bet is equivalent to a $1 bet. Holding a small share of many large bets essen- tially allows you to replace a stake in one large bet with a diversified portfolio of manage- able bets. How does this apply to insurance companies? Investors can purchase insurance com- pany shares in the stock market, so they can choose to hold as small a position in the over- all risk as they please. No matter how great the risk of the policies, a large group of individual small investors will agree to bear the risk if the expected rate of return exceeds the risk-free rate. Thus it is the sharing of risk among many shareholders that makes the in- surance industry tick. II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001           254 PART II Portfolio Theory     APPENDIX C: THE FALLACY OF TIME DIVERSIFICATION   The insurance story just discussed illustrates a misuse of rate of return analysis, specifically the mistake of comparing portfolios of different sizes. A more insidious version of this er- ror often appears under the guise of "time diversification." Consider the case of Mr. Frier, who has $100,000. He is trying to figure out the appro- priate allocation of this fund between risk-free T-bills that yield 10% and a risky portfolio that yields an annual rate of return with E(rP) 15% and P 30%. Mr. Frier took a course in finance in his youth. He likes quantitative models, and after careful introspection estimates that his degree of risk aversion, A, is 4. Consequently, he calculates that his proper allocation to the risky portfolio is E(rP) rf   15 10 y .01 A 2 .01 4 302 .14   that is, a 14% investment ($14,000) in the optimal risky portfolio. With this strategy, Mr. Frier calculates his complete portfolio expected return and stan- dard deviation as   E(rC) rf y[E(rP) rf] 10.70% C y P 4.20%   At this point, Mr. Frier gets cold feet because this fund is intended to provide the main- stay of his retirement wealth. He plans to retire in five years, and any mistake will be bur- densome. Mr. Frier calls Ms. Mavin, a highly recommended financial adviser. Ms. Mavin explains that indeed the time factor is all-important. She cites academic research