Risk in Public Sector Project Appraisal: It Mostly Does Not Matter!

Abstract

Public project appraisal using cost-benefit analysis (CBA) requires analysts to project risky net benefits and to convert these into present values using a social discount rate (SDR). We consider which types of risk matter for CBA. For small projects with only idiosyncratic risks, expected net benefits should be discounted at a risk-free SDR. If projects are large or expected net benefits are correlated with aggregate consumption, the alternatives are to replace expected net benefits with their certainty equivalents (CEs) and to discount these at a risk-free SDR, or to discount expected net benefits using a higher SDR that includes a risk premium. These methods are equivalent under special circumstances that are unlikely; the first approach is the correct one. We examine when replacing expected values with CEs will matter, and how this might be done. For most projects, analysts should discount expected values of net benefits at a risk-free SDR.

Keywords

certainty equivalent social discount rate cost-benefit analysis idiosyncratic risk systematic risk

Introduction

Three major factors affect estimation of the present values of the social costs and benefits of public projects that accrue over time. These three factors are (a) the relative valuation of consumption in present versus future time periods (including intergenerational time periods), (b) how to account for the opportunity cost of real resource flows over time that displace private investment as against those that displace private consumption, and (c) the appropriate treatment of risk from a social perspective. Each factor raises important and controversial issues about how the social discount rate (SDR) should be estimated and how the resulting rate differs from those used by private individuals and firms. Moore, Boardman, Greenberg, Vining, and Weimer (2004); Boardman, Moore, and Vining (2010); and Moore, Boardman, and Vining (2013a, 2013b) among others analyze the first two issues in detail. This article focuses on the third factor, risk appraisal.

The appropriate treatment of project risk is an important topic because there is considerable explicit and implicit disagreement in project appraisal generally, and in cost-benefit analysis (CBA) specifically, about how to treat project risk and when and how to undertake adjustments for risk. The treatment of risk is particularly material for large infrastructure projects, whether they are directly funded through government procurement or indirectly through contracting or via hybrid public–private organizations, including public–private partnerships (P3s). Many practitioners think that the most serious risk in project appraisal is optimism bias, which refers to the overestimation of benefits and underestimation of costs, especially capital costs (Flyvbjerg, Bruzelius, & Rothengatter, 2003). Optimism bias is certainly an important issue, but it is not the focus of this article. Here, the assumption for the most part is that analysts derive unbiased estimates of the expected net benefits, and our focus is on the risk that actual net benefits differ from the expected values, that is, the variance.

We argue that the correct way to adjust for risk in project appraisal is to discount certainty equivalents at a risk-free rate. Analysts and governments should not include a risk premium in the SDR. Given the importance of certainty equivalents, we show how to compute them. In practice, the most important issue is the extent of any divergence between certainty equivalents and expected values, that is, the magnitude of the risk premium. Risk premia are significant only for projects that affect a very large share of some individuals’ consumption. For almost all public sector projects, however, the net benefits are less than 10% of an individual’s consumption. So, in most cases, it is not necessary to adjust estimates of the expected net benefits to account for risk.

The rest of the article is organized as follows. The second section summarizes what discount rates government and major agencies currently mandate or recommend. The third section reviews how to estimate the SDR, ignoring risk and uncertainty. The fourth section presents the basic model that frames the subsequent analysis: It treats project risks as different possible outcomes at sequential project stages, using an event tree, and includes an example. It also discusses the four major elements necessary to analyze decision making under risk: expected values, variances (and standard deviations), certainty equivalents and risk premia. The fifth section makes the case that, to incorporate risk into project analysis, analysts should use certainty equivalents of net benefits, rather than expected values, before discounting at a risk-free SDR. The sixth section considers the three most important categories of risk: optimism bias, project-specific risk (also called idiosyncratic or nonsystematic risk), and systematic risk, that is, risk that is correlated with overall consumption. The seventh section explains how to obtain certainty equivalents, which is rarely done in the cost-benefit literature. The eighth section uses the example to show that for projects whose net benefits are a small percentage (certainly less than 10%) of an individual’s consumption, the certainty equivalent values will be essentially the same as the expected values. In such circumstances, it is reasonable to discount the expected values at the risk-free rate. The ninth section discusses the extremely unlikely circumstances under which discounting expected values at a higher risk-adjusted rate gives the same result as discounting certainty equivalents at a risk-free SDR. The final section argues that private sector discount rates, which include significant risk premia, should not be used in public project appraisal.

How Do Analysts and Government Agencies Treat Risk?

Currently, analysts and governments have proposed a number of different ways of dealing with risk. The most common approach is to initially predict a base case (i.e., the most likely) net benefits or expected net benefits and use the base case to compute the net present value (NPV). Then, to account for the risk that the projected net benefits may turn out to be different from the base case, the usual advice is to perform sensitivity analysis (on outcomes) and scenario analysis (on states of the world), preferably using Monte Carlo analysis (see Boardman, Greenberg, Vining, & Weimer, 2011; Campbell & Brown, 2003; de Rus, 2010; Florio, Forte, Pancotti, Sirtori, & Vignetti, 2016; Miller & Szimba, 2015). However, it is almost always unclear as to which of these techniques are necessary or how analysts should use them. Furthermore, there is often little guidance on decision-making procedures subsequent to the sensitivity analysis: For example, how should one choose between one project and another that has a larger expected NPV but higher variance? Some authors suggest using certainty equivalents rather than expected values for net benefits (Campbell & Brown, 2003; Boardman et al., 2011), but there is then little guidance on how to do this.

A second problem with most treatments of risk is that each project is typically analyzed in isolation. Most analyses do not consider the incremental risk to society when a new project is adopted. For government-funded projects, this incremental risk is what matters. This article shows that, for “small” projects, this risk can effectively be ignored.

A third problem is that, in practice, many project appraisal authors recommend mimicking private sector appraisal by including a significant risk premium in the SDR. This adjustment results in either project-specific discount rates (Campbell & Brown, 2003; Dixit & Williamson, 1989; Hultkrantz, Kruger, & Mantalos, 2014; van Ewijk & Tang, 2003) or in a single rate for all projects (Harrison, 2010; Office of Management and Budget [OMB], 1992; Organisation for Economic Co-operation and Development [OECD], 2008). However, this article argues that private sector risk premia almost always include adjustments for project-specific or idiosyncratic risk. Yet, once one accepts that individuals or firms cannot spread idiosyncratic risk as well as the public sector can, then one further implicitly accepts that the SDR is quite different from the myriad private sector discount rates.

Most governments recommend the use of a single SDR for all project appraisals, whether they use a risk-free rate or a risk-adjusted one. Table 1 summarizes the recommended SDRs in a range of countries (European Commission, 2016; Harrison, 2010; OECD, 2004; Zhuang, Liang, Lin, & De Guzman, 2007). The United Kingdom, Germany, and France, for example, all recommend a risk-free rate. In the United States, the Environmental Protection Agency (EPA) also basically recommends a risk-free SDR. However, Canada, Australia, New Zealand, and the OMB in the United States all use much higher rates based on private sector returns to equity that implicitly include significant risk premia. The OECD (2008) also explicitly argues that discount rates should be adjusted as a “way of recognizing . . . risk . . .” The range of values summarized in Table 1 demonstrates that there remains considerable disagreement about how to (or not to) adjust for project risk in the appraisal of infrastructure investments. Perhaps as concerning as the differences in rates shown in the table, many other countries have no clear stated policy on either the appropriate level of the SDR or on how to treat risk (OECD, 2004).

Table 1.

Recommended Social Discount Rates by Country: Risk-Free Versus Risk-Adjusted.

Country or agency	SDR (%)	Comment
Risk-free rates
USA (EPA)	2-3
Germany	3
United Kingdom	3.5	Declining rates over time for long-lived projects
Norway	3.5	May add a risk premium
France	4
Netherlands	4
European Commission	4
Risk-adjusted rates
Australia	8
USA (OMB)	7
Canada	8	Sensitivity analysis usually 7.5%-12%. 3.5% in some circumstances
New Zealand	8
China	8
South Africa	8
World Bank	10-11
India	12
Philippines	15

Note. EPA = Environmental Protection Agency; OMB = Office of Management and Budget.

The disagreements and ambiguities on the treatment of risk have important consequences for all types of public project appraisal. Many countries, for example, now encourage the use of public–private partnerships (P3s) to provide public infrastructure (Hodge & Greve, 2016; Siemiatycki & Friedman, 2012; Snyder & Luby, 2012). Boardman and Hellowell (2016) document, for example, that almost all jurisdictions adjust P3 costs downward because they assume a transfer of risk from the public to the private sector. However, these jurisdictions do not discuss the consequences of this assumption on the level of risk that will be borne by society (see also Morallos & Amekudzi, 2008).

Estimating the SDR While Ignoring Risk

Most policy analysts assume that the goal of government policy should be to maximize social welfare (Snyder & Luby, 2012). Analysts model this goal as the maximization over time of the expected utility of a representative individual (RI) who receives a per capita share of aggregate consumption in each time period. In a certain world, both future aggregate consumption growth and all net benefits of a potential project are known with certainty. If we assume that the RI behaves rationally, future consumption is discounted at the SDR.¹ For commonly used utility functions, maximization of utility occurs if consumption is discounted at r (Ramsey, 1928)²:

r = δ + g \times η .

Ignoring project risk and risk to consumption growth, net benefits that occur with certainty t years in the future should be multiplied by the discount factor ${(1 / (1 + r))}^{t}$ to express them as NPVs. If the NPV (the sum of current and all discounted future net benefits) is positive, then the project would increase social welfare.

The first term in Equation 1, δ, is the pure rate of time preference—the rate at which society discounts future utility. The second term equals the product of economic growth, g, and the elasticity of the marginal utility of consumption, η. An assumption of decreasing marginal utility of consumption implies that any increment to consumption in a wealthier future is worth less than the same increment today. Higher values of η indicate more risk aversion.

Ramsey argued that δ should be set to zero because he saw no moral justification for the current generation to discount the welfare of future generations. However, setting δ to zero implies high rates of saving (that are not observed), so most analysts conclude that it should be set closer to 1 (Arrow, 1995). Most estimates of η are in the range of 1 to 2. If one uses revealed preferences for intra-generational inequality in tax data to estimate η, one gets 1.35 for the United States. Using historical U.S. data to forecast g to be 1.9% annually results in a risk-free SDR of around 3.5% (Boardman et al., 2010; Moore et al., 2004; Moore et al., 2013a, 2013b).³

Measuring and Incorporating Risk Into Project Appraisal

Consider a government infrastructure project, such as a new road or bridge. This project has an initial design and construction phase during which costs are incurred. These costs vary with the prices of various inputs and may be positively correlated with economic growth. Costs may also depend on idiosyncratic (i.e., project-specific or nonsystematic) factors, such as the weather, work interruptions due to labor action or geological conditions encountered. These idiosyncratic events are unlikely to be correlated with the macroeconomic environment.

Once the infrastructure is built, there can still be risky costs for maintenance and operation.⁴ Again, some of these costs may be correlated with the macroeconomy, while others will be idiosyncratic. The project will also produce benefits that are uncertain, such as the volume of reduced travel time and the number of lives saved. These benefits might be positively or negatively correlated with the macroeconomy, depending largely on whether they are normal or inferior goods. The impact of economic growth on net benefits depends on how the benefits and costs in each period are affected. Finally, infrastructure decommissioning usually entails risky net costs, some of which are idiosyncratic while others are systematic.

Initially, analysts should incorporate risk into project appraisal in three steps: (a) Specify the set of possible events that might occur at each project stage, (b) predict and monetize the outcome under each possible event, and (c) attach probabilities to each event/outcome. These three steps are illustrated using a hypothetical infrastructure project with only two periods: a construction period (Period 0) and an operational period (Period 1). (There are no decommissioning costs.)

Suppose that there are three possible events during the initial Period 0 and that the project’s construction costs may take one of three values, depending on which event occurs. In the most likely base case, the costs are 1,000; in the worst case, they are 50% higher at 1,500, and in the best case, costs are 750 or 25% lower than the base case. In the absence of known probability distributions or observed frequency data, analysts should attach their best guesses as to the probability of each outcome (i.e., specify their subjective probabilities). Assume that the probability of the base case cost is .6 and the probability of both the best and worst case costs are .2. Now suppose that there are also three possible outcomes for the net benefits received during Period 1, which depend on demand and operating costs. In the base case (probability = .4), the net benefits are 1,200; in the best case (probability = .3), they are 25% higher at 1,500, and in the worst case (probability = .3), net benefits are 750.

Figure 1 summarizes the possible events and resulting states using an event tree. If the cost and demand events in Period 1 are independent of Period 0 construction costs, then the probabilities of each of the nine states are obtained by multiplying the probability of the events at each stage (e.g., the probability that both costs in Period 0 and the cost and demand conditions in Period 1 are at their most likely values is: 0.6 × 0.4 = 0.24, that is, 24%). The terminal nodes of the tree show the NPVs for each of the nine possible outcomes using a 3.5% discount rate. In reality, of course, projects last longer than two periods. Quiggin (2005) provides practical guidance on extending event trees over multiple periods.

Figure 1.

Event tree for a hypothetical project with only idiosyncratic risk.

The expected net benefit in period t is the probability-weighted average of the net benefits across all the states in that period:

E (N B_{t}) = \sum_{i = 1}^{i = S} p_{i t} N B_{i t},

where E( ) is the expected value; NB_it is the net benefit predicted in period t under state i; p_i is the probability that state i will occur in period t; and the sum of the p_i over all (S) states is equal to one. (Subsequently, the subscript t is usually suppressed for simplicity.) In our example, E(NB₀) = −1,050, E(NB₁) = 1,155, and the E(NPV) = 66.

Even if the expected values are estimated correctly, the project is risky because it is impossible to know with certainty which event will actually occur in any period. Consequently, the actual net benefits may be higher or lower than the expected value. (Total) economic risk in any period is measured by:

V a r (N B) = \sum_{i = 1}^{i = S} p_{i} {(N B_{i} - E (N B))}^{2},

where Var( ), the variance of the net benefits, measures the variation around the expected value. Greater variance means greater risk. Risk may change from one period to the next, as the probabilities and net benefits of the outcomes can vary across time. To express risk in the same units as expected values, the standard deviation (SD), the square root of the variance, is often used. In our example, SD(NB₀) = 245 and SD(NB₁) = 293. To compare riskiness across periods, the standard deviation is divided by the (absolute value of) the expected value to find the coefficients of variation (CV). In our example, CV(NB₀) = 0.23 and CV(NB₁) = 0.25, which implies that the risk in Period 1 is slightly larger than the Period 0 risk.

Evaluating Projects While Incorporating Risk

As explained above, analysts model the maximization of social welfare as the maximization over time of the expected utility (or well-being) of the RI, who we assume is risk averse. Suppose that the individual will receive an unknown risky net benefit of NB, and that she is indifferent between receiving this and receiving a net benefit of CE with certainty. CE is the certainty equivalent of the risky net benefit. Generally, individuals are risk averse. Risk aversion implies that CE < E(NB). The difference between the expected value and the certainty equivalent is the risk premium, RP:

R P = E (N B) - C E .

As risk (variance in net benefits) increases, the CE decreases and the RP increases.

Expected utility in a period equals the probability-weighted average of the utility received from the consumption in that period across all possible states:

E (U) = \sum_{i = 1}^{i = S} p_{i} U (c_{i}),

where U( ) is the RI’s utility function and c_i is the RI’s per capita share of consumption, including her share of any net benefits from a public project. Suppose a project costs the RI one dollar of foregone consumption today (Period 0) and yields the RI a risky net benefit of NB next period (Period 1). This investment improves social welfare if the discounted value of the RI’s current and expected future utility increases. Suppose that in exchange for this risky net benefit in Period 1, the RI would be willing to accept a certainty equivalent, CE, which yields the same increase in future expected utility without any risk. Social welfare improves if the sacrifice of one dollar now in exchange for CE in the future increases discounted utility, that is, if the discounted value of the CE exceeds one dollar. In general, projects will increase social welfare if the sum of current and all discounted future CEs is positive. Consequently, the correct way to incorporate project risk is to discount the certainty equivalents of risky net benefits at the risk-free SDR (Boardman et al., 2011; Gollier, 2012; Little & Mirrlees, 1974; Quiggin, 2005).

Types of Project Risk: Optimism Bias, Idiosyncratic and Systematic Risk

The analysis so far assumes that estimates of p_i and NB_i, and therefore of E(NB), are unbiased. However, analysts may suffer from various types of biases. They might intentionally overestimate expected net benefits for political or strategic reasons (Flyvbjerg, Holm, & Buhl, 2002; Milligan & Smart, 2005). Or, they might unconsciously err on the positive side and overestimate expected net benefits (Flyvbjerg et al., 2003). Unconscious optimism bias is less likely if analysts use the event tree approach because it requires analysts to explicitly specify the different potential outcomes, and to think carefully about the net benefits under each outcome and their probabilities.

The most common alternative approach is simply to pick a most likely estimate of the net benefit in each period. Note that in the infrastructure example, the most likely Period 0 costs are 1,000, but the expected cost is 1,050, and the most likely Period 1 net benefits are 1,200, while the expected net benefits are only 1,155. Discounting these most likely values results in a base case NPV of 159, whereas the expected NPV (the probability-weighted average of the nine possible outcomes) is only 66.⁵ If optimism bias remains despite careful analysis, analysts may reduce their expected values by the observed average difference between expected and actual outcomes in similar past project appraisals.

Even if optimism bias is eliminated, the project’s actual net benefits in any period may differ from the expected values. (Total) economic risks can be divided into those that are specific to the project (idiosyncratic or nonsystematic risks) and those that are correlated with risks to the overall economy (systematic risks). Only systematic risks matter for small projects. To understand why, consider the effect of a single government project on the RI. As discussed above, what matters for social welfare is the effect on the RI’s consumption: both the level of consumption and the riskiness of the overall consumption stream. Suppose that, ignoring the project, in any year the RI expects to have a risky amount of consumption equal to c. In aggregate, this is nonproject consumption. Furthermore, suppose that she receives a fraction λ of the net benefits of the project. The extra risk that the project creates in a period is given by the change in the variance of her consumption in that period:

V a r (c + λ N B) - V a r (c) = λ^{2} V a r (N B) + 2 λ C o v (c, N B),

where Cov( ) represents the covariance between the two risky amounts.⁶

If λ is small because the project’s net benefits are spread broadly across members of society, then the first term in Equation 6 can effectively be ignored. If, in addition, the risks are purely idiosyncratic (Cov(c,NB) = 0), then the second term equals zero. In these circumstances, the project does not significantly affect the variance in consumption and the expected net benefit approximately equals its certainty equivalent. Thus, for purely idiosyncratic risks, the change in aggregate risk is insignificant provided the benefits and costs are spread broadly so that λ is small (Arrow & Lind, 1970). How small is small enough? Given that a λ of 10% (0.1) would imply that project variance is multiplied by one one-hundredth, almost all projects would meet this small enough criterion. Almost no plausible public projects would concentrate more than 10% of the net benefits on a single individual.

The Arrow and Lind result is based on the assumption that λ goes to zero as the population of the relevant jurisdiction increases. This assumption clearly applies to rivalrous goods. However, it is not clear that it applies to public goods or, more generally, to nonrivalrous goods, such as the reduction of a negative externality like pollution (Fisher, 1973; Foldes & Rees, 1977). This concern about its applicability would be valid if one considers each public project in isolation, as many analyses seem to do. But most governments hold a large portfolio of investments that provide a mix of both rivalrous and nonrivalrous goods, and citizens have many private sources of consumption. From the social perspective, the effect of an additional project depends on its overall effects on the RI’s consumption. A straightforward application of portfolio diversification theory shows that idiosyncratic risks will be almost completely eliminated if the total portfolio of government spending (and a fortiori, overall consumption sources) is spread across many investment projects (Markowitz, 1952). So, Equation 6 can be interpreted as measuring the effect on overall risk of adding one more risky asset to an already risky market portfolio, where the additional asset is the government project under consideration, the market portfolio consists of all the assets that produce per capita consumption, and λ is the weight of the project’s net benefits in the RI’s consumption. Analogously to the risk spreading argument above, if the project’s net benefits accruing to the RI represent less than 10% of her per capita consumption, then the first term in Equation 6 can be ignored. So, whether a small project provides rivalrous or nonrivalrous output, if there are only idiosyncratic risks, then analysts can ignore project risk and proceed to discount the expected values of the net benefits without adjustments. If all the risks in our hypothetical project are uncorrelated with per capita consumption, then it should proceed because it has a positive expected NPV (of 66) when the expected values of net benefits are discounted at the risk-free rate of 3.5%.

However, even for a small project, Equation 6 shows that the RI’s risks will increase if the project’s net benefits are correlated with aggregate consumption. In this case, there is systematic or nondiversifiable (and unspreadable) risk. The correlation could be either positive or negative. Note that it is the correlation between the net benefits and consumption within each period that matters. For a project producing a normal good, it may well be the case that both benefits and operating and maintenance costs are positively correlated with consumption (as input costs may rise during boom periods). Then, it is not obvious that the net benefits will exhibit systematic risk, as the effect on benefits may be offset by the effect on costs.

If the covariance of net benefits and consumption is positive, then the project adds to overall consumption risk. In turn, this reduces the certainty equivalent of the risky net benefits, and so CE < E(NB). If the covariance is negative, then the project reduces overall consumption risk, which increases the certainty equivalent of the risky net benefits and thus CE > E(NB). In the latter situation, using expected values instead of certainty equivalents gives a lower bound to the project’s NPV (Howarth, 2003).

In those few cases where a project is large and a significant part of the net benefits go to some individuals (i.e., λ is a very large fraction of a very large NB), then the project would increase the risk in aggregate consumption, mainly per the first term in Equation 6.⁷ However, a project would have to provide over 30% of some individual’s consumption for this term to be multiplied by a fraction greater than 0.1. This might be the case where the government project would provide electricity, water, critical transportation, or health care to a particular segment of the population and would represent an extremely large share of the consumption of the members of this segment. In that case, if the project fails to deliver benefits, then the welfare of the affected individuals may be seriously reduced. Examples in the United States might include the Tennessee Valley Authority, the interstate highway system, or the Affordable Health Care Act. In such situations, analysts should calculate certainty equivalents for the net benefits to account for the direct project risk, which is discussed in the next section.

Calculating Certainty Equivalents

Certainty equivalents are calculated with respect to aggregate consumption, that is, consumption if there were no project, c, plus the net benefits of the project, NB, if applicable. Building on our hypothetical example, suppose that in the Period 0 the nonproject base case consumption (i.e., consumption if there were no project) would be 100,000. Given the base case project cost of 1,000, the base case aggregate consumption in Period 0 would equal 99,000. Now we introduce systematic risk. Suppose that there is a 20% chance of faster growth of the economy and consumption and, if so, project costs would be higher, due to more rapidly increasing labor and raw material costs. Assume, for example, that with faster economic growth, consumption (excluding the project) will rise to 103,000 but project costs will also rise to 1,500, yielding aggregate consumption of 101,500. Further suppose that there is a 20% chance that growth will decline and, in this case, consumption will be 97,000, project costs will be 750, yielding aggregate consumption of 96,250. Note that this assumes the same probabilities as before. Under these assumptions, the Period 0 net benefits of the project (which are negative) are negatively correlated with nonproject consumption (the correlation coefficient is −0.97). The standard deviation of nonproject consumption alone is 1,897 and the coefficient of variation is 0.019.⁸ Figure 2 shows the three nonproject consumption levels along with the associated project costs.

Figure 2.

Event tree for a hypothetical project with systematic risk.

In Period 1, we again introduce systematic risk. Suppose in the most likely case the economy and nonproject consumption grow at 2.5% per year and project net benefits are 1,200. In the best case, nonproject consumption grows faster, at 5.5% per year and net benefits grow to 1,500 due to greater demand (the project produces a normal good). In the worst case, nonproject consumption falls by 0.5% and net benefits are only 750. Figure 2 shows all nine possible Period 1 nonproject consumption levels and their associated project net benefits. Note that the Period 1 net benefits are positively correlated with nonproject consumption (the correlation coefficient is 0.76.) Again assuming independent probabilities for simplicity, Period 1 average consumption is 102,500, with a standard deviation of 3,031 and a coefficient of variation of 0.03.

The analyst can now proceed to calculate the certainty equivalents of the net benefits for each period. First, assume a utility function for the RI. We use the standard iso-elastic function:

\begin{array}{l} u (c) = k \frac{c^{1 - n}}{1 - η}, η = 1, \\ u (c) = k \ln (c), η = 1 \end{array}

where u is utility, c is per capita consumption, η is the (constant) elasticity of the marginal utility of consumption, ln( ) is the natural logarithm function and k is a constant. Second, use Equation 7 to calculate for each event in each period the utility of aggregate consumption with the project (i.e., with project net benefits included in consumption) and consumption without the project. Third, using the probabilities for the various events/outcomes, calculate the expected utility across all events, both for the case with the project net benefits included in consumption, and for the case without the project. Finally, convert the expected utility values back into monetary values (the certainty equivalents) using the function in Equation 7. The certainty equivalent of the net benefits in each period is equal to the difference between the certainty equivalent of (risky) consumption without the project’s net benefits in that period, and the certainty equivalent of (risky) consumption including the project’s net benefits.

To illustrate we set η = 1.5 and calculate the CE of the net benefits (actually costs) in Period 0. These costs may be 750, 1,000, or 1,500, yielding aggregate consumption of 96,250, 99,000, or 101,500 with probabilities 0.2, 0.6, and 0.2, respectively. With η = 1.5 (and setting k = 1), u(c) = −2/√c.⁹ Therefore, the utilities of the three outcomes are: −0.00627765, −0.00635642, and −0.00644658, respectively, with EU = −0.006359. Substituting this expected utility value into Equation 7 yields the CE of aggregate consumption in Period 0 = 98,929. Similarly, the CE of aggregate consumption in Period 0 if there were no project = 99,972. Therefore, the CE of the net benefits of the project in Period 0 equals −1,044. By following a similar process, the CE of the net benefits of the project in Period 1 equals 1,145.

Risk Premia for Small Projects

Calculating CEs is a time-consuming process. As discussed above, it may be necessary to calculate them for projects where the net benefits constitute a large portion of some individuals’ consumption. However, it is unnecessary for small projects with only nonsystematic risk. We now demonstrate that, even in the presence of systematic risk, the risk premia are so small that it is unnecessary to calculate CEs for small projects.

For each period, we calculate the RP for the net benefits and express it as a percentage of the absolute value of the EV of the corresponding net benefits. Thus, it is a relative amount. Because this RP reflects both idiosyncratic risk and systematic risk, we need to remove the part associated with idiosyncratic risk. To do this, we subtract the relative risk premium obtained by repeating the procedure with no variance in consumption.

Table 2 shows the resulting relative risk premia for each period for η = 1, 1.5, and 2, and for both small projects (i.e., with high levels of nonproject certain consumption: 100,000 in Period 0 and of 102,500 in Period 1, implying the project accounts for about 1% of consumption) and for large projects (i.e., with low levels of nonproject certain consumption: 10,000 in Period 0 and 10,250 in Period 1 implying that the project accounts for about 10% of consumption). The relative systematic risk premia are negative in Period 0, as the net benefits are negatively correlated with consumption. In this case, the project has an insurance value in that period (net benefits are less negative when consumption is lower). Even though we allow for high correlations between net benefits and consumption, and even though we assume very large variations in project net benefits, it is only for the highest value of η that the relative systematic risk premia rise above 1%. Generally, therefore, these risk adjustments are so small that they can be ignored in almost all cases.

Table 2.

Relative Systematic Risk Premia for a Hypothetical Project.^a.

Project size	Period	Elasticity of marginal utility of consumption
Project size	Period	η = 1	η = 1.5	η = 2
Small	0	−0.41 (−0.43)	−0.62 (−0.64)	−0.82 (−0.86)
Small	1	0.52 (0.57)	0.78 (0.86)	1.05 (1.14)
Large	0	−0.46 (−0.43)	−0.70 (−0.64)	−0.93 (−0.86)
Large	1	0.48 (0.57)	0.72 (0.86)	0.96 (1.14)

Note. Period 0: Coefficient of variation for net benefits = −0.233. CV for consumption = 0.019. Correlation coefficient between Period 0 costs and consumption = −0.968. Period 1: CV(NB) = 0.253. CV(c) = 0.0296. Correlation coefficient between Period 1 benefits and consumption = 0.761.

Risk premia are reported as percentages of the expected value of the net benefit in each period.

Approximations (η × CV(NB) × CV(c) × ρ) appear underneath in parentheses.

The method described above for computing the systematic risk premia as a percentage of expected net benefits is not straightforward. However, for small projects, it is approximately equal to

\frac{R P}{E (N B)} \approx η \times ρ \times \frac{S D (N B)}{E (N B)} \times \frac{S D (c)}{E (c)},

where $ρ = C o v (N B, c) / S D (N B) S D (c)$ measures the correlation between the project’s net benefits and consumption; η is discussed above; and the last two terms are the coefficients of variation for the project’s net benefits and for consumption (Quiggin, 2005). For the United States, Moore et al. (2013b) estimated η = 1.35. Using 1947-2009 data on per capita consumption (Shiller, 2005), we estimate the last term in Equation 8 as 0.018. Under the worst case, ρ = 1. If, in addition, net benefits are log-normally distributed, then even if the rate of return of the project had a standard deviation as large as 40%, this would still require less than a 1% adjustment for systematic risk to the expected net benefit.¹⁰ In practice, ρ will usually be less than one. Indeed, it is unlikely that many projects have standard deviations of returns as large as 40%, but, even if so, the adjustments would be negligible.¹¹ Table 2 shows these approximations for the relative systematic risk premia in parentheses under our calculated values. These approximations are remarkably close to our values when the project is small (accounts for 1% of consumption), and are still close even when it accounts for 10%.

Using a Risk-Adjusted SDR Based on the CAPM: Not!

Many economists propose that the public sector should appraise projects in the same way as the private sector, using the capital asset pricing model (CAPM) (Dixit & Williamson, 1989; Hultkrantz et al., 2014; van Ewijk & Tang, 2003). The CAPM proposes that to compensate investors for making risky investments, one should discount the expected net benefits (equal to the cash flows of investments for a private project) at a rate that includes a premium for risk. As previously discussed, the risk that any investment adds to the investor’s portfolio depends only on the systematic risk. In a financial context, systematic risk is typically measured by the investment’s beta, defined as

β_{j} = \frac{C o v (R_{j}, R_{m})}{V a r (R_{m})},

where R_j denotes returns to a particular investment j and R_m denotes the returns to the overall market for equities. β_j can be interpreted as the ratio of the volatility of a given investment (due to stock market risk) to the volatility of the stock market as a whole.

For a public project, the CAPM proposes discounting the expected values of the net benefits by a SDR that equals the risk-free rate, plus a premium for systematic project risk. Gollier (2012) shows that, provided the assumptions underlying the CAPM are valid, then this method is equivalent to discounting CEs at the risk-free rate if the project-specific discount rate risk premium, denoted π(β), equals

π (β) = η \times β \times V a r (g),

where η is defined above, β equals the covariance of the project’s rate of return and the growth rate of consumption divided by Var(g), where Var(g) is the variance of the growth rate g of per capita consumption. However, this adjustment is likely to be very small. Using our previous estimates for η and for the variance of the growth of U.S. per capita consumption then, if the project has a β of 1, this implies π(β) equals about 4.4 basis points (0.044%). Even if β = 1.72, which Kocherlakota (1996) estimates would represent the systematic risk of holding the S&P 500 Index, π(β) = 7.6 basis points. These are inconsequential adjustments to a risk-free discount rate of 3.5%.

More fundamentally, though, the CAPM is not appropriate for public sector projects. It was originally formulated to model a single period of investment followed by a one-period return. To extend CAPM to multiple periods, which characterize most government projects, one must assume that the project returns and consumption growth are jointly (and normally) distributed as a random walk. This implies that the variance of the net benefits increases linearly with time (Gollier, 2012). Using our recommended approach, CEs are worth less than the expected value of net benefits (if net benefits are positively correlated with consumption.) So the CAPM only produces approximately equivalent results if the systematic riskiness of the project net benefits increases in proportion to time (since the discount factors will decrease in proportion to time). But actually risks often can and do vary over time, and may affect costs and benefits (and so net benefits) differently at different points over a project’s life.

In particular, raising the discount rate will not change the valuation of the net benefits in the initial investment period. Analysts often take “project risk” to mean the risk of initial construction cost overruns, but the valuation of these will not be affected much, if at all, by raising the discount rate. And if there are large terminal costs (for decommissioning, for example), then raising the discount rate will reduce the NPV of these costs, making the project appear more, not less, attractive.

Thus, even if one is willing to assume that project returns and consumption growth are jointly distributed as a random walk, then for the CAPM approach to be valid, the timing of all projects’ costs and benefits must be identical and must be such that systematic risk increases linearly with time in exact proportion to the decrease in the discount factors. The only plausible case meeting these conditions would be a project that consists of a single investment period, followed by many periods of net benefits, where those net benefits are systematically risky and the risk grows smoothly over time. This might occur in a project where all the systematic risk is to demand, and the risk of the demand forecasts grows steadily with the forecast period. For almost any other pattern of net benefits and any other sources of systematic risk, the use of the CAPM approach will not be equivalent to our recommended approach of discounting CEs at the risk-free rate. Instead, upward adjustment of the SDR would favor projects with short-lived benefits and long-lived costs.

Furthermore, adding the same risk premium for all projects is clearly incorrect. Even if all the assumptions of the CAPM do hold, it would still not justify the use of single risk-adjusted SDR for all project appraisals.¹² The covariance of the project returns and the growth of consumption will vary across projects. If a project produces an inferior good, then the covariance of net benefits and consumption, and so the project’s β and the risk premium, will all be negative, implying that the project’s net benefits should be discounted at a rate below the risk-free rate.

Using a Risk-Adjusted SDR Based on Private Sector Rates

Many economists continue to advocate the use of a high SDR based on private sector discount rates and returns that include risk premia. Some authors do so because they believe that the use of a risk-free SDR underestimates the true financing costs for the public sector (Boyer, Gravel, & Mokbel, 2013; Klein, 1997; Lucas, 2014). They argue that taxpayers implicitly provide governments with a guarantee to provide additional funds should they be necessary. If this risk to taxpayers were included, then government borrowing rates would not be any lower than those of the private sector. In other words, the SDR should be set above the government’s seemingly risk-free borrowing rate so as to include a premium for the implicit default risk.

However, there is no reason to believe that financing costs for the private sector exceed observed public sector borrowing rates only because of unpriced public default risk. As Jenkinson (2003) points out, the higher costs of private financing of government projects may well reflect the reality of unavoidable contractual incompleteness and greater transactions costs. Given the reality of imperfect capital markets, governments may enjoy advantages in spreading diversifiable risks and creating liquid assets that are simply not available to private sector actors (Arrow, 1995). Furthermore, we note that the average yield difference between government and highly rated private debt for the last 40 years is only 80 basis points, which is much smaller than the risk premium that advocates of risk-adjusted discount rates seem to promote.¹³

Discounting CEs at the risk-free SDR is the correct way to handle systematic risk under all circumstances. However, one might argue that analysts are unlikely to calculate CEs, so that it is better to use the flawed procedure of discounting expected values at a risk-adjusted rate, rather than simply ignoring risk in project appraisal. But even if one were to add a risk premium to the risk-free rate as per Equation 10, the adjustment would be negligible, as shown above.

The negligible effect of the potential adjustments follows from estimates of the extent of individual risk aversion, and from the observed volatility of average consumption growth. These observations also drive the “equity premium puzzle” (Kocherlakota, 1996; Mehra & Prescott, 1985). That is, why is the premium that investors require to hold risky equities in the range of 4% to 6%, rather than less than 1%?¹⁴

One explanation for the equity risk premium is that investors are actually much more risk averse than a value for η of between 1 and 2 would indicate.¹⁵ Alternatively, investors may perceive much more risk to the growth rate of consumption than is evident from the historical data. Using Equation 10, if we set the risk premium π(β) in the 4% to 6% range, then using our estimates of β = 1.72 and η = 1.35 implies a standard deviation of annual consumption growth of between 13% and 16%, as compared with observed values of less than 3%. With a normal distribution, 95% of the growth rates should fall between plus or minus 1.96 times the standard deviation, and with any distribution, Chebyshev’s inequality implies that 75% of the growth rates should be within plus or minus two times the standard deviation. Taking 15% as an estimate of the standard deviation implied by the observed equity premium, this means that equity investors believe that there is somewhere between a 5% and a 25% chance that the growth rate of per capita consumption can fluctuate by more than 30% around an average value of about 2%! This seems highly unlikely.

While the equity risk premium puzzle is still unresolved, one possibility is that individuals are not able to diversify away all idiosyncratic risk, as the CAPM assumes.¹⁶ Another is that many private sector firms may be trying to avoid idiosyncratic as well as systematic risk in their investment decisions. This could be the case if owners or managers cannot easily diversify their wealth. Most publicly traded firms are managerially controlled, and managers likely face firm-specific risks since their human wealth (and often their financial wealth) is highly concentrated in one firm. If so, there are good reasons why the private sector requires higher returns on investment (and uses higher discount rates) than the CAPM model implies they should.

Grant and Quiggin (2003) argue that, because the tax system can spread risks that the private sector cannot, there is no valid reason to lean on observed private sector prices for risk when evaluating public sector projects. We are in a second-best world, where it is appropriate for the government to derive shadow prices for risk and time based on maximization of social welfare (Snyder & Luby, 2012). The use of risk-adjusted SDRs based on observed private sector risk premia is not the correct approach for public project appraisal.¹⁷

Conclusion

In project appraisal, it is important to explicitly model risk by projecting outcomes (net benefits) under various possible events or scenarios and by assigning probabilities in order to calculate expected values of net benefits in as unbiased a manner as possible. We demonstrate the use of event trees. Further adjustments to expected values may be made to account for any lingering optimism bias.

To account for risk in project appraisal and CBA, the expected values of net benefits (derived from the event tree) should be replaced by their certainty equivalents, and these should be discounted at a risk-free SDR. The resultant NPV provides a clear indicator of the change in social welfare. For small projects, when the project outcomes constitute a small part (certainly less than 10%) of individual consumption, then whether risks are purely idiosyncratic or whether they are systematic, the difference between the certainty equivalents and expected values will be negligible and may be ignored. Analysts may simply discount expected net benefits at the risk-free rate.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

Author Biographies

Mark A. Moore is a Senior Lecturer in the Beedie School of Business at Simon Fraser University. His research interests include cost-benefit analysis, social discounting, privatization, and public policy analysis.

Anthony E. Boardman is the Van Dusen Professor of Business Administration in the Sauder School of Business at the University of British Columbia. His research interests include cost-benefit analysis, privatization, and public–private partnerships.

Aidan R. Vining is the CNABS Professor of Business and Government Relations in the Beedie School of Business at Simon Fraser University. His research interests include policy analysis, cost-benefit analysis, privatization, and public–private partnerships.

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