The certainty equivalent method is an adjustment to the numerator of the basic valuation model to account for risk. Under the certainty equivalent approach, decision makers specify the certain sum that they regard comparable to the expected value of a risky investment alternative. The certainty equivalent of an expected risk amount typically differs in dollar terms but not in terms of the amount of utility provided. To illustrate, suppose that you face the following choices:

• Invest \$100,000. From a successful project, you receive \$1,000,000; if it fails, you receive nothing. If the probability of success is 0.5, or 50 percent, the investment's expected payoff is \$500,000 (= 0.5 X \$1,000,000 + 0.5 X \$0).

• You do not make the investment; you keep the \$100,000.

If you find yourself indifferent between the two alternatives, \$100,000 is your certainty equivalent for the risky expected return of \$500,000. In other words, a certain or riskless amount of \$100,000 provides exactly the same utility as the 50/50 chance to earn \$1,000,000 or \$0. You are indifferent between these two alternatives.

Ratio of a certain sum divided by an expected risky amount, where both dollar values provide the same level of utility

In this example, any certainty equivalent of less than \$500,000 indicates risk aversion. If the maximum amount that you are willing to invest in the project is only \$100,000, you are exhibiting very risk-averse behavior. Each certain dollar is "worth" five times as much as each risky dollar of expected return. Alternatively, each risky dollar of expected return is worth only 20tf in terms of certain dollars. In general, any risky investment with a certainty equivalent less than the expected dollar value indicates risk aversion. A certainty equivalent greater than the expected value of a risky investment indicates risk preference.

Any expected risky amount can be converted to an equivalent certain sum using the certainty equivalent adjustment factor, a, calculated as the ratio of a certain sum divided by an expected risky amount, where both dollar values provide the same level of utility:

Certainty Equivalent _ _ Equivalent Certain Sum Adjustment Factor Expected Risky Sum

The certain sum numerator and expected return denominator may vary in dollar terms, but they provide the exact same reward in terms of utility. In the previous investment problem, in which a certain sum of \$100,000 provides the same utility as an expected risky return of \$500,000, the certainty equivalent adjustment factor a = 0.2 = \$100,000/\$500,000. This means that the "price" of one dollar in risky expected return is 20tf in certain dollar terms.

The following general relations enable managers to use the certainty equivalent adjustment factor to analyze risk attitudes:

Then

### Implies

Equivalent certain sum < Expected risky sum Equivalent certain sum = Expected risky sum Equivalent certain sum > Expected risky sum a < 1 a = 1 a > 1

Risk aversion Risk indifference Risk preference risk-adjusted valuation model

Valuation model that reflects time-value and risk considerations

The appropriate a value for a given managerial decision varies according to the level of risk and degree of the decision maker's risk aversion.

The basic valuation model (Equation 14.7) can be converted into a risk-adjusted valuation model, one that explicitly accounts for risk:

In this risk-adjusted valuation model, expected future profits, E(nt), are converted to their certainty equivalents, aE(nt), and are discounted at a risk-free rate, i, to obtain the risk-adjusted present value of a firm or project. With the valuation model in this form, one can appraise the effects of different courses of action with different risks and expected returns.

To use Equation 14.9 for real-world decision making, managers must estimate appropriate as for various investment opportunities. Deriving such estimates can prove difficult, because a varies according to the size and riskiness of investment projects as well as according to the risk attitudes of managers and investors. In many instances, however, the record of past investment decisions offers a guide that can be used to determine appropriate certainty equivalent adjustment factors. The following example illustrates how managers use certainty equivalent adjustment factors in practical decision making. 