3.1Meaning and Measurement of Risk

Chapter 3

Risk Analysis

3.1MEANING AND MEASUREMENT OF RISK

In many managerial decisions, the manager does not know the exact outcome of each possible course of action. In such cases, we say that the firm faces risk or uncertainty. Risk refers to the situation in which there is more than one possible outcome to a decision and the probability of each specific outcome is known or can be estimated. If such probabilities are not known and cannot be estimated, we have uncertainty. In evaluating and comparing investment projects that are subject to risk, we use the concepts of expected value, standard deviation, and coefficient of variation.

The expected profit of a project subject to risk is obtained by multiplying each possible outcome or profit from the project by its probability of occurrence, and then adding these products (see Example 1). That is,

(3-1)

where i is the profit level associated with outcome i, Piis the probability or chance that outcome i will occur, i = 1 to n refers to the number of possible outcomes or states of nature, and  refers to the “sum of.” For investment projects facing equal risk, the firm will choose the project with the largest expected profit.

The absolute risk of an investment project can be measured by the standard deviation of the possible profits from the project. The standard deviation,  (read “sigma”), is given by

(3-2)

The greater is the possible dispersion of the profits from a project, the greater is the project's standard deviation and risk.

To compare the relative dispersion of the possible profits, or risk, of two or more projects, we will use the coefficient of variation (). This is given by the ratio of the standard deviation to the expected profit of each project. That is, an investor will usually prefer a more risky project only if its expected profit is sufficiently higher than that of a less risky project (see Example 2).

EXAMPLE 1. Suppose that the possible profits of two investment projects (A and B) under three possible states of the economy (boom, normal, and recession) are those of column (3) in Table 3.1. To obtain the expected profit from each project in column (4) of the table, multiply the probability of occurrence of each state of the economy by the profit from the project under that particular state, and then add these products. Note that the sum of the probabilities for the possible profits from each project in column (2) is I or 100 percent, since one of the three states of the economy must occur with certainty. In column (4) we see that the expected profit from each project is $400. However, in column (3) we see that the range of profits for project A (from $500 to $300) is smaller than for project B (from $600 to $200). Figure 3-1 shows the probability distribution of profits from the two projects.

Table 3.1

State of
Economy
(1) / Probability of Occurrence
(2) / Profit
(3) /

Expected Value

(2)  (3)
(4)
Project A
Boom / 0.25 / $ 500 / $ 125
Normal / 0.50 / 400 / 200
Recession / 0.25 / 300 / 75
Expected profit from project A = $400
Project B
Boom / 0.25 / $ 600 / $ 150
Normal / 0.50 / 400 / 200
Recession / 0.25 / 200 / 50
Expected profit from project A = $400

EXAMPLE 2. Table 3.2 shows how to calculate the standard deviation () of the probability distribution of profits in projects A and B. In Table 3.2, we see that for project A,  = $70.71, while for project B,  = $141.42. These values provide a numerical measure of the absolute dispersion of profits in each project from the project mean and confirm the smaller dispersion of profits and risk for project A than for project B, which is shown graphically in Fig. 3-1. Note in Table 3.2 that a is the square root of the variance (2)

Fig. 3-1

Table 3.2

Deviation

/ Deviation Squared
/ Probability
(Pi) / Deviation Squared Times Probability
. PI
Project A
$500 – $400 = $100 / $10,000 / 0.25 / $ 2,500
400 –400 =0 / 0 / 0.50 / 0
300 –400 = –100 / 10,000 / 0.25 / 2,500
Variance=2=$5,000
Standard deviation=== =$70.71
Project B
$600 – $400 = $200 / $40,000 / 0.25 / $ 10,000
400 –400 =0 / 0 / 0.50 / 0
200 –400 = –200 / 40,000 / 0.25 / 10,000
Variance=2=$20,000
Standard deviation=== =$141.71

To measure relative dispersion or risk, we use the coefficient of variation ():

Since the expected profit is the same for the two projects, but project A has a smaller relative dispersion (v), or risk, than project B, a risk-averse investor would choose project A. (Note that in this case, since the expected profit from each project is the same, this conclusion can be inferred from the value of u for each project, without the need to calculate the ’s.)

3.2INCORPORATING RISK INTO MANAGERIAL DECISIONS

Most managers are risk-averse and face diminishing marginal utility for money. That is, doubling money or profits less than doubles their total utility or satisfaction, so that the marginal or extra utility diminishes. Under such conditions, a manager will not undertake an investment project even though it has a positive expected profit, if its expected utility is negative (see Example 3). While some managers are risk neutral or risk seekers, most are risk-averse. A risk-averse individual will not accept a fair bet (i.e., one with a 50-50 chance of winning or losing a specific sum of money) because the utility gained by winning the bet is smaller than the utility lost by losing the bet.

Risk can also be incorporated into decision-making by the risk-adjusted discount rate approach or by the certainty-equivalent approach. According to the first, a risk-averse manager adds a risk premium to the risk-free discount rate (as shown by a risk-return trade-off junction, or indifference curve) in calculating the present value of the expected profits for a risky investment. Alternatively, the risk-averse manager can substitute equivalent certain sums in place of larger but risky sums (net cash flows) and use the risk-free discount rate to calculate the present value of the project (see Example 4). The ratio of an equivalent certain sum to a larger risky sum is called the certainty-equivalent coefficient (  ).

Managerial decisions involving risk are often made in stages, with subsequent decisions and events depending on the outcome of earlier decisions and events. The sequence of possible managerial decisions and their expected outcomes under each set of circumstances or states of nature can be represented graphically by decision trees (see Example 5). Risk can also be incorporated into managerial decisions by simulation (see Problem 3.11).

EXAMPLE 3. Suppose that a manager wants to determine whether or not to undertake an investment project that has 840 percent probability of providing a profit of $40,000 and a 60 percent probability of producing a loss of $20,000. Suppose also that the manager's utility function is that shown in Fig. 3-2, with money measured along the horizontal axis and the utility of money (measured in arbitrary or fictitious units called utils) measured along the vertical axis. Since the utility function for money is concave, or faces down, its slope, or marginal utility, is diminishing, and so the manager is risk-averse. In column (4) of Table 3.3, we see that the expected profit from the project is positive ($4,000), but the manager will not undertake the project because the corresponding expected utility of the project (column 5) is negative (-3 utils).

Fig. 3-2

Tables 3.3

State
of
Nature / Probability
( 1 ) / Monetary
Outcome
( 2 ) / Associated
Utility
( 3 ) / Expected
Profit
( 4 ) / Expected
Utility (utils)
( 5 )
Success / 0.40 / $40,000 / 15 / $16,000 / 6
Failure / 0.60 / –20,000 / –15 / –12,000 / –9
$4,000 / –3

EXAMPLE 4. Suppose that the risk-return trade-off function of a manager is that shown in Fig. 3-3. It indicates that the manager is indifferent between a 10 percent rate of return on a risk-free investment with g = 0 (point A) and a 14 percent rate of return on an investment with risk given by o' = 0.5 (point B), so that the risk premium is 4 percent. Suppose also that the firm is contemplating an investment project that is expected to generate a net cash flow of $20,000 per year for three years and to cost initially $48,000. From Table 3.4, we see that the firm would undertake the project at the risk-free rate of interest, or discount, of 10 percent because the present value of the project is positive ($1,737.05), but not at the rate of 14 percent because the present value of the project is negative (-$1,567.36).

Fig. 3-3

Alternatively, if the manager regarded the certain sum of $18,600 as equivalent to the risky net cash flow of $20,000 (implying a certainty-equivalent coefficient,
 = $18,600/$20,000 = 0.93) and used the certain sum of $18,600 and the risk-free discount rate of 10 percent, the net present value of the project would be –$1,744.55 (the calculations are left to the reader). This result is similar to that obtained by using the risky net cash flow of $20,000 and the risk-adjusted discount rate of 14 percent, and so the firm would not undertake the project. While the choice of the risk-adjusted discount rate and the certainty-equivalent coefficient is subjective, the latter is somewhat superior because it explicitly considers the manager's attitude toward risk.

Table 3.4 Present Value (PV) of a Project

At a Rate of Return of 10 Percent / At a Rate of Return of 14 Percent

–$48,000

–$48,000

+ $15,026.30 – $48,000
/
–$48,000

–$48,000

+ $13,499.43 – 13,499.43

EXAMPLE 5. Figure 3-4 shows a decision tree that a firm can use to determine whether to build a $2 million plant or a $1 million plant [section (1) of the figure]. Three states of the economy (boom, normal, or recession) can occur [section (2)]. Thus, we have six possible outcomes, each with its probability of occurrence [section (3)] and present value of net cash flows [section (4)]. Multiplying the probability of each outcome by the present value of its net cash flow, we get the corresponding expected net cash flow [section (5)]. Adding the expected cash flows from each strategy, we get $3.4 million for the $2 million plant and $2.2 million for the $1 million plant. Thus, the firm should build the larger plant because its profit of $1.4 million is larger than the $1.2 million profit for the smaller plant (which is, therefore, crossed off in section (1) of the figure as suboptimal).

Fig. 3-4

3.3DECISION MAKING UNDER UNCERTAINTY

Uncertainty exists when the decision maker does not know and cannot estimate the probability of occurrence of each specific outcome. Two decision rules applicable under uncertainty ate the maximin criterion and the minimax regret criterion.

The maximin criterion postulates that the decision maker should determine the worst possible outcome of each strategy and then pick the strategy that provides the best of the worst possible outcomes (see Example 6). This very conservative criterion is appropriate when the firm has a very strong aversion to risk, as for example, when the survival of a small firm depends on avoiding losses.

The minimax regret criterion postulates that the decision maker should select the strategy that minimizes the maximum regret or opportunity cost of the wrong decision, whatever the state of nature that actually occurs. Regret is measured by the difference between the payoff of a given strategy and the payoff of the best strategy under the same state of nature (see Example 7). Which of the above two decision rules a firm might apply depends on its objectives and circumstances. (The maximin criterion and the minimax regret criterion very occasionally lead to the same conclusion, i.e., both may lead either to a decision to invest or a decision not to invest.)

EXAMPLE 6. Suppose that a manager wants to determine whether to undertake an investment project that provides a profit of $40,000 if successful and a loss of $20,000 if a failure (as in Example 3), but now does not know and cannot estimate the probability of success or failure. To apply the maximin criterion, the manager first determines the worst possible outcome of each strategy (row) in Table 3.5. This is
–$20,000 for the strategy of investing and 0 for the strategy of not investing (shown in the "Failure" column of the table). Hence, the manager picks the strategy of not investing, which is indicated by an asterisk next to its zero return, because it provides the best (maximum) of the worst (minimum) possible outcomes (i.e., maximin).

Table 3.5

Strategy / State of Nature / Maximin
Success / Failure
Invest / $40,000 / –$20,000 / –$20,000
Do not invest / 0 / 0 / 0*

* Maximin choice.

EXAMPLE 7. Table 3.6 presents the payoff and regret matrices for the investment project that we examined in Example 6. The regret matrix is constructed by determining the maximum payoff for each state of nature [columns (1) and (2)] and then subtracting the payoff of the strategy chosen from the maximum payoff. For example, if the manager chooses to make the investment and the state of nature that actually occurs is one of success, the manager has no regret because that was the correct strategy. Thus, the regret value of zero is entered at the top of the first column in the regret matrix. However, if the firm had chosen not to invest, the regret value would be $40,000, which is entered at the bottom of the first column of the regret matrix. The payoffs in the second column of the regret matrix under the state of nature of failure are similarly determined. The manager then chooses the strategy of investing because it provides a minimum regret value of $20,000 (indicated by the asterisk in the last column of the table), as compared with the maximum regret of $40,000 that results from the strategy of not investing.

Table 3.6

Strategy / State of Nature / Regret Matrix / Maximin
Regret
( 5 )
Success
( 1 ) / Failure
( 2 ) / Success
( 3 ) / Failure
( 4 )
Invest / $40,000 / –$20,000 / $0 / $20,000 / $20,000*
Do not invest / 0 / 0 / 40,000 / 0 / 40,000

* Minimax regret choice.

Glossary

Certainty The situation in which a decision has only one possible outcome and this outcome is known precisely; that is, the decision is risk-free.

Certainty-equivalent approach The method of using the risk-free discount rate and equivalent smaller sums in place of larger but risky sums or profits to adjust the valuation model for risk.

Certainty-equivalent coefficient () The ratio of the certain sum equivalent to the expected risky sum, or profit, from an investment, that is used to adjust the valuation model for risk.

Coefficient of variation () The ratio of the standard deviation to the expected value.

Decision tree A graphical technique for showing and analyzing the sequence of possible managerial decisions and their expected outcomes under each set of circumstances or states of nature.

Diminishing marginal utility for money The decline in the extra utility received for each dollar increase in income.

Expected profitThe sum of the products of each possible profit from an investment times the probability of its occurrence.

Expected utility The sum of the products of the utility of each possible outcome times its probability of occurrence.

Maximin criterion Thedecision rule under uncertainty that postulates that the decision maker should determine the worst possible outcome of each strategy and then pick the strategy that provides the best of the worst possible outcomes.

Minimax regret criterionThe decision rule under uncertainty that postulates that the decision maker should select the strategy that minimizes the maximum regret, or opportunity cost, of the wrong decision, whatever the state of nature that occurs.

Risk The situation in which a decision has more than one possible outcome and the probability of each possible outcome is known or can be estimated.

Risk-adjusted discount rate approach The method of using a higher rate of discount to calculate the present value of the net cash flows or profits of an investment project in order to compensate for risk.

Risk-averseTerm applied to someone who has a diminishing marginal utility for money. Risk-neutral Term applied to someone whose marginal utility for money is constant.

Risk premium The excess in the expected or required rate of return on a risky investment over the rate of return on a riskless asset; the premium compensates for risk.

Risk-return trade-off function Acurve showing the various risk-return combinations among which a manager or investor is indifferent.

Standard deviation () Ameasure of the amount of difference between possible outcomes and the expected value of the outcome.

UncertaintyThe case in which there is more than one possible outcome to a decision and the probability of occurrence of each specific outcome is not known and cannot be estimated.

Review Questions

1.Risk refers to the situation in which there is

(a)one possible outcome to a decision.

(b)more than one possible outcome to a decision.

(c)more than one possible outcome to a decision and the probability of each specific outcome is known or can be estimated.

(d)more than one possible outcome to a decision and the probability of each specific outcome is not known and cannot he estimated.

Ans. (c) See Section 3.1.

2.If the profit associated with project A under conditions of boom in Table 3.1 had been $600 instead of $500, the expected profit from project A would have been

(a)$425.

(b)$450.

(c)$500.

(d)$600.

Ans. (a) See Example 1.

3.Which of the following measures can be used to compare the risk of two or more investment projects?

(a)Expected profit.

(b)Standard deviation of possible profits.

(c)Coefficient of variation.

(d)Variance.

Ans. (e)See Section 3.1.

4.The coefficient of variation for a given project is given by the

(a)ratio of the standard deviation to the expected profit of the project.

(b)ratio of the expected profit to the standard deviation of the project.

(c)expected profit of the project.

(d)standard deviation of the actual profits of the project.

Ans. (a) See Section 3.1.

5.If the utility function of a manager is concave, or faces down, the (a) manager is a risk seeker.