report

THE ENGINEERING ECONOMIST , VOL. , NO. , – https://doi.org/./X..

Adjustments within discount rates to cater for uncertainty—Guidelines

David G. Carmichael

School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia

ABSTRACT Deterministic discounted cash flow (DCF) analysis is a well-accepted technique in engineering appraisals. Common practice is to incorpo- rate all uncertainty influences within a single variable—namely, the discount rate—which also represents the time value of money. Com- mentary already exists in the literature that such a practice is expedi- ent but not rational and has shortcomings. This article examines the error involved in this practice and provides guidelines and precautions for using blanket or constant discount rates in dealing with uncertain- ties. It shows the adjustments necessary for any given investment sce- nario. This is done through establishing equivalence of the expected utility of deterministic and probabilistic present worth, allowing a rate adjustment to be calculated. Numerical studies look at the relationship or trends of this rate adjustment to the key analysis variables. Gener- ally, it is found that the rate adjustment should be decreased as the timing of a cash flow’s occurrence increases, increased as the variance of the cash flow increases, kept almost as an additive constant as the base rate increases, and increased as the investor’s level of risk aver- sion increases. The article provides practitioner-friendly usable guide- lines for adjusting rates, something that is unavailable elsewhere in the literature.

Introduction

Deterministic discounted cash flow (DCF) analysis is a well-accepted technique in engineer- ing appraisals, despite the general acknowledgement that uncertainty exists in the analysis variables, namely, cash flows, cash flow timing, and interest rate. (The term “uncertainty” here is used in the sense of implying probability, likelihood, or frequency of occurrence, in contrast to determinism; Carmichael 2016b; Carmichael and Balatbat 2008.) Common practice is to incorporate all uncertainty influences within a single parameter—namely, a discount rate— which also represents the time value of money (Robichek and Myers 1966). And this might be supplemented with a sensitivity-style analysis. This practice is expedient, but incorporating both uncertainty and money time value within the one variable is not without its troubles; in particular, a consistent rate encompassing both matters may not be able to be found and users are confronted with the difficulty of determining an appropriate discount rate to use (Espinoza 2014; Zinn et al. 1977). The practice relocates the analysis within the comfort zone

CONTACT David G. Carmichael [email protected] School of Civil and Environmental Engineering, The Uni- versity of New South Wales, Sydney, NSW , Australia.

©  Institute of Industrial & Systems Engineers

THE ENGINEERING ECONOMIST 323

of determinism and away from a more mathematical but also more realistic probabilistic DCF analysis. Users forego accuracy in their investment models in return for ease of use. Robichek and Myers (1966), Fama (1996), Halliwell (2001), Cheremushkin (2009), and others express concern that a single variable cannot capture both the effect of uncertainty and the time value of money.

Despite the well-known shortcomings of discount rates, (deterministic) discounted cash flow analysis is increasingly being adopted (Block 2005). According to a recent survey, the (deterministic) discounted cash flow approach is a dominant methodology used (KPMG 2013). A potential explanation for this is the simplicity of the calculations, though they may not be intuitive. In addition, the inertia of using discount rates, the comfort of not doing differently to others, together with human nature’s resistance to change, may explain its pop- ularity. However, these and other surveys do not comment in any depth on the implications of the incorrect modeling of uncertainty within conventional (deterministic) discounted flow analysis.

The literature gives various ways by which discount rates might be established, such that multiple influences (and not just uncertainty) are accommodated. This article only addresses and quantifies the influence of uncertainty in the choice of discount rate and only examines discount rates in the context of assets and infrastructure; that is, investments where no compa- rable markets exist. It examines the error involved in the practice of incorporating uncertainty within a blanket or constant discount rate and provides guidelines and precautions for using blanket discount rates in dealing with uncertainties. This is done through establishing equiv- alence of the expected utility of deterministic and probabilistic present worth, allowing a rate adjustment to be calculated. Numerical studies look at the relationship of this rate adjustment to the key analysis variables—namely, the timing of the investment’s cash flows, the variances of the cash flows, the base rate assumed, and the risk attitude of the investor.

The approach of this article is distinguished from other commentaries on rates and that also use utility, particularly in the area of certainty equivalence. Such commentaries include Robichek and Myers (1966), Berry and Dyson (1980), Dyson and Berry (1983), Baker and Fox (2003), and Cheremushkin (2009). In contrast with this article, Robichek and Myers (1966) work with individual certainty equivalent cash flows for cash flows occurring in the future but do not inform on base rate effects, cash flow variance effects, multiple cash flows and levels of risk aversion, and use the risk-free rate rather than a general base rate. Berry and Dyson (1980) and Dyson and Berry (1983) do similarly but do now allow cash outflows and investment markets, whereas Baker and Fox (2003) and Cheremushkin (2009) model uncertain cash flows as time series, which do not apply to non-market-oriented investments. This article explicitly uses utility functions on the present worth rather than on cash flows, because it is the collective present worth of all cash flows (of any sign) that is of paramount concern to the investor, rather than individually occurring cash flows at different points in time with differing uncertainty and magnitude; this article also goes directly to the rate adjustment rather than having to infer the adjustment, and because no markets are involved, it makes no distinction among sources of uncertainty.

The article is structured in the following way. The article first reviews suggestions for how discount rates might be established. Demonstration results are given whereby a rate adjust- ment is established for a range of values of the underlying analysis variables of cash flow vari- ance, cash flow timing, base rate, and risk attitude. This is followed by summarized guidelines. The method behind the equivalence calculations is provided in the Analysis and Numerical Studies sections.

324 D. G. CARMICHAEL

The article’s results will be of interest to anyone who uses discounted cash flow analysis. The article provides practitioner-friendly usable guidelines for adjusting rates, something that is unavailable elsewhere in the literature.

Background

The literature gives various ways by which discount rates might be established. This arti- cle only examines discount rates in the context of assets and infrastructure—that is, invest- ments where no comparable markets exist—and looks at uncertainty in isolation from other influences.

Sensitivity-style analyses, scenario testing, and Monte Carlo simulation can assist in under- standing the influence of uncertainty but do not say anything directly about the choice of an appropriate discount rate. Uncertainty could be double counted in these approaches if the discount rate used already has been adjusted for uncertainty.

Rate adjustment

Common deterministic DCF analysis incorporates uncertainty using risk-adjusted discount rates. Rates are adjusted to take into account uncertainty, among other things. A premium or loading is added to a base rate to give a discount rate. The intent is that the premium should compensate the investor for investment uncertainty. Generally, premiums and dis- count rates are increased in line with greater perceived uncertainty. This leads to lower cal- culated present worths (or net present value), lower calculated profitability of the investment, and lower weight being given to cash flows further into the future.

A risk-adjusted discount rate oversimplifies an investment model. Among other things, it does not represent uncertainty well throughout the duration of an investment. It would be possible to change the rate over time or to use a different rate for each cash flow, for example, but this appears to be rarely done (Fama 1977). Such an adjusted rate approach disconnects uncertainties from their actual sources and implicitly assumes that uncertainty and time are interchangeable. This can lead, among other things, to cash flows being wrongly valued and weakens an investor’s ability to connect an investment’s return with its source of uncertainty. Some attempt in the literature has been made to separate uncertainty and the time value of money; for example, through adjusting the cash flows rather than the base rate (Baker and Fox 2003; Berry and Dyson 1980; Cheremushkin 2009; Dyson and Berry 1983; Robichek and Myers 1966).

Halliwell (2001) comments that the current use of adjustments for uncertainty to calculate discount rates is subjective and inconsistent, whereas Baker and Fox (2003) comment that the current selection of the magnitude of the rate adjustment appears somewhat arbitrary and varies among investors. In the survey of Block (2005, p. 62), when adjusting for uncertainty, some firms considered “risk to be a concept that cannot be appropriately quantified and sim- ply use a subjective approach.” Uncertainty and the time value of money are separate issues and hence there can be no unique adjustment (Robichek and Myers 1966), with investors consequently adopting different, and inconsistent, practices.

When dealing simultaneously with both positive and negative cash flows, the usual notion of using higher discount rates for cash flows with higher uncertainty may lead to anomalous practices. Some investors discount negative cash flows at a lower rate than the risk-adjusted rate and positive cash flows at the risk-adjusted rate, but this is an inconsistent treatment of

THE ENGINEERING ECONOMIST 325

uncertainty. The choice of discount rate for negative cash flows should reflect their uncertainty and not be because they are negative (Ariel 1998; Cheremushkin 2009). The uncertainty fun- damentally lies in the cash flows, not the discount rate.

Other methods of establishing discount rates

There exist methods for establishing discount rates, peculiar to investments where com- parable markets exist, rather than being applicable to asset and infrastructure investments other than by analogy. Some do not incorporate uncertainty effects. Most of these meth- ods relying on markets do not transfer well to non-market-oriented investments, because of issues with finding comparable proxies and separating diversifiable from nondiversifi- able uncertainties (Espinoza 2014). However, having said that, many investors are com- fortable using analogies between the two. There also exist methods for establishing dis- count rates, either directly or indirectly, incorporating multiple influences beyond uncer- tainty or not addressing uncertainty and hence not directly applicable to this article. Some of these other methods, commonly mentioned in books—for example Damodaran (2001, 2007a, 2007b), Bodie (2011), Brealey et al. (2011), and Ross et al. (2013)—as well as other citations in this article include implied discount rates; capital asset pricing model (Bren- nan 1997; Fama 1996; Fama and French 2004; Lintner 1965; Sharpe 1964); opportunity cost of capital; cost of debt/funds; cost of equity, dividend growth model; weighted average cost of capital (WACC; Block 2011); social time preference (National Oceanic and Atmo- spherica Administration 2014); past projects or real investments; post valuation adjust- ment; illiquidity discount; rates related to the investment time horizon (Gollier 2002); syn- thetic insurances (Espinoza 2014; Espinoza and Morris 2013); and the certainty equivalent method.

Espinoza (2014) gives commentary on some of these methods. Generally, the methods lead to different values for the discount rate (Bruner et al. 1998; JPMorgan 2008). Commentary on industry adoption of these approaches is given, for example, in Block (2005) and KPMG (2013). The survey of Block (2005, pp. 60, 62) gives:

Although [deterministic] discounted cash flow methods (based on NPV or IRR) are almost uni- versal, the same cannot be said for the discount rate. There are a number of approaches for adjust- ing for risk. … The most common is to adjust the discount rate for risk [according to] low-risk projects are assigned the minimum discount rate and high-risk projects the maximum rate.

Analysis

In order to establish the relationship between rate adjustment and uncertainty, equivalence between deterministic DCF and probabilistic DCF results is used here in conjunction with expected utility. The following develops the analysis in terms of its components: present worth, utility, and expected utility. The notation used in the article is as follows:

bi constants COV coefficient of variation E[] expected value

Var[] variance PW present worth

r interest or discount rate RA risk aversion coefficient

326 D. G. CARMICHAEL

u, U utility Xi cash flow in period i, i = 0, 1, 2, …, n

α, β, γ constants ρi j correlation coefficients between Xi and Xj

Consider a general set of cash flows over time. Let the net cash flow at period i, i = 0, 1, 2, …, n, be Xi, characterized by its expected value E[Xi] and variance Var[Xi]. Then,

E[PW] = n∑

i=0

biE[Xi] (1 + r)i (1a)

Var[PW] = n∑

i=0

b2iVar[Xi] (1 + r)2i + 2

n−1∑

i=0

n∑

j=i+1

bibjρi j √ Var[Xi]

√ Var[Xj]

(1 + r)i+ j , (1b)

where PW is present worth, r is the interest or discount rate, bi are constants (typically, +1 and −1), and ρi j are the correlation coefficients between Xi and Xj. For the deterministic case, the variances are set to zero. Monte Carlo simulation could also be used to get information on present worth, alternative to the second-order moment results (1a, 1b), but in numerical form only. To include variability in the interest rates, in addition to cash flows, see Carmichael and Bustamante (2014).

Utility is a measure of value and preferences, which may be represented in a utility func- tion. Value and preference may depend not only on the magnitude of a return, but also its probability as well as the financial status of the investor. Each investor could be anticipated to have its own utility function, but commonly these functions might be grouped as being risk averse, risk neutral, or risk seeking. Risk aversion is associated with accepting lower but more certain investment returns compared to higher but less certain returns (Ang and Tang 1984). Commonly, investors are considered risk averse (Brealey et al. 2011).

Consider a general utility function for present worth, applicable over the range of present worths anticipated in the investment. Here a quadratic is used, but other forms (for example, exponential and logarithmic) are possible (Ang and Tang 1984):

u(PW ) = αPW2 + βPW + γ , (2)

where u is utility, and α, β, and γ are constants, different for each investor and investment circumstances. The establishment of utility functions is well documented in the literature and is not repeated here. Ang and Tang (1984, p. 74) note that

the expected utility is relatively insensitive to the form of the utility function at a given level of risk-aversion, and that the expected utility does not change significantly over a wide range of risk- aversion coefficients. Hence, the exact form of the utility function may not be a crucial factor in the computation of an expected utility. Moreover, the risk-aversiveness coefficient in the utility function need not be very precise; that is, any error in the specification of the risk-aversiveness coefficient may not result in a significant difference in the calculated expected utility.

Further comment is given below on utility functions. The second derivative of u gives an indication of the risk attitude. The degree of risk aver-

sion, RA, is sometimes measured by

RA(PW ) = −u ′′(PW ) u′(PW )

. (3)

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Expected utility becomes, using a Taylor series expansion of u about E[PW],

E[U] ∼= αE2[PW] + βE[PW] + γ + αVar[PW] (4)

The more general version of this can be found in Benjamin and Cornell (1970), Ang and Tang (1984), and Carmichael (2014). Such an expansion is valid for usual utility function shapes. Markowitz (2014) gives portfolio commentary on mean–variance approximations to expected utility.

Expected utility is used in the following to establish the rate adjustment necessary for the deterministic DCF analysis (using discount rates) to give equivalence with the true proba- bilistic DCF analysis. Two scenarios are considered:

1. A base interest rate together with probabilistic cash flows. 2. A discount rate (including the base rate and an adjustment) together with deterministic

cash flows. Equivalence between these two scenarios is established through expected utility. That is,

the analysis is addressing the question: If investors wish to use discount rates and assume deterministic cash flows (as is common practice) as a substitute for base interest rates and true probabilistic cash flows, how different should base interest rates and discount rates be? Alternatively, by how much does the base interest rate need to be adjusted in order to get a correct discount rate? This is established by using expected utility as a measure of the goodness of an investment. Expected utility for the two above scenarios is equated.

For the same cash flows and the same rate, the expected utility for the deterministic case will be different from that for the probabilistic case. To make them equivalent requires an adjusted rate or adding a premium or loading to the base rate for the deterministic case.

Note that, with the second-order approach given above, no assumptions are being made on the probability distributions of any of the variables and, in particular, of the cash flows or the present worth. The base rate can be determined by any means that the user wishes, such as using a risk-free rate or WACC; the article’s results do not depend on this choice. The cash flow variance can also be established by any means that the user wishes; the article’s results do not depend on this choice.

Numerical studies

Introduction

A range of numerical studies, covering example cash flows, cash flow uncertainty, cash flow timing, base interest rates, and typical utility functions, is given. These are example results of a much larger numerical experimentation but represent typical results. The range of values used in the experimentation covers typical commercial values. To establish how much base rates should be adjusted to give discount rates, in the presence of uncertainty, the approach to the studies proceeds as in the following. Each analysis includes

1. A utility function is chosen over the possible range of PW. This gives α, β, and γ in Equation (2).

2. A cash flow regime is chosen, with each cash flow, Xi, occurring at time i, characterized by its moments, E[Xi] and Var[Xi].

3. A base interest rate is chosen. This gives E[PW], Var[PW], and E[U] for the general probabilistic case, according to Equations (1a), (1b), and (4).

328 D. G. CARMICHAEL

Figure . Utility functions used in the study. Risk aversion coefficients shown in the legend. Increasing risk aversiveness from bottom to top.

4. Also using Equations (1a) and (4), an adjusted rate (discount rate; leading to E[PW] for the deterministic case) is calculated such that E[U] for both the probabilistic and deterministic cases is the same.

Two cases are not considered. First, risk seeking attitude is not given as “such preference behavior is ordinarily not realistic” (Ang and Tang 1984, p. 72). Comments only are given on the influence of risk seeking attitudes. Second, negative present worths are not examined on the basis that investors would ordinarily desire positive present worths.

The following variables and associated numerical ranges are considered in the numerical experimentation:

� Levels of uncertainty in the cash flow. This is measured in terms of cash flow coefficient of variation (COV). Cash flow COV values range from 0.05 to 0.15.

� Singly occurring cash flows and uniform series of cash flows. � Time into the future at which the cash flow occurs. Times range from 1 to 10 years. � The base interest rate, to which an additive adjustment is made to give a discount rate. Base interest rates range from 0.05 (5% per annum) to 0.15 (15% per annum).

� The level of risk aversion of the investor. Figure 1 shows the utility functions used in the analysis, ranging from risk neutral to risk averse. (Comments only are given on risk seeking attitudes.) These are typical utility functions, representative of different levels of risk aversion. The risk aversion coefficients, RA, given in Figure 1 are those evaluated at E[PW]. The risk aversion coefficient for a quadratic utility function varies with the value of PW used in its evaluation.

Demonstration of adjustments needed

Using the above analysis, Figures 2 to 7 illustrate the type of adjustment that needs to be made to blanket or constant discount rates where uncertainty is present. By implication, they also show the errors involved in assuming constant discount rates in the presence of uncertainty. All rate adjustments shown are additive to the base rate.

Figures 2 to 7 are examples of a much larger numerical experimentation but represent typi- cal results. The range of values used in the experimentation covers typical commercial values.

Single cash flow Figures 2 to 4 show typical results for a single cash flow occurring at a future time. RA is the level of risk aversion defined in Equation (3). COV is the coefficient of variation. Base rate is the constant discount rate to which an adjustment is applied.

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Figure . Example adjustment versus year of cash flow; base rate = .; cash flow COV = ..

Figure . Example adjustment versus cash flow COV; base rate = .; year of cash flow = .

Although not presented here, adjustments for the risk-seeking case are opposite in sign to those for the risk averse case.

Uniform cash flows Figures 5 to 7 show typical results for a uniform series of cash flows starting in year 1 and proceeding variously up to year 1, 4, 7, and 10. The coefficient of variation, COV, refers to

Figure . Example adjustment versus base rate; cash flow COV = .; year of cash flow = .

330 D. G. CARMICHAEL

Figure . Example adjustment versus years of cash flow; base rate = .; cash flow COV = ..

Figure . Example adjustment versus cash flow COV; base rate = .; years of cash flow = .

the cash flow at each year. Figures 5 to 7 are given for cash flows in each year being perfectly correlated; lesser correlation (including independence) leads to lower adjustments.

General collection of cash flows For a general collection of cash flows and correlation assumptions, the analysis does not change; however, it can be harder to isolate the influence of a mixture of analysis inputs.

Figure . Example adjustment versus base rate; cash flow COV = .; years of cash flow = .

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Summary

In summary, the numerical experimentation shows that deterministic DCF analysis using blanket discount rates does not accurately incorporate an investment’s uncertainty. This affects any conclusions on investment viability. The resultant present worth calculated will be wrongly valued.

The numerical study results can be summarized as follows: � Risk-neutral attitudes lead to no adjustment of the base rate. Risk-averse attitudes require rate adjustment according to the following points.

� With increasing base interest rate, the additive adjustment decreases slightly but is almost constant. (As a proportion of the base rate, it decreases.)

� With increasing time, i, into the future at which the cash flow occurs, the adjustment decreases (with the rate of adjustment decreasing with time). No adjustment is necessary at long times into the future.

� With increasing uncertainty in the cash flow (as measured by the cash flow COV), the adjustment increases (with the rate of adjustment increasing with COV).

� With increasing level of risk aversion, the adjustment increases. Depending on where the present worth expected value lies within the utility function, the

results will change but still the trends will remain. Accordingly, the values given in Figures 2 to 7 are not to be taken as definitive but rather as indicating trends. To establish specific numer- ical values, rather than trends, each investor needs its own utility function and analysis for each investment case.

For identical utility functions, but applying over different magnitudes of present worth, the form of adjustment remains unchanged for different magnitudes, only varying with the cash flow coefficient of variation. However, it is anticipated that investors’ utility functions would change, depending on the degree of risk aversion exhibited, with increasing magnitudes of present worth involved.

The analysis for multiple cash flows is no different than that for a single cash flow or uni- form series of cash flows. Some extensions can be argued (by comparison with a single cash flow) using Equations (1a) and 1(b) and Figures 2 to 7 to apply to multiple cash flows.

Comparison with the literature

Existing literature analyzing the influence of uncertainty on discount rates tends to be directed at market-oriented investments rather than real assets as in this article. Hence, the treatment and categorization of uncertainty is different. The present article looks at an investment’s total cash flow uncertainty, rather than components of uncertainty. In addition, different from existing methods is the choice of the base interest rate. Generally, market-oriented treatments use a risk-free rate as a base. In the present article, the user is able to select any base rate that is considered appropriate, including the risk-free rate, WACC, or other. The results are not dependent on what this base rate is, unlike market-oriented treatments.

It is believed that the present article provides a more complete understanding of discount rate determination in the presence of cash flow uncertainty for real assets compared to existing methods. Existing methods of establishing a risk-adjusted discount rate tend to be subjective and inconsistent (Halliwell 2001). Users attempt to acknowledge the uncertainty associated with cash flows with a rate adjustment that does not accurately reflect an investment’s uncer- tainty. A strong argument against the current practice of using risk-adjusted discount rates is that the adjustment adopted is a constant over time and over different cash flows and does not

332 D. G. CARMICHAEL

Table . Guidelines for adjusting rates.

Increases in Will lead to adjustments being

Base interest rate Slightly lower; almost constant (but proportionally lower) The future timing of cash flows Lower Cash flow uncertainty Higher Risk aversion Higher

reflect the true underlying cash flow uncertainty. This article’s results show the deficiencies in such an approach; here it is shown that the adjustment needs to change with level of cash flow uncertainty, cash flow timing, and degree of risk aversion but only mildly with base rate. With possibly the exception of a change in the base rate, the rate adjustments are not constant over these variables.

The time variability of adjustments shown in Figure 2 is in agreement with the arguments of Weitzman (1998) and Gollier (2002), the implied rates in Espinoza (2014) using synthetic insurances, and the real option results of Carmichael et al. (2011) and Carmichael (2014, 2016a). The time influences in Figure 2 also agree in form with Robichek and Myers (1966), Berry and Dyson (1980), Dyson and Berry (1983), Baker and Fox (2003), and Cheremushkin (2009). However, these papers are silent on the influence of the base rate and cash flow vari- ance. On the influence of the level of risk aversion, the qualitative comments in these papers agree with Figures 3 and 6.

The negative adjustment comment is consistent with Berry and Dyson (1980) and Dyson and Berry (1983).

Guidelines

In using discount rates and a deterministic DCF analysis, Table 1 gives guidelines for adjusting the rate to take care of uncertainty in the cash flows.

These guidelines provide a rate adjustment that more accurately represents an investment’s cash flow uncertainty over existing methods. In order to use the article’s findings, it is neces- sary that users only understand their level of risk aversion (ranging from risk neutral through to low, medium, and high risk aversion), not that they be able to generate their own utility functions. For accurate adjustments for any given situation, users should develop their own utility function and apply their own values substituted into Equations (1) to (4).

Conclusions

A single blanket or constant discount rate is not able to simultaneously represent the time value of money and uncertainty. The article shows the limitations and errors involved in requiring the discount rate to do this but also provides, on a case-by-case basis, a way of adjusting rates such that the errors are minimized. The requirement for adjustments demon- strates that errors are involved in using blanket discount rates. The article’s results show that the adjustment varies with timing of the cash flow, cash flow uncertainty, and level of risk aversion but only mildly with the base rate. By not appropriately acknowledging investment uncertainty, any conclusions on investment viability can be questioned.

The article showed trends relating to the influence of the underlying analysis variables. It showed the quantitative adjustments necessary for any given investment scenario. Generally, it is found that the rate adjustment should be decreased as the cash flow timing increases,

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increased as the variance of the cash flows increases, kept almost constant as the base rate increases, and increased as the investor’s level of risk aversion increases.

In the absence of a full probabilistic analysis, the guidelines presented here represent a way forward if deterministic analysis is pursued, as is the current custom. Users are now able to make a more informed approach to rate adjustment, rather than it being arbitrary. The article’s results will be useful not only for single investments but also in the comparison of multiple investments involving uncertain cash flows, where the cash flow timing and uncertainty differ across the different investments.

The numerical results are based on assumptions regarding utility. Each person and organi- zation has its own utility function and this can change depending on the type and magnitude of an investment and the range of present worth anticipated. There is no standardized util- ity function that can be applied to all investments. Here, typical utility functions for different levels of risk aversion are used. Using the theory presented in this article, each person or orga- nization could incorporate its own utility function and cash flows and derive a specific rate adjustment. The trends demonstrated in this article are not anticipated to change, though particular numerical values will. Utility is not totally embraced by everyone, but it has strong support and appears frequently in the commerce literature. Accordingly, it is emphasized that the article’s results have this qualification.

Further research

More extensive numerical studies could be performed to verify the article’s results, in partic- ular, looking at the influence of the investor’s degree of risk aversion as represented by utility functions. Ultimately, an aim of further research might be to develop a function for the rate adjustment that incorporates all of the key investment variables. The research only accounted for uncertainty in cash flows and assumes no uncertainty in the base rate uncertainty and no uncertainty in the cash flow timing. Uncertainty in the interest rate could be included through the results of Carmichael and Bustamante (2014). See also Carmichael and Handford (2015).

With present worth being calculated from a nonlinear expression and with the quadratic used for utility, the combined nonlinearity prevented obtaining any closed-form result. The rate adjustment occurs within the denominator raised to a power. Restricted closed-form results may, however, be possible using an exponential utility curve.

Notes on contributor

David G. Carmichael is a Professor of Civil Engineering and former Head of the Department of Engi- neering Construction and Management at the University of New South Wales, Australia. He is a grad- uate of the Universities of Sydney and Canterbury; a Fellow of the Institution of Engineers, Australia; a Member of the American Society of Civil Engineers; and a former graded arbitrator and mediator. Pro- fessor Carmichael publishes, teaches, and consults widely in most aspects of project management, con- struction management, systems engineering, and problem solving. He is known for his leftfield thinking on project and risk management (Project Management Framework, A. A. Balkema, Rotterdam, 2004), project planning (Project Planning, and Control, Taylor and Francis, London, 2006), problem solving (Problem Solving for Engineers, CRC Press, Taylor and Francis, London, 2013), and infrastructure invest- ment (Infrastructure Investment: An Engineering Perspective, CRC Press, Taylor and Francis, London, 2014).

334 D. G. CARMICHAEL

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