooling (i.e., accumulating independent risky prospects) does not act to eliminate risk. Samuelsons colleague should have counteroffered: "Lets make 1,000 bets, each with your $2 against my $1." Then he would be holding a portfolio of fixed size, equal to $1,000, which is diversified into 1,000 identical independent prospects. This would make the insurance principle work. Another way for Samuelsons colleague to get around the riskiness of this tempting bet is to share the large bets with friends. Consider a firm engaging in 1,000 of Paul Samuel- sons bets. In each bet the firm puts up $1,000 and receives $3,000 or nothing, as before. Each bet is too large for you. Yet if you hold a 1/1,000 share