■ Definition: The certainty equivalent of a decision alternative with more than one possible outcome is the sum of money, available with certainty, that would cause the decision maker to be indifferent between that decision alternative and accepting the certain sum of money.
Example: Suppose that I offer to give you either $0.50 or “toss you” for double or nothing. If you accept the $0.50, that means you prefer the certain sum ($0.50) to the gamble which has two possible outcomes ($1.00 or nothing). To find your point of indifference I would now offer you the choice of $0.30 with certainty or the toss for $1.00 or nothing. Suppose you now prefer to take the gamble. Somewhere between $0.30 and $0.50 is your certainty equivalent of the gamble. Suppose that after several more offers I find that you are indifferent between accepting $0.42 with certainty and taking the gamble. Thus, your certainty equivalent (CE) for that gamble is $0.42. In effect, the CE is the sum of money that will almost bribe you to give up the gamble.
The notion of an individual’s certainty equivalent involves the concept of utility and indifference, and it incorporates the decision maker’s degree of risk aversion (or risk preference). The more risk-averse decision maker would exhibit a lower CE than a less risk-averse (or a risk-preferring) decision maker for the same gamble. Moreover, it can incorporate other considerations as well. If the gamble is to be repeated many times, we would expect the CE to be equal to the expected value (in this case $0.50). If the gamble is to be taken just once or a few times, we expect a risk averter’s CE to be less than the EV. In effect, the risk averter is trading off expected return for removal of the risk. Conversely, a risk preferrer’s CE is expected to exceed the EV, since this
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Introduction
person gets utility from risk and will require more than the gamble s EV to be bribed into giving it up. Finally, suppose a person is desperate for a given sum of money. For example, a student needs $0.75 to wire home and ask for more money: That individual’s CE is likely to be $0.75. Any lesser sum doesn’t solve the problem, and the student will be prepared to gamble for the best payoff unless assured of the $0.75 with certainty. This person, whatever his or her usual attitude toward risk, would be acting like a risk preferrer in this instance.
Expressed in terms of indifference-curve analysis, the certainty equivalent of a decision alternative can be shown as the point on the vertical axis joined by an indifference curve to the point representing the decision alternative under consideration.
Example: In Figure 2-4 we show the two decision alternatives introduced as the large-machine/small-machine decision problem earlier. Note that we have shown the small machine as the preferred alternative, since it lies on the higher indifference curve. The point CE S on the expected-value axis is the certainty equivalent of the combination of expected value and risk represented by point S. It is the certainty equivalent because risk, on the horizontal axis, has been reduced to zero and, thus, the decision maker is indifferent between $CEs with certainty and $6,346.60 with risk (standard deviation of $2,903.60). Similarly, the certainty equivalent of the large machine, point L, is $CE t .
Rule: The certainty-equivalent criterion involves selecting the decision alternative which has the highest certainty equivalent. Note that since indifference curves neither meet nor intersect and since higher curves are preferred to lower curves, the certainty-
FIGURE 2-4.
Certainty Equivalents of the Large-Machine/Smali-Maehine Decision
equivalent criterion is entirely consistent with the utility theory of rational consumer (or decision-making) behavior.
Unfortunately, we are now in a position to see that the earlier methods of adjusting for risk, namely, the coefficient-of-variation and the EPV criterion using different discount rates, are not entirely consistent with utility theory and may sometimes indicate the choice of a decision alternative that would not be optimal in terms of the decision maker’s utility. Consider Figure 2-5 in which the points S and L for the small and large machine alternatives are shown on rays emanating from the origin. The slopes of these rays reflect the ratio of expected value to standard deviation, or what we have earlier called risk-adjusted return. The slope of each ray is equal to the reciprocal of the coefficient of variation. Thus, as we saw earlier, the small machine, point S , is preferred to the large machine, point L , on the basis of both the coefficient-of-variation and the certainty-equivalent criteria.
But consider points M and P which have the same value for their coefficient of variation as point S , yet are ranked differently in terms of their certainty equivalents: M is preferred to S , which in turn is preferred to P. The coefficient-of-variation criterion is unable to distinguish between M, S, and P and ranks them as equals. In effect, it says that the ray from the origin 0 PSM is an indifference curve, with all points on it giving equal utility (or certainty equivalent) to the decision maker. Similarly, it effectively says that the ray 0 TRL is a lower indifference curve, with all points on that ray regarded as equal, but inferior to any point on the higher ray 0 PSM. The curvilinear indifference curves shown in Figure 2-5 indicate that this is untrue for the case presented: L is preferred to R, which in turn is preferred to T, and, moreover, L is pre-
FIGURE 2-5. Conflict between the Coefficient-of-Variation Criterion and the Certainty-Equivalent Criterion
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Introduction
ferred to P, and R is equivalent to P, despite P’s lower coefficient of variation (or higher risk-adjusted return).
Thus, although the coefficient-of-variation criterion correctly ranked points S and L, it incorrectly ranked most of the other points shown! In fact, the coefficient-of- variation criterion may incorrectly rank S and L for a particular decision maker. In Figure 2-6 we show a decision maker who is only slightly risk averse (with very low marginal rate of substitution of expected return for risk) who prefers point L over point S. Point L lies on a higher indifference curve for this individual and, accordingly, has a higher certainty equivalent. Note that the expected-value criterion with different discount rates also incorrectly ranked the small versus large machine alternatives for this particular decision maker and would similarly make mistakes between other pairs of decision alternatives.
Why does this problem arise? Essentially, the coefficient-of-variation criterion and the EV criterion using different ODRs assume a linear and constant tradeoff between risk and return. In fact, most risk averters exhibit a nonlinear risk-return tradeoff. In economic terms we explain this as diminishing marginal utility of wealth and increasing marginal disutility of risk. The more wealth (expected return) one is offered, the less utility one receives from the marginal dollar, and the more risk one is offered, the greater the disutility one receives from the marginal unit of risk.
Should we throw away the coefficient-of-variation and the EV criterion using different ODRs and simply use the certainty equivalent? Without doubt the certainty equivalent gives the better answers for maximizing the decision maker’s utility function. This might be quite different from maximizing the firm’s objective function, however. For the firm with a single owner-manager, maximizing net worth and maxi-
FIGURE 2-6. Cocfficient-of-Variation Criterion Refuted for a Slightly Risk-Averse Decision Maker
mizing the decision maker’s utility may well amount to the same thing, but for a widely held firm with a variety of decision makers within the management team, it is not at all clear that maximizing managerial utility will lead to the maximization of net worth. One could argue that the firm does have a constant tradeoff between risk and return. More importantly, perhaps, the coefficient-of-variation criterion, and the EV using different ODRs to a lesser extent, can be calculated and subjected to scrutiny, whereas certainty equivalents and utility are more cerebral and intuitive notions that are more difficult to defend quantitatively.
Rule: The solution to this apparent impasse is to use all three methods with caution and with reservations. For example, one would say that the small machine appears to be preferable on a risk-adjusted basis, unless the firm is only very slightly risk averse, to the extent that the firm would be willing to accept an additional $2,492.10 standard deviation (that is $5,395.70 — $2,903.60) for an additional $54.90 expected return (that is, $6,401.50 — $6,346.60). The caveat attached to this recommendation can be settled by management consensus (or by the boss’s opinion). If the consensus is that the extra risk should be undertaken for the extra expected return, the simple risk- adjustment criterion is overturned. Alternatively, if the consensus (or the owner’s opinion) is not to accept the extra risk for the extra return, the simple risk-adjustment criterion is supported.
Note: In many cases there will be no doubt which decision alternatives dominate others for the risk-averse decision maker or firm. Consider Figure 2-7 in which we show one of several possible outcomes, labeled point A. Relative to point A, we show quadrants I, II, III, and IV. Any point in quadrant IV is unambiguously superior to point A (for a risk averter) since it has either more return for the same or less risk, or less risk for the same return. Similarly, any point in quadrant II is unambiguously
oi
IV
Superior
I
Ambiguous
II
Inferior
Risk (cr)
FIGURE 2-7.
Superior and Inferior Quadrants with Respect to a Particular Decision Alternative (Point A)
inferior to point A, having either less return for the same or more risk, or more risk for the same return. Quadrants I and III, relative to point/4, on the other hand, do represent potential areas of conflict between the coefficient-of-variation and the EV criteria using different ODRs, on the one hand, and the certainty-equivalent criterion on the other. Points in quadrant I have both more risk and more return, and in quadrant III there is both less risk and less return. When decision alternatives fall in quadrants I and III with respect to each other, one must use the “caution and reservation rule suggested above.
The Certainty-Equivalent Factor. To find the certainty equivalent of an uncertain venture without involving indifference curves, decide what fraction of the EPV would make you indifferent between (a) the EPV of the uncertain venture and (b) that fraction of the EPV if it were available with certainty. One way to do this is to decide “how many cents in the dollar” you would consider to be equivalent if these were available with certainty. This fraction, say 0.75, is known as the certainty-equivalent factor (CEF). Multiplying the CEF by the EPV will give you the CE of the gamble. Riskier ventures will have smaller CEFs, but the CE is the product of the CEF and the EPV, such that the CE may be greater or smaller for the more risky venture. The decision rule remains the same: choose the alternative with the greatest (positive valued) certainty equivalent.