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Suppose an investor is offered an investment portfolio with a payoff in one year described by a simple prospect. How can you evaluate this portfolio?

First, try to summarize it using descriptive statistics. For instance, the mean or expected end-of-year wealth, denoted E(W), is

The expected profit on the $100,000 investment portfolio is $22,000: 122,000 - 100,000. The variance, ct2, of the portfolio's payoff is calculated as the expected value of the squared deviation of each possible outcome from the mean:

= .6(150,000 - 122,000)2 + .4(80,000 - 122,000)2 = 1,176,000,000

The standard deviation, ct, which is the square root of the variance, is therefore $34,292.86.

Clearly, this is risky business: The standard deviation of the payoff is large, much larger than the expected profit of $22,000. Whether the expected profit is large enough to justify such risk depends on the alternative portfolios.

1 Chapters 6 through 8 rely on some basic results from elementary statistics. For a refresher, see the Quantitative Review in the

### Appendix at the end of the book.

Let us suppose Treasury bills are one alternative to the risky portfolio. Suppose that at the time of the decision, a one-year T-bill offers a rate of return of 5%; $100,000 can be invested to yield a sure profit of $5,000. We can now draw the decision tree.

A. Invest in risky prospect

A. Invest in risky prospect profit profit profit profit

B. Invest in risk-free T-bill profit = $5,000

Earlier we showed the expected profit on the prospect to be $22,000. Therefore, the expected marginal, or incremental, profit of the risky portfolio over investing in safe T-bills is

meaning that one can earn a risk premium of $17,000 as compensation for the risk of the investment.

The question of whether a given risk premium provides adequate compensation for an investment's risk is age-old. Indeed, one of the central concerns of finance theory (and much of this text) is the measurement of risk and the determination of the risk premiums that investors can expect of risky assets in well-functioning capital markets.

CONCEPT CHECK ^ QUESTION 1

CONCEPT CHECK ^ QUESTION 1

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