Certainty Equivalent Adjustment Example

Assume that operations at Burns & Allen Industries have been seriously disrupted by problems with a faulty boiler at its main fabrication facility. In fact, state fire marshals shut the facility down for an extended period recently following repeated overheating and minor explosions. The boiler problem was solved when it was discovered that a design flaw had made the pilot light safety switch inoperable.

Burns & Allen retained the Denver law firm of Dewey, Cheetum & Howe to recover economic damages from the boiler manufacturer. The company has filed suit in state court for $250,000 in damages. Prior to filing suit, the attorney estimated legal, expert witness, and other litigation costs to be $10,000 for a fully litigated case, for which Burns & Allen had a 10 percent chance of receiving a favorable judgment. For simplicity, assume that a favorable judgment will award Burns & Allen 100 percent of the damages sought, whereas an unfavorable judgment will result in the firm receiving zero damages. Also assume that $10,000 is the most Burns & Allen would be willing to pay to sue the boiler manufacturer.

In filing suit against the boiler manufacturer, Burns & Allen has made a risky investment decision. By its willingness to bear litigation costs of $10,000, the company has implicitly stated that it regards these out-of-pocket costs to be at least equivalent to the value of the risky expectation of receiving a favorable judgment against the boiler manufacturer. In other words, Burns & Allen is willing to exchange $10,000 in certain litigation costs for the possibility of receiving a $250,000 judgment against the boiler manufacturer.

Burns & Allen's investment decision can be characterized using the certainty equivalent adjustment method. To do this, it is important to realize that the $10,000 in litigation costs is incurred irrespective of the outcome of a fully litigated case. This $10,000 represents a certain sum that the company must value as highly as the expected risky outcome to be willing to file suit. The expected risky outcome, or expected return from filing suit, is

Expected Return = Favorable Judgment Payoff X Probability

+ Unfavorable Judgment Payoff X Probability

To justify filing suit, Burns & Allen's certainty equivalent adjustment factor for investment projects of this risk class must be

Certain Sum Expected Risky Sum

Litigation Costs Expected Return $10,000 $25,000 0.4

Therefore, each risky dollar of expected return from the litigation effort is worth, in terms of utility, at least 40tf in certain dollars. Alternatively, $10,000 is the certain sum equivalent of the risky expected return of $25,000.

Now assume that after Burns & Allen goes to court, incurring $5,000 in litigation costs, especially damaging testimony by an expert witness dramatically changes the outlook of the case in Burns & Allen's favor. In response, the boiler manufacturer's attorney offers an out-of-court settlement in the amount of $30,000. However, Burns & Allen's attorney recommends that the company reject this offer, estimating that it now has a 50/50 chance of obtaining a favorable judgment in the case. Should Burns & Allen follow the attorney's advice and reject the settlement offer?

In answering this question, one must keep in mind that having already spent ("sunk") $5,000 in litigation costs, Burns & Allen must consider as relevant litigation costs only the additional $5,000 necessary to complete litigation. These $5,000 litigation costs, plus the $30,000 out-of-court settlement offer, represent the relevant certain sum, because proceeding with the suit will require an "investment" of these additional litigation plus opportunity costs. Given the revised outlook for a favorable judgment, the expected return to full litigation is

Expected Return = ($250,000)(0.5) + ($0)(0.5) = $125,000

In light of Burns & Allen's earlier decision to file suit on the basis that each dollar of expected risky return was "worth" 40tf in certain dollars, this expected return would have a $50,000 (=$125,000 X 0.4) certainty equivalent value. Because this amount exceeds the settlement offer plus remaining litigation costs, the settlement offer seems deficient and should be rejected. On the basis of Burns & Allen's revealed risk attitude, an out-of-court settlement offer has to be at least $45,000 to receive favorable consideration. At that point, the settlement plus saved litigation costs of $5,000 would equal the certainty equivalent value of the expected return from continuing litigation.

This simple example illustrates that historical investment decisions offer a useful guide to current decisions. If a potential project's required investment and risk levels are known, the a implied by a decision to accept the investment project can be calculated. This project-specific a can then be compared with as for prior projects with similar risks. Risk-averse individuals should invest in projects if calculated as are less than or equal to those for accepted historical projects in the same risk class. Furthermore, given an estimate of expected return and risk, the maximum amount that the firm should be willing to invest in a given project can also be determined from the certainty equivalent adjustment factor. Risk-averse management will accept new projects if the level of required investment per dollar of expected return is less than or equal to that for historical projects of similar risk.

risk-adjusted discount rate

Risk-free rate of return plus the required risk premium risk premium

Added expected return for a risky asset over that of a riskless asset

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