express great en- thusiasm for highly positively skewed lotteries. This hypothesis is, however, ex- tremely difficult to prove with properly controlled experiments. A.2. The better diversified the portfolio, the smaller is its standard deviation, as the sample standard deviations of Table 6A.1 confirm. When we draw from distribu- tions with smaller standard deviations, the probability of extreme values shrinks. Thus the expected smallest and largest values from a sample get closer to the mean value as the standard deviation gets smaller. This expectation is confirmed by the samples of Table 6A.1 for both the sample maximum and minimum annual rate.       APPENDIX B: RISK AVERSION, EXPECTED UTILITY, AND THE ST. PETERSBURG PARADOX   We digress here to examine the rationale behind our contention that investors are risk averse. Recognition of risk aversion as central in investment decisions goes back at least to 1738. Daniel Bernoulli, one of a famous Swiss family of distinguished mathematicians, spent the years 1725 through 1733 in St. Petersburg, where he analyzed the following coin- toss game. To enter the game one pays an entry fee. Thereafter, a coin is tossed until the first head appears. The number of tails, denoted by n, that appears until the first head is tossed is used to compute the payoff, $R, to the participant, as   R(n) 2n   The probability of no tails before the first head (n 0) is 1⁄2 and the corresponding payoff is 2 0 $1. The probability of one tail and then heads (n 1) is 1⁄2 1⁄2 with payoff 21 $2, the probability of two tails and then heads (n 2) is 1⁄2 1⁄2 1⁄2, and so forth. The following table illustrates the probabilities and payoffs for various outcomes:     Tails Probability Payoff $R(n) Probability Payoff   0 1⁄2 $1 $1/2 1 1⁄4 $2 $1/2 2 1⁄8 $4 $1/2 3 1⁄16 $8 $1/2 . . . . . . . . . . . . n (1/2)n 1 $2n $1/2     The expected payoff is therefore   q E(R) Pr(n)R(n) 1/2 1/2 % q n 0