## Risk Assessment

Many of the inputs to the above evaluation methods will be highly uncertain at the time an investment decision must be made. To make the most informed decision possible, an investor should employ these methods within a framework that explicitly accounts for risk and uncertainty.

Risk assessment provides decision makers with information about the "risk exposure" inherent in a given decision, i.e., the probability that the outcome will be different from the "best-guess" estimate. Risk assessment is also concerned with the risk attitude of the decision maker that describes his/her willingness to take a chance on an investment of uncertain outcome. Risk assessment techniques are typically used in conjunction with the evaluation methods outlined earlier; and not as stand-alone evaluation techniques.

The risk assessment techniques range from simple and partial to complex and comprehensive. Though none takes the risk out of making decisions, the techniques—if used correctly—can help the decision maker make more informed choices in the face of uncertainty.

This chapter provides an overview of the following probability-based risk assessment techniques:

• Expected value analysis

• Mean-variance criterion and coefficient of variation

• Risk-adjusted discount rate technique

• Certainty equivalent technique

• Monte Carlo simulation

• Decision analysis

• Sensitivity analysis

There are other techniques that are used to assess risks and uncertainty (e.g., CAP_M, and break-even analysis), but those are not treated here.

### 3.4.1 Expected Value Analysis

Expected value (EV) analysis provides a simple way of taking into account uncertainty about input values, but it does not provide an explicit measure of risk in the outcome. It is helpful in explaining and illustrating risk attitudes.

### 3.4.1.1 How to Calculate EV

An expected value is the sum of the products of the dollar value of alternative outcomes and their probabilities of occurrence. That is, where a,- (i = 1,...,n) indicates the value associated with alternative outcomes of a decision, and pi indicates the probability of occurrence of alternative a,-, the EV of the decision is calculated as follows:

3.4.1.2 Example of EV Analysis

The following simplified example illustrates the combining of EV analysis and NPV analysis to support a purchase decision.

Assume that a not-for-profit organization must decide whether to buy a given piece of energy-saving equipment. Assume that the unit purchase price of the equipment is $100,000, the yearly operating cost is $5,000 (obtained by a fixed-price contract), and both costs are known with certainty. The annual energy cost savings, on the other hand, are uncertain, but can be estimated in probabilistic terms as shown in Table 3.1 in the columns headed au pu a2, and p2. The present-value calculations are also given in Table 3.1.

If the equipment decision were based only on NPV, calculated with the "best-guess" energy savings (column a ¡), the equipment purchase would be found to be uneconomic. But if the possibility of greater energy savings is taken into account by using the EV of savings rather than the best guess, the conclusion is that, over repeated applications, the equipment is expected to be cost effective. The expected NPV of the energy-saving equipment is $25,000 per unit.

3.4.1.3 Advantages and Disadvantages of the EV Technique

An advantage of the technique is that it predicts a value that tends to be closer to the actual value than a simple "best-guess" estimate over repeated instances of the same event, provided, of course, that the input probabilities can be estimated with some accuracy.

A disadvantage of the EV technique is that it expresses the outcome as a single-value measure, such that there is no explicit measure of risk. Another is that the estimated outcome is predicated on many replications of the event, with the EV, in effect, a weighted average of the outcome over many like events. But the EV is unlikely to occur for a single instance of an event. This is analogous to a single coin toss: the outcome will be either heads or tails, not the probabilistic-based weighted average of both.

TABLE 3.1 Expected Value Example

Year Equipment Operating Energy Savings Purchase Costs ($1000) ($1000) -

TABLE 3.1 Expected Value Example

Year Equipment Operating Energy Savings Purchase Costs ($1000) ($1000) -

($1000) |
Pi |
A 2 ($1000) |
P 2 |
PV* Factor |
($1000) | ||

0 -100 |
— |
— |
— |
— |
— |
1 |
-100 |

1 |
- 5 |
25 |
0.8 |
50 |
0.2 |
0.926 |
23 |

2 |
- 5 |
30 |
0.8 |
60 |
0.2 |
0.857 |
27 |

3 |
- 5 |
30 |
0.7 |
60 |
0.3 |
0.794 |
27 |

4 |
- 5 |
30 |
0.6 |
60 |
0.4 |
0.7354 |
27 |

5 |
- 5 |
30 |
0.8 |
60 |
0.2 |
0.681 |
21 |

Expected Net Present Value: |
25 |

a Present value calculations are based on a discount rate of 8%. Probabilities sum to 1.0 in a given year.

a Present value calculations are based on a discount rate of 8%. Probabilities sum to 1.0 in a given year.

### 3.4.1.4 Expected Value and Risk Attitude

Expected values are useful in explaining risk attitude. Risk attitude may be thought of as a decision maker's preference between taking a chance on an uncertain money payout of known probability versus accepting a sure money amount. Suppose, for example, a person were given a choice between accepting the outcome of a fair coin toss where heads means winning $10,000, and tails means losing $5,000 and accepting a certain cash amount of $2,000. EV analysis can be used to evaluate and compare the choices. In this case, the EV of the coin toss is $2,500, which is $500 more than the certain money amount. The "risk-neutral" decision maker will prefer the coin toss because of its higher EV. The decision maker who prefers the $2,000 certain amount is demonstrating a "risk-averse" attitude. On the other hand, if the certain amount were raised to $3,000 and the first decision maker still preferred the coin toss, he or she would be demonstrating a "risk-taking" attitude. Such tradeoffs can be used to derive a "utility function" that represents a decision maker's risk attitude.

The risk attitude of a given decision maker is typically a function of the amount at risk. Many people who are risk averse when faced with the possibility of significant loss become risk neutral—or even risk taking—when potential losses are small. Because decision makers vary substantially in their risk attitudes, there is a need to assess not only risk exposure (i.e., the degree of risk inherent in the decision) but also the risk attitude of the decision maker.

3.4.2 Mean-Variance Criterion and Coefficient of Variation

These techniques can be useful in choosing among risky alternatives, if the mean outcomes and standard deviations (variation from the mean) can be calculated.

Consider a choice between two projects—one with higher mean NB and a lower standard deviation than the other. This situation is illustrated in Figure 3.4. In this case, the project whose probability distribution is labeled B can be said to have stochastic dominance over the project labeled A. Project B is preferable to project A, both on grounds that its output is likely to be higher and that it entails less risk of loss. But what if project A, the alternative with higher risk, has the higher mean NB, as illustrated in Figure 3.5? If this were the case, the mean-variance criterion (MVC) would provide inconclusive results.

When there is not stochastic dominance of one project over the other(s), it is helpful to compute the coefficient of variation (CV) to determine the relative risk of the alternative projects. The CV indicates which alternative has the lower risk per unit of project output. Risk-averse decision makers will prefer the alternative with the lower CV, other things being equal. The CV is calculated as follows:

where CV is the coefficient of variation, a is the standard deviation, and is the mean.

The principal advantage of these techniques is that they provide quick, easy-to-calculate indications of the returns and risk exposure of one project relative to another. The principal disadvantage is that the

Project A

Project B

Project A

Project B

Net present value

FIGURE 3.4 Stochastic dominance as demonstrated by mean-variance criterion.

Net present value

FIGURE 3.4 Stochastic dominance as demonstrated by mean-variance criterion.

Project B

Project B

M2 Mi Net present value

FIGURE 3.5 Inconclusive results from mean-variance criterion.

M2 Mi Net present value

MVC does not provide a clear indication of preference when the alternative with the higher mean output has the higher risk, or vice versa.

3.4.3 Risk-Adjusted Discount Rate Technique

FIGURE 3.5 Inconclusive results from mean-variance criterion.

The risk-adjusted discount rate (RADR) technique takes account of risk through the discount rate. If a project's benefit stream is riskier than that of the average project in the decision maker's portfolio, a higher than normal discount rate is used; if the benefit stream is less risky, a lower than normal discount rate is used. If costs are the source of the higher-than-average uncertainty, a lower than normal discount rate is used and vice versa. The greater the variability in benefits or costs, the greater the adjustment in the discount rate.

The RADR is calculated as follows:

where RADR is the risk-adjusted discounted rate, RFR is the risk-free discount rate, generally set equal to the treasury bill rate, NRA is the "normal" risk adjustment to account for the average level of risk encountered in the decision maker's operations, and XRA is the extra risk adjustment to account for risk greater or less than normal risk.

An example of using the RADR technique is the following: A company is considering an investment in a new type of alternative energy system with high payoff potential and high risk on the benefits side. The projected cost and revenue streams and the discounted present values are shown in Table 3.2. The treasury bill rate, taken as the risk-free rate, is 8%. The company uses a normal risk adjustment of 4% to account for the average level of risk encountered in its operations. The revenues associated with this investment are judged to be more than twice as risky as the company's average investment, so an additional risk adjustment of 6% is added to the RADR. Hence, the RADR is 18%. With this RADR, the NPV of the investment is estimated to be a loss of $28 million. On the basis of this uncertainty analysis, the company would be advised to not accept the project.

Advantages of the RADR technique are that it provides a way to account for both risk exposure and risk attitude. Moreover, RADR does not require any additional steps for calculating NPV once a value of the RADR is established. The disadvantage is that it provides only an approximate adjustment. The value of the RADR is typically a rough estimate based on sorting investments into risk categories and adding a

TABLE 3.2 |
Risk-Adjusted Discount Rate Example | ||||

Year |
Costs ($M) |
Revenue ($M) |
PV Costs" ($M) |
PV Revenue1, ($M) |
NPV ($M) |

0 |
80 |
— |
80 |
— |
- 80 |

1 |
5 |
20 |
4 |
17 |
13 |

2 |
5 |
20 |
4 |
14 |
10 |

3 |
5 |
20 |
4 |
12 |
8 |

4 |
5 |
20 |
3 |
10 |
7 |

5 |
5 |
20 |
3 |
9 |
6 |

6 |
5 |
20 |
3 |
7 |
4 |

7 |
5 |
20 |
2 |
6 |
4 |

Total NPV |
- 28 |

a Costs are discounted with a discount rate of 12%. b Revenues are discounted with the RADR discount rate of 18%.

a Costs are discounted with a discount rate of 12%. b Revenues are discounted with the RADR discount rate of 18%.

"fudge factor" to account for the decision maker's risk attitude. It generally is not a fine-tuned measure of the inherent risk associated with variation in cash flows. Further, it typically is biased toward investments with short payoffs because it applies a constant RADR over the entire analysis period, even though risk may vary over time.

### 3.4.4 Certainty Equivalent Technique

The certainty equivalent (CE) technique adjusts investment cash flows by a factor that will convert the measure of economic worth to a "certainty equivalent" amount—the amount a decision maker will find equally acceptable to a given investment with an uncertain outcome. Central to the technique is the derivation of the certainty equivalent factor (CEF) that is used to adjust net cash flows for uncertainty.

Risk exposure can be built into the CEF by establishing categories of risky investments for the decision maker's organization and linking the CEF to the CV of the returns—greater variation translating into smaller CEF values. The procedure is as follows:

1. Divide the organization's portfolio of projects into risk categories. Examples of investment risk categories for a private utility company might be the following: low-risk investments: expansion of existing energy systems and equipment replacement; moderate-risk investments: adoption of new, conventional energy systems; and high-risk investments: investment in new alternative energy systems.

2. Estimate the coefficients of variation (see the section on the CV technique) for each investment-risk category (e.g., on the basis of historical risk-return data).

3. Assign CEFs by year, according to the coefficients of variation, with the highest-risk projects being given the lowest CEFs. If the objectives are to reflect only risk exposure, set the CEFs such that a risk-neutral decision maker will be indifferent between receiving the estimated certain amount and the uncertain investment. If the objective is to reflect risk attitude as well as risk exposure, set the CEFs such that the decision maker with his or her own risk preference will be indifferent.

To apply the technique, proceed with the following steps:

4. Select the measure of economic performance to be used, such as the measure of NPV (i.e., NB).

5. Estimate the net cash flows and decide in which investment-risk category the project in question fits.

6. Multiply the yearly net cash flow amounts by the appropriate CEFs.

7. Discount the adjusted yearly net cash flow amounts with a risk-free discount rate (a risk-free discount rate is used because the risk adjustment is accomplished by the CEFs).

8. Proceed with the remainder of the analysis in the conventional way.

In summary, the certainty equivalent NPV is calculated as follows:

where NPVCE is the NPV adjusted for uncertainty by the CE technique, Bt is the estimated benefits in time period t, Ct is the estimated costs in time period t, and RFD is the risk-free discount rate.

Table 3.3 illustrates the use of this technique for adjusting net present-value calculations for an investment in a new, high-risk alternative energy system. The CEF is set at 0.76 and is assumed to be constant with respect to time.

A principal advantage of the CE Technique is that it can be used to account for both risk exposure and risk attitude. Another is that it separates the adjustment of risk from discounting and makes it possible to make more precise risk adjustments over time. A major disadvantage is that the estimation of CEF is only approximate.

TABLE 3.3 Certainty Equivalent (CE)

Yearly Net Cash CV CEF RFD Discount NPV ($M)

TABLE 3.3 Certainty Equivalent (CE)

Yearly Net Cash CV CEF RFD Discount NPV ($M)

1 |
-100 |
0.22 |
0.76 |
0.94 |
- 71 |

2 |
-100 |
0.22 |
0.76 |
0.89 |
- 68 |

3 |
20 |
0.22 |
0.76 |
0.84 |
13 |

4 |
30 |
0.22 |
0.76 |
0.79 |
18 |

5 |
45 |
0.22 |
0.76 |
0.75 |
26 |

6 |
65 |
0.22 |
0.76 |
0.7 |
35 |

7 |
65 |
0.22 |
0.76 |
0.67 |
33 |

8 |
65 |
0.22 |
0.76 |
0.63 |
31 |

9 |
50 |
0.22 |
0.76 |
0.59 |
22 |

10 |
50 |
0.22 |
0.76 |
0.56 |
Total NPV a The RFD is assumed equal to 6%. ## 3.4.5 Monte Carlo SimulationA Monte Carlo simulation entails the iterative calculation of the measure of economic worth from probability functions of the input variables. The results are expressed as a probability density function and as a cumulative distribution function. The technique thereby enables explicit measures of risk exposure to be calculated. One of the economic-evaluation methods treated earlier is used to calculate economic worth; a computer is employed to sample repeatedly—hundreds of times—from the probability distributions and make the calculations. A Monte Carlo simulation can be performed by the following steps: 1. Express variable inputs as probability functions. Where there are interdependencies among input values, multiple probability density functions, tied to one another, may be needed. 2. For each input for which there is a probability function, draw randomly an input value; for each input for which there is only a single value; take that value for calculations. 3. Use the input values to calculate the economic measure of worth and record the results. 4. If inputs are interdependent, such that input X is a function of input Y, first draw the value of Y, then draw randomly from the X values that correspond to the value of Y. 5. Repeat the process many times until the number of results is sufficient to construct a probability density function and a cumulative distribution function. 6. Construct the probability density function and cumulative distribution function for the economic measure of worth, and perform statistical analysis of the variability. The strong advantage of the technique is that it expresses the results in probabilistic terms, thereby providing explicit assessment of risk exposure. A disadvantage is that it does not explicitly treat risk attitude; however, by providing a clear measure of risk exposure, it facilitates the implicit incorporation of risk attitude in the decision. The necessity of expressing inputs in probabilistic terms and the extensive calculations are also often considered disadvantages. ## 3.4.6 Decision AnalysisDecision analysis is a versatile technique that enables both risk exposure and risk attitude to be taken into account in the economic assessment. It diagrams possible choices, costs, benefits, and probabilities for a given decision problem in "decision trees," which are useful in understanding the possible choices and outcomes. $50 M Revenue loss $50 M Revenue loss Denotes chance node (expected value noted inside) Denotes probability // Deleted alternative because inferior choice 3d with a decision to lease or build a facility, the le with an expected cost of $6.26 million based on the above costs and probabilities of outcomes. High O&M costs $3 M(PV) O&M $5 M(PV) O&M Faced with a decision to lease or build a facility, the least-cost choice is to build the facility, /060\ Denotes chance node (expected value noted inside) Denotes probability // Deleted alternative because inferior choice 3d with a decision to lease or build a facility, the le with an expected cost of $6.26 million based on the above costs and probabilities of outcomes. High O&M costs $3 M(PV) O&M $5 M(PV) O&M Faced with a decision to lease or build a facility, the least-cost choice is to build the facility, /060\ FIGURE 3.6 Decision tree: build versus lease. Although it is not possible to capture the richness of this technique in a brief overview, a simple decision tree, shown in Figure 3.6, is discussed to give a sense of how the technique is used. The decision problem is whether to lease or build a facility. The decision must be made now, based on uncertain data. The decision tree helps to structure and analyze the problem. The tree is constructed left to right and analyzed right to left. The tree starts with a box representing a decision juncture or node—in this case, whether to lease or build a facility. The line segments branching from the box represent the two alternative paths: the upper one the lease decision and the lower one the build decision. Each has a cost associated with it that is based on the expected cost to be incurred along the path. In this example, the minimum expected cost of $6.26 M is associated with the option to build a facility. An advantage of this technique is that it helps to understand the problem and to compare alternative solutions. Another advantage is that, in addition to treating risk exposure, it can also accommodate risk attitude by converting benefits and costs to utility values (not addressed here). A disadvantage is that the technique, as typically applied, does not provide an explicit measure of the variability of the outcome. ## 3.4.7 Real Options AnalysisReal options analysis (ROA) is an adaptation of financial options valuation techniques4 to real asset investment decisions. ROA is a method used to analyze decisions in which the decision maker has one or more options regarding the timing or sequencing of an investment. It explicitly assumes that the investment is partially or completely irreversible, that there exists leeway or flexibility about the timing of 4Financial options valuation is credited to Fisher Black and Myron Scholes who demonstrated mathematically that the value of a European call option—an option, but not the obligation, to purchase a financial asset for a given price (i.e., the exercise or strike price) on a particular date (i.e., the expiry date) in the future—depends on the current price of the stock, the volatility of the stock's price, the expiry date, the exercise price, and the risk-free interest rate (see Black and Scholes 1973). the investment, and that it is subject to uncertainty over future payoffs. Real options can involve options (and combinations) to defer, sequence, contract, shut down temporarily, switch uses, abandon, or expand the investment. This is in contrast to the NPV method that implies the decision is a "now or never" choice. The value of an investment with an option is said to equal the value of the investment using the traditional NPV method (that implicitly assumes no flexibility or option) plus the value of the option. The analysis begins by construction of a decision tree with the option decision embedded in it. There are two basic methods to solve for the option value: the risk-adjusted replicating portfolio (RARP) approach and the risk-neutral probability (RNP) approach. The RARP discounts the expected project cash flows at a risk-adjusted discounted rate, whereas the RNP approach discounts certainty-equivalent cash flows at a risk-free rate. In other words, the RARP approach takes the cash flows essentially as-is and adjusts the discount rate per time period to reflect that fact that the risk changes as one moves through the decision tree (e.g., risk declines with time as more information becomes available). In the RNP approach, the cash flows themselves essentially are adjusted for risk and discounted at a risk-free rate. Copeland and Antikarov provide an overall four-step approach for ROA:5 1. Step 1: Compute a base-case traditional NPV (e.g., without flexibility). 2. Step 2: Model the uncertainty using (binominal) event trees (still without flexibility; e.g., without options)—although uncertainty is incorporated, the "expected" value of Step 2 should equal that calculated in Step 1. 3. Step 3: Create a decision tree incorporating decision nodes for options, as well as other (nondecision and nonoption decisions) nodes. 4. Step 4: Conduct a ROA by valuing the payoffs, working backward in time, node by node, using the RARP or RNP approaches to calculate the ROA value of the investment. ## 3.4.8 Sensitivity AnalysisSensitivity analysis is a technique for taking into account uncertainty that does not require estimates of probabilities. It tests the sensitivity of economic performance to alternative values of key factors about which there is uncertainty. Although sensitivity analysis does not provide a single answer in economic terms, it does show decision makers how the economic viability of a renewable energy or efficiency project changes as fuel prices, discount rates, time horizons, and other critical factors vary. Figure 3.7 illustrates the sensitivity of fuel savings realized by a solar energy-heating system to three critical factors: time horizons (zero to 25 years), discount rates (D equals 0%, 5%, 10%, and 15%), and energy escalation rates (E equals 0%, 5%, 10%, and 15%). The present value of savings is based on yearly fuel savings valued initially at $1,000. Note that, other things being equal, the present value of savings increase with time—but less with higher discount rates and more with higher escalation rates. The huge impact of fuel price escalation is most apparent when comparing the top line of the graph (D = 0.10, E=0.15) with the line next to the bottom (D = 0.10, E=0). The present value of savings at the end of 25 years is approximately $50,000 with a fuel escalation rate of 15%, and only about $8,000 with no escalation, other things equal. Whereas the quantity of energy saved is the same, the dollar value varies widely, depending on the escalation rate. This example graphically illustrates a situation frequently encountered in the economic justification of energy efficiency and renewable energy projects: the major savings in energy costs, and thus the bulk of the benefits, accrue in the later years of the project and are highly sensitive to both the assumed rate of fuel-cost escalation and the discount rate. If the two rates are set equal, they will be offsetting as shown by the straight line labeled D=0, E=0 and D = 0.10, E = 0.10. 5See Dixit Avinash and Pindyck (1994), which is considered the "bible" of real options, and Copeland Antikarov (2001), which offers more practical spreadsheet methods. FIGURE 3.7 Sensitivity of present value energy savings to time horizons, discount rates, and energy price escalation rates. Years FIGURE 3.7 Sensitivity of present value energy savings to time horizons, discount rates, and energy price escalation rates. |

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