an opposite nature. A rainy summer lowers the return on the suntan-lotion firm but raises it on the umbrella firm. Shares of the umbrella firm act as "weather insurance" for the suntan-lotion firm shareholders in the same way that fire in- surance policies insure houses. When the lotion firm does poorly (bad weather), the "in- surance" asset (umbrella shares) provides a high payoff that offsets the loss. Another means to control portfolio risk is diversification, whereby investments are made in a wide variety of assets so that exposure to the risk of any particular security is limited. By placing ones eggs in many baskets, overall portfolio risk actually may be less than the risk of any component security considered in isolation. To examine these effects more precisely, and to lay a foundation for the mathematical properties that will be used in coming chapters, we will consider an example with less than perfect hedging opportunities, and in the process review the statistics underlying portfolio risk and return characteristics. II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001           CHAPTER 6 Risk and Risk Aversion 163     A Review of Portfolio Mathematics   Consider the problem of Humanex, a nonprofit organization deriving most of its income from the return on its endowment. Years ago, the founders of Best Candy willed a large block of Best Candy stock to Humanex with the provision that Humanex may never sell it. This block of shares now comprises 50% of Humanexs endowment. Humanex has free choice as to where to invest the remainder of its portfolio.2 The value of Best Candy stock is sensitive to the price of sugar. In years when world sugar crops are low, the price of sugar rises significantly and Best Candy suffers considerable losses. We can describe the fortunes of Best Candy stock using the following scenario analysis:   Normal Year for Sugar Abnormal Year   Bullish Bearish Stock Market Stock Market Sugar Crisis   Probability .5 .3 .2 Rate of return 25% 10% 25%     To summarize these three possible outcomes using conventional statistics, we review some of the key rules governing the properties of risky assets and portfolios.   Rule 1 The mean or expected return of an asset is a probability-weighted