An individual's or company's aversion to risk taking can be quantified and measured. The concept works as follows. If decision makers did not care about risk, they would want to "go with the odds;" that is, they would want to make decisions so as to maximize expected value. The expected value is defined as the probability-weighted sum of the possible uncertain outcomes. Decision makers unconcerned about risk would want to maximize expected value because the expected value is the amount that they would obtain on average each time the uncertainty is faced. As an example, the expected value of a coin flip that pays $1 on "heads" and zero on "tails" is 50 cents.
For substantial risks, organizations (and individuals) are risk averse, meaning that they value uncertainties at less than their expected values. The certain equivalent is defined as the amount of money for which a decision maker would be indifferent between receiving that amount for certain and receiving the uncertain outcomes of the gamble. For example, a risk-averse decision maker might assign a certain equivalent of $500,000 to a risky project with equal chances of yielding $0 and $2,000,000, even though the expected value for this alternative is $1,000,000. Note that this same logic means that a gamble with negative expected value (large downside risk) has a certain equivalent that is even more negative than its expected value (which is why individuals and organizations are willing to pay more for insurance premiums than the expected loss that they are eliminating). The goal of a risk averse decision maker is to maximize the certain equivalent.
For risks with complex payoff distributions, it is generally difficult to determine the certain equivalent. However, the certain equivalent can be estimated for a simple gamble and the results used to infer the certain equivalents of more complicated risks. The approach involves constructing a utility function that represents the degree of aversion to taking risks.
Figure 34 illustrates the form (exponential) often chosen for the utility function. The horizontal axis shows possible values or certain equivalents. The vertical axis shows the corresponding "utility," where utility is a numerical rating assigned to every possible value.
Figure 34: The exponential utility function is often used to model risk aversion.
The shape of the utility function determines the degree of aversion to taking risks. The more the plot curves or bends over, the more risk aversion is represented. With the exponential utility function, the degree of curvature is determined by the parameter R, known as the risk tolerance. Thus, risk tolerance is an indicator of a decision maker's or organization's willingness to accept risk. Risk tolerance, as defined here, is not the maximum amount that the decision maker can afford to lose, although decision makers and organizations with greater wealth generally have larger risk tolerances.
Once the risk tolerance is set, the utility function may be used to compute a certain equivalent as follows. First, locate each possible payoff x on the horizontal axis and determine the corresponding utility U(x) on the vertical axis. For example, if risk tolerance is $1 million and the risk is 50% chance of $0 or $2 million, the corresponding utilities (from Figure 10) are 0 and approximately 0.8. Second, compute the expected utility by multiplying each utility by its probability and summing the products. For the example, the expected utility is roughly 0.5×0 + 0.5×0.8 = 0.4. Third, locate the expected utility on the vertical axis and determine the corresponding certain equivalent on the horizontal axis. The result for the example is approximately $500,000.
There are several ways to determine the risk tolerance for an organization. One is to ask senior decision makers (ideally, the CEO) to answer the following hypothetical question. Suppose you have an opportunity to make a risky, but potentially profitable investment. The required investment is an amount R that, for the moment, is unspecified. The investment has a 50-50 chance of success. If it succeeds, it will generate the full amount invested, including the cost of capital, plus that amount again. In other words, the return will be R if the investment is successful. If the investment fails, half the investment will be lost, so the return is minus R/2. Figure 35 illustrates the opportunity. Note that the expected value of the investment is R/4.
Figure 35: What is the maximum amount R you would accept in this gamble?
If R were very low, most CEOs would want to make the investment. If R were very large, for example, close to the market value of the enterprise, most CEOs would not take the investment. The risk tolerance is the amount R for which decision makers would just be indifferent between making and not making the investment. In other words, the risk tolerance is the value of R for which the certain equivalent of the investment is zero.
Various studies have been conducted to measure organizational risk tolerances. The results show that risk tolerances obtained from different executives within the same organization vary tremendously. Generally, those lower in the organization have lower risk tolerances. As a rough rule of thumb, for publicly traded firms, typical risk tolerances at the CEO or Board level are often equal to about 20% of the organization's market value.
Once risk tolerance has been established, the certain equivalent for any risky project or project portfolio can be obtained via the utility function. The effect, as illustrated in Figure 36, is to subtract a risk adjustment factor from the expected value (if projects allow you to avoid risks, the effect is to add, rather than subtract, adjustment factors). The risk adjustment depends on the risk tolerance and the amount of risk. If the projects are independent (i.e., their risks are uncorrelated), then the certain equivalent of the project portfolio will be the sum of the certain equivalents of the individual projects. If project risks are correlated, the certain equivalent for the portfolio can be obtained once the distribution of payoffs for the portfolio are computed (accounting for correlations as described above).
Figure 36: Adjusting project value for risk
An advantage of this approach is that a single risk tolerance can be established for the organization. Use of the common risk tolerance ensures that risks are treated consistently, thus avoiding the common bias in which greater levels of risk aversion tend to be applied by lower-level managers.
For a demonstration of the importance in the context of project prioritization of addressing risk and risk tolerance, see the Risk Demo.
References for Part 4