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DNPV: a valuation methodology for infrastructure and Capital investments consistent with prospect theory

David Espinoza, Javier Rojo, Arturo Cifuentes & Jeremy Morris

To cite this article: David Espinoza, Javier Rojo, Arturo Cifuentes & Jeremy Morris (2020) DNPV: a valuation methodology for infrastructure and Capital investments consistent with prospect theory, Construction Management and Economics, 38:3, 259-274, DOI: 10.1080/01446193.2019.1648842

To link to this article: https://doi.org/10.1080/01446193.2019.1648842

Published online: 08 Aug 2019.

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DNPV: a valuation methodology for infrastructure and Capital investments consistent with prospect theory

David Espinozaa, Javier Rojob, Arturo Cifuentesc,d and Jeremy Morrisa

aGeosyntec Consultants, Washington, DC, USA; bSustainability Strategic Advisors, London, UK; cColumbia University, New York, NY, USA; dCLAPES-Universidad Catolica, Santiago, Chile

ABSTRACT Traditional valuation methods such as net present value (NPV) utilize increased discount rates to account for risk, in the process introducing a time bias effect that promotes short-termism. Application of NPV often discourages much needed infrastructure projects that require large capital investments yet are slow to generate positive cashflows. NPV also downplays the signifi- cance of future liabilities and can lead to risk misallocation amongst investment partners and stakeholders. The decoupled net present value (DNPV) method introduces the risk-as-a-cost con- cept that prices the risk of obtaining lower-than-expected cashflows and thus represents invest- ors’ compensation for bearing such risks. Capturing the loss-aversion attitudes described by prospect theory, DNPV provides a transparent and consistent valuation framework for long-term investments by: (i) calculating expected values of cashflow components using their probability characterizations, (ii) defining the cost of risk (market and non-market) as the expected down- side value, (iii) subtracting/adding the cost of risk from/to expected revenues/expenditures, and (iv) discounting the results using risk-free rates. DNPV’s power is illustrated by re-analyzing a 42- year toll-road concession initially evaluated using NPV and real options. The case study shows how explicit risk quantifications could be used to better structure the concession and reallocate risks among stakeholders.

ARTICLE HISTORY Received 18 October 2018 Accepted 19 July 2019

KEYWORDS Real options; derivatives; valuation; NPV; DNPV; prospect theory

Introduction

Myopic investment mindsets have encouraged corpo- rations and governments to select projects with quick payoffs over those that would better increase long- term stakeholder value and security (Lundstrum 2002, Antia et al. 2010). Key reasons suggested for such short-termism vary from upper management oppor- tunism in the face of shorter tenures for C-suite execu- tives to investors’ increased focus on short-term earnings (Repenning and Henderson 2010). This prob- lem is further exacerbated by widespread use of the risk-adjusted discount rate (RADR) concept together with valuation methods such as net present value (NPV) and the internal rate of return (IRR). The RADR concept consists of accounting for risk in the discount rate, ineffectively and incorrectly commingling risk with the time value of money.

Hamada (1977) voiced concerns on RADR best when he wrote: “Why is the financial profession seem- ingly obsessed with the risk-adjusted cost of capital approach to project selection when all the theoretical

considerations point to using a certainty-equivalent approach? All kinds of simplifying approximations, con- tortions, and mathematical manipulations are employed merely to fit the risk adjusted cost of capital format.” Zeckhauser and Viscusi (2008) later stated: “Risk is risk, and time preference is time preference. Though risk and time are inevitably entangled as we confront them, they should be separated as best as possible when we ana- lyze them.” Efforts to move away from the practice of arbitrarily adjusting discount rates to account for risk should be welcomed, especially for long-term invest- ments evaluations. The economic impact of unduly penalizing long-term projects when using RADRs is not trivial, particularly considering the limited resour- ces for infrastructure investments of the public sector who may seek private capital through public-private partnerships (PPPs). Commingling risk with the time value of money (represented by the risk-free rate) makes it difficult to evaluate the actual effect of differ- ent risk-sharing mechanisms on the performance of PPPs (Ashuri et al. 2012), which may result in unfair

CONTACT David Espinoza despinoza@geosyntec-cat.com Civil Engineering, Geosyntec Consultants, 10220 Old Columbia Road, Suite A, Columbia, MD 21046, USA � 2019 Informa UK Limited, trading as Taylor & Francis Group

CONSTRUCTION MANAGEMENT AND ECONOMICS 2020, VOL. 38, NO. 3, 259–274 https://doi.org/10.1080/01446193.2019.1648842

compensation for the risk borne by some parties who unsuspectingly accept long-term contracts with unfavorable terms.

Because NPV results are extremely sensitive to the selection of discount rates, investors’ use of fragile heuristics, rules of thumb, and top-down mandates may lead to their demanding unnecessarily high pre- miums (i.e., discount rates). This renders potentially good projects unattractive as cashflows that occur far in the future appear less valuable (Davies et al. 2012). As a result, projects that need several years to start generating positive cashflows are rejected, regardless of the reliability of those cashflows. Conversely, using high RADRs may make projects with significant long- term liabilities look more attractive than they really are. Worse yet, this may lead to insufficient accrual of funds to cope with known future costs (e.g., asset decommissioning, site reclamation) and/or potential contingent liabilities (e.g., flooding, drought).

Recently, Espinoza and Morris (2013) and Espinoza (2014) introduced the decoupled net present value (DNPV) method along with the risk-as-a-cost concept to assess the value of long-term infrastructure proj- ects. Conceptually, DNPV combines features of cer- tainty equivalent (Robichek and Myers 1966) and real options (Myers 1984) methods to address many of NPV’s shortcomings. DNPV accounts for the uncertain nature of cashflows by describing them probabilistic- ally while retaining the simplicity of the NPV format that has led to its popularity. The main objective of this paper is to further the use of DNPV as a robust investment valuation method that separates the time value of money from risk by demonstrating how the risk-as-a-cost concept aligns with behavioral econom- ics (i.e., prospect theory) as well as more established financial derivative concepts such as put options. The linking of prospect theory and financial derivative con- cepts provides a solid framework for financial risk quantification regardless of its source (i.e., market and non-market), which is a departure from the wide- spread practice of pricing only systematic (i.e., market) risk using NPV along with RADRs. This paper is organ- ized as follows. First, a review of the mainstream valu- ation methods including certainty equivalent and real options is presented. Second, the paper discusses DNPV through the lens of prospect theory and loss- aversion concepts. Finally, a troubled toll road project taken from the literature is used as a case study to illustrate how explicit identification and quantification of risks would allow stakeholders to better understand the project’s sources of risk and their impact on value.

Review of mainstream valuation methods and shortcomings

Central to any investment decision is assessment of future cashflow risks and how these risks may affect the investment’s value. With the advent of modern portfolio theory (Markowitz 1952), academic research focused on assessing the market risk of bundled rather than individual securities. Furthermore, to facilitate comparison among differently traded securities, asset returns rather than values were analysed. From this work, the annualized standard deviation of returns (i.e., volatility) for a portfolio of securities was used as a proxy for market risk. Hence, it is the contribution (i.e., correlation) of a single security to a well-diversi- fied portfolio that is important rather than the returns distribution of the individual security itself. Complementing Markowitz’s work, Sharpe (1964) intro- duced the Capital Asset Pricing Model (CAPM) to esti- mate the additional expected return over the risk-free rate (r) per unit of market risk (rm-r) borne. In other words, investments’ total expected return (a) were expressed as the risk-free rate plus a risk premium, effectively combining the time value of money with risk:

a ¼ r þ b rm�rð Þ (1) where rm is the market return, and b is a parameter that correlates a and (rm � r). Ensuing criticism regard- ing CAPM’s validity fuelled the development of alter- native models (e.g., Ross 1976, Fama and French 1997). However, despite added complexity, these alter- native models still relied on risk premiums (albeit cal- culated using more parameters) added to the risk-free rate. Although volatility of returns is widely accepted as a proxy for risk, increasing evidence indicates that this metric alone does not fully capture financial risk (Taleb 2007) nor investor’s risk preference (e.g., Kahneman and Tversky 1979, Chiu 2005, Ang et al. 2005).

Subsequent work on capital budgeting and project investment assumed that cashflow risk evolved in a manner consistent with CAPM (e.g., Brennan 1973, Bogue and Roll 1974, Fama 1977). As a result, NPV cal- culations expressed uncertain cashflows in terms of expected values and their risks were included in the discount rate. Conceptually, this easy-to-grasp concept of adjusting discount rates to account for risk cap- tured the risk-averse nature of investors who would pay less for a stream of risky future cashflows than for riskless ones (i.e., demand higher returns for riskier projects than for comparable safer investments). However, adding a constant risk premium to the

260 D. ESPINOZA ET AL.

risk-free rate to estimate the discount rate implicitly assumes that cashflows are always positive1 and fol- low a random walk process (Bhattacharya 1978, Giacotto 2007), which is not always the case. Hence, extending CAPM (or its alternatives) to value risky cashflows for investments other than traded securities is questionable at best. Despite well-known limitations (e.g., Halliwell 2011), the use of RADR combined with NPV has gained significant popularity for project finan- cial analysis and corporate capital budgeting (Graham and Harvey 2001, Triantis 2005, Block 2007) and valu- ing infrastructure investments (e.g., Ariel 1998, Froth and Stein 1999, Garvin and Cheah 2004, Parrino et al. 2005, Wibowo 2006, Liou and Huang 2008). Although an agreement regarding how to estimate the appro- priate discount rate is yet to be reached, most NPV users relies on CAPM or the weighted average cost of capital to estimate the discount rate (Ashuri et al. 2012).

The problem is further aggravated when consider- ing non-market risks of one-off projects like those found in large infrastructure projects (e.g., toll roads, utilities, airports) where these risks cannot be diversi- fied into a portfolio of similar projects. To account for risks other than market risk, additional risk premiums are added to discount rates calculated using CAPM, the weighted average cost of capital, or similar alter- natives. However, there is no consistent methodology to guide selection of additional risk premiums to account for non-market risks (Ashuri et al. 2012, Zeckhauser and Viscusi 2008). Adding to the confu- sion, projects fully financed by public funds and inter- national multilateral agencies are often discounted using mandated discount rates independently of the actual project risk (e.g., the Inter-American Development Bank typically uses a 12% real discount rate). In summary, selected discount rates rarely reflect the project risk characteristics.

Recently, applications of Monte Carlo simulations to estimate a project’s NPV probability distribution have become ubiquitous in the literature (e.g., Chiara and Garvin 2008, Liou and Huang 2008, Girmscheid 2009). However, simulation techniques are infrequently used in practice (Graham and Harvey 2001). In addition, simulation examples found in the literature typically discount cashflows using RADRs, effectively double accounting for risk. In theory, discount rates are adjusted to account for cashflow risks thus making redundant the need to describe the cashflows prob- abilistically beyond the expected value (Myers 1976). Worse yet, the RADR is often treated as a random vari- able itself, adding difficulty to interpretation of the

results. Although consistent probabilistic methods that capture cashflow uncertainties through their means, standard deviations and correlations, and use risk-free rates to discount future cashflows, are found in the lit- erature (e.g., Cheah and Liu 2006, Hawas and Cifuentes 2017), implementing such methods and interpreting their results remain difficult tasks restricted to experts.

The certainty equivalent method (CEM) estimates the expected revenues (~S), reduces them to account for risk (i.e., k~S, where k is a risk reduction factor2

such that 0�k � 1), and then discounts these reduced revenues by the risk-free rate, essentially separating risk from the time value of money. Using CEM, risk- averse investors are asked for the value of the smallest certain cashflow that they would accept at a future time in exchange for an uncertain expected future cashflow. As the name suggests, the expected value of uncertain revenues needs to be reduced (i.e., k < 1) to be considered equivalent to a certain amount (i.e., without risk). Freeing risk from the time value of money allows for less restrictive assumptions regard- ing risk as a function of time. Unfortunately, practical implementation of CEM is not straightforward as gen- eral methodologies for calculating k were not pro- vided except for particular cases where k was back- calculated from RADRs, thus forcing the results of CEM and NPV to be equivalents (Robichek and Myers 1966, Fama 1977, Constantinides 1978, Sick 1986), further cementing NPV’s popularity.

Recognizing RADRs’ shortcomings for capital budg- eting, Myers (1984) introduced the real options valu- ation (ROV) concept to value the right—but not the obligation—to undertake specific project initiatives (e.g., deferring, abandoning, expanding, staging, or contracting). Despite its academic popularity, ROV has not become mainstream practice (Triantis 2005, Block 2007). Reasons cited for this are lack of expertise as well as implementation complexity (Baker et al. 2011). To increase ROV’s appeal among NPV users, the rec- ommended practice is to consider real options as a component of an expanded NPV (e.g., Luehrman 1998, van Putten and MacMillan 2004, Copeland and Antikarov 2005, Denison, et al. 2012). For instance, Copeland and Antikarov (2005) suggested: “Whichever approach is used, the value of the project with flexibility should be equal to the NPV of the base case cashflows (the value of the project without flexibility) plus the (option) value of the management’s ability to respond to change.” However, this approach ties ROV’s results to a questionable technique. In addition, because options are always positive, the expanded NPV

CONSTRUCTION MANAGEMENT AND ECONOMICS 261

concept would invariably calculate investment values greater than the project value without options (i.e., traditional NPV) implying that the options would be obtained for free.3 Furthermore, because options are more valuable with increased uncertainty, it follows that higher values would be obtained using the expanded NPV concept. However, higher uncertainty should result in higher RADR which in turn should result in lower NPV thus neglecting the increased option value. These and other inconsistencies have fuelled detractors’ arguments that ROV methods are simply used to justify investments not supported by traditional NPV methods (Damodaran 2000).

The DNPV method

Connecting the cost of risk with loss aversion

Like CEM, DNPV decouples risk from the time value of money and discounts future cashflows using the risk- free rate. However, while CEM focuses on investors’ risk preference to estimate risk reduction factors, DNPV focuses on cashflows characteristics to assess the investment’s riskiness and consistently defines risk as fair insurance premiums (i.e., cost of risk) designed to protect investors from cashflow shortfalls below their expected values. Because premiums are derived from the cashflow variability, the cost-of-risk is con- ceptually equivalent to a financial derivative. Furthermore, DNPV is consistent with prospect theory, a descriptive theory of decision-making under uncer- tainty introduced by Kahneman and Tversky (1979) to better explain the observed behaviour of investors that is not captured by standard financial concepts (e.g., utility theory). Based on prospect theory, Benartzi and Thaler (1999) postulated that investors are loss averse (i.e., averse to wealth reduction) rather than risk averse (i.e., averse to variability of returns). Loss

aversion refers to individuals’ tendency to be more sensitive to wealth reduction than to wealth increase. Empirical evidence indicates that, on average, losses weigh twice as much as gains on investors’ minds (Tversky and Kahneman 1991). Figure 1 illustrates how loss aversion would affect gambling patterns on a sin- gle fair coin flip (50–50 chance). A risk-neutral gambler (Figure 1a) would be indifferent to paying a certain amount X (¼ So) in exchange of an uncertain future revenue (~S) with an expected value ~S ¼ So, that is, ~S – X ¼ 0. On the other hand, an average loss-averse individual (Figure 1b) would not take such a gamble unless the expected value of the uncertain revenue is ~S¼ 1.5 So, that is, ~S – X ¼ 0.5 So (or ~S – 0.5 So � X, that is, average loss-averse individuals reduce the expected value of uncertain revenues to take on that risk). Although individuals may be more or less loss- averse than the average loss-averse investor (gambler), understanding the behaviour of average loss-averse investors provides a useful benchmark for invest- ment analysis.

As discussed above, rational investors generally have loss-aversion mindsets (Kahneman and Tversky 1979, Benartzi and Thaler 1999, Chiu 2005). DNPV takes advantage of the probabilistic descriptions of cashflows that can often be developed from available data and/or industry experience to consistently and transparently capture the loss-aversion attitudes of average investors. Loss-aversion is characterized by the potential downside, which is defined as the likeli- hood of obtaining lower-than-expected revenues and/ or incurring higher-than-expected expenditures. Figure 2a shows the revenue probability distribution where the revenues downside4 (i.e., the probability that actual revenues will be lower than the expected reve- nues, ~St) is represented by the area to the left of ~St denoted as Ut. The value of ~St is the centre of gravity of the probability density function (PDF) describing

Figure 1. Representation of loss-aversion on a single coin flip gamble.

262 D. ESPINOZA ET AL.

the revenues in Figure 2a. As before, the expected value of the revenue downside ð~S0tÞ corresponds to the centre of gravity of the area Ut which can be used to objectively characterize the downside potential. DNPV derives the cost of risk for revenues directly from the revenues’ PDF, /(St), as the difference between Ut~St and ~S

0 t which is Ut

~St�~S0t � �

as depicted in Figure 2b. It follows from Figure 2 that higher nega- tive skewness would simply shift the centre of gravity of the downside further to the left, thus leading to higher cost of risk. Hence, it is apparent that the cost of risk depends upon the PDF’s skewness. A mirror- image PDF geometry can be applied to expenditures.

To link investors’ risk preferences to cashflow risk profiles, the concept of “Risk Neutrality Level” is intro- duced herein and it is defined as the level of risk an investor would be willing to accept to take on a given investment (gamble) with a known PDF. Investors that accept the value of the expected revenues/expendi- tures without requiring compensation for taking on risks associated with their uncertainty (Figure 2a) are defined as Risk Neutral Level 0 (or simply risk neutral) investors. The next level (Risk Neutral Level I) refers to those investors who are loss-averse and consider Ut~St�~S0t

� � as appropriate compensation for the risk

they are bearing (i.e., they are risk neutral with respect to the downside).5 The cost of risk concept coupled with the Risk Neutrality Level can be useful to bench- mark how risk averse an investor is, particularly when

dealing with non-market risks. Investors can set objective targets for risks that they are unfamiliar with or are lacking data, or that they simply cannot afford. Furthermore, because most individuals are familiar with the notion of insurance, investors should find the cost of risk concept a relatively easy way of capturing the riskiness of cashflows.

A framework for prospect theory

The proposed cost of risk (i.e., synthetic insurance pre- mium) has features that are consistent with prospect theory (e.g., loss aversion, certainty effect, narrow framing, status-quo bias). DNPV’s idea is that a loss- averse individual would be willing to exchange a cer- tain amount ~St for an uncertain revenue stream (~St�RS), where RS represents the expected (i.e., actuari- ally fair) value of the revenue downside defined as a drop below the expected revenues. Because RS is con- sidered a cost (albeit a synthetic one), it is subtracted from the expected revenues. Thus, the resulting reduced revenue stream (~St� RS) is considered a cer- tainty equivalent (Figure 3). Similarly, the calculated cost of risk for expenditures (RX) is also treated as a cost and added to the expected expenditures, with the resulting increased expenditure (~Xt þ ~RX) consid- ered a certainty equivalent for expenditures (Figure 3). This feature of DNPV is consistent with the certainty effect property of prospect theory, which turns

Figure 2. Synthetic insurance premium calculation for uncertain revenues (S).

CONSTRUCTION MANAGEMENT AND ECONOMICS 263

investors from risk averse when facing choices with positive outcomes (i.e., preference of sure gains over uncertain larger gains) to risk seeking when facing choices with negative outcomes (i.e., preference of uncertain loss over a smaller but certain loss). Accordingly, in the DNPV methodology, risk reduces/ increases the value of the certainty equivalent reve- nues/expenditures.

Following this approach, if all identified risks (mar- ket and non-market) that could affect future revenues and costs are accounted for and subtracted/added accordingly, then consistent with CEM the resulting cashflows can be considered riskless. Since revenues and expenses can take place at different times, pre- sent values that are certain need to be scaled-up to their future value using the risk-free rate (r) to estab- lish the future reference level for the status-quo (Barberis et al. 2001) or, alternatively, future riskless certainty equivalents need to be discounted using the risk-free rate. Thus, it follows that DNPV can be expressed in a familiar form as:

DPNV ¼ Xn t¼0

~St�~Xt�~Rt 1 þ rð Þt

¼ Xn t¼0

DNPVt (2)

where ~Rt ¼ ~RS þ ~RX. This feature is paramount for val- uing projects with long service lives as future revenues

and expenses are not reduced to negligible values by the process of discounting using inflated RADRs. Figure 4 graphically represents the DNPV method for a single period. If the cost of risk (~Rt) was truly an insurance product, negative cashflows would be compensated by said insurance product and the cashflow line would be represented by Line 1 in Figure 4. However, to obtain real downside protection, an insurance product would need to be purchased. In the DNPV concept, the cost of risk simply captures the value of the downside and it represents the monetary compensation for investors who take on that risk.6 Thus, it follows from Figure 4 that the investment should not be considered unless revenues exceed expenditures plus the cost of risk (Line 2), that is, DNPV � 0.

Validation of the cost of risk concept

To validate that the proposed cost of risk in the DNPV method is indeed reasonable compensation for risks taken, consider the simplest comparison between a certain amount versus a risky choice consists of a sin- gle fair coin flip (50-50 chance). The average loss aver- sion associated with this gamble under the DNPV method is illustrated in Figure 5. As shown, the value of the downside is 0.5So. From the DNPV definition, the certainty equivalent for the average loss-averse individual would be the expected value ~S (¼ So) minus the downside 0.5So (i.e., ~S – 0.5So ¼ 0.5So). Hence, X ¼ 0.5So is the maximum value that average loss- averse individuals would be willing to pay for this gamble. It follows that the expected return ~S – X ¼ 0.5So. This value is consistent with the findings of prospect theory. However, as illustrated in the example below, DNPV allows practitioners to imple- ment a process that is consistent with prospect theory more easily. Continuing with the coin-flip example, the downside (i.e., the cost of risk) decreases as the number of coin flips increases. An example for mul- tiple coin flips is illustrated in Figure 6 where the cost

Figure 3. Certainty equivalent method within the DNPV framework.

Figure 4. Graphical representation of DNPV.

264 D. ESPINOZA ET AL.

of risk becomes vanishingly small for a large number of coin flips (e.g., casinos). Hence, it follows that only risk-neutral individuals would participate in actuarially fair gambles as casinos that wish to remain in business would not give favorable odds to customers. In this simple example, casinos represent the market that makes the cost of risk for this gamble nearly zero.

Furthermore, assuming there is a one-year period between taking on the bet and the resolution of the uncertainty, then the maximum value (X) an average loss-averse investor would pay at time 0 to take on such bets can be calculated from the equation below.

DPNV ¼ So�0:5So 1 þ r � X ¼ 0 (3)

Capital markets also provide a social laboratory where multiple participants exchange opinions on the

value of risk through their pricing of stocks and corre- sponding put options,7 thereby rendering an unbiased, transparent, measurable, and observable verdict on how much the cost of risk is worth (Appendix A). Different from commercial insurance products where true risk premiums are not observable because they include operating expenses and profits, options give us the opportunity to assess the expected (actuarily fair) market value of the cost of risk. As in the coin flip example, the value of the stock price minus the put option should return, on average, the risk-free rate. Like in a casino, it is the average loss-averse attitude of the market participants not of an individual that is important.

In summary, the cost-of-risk concept can accommo- date loss aversion from one-off bets such as invest- ments in real projects all the way to multiple bets

Figure 6. Multiple coin-flip gamble using DNPV method.

Figure 5. Valuation of a coin-flip gamble for loss-aversion case using DNPV method.

CONSTRUCTION MANAGEMENT AND ECONOMICS 265

regardless of the risk source (market or non-market). In all cases, the expected value of the downside repre- sents a defensible quantification of risk compensation and it is consistent with prospect theory.

Infrastructure risks

The proposed cost of risk concept is applicable for any uncertain cashflows, including those affecting infrastructure projects (e.g., toll roads) as a result of natural hazards (e.g., temporary or permanent shut- down due to flooding), political risks (e.g., unexpected tariffs), technical risks (e.g., pavement materials wear and tear), operational risks, or a host of other issues. In some cases, cashflows are described by well-defined PDFs with significant amount of data (e.g., commod- ities quoted prices, solar data for renewable energy, traffic loads for existing roads). In other cases, cash- flow variability is not well-defined because data is sparse and/or is based on expert opinions (e.g., traffic demand for new roads, political risk). However, whether or not cashflows are described by well- defined PDFs, Figure 2b can be used to conceptually represent the cost of risk. As discussed above, for cashflows characterized by well-defined PDFs, the cost of risk should be relatively accurate and, in competi- tive markets, investors should be Risk Neutral Level I. However, depending on risk tolerance levels, individ- ual investors may be more or less loss averse than average. However, it is apparent that investors that demand higher compensation to take on an invest- ment risk would be less competitive. In general, Investors may accept lower compensation if the risk profile is well understood and well defined, losses would not jeopardize their overall wealth, or they per- ceive a strategic advantage in making the investment.

Valuation of a toll road: case study

Project background

To illustrate the application of DNPV, a case study taken from the literature (Garvin and Cheah 2004) is presented. A 23-km, 8-lane Dulles Greenway Highway project in Fairfax County, Virginia, USA, was to be delivered under a 42-year build/own/transfer (BOT) concession. The toll road was originally scheduled to open in 1992 following two years of construction. The initial average daily traffic (ADT) projection was 20,000, with an average annual increase of 14% for the first five years and 7% thereafter. Tolls were sched- uled to increased gradually from $2.00 to $3.00 during the first 15 years of operations (Table 1). Operating

expenditures (OPEX) started at $9 million in Year 1 and increasing at 5% annually. Using Equation (1) along with b ¼ 1.2, r ¼ 6%, and a market risk premium of 8%, a discount rate of 15.6% was estimated and used to calculate the project’s NPV. The initial 2-year capital expenditures (CAPEX) was estimated at $279 million. The project developer anticipated four capital injections of $3.0, $1.7, $9.4 and $10.4 million in Years 10, 12, 16, and 20, respectively. Using this information, the project cashflows are replicated in Table 2 for Years 1 to 42. The NPV for the project is calculated at -$86.3 million.

The project was ultimately delayed by four years. Subsequently, the project developer reportedly used the projected ADT in Year 4 of 34,000 as the initial ridership estimate while leaving all other input assumptions unchanged. Obtaining a positive NPV of $56.0 million as a result provided the impetus to exe- cute the project. The project was in financial distress within six months of operations as the initial ridership (the only source of revenues) was 10,500, less than a third of the expected value.

Real option valuation

Acknowledging the limitations of NPV analysis, Garvin and Cheah (2004) analysed the investment opportun- ity considering a hypothetical option to defer the deci- sion to proceed with construction for five years to allow collection of better information on traffic demand. Assuming pessimistic, expected, and optimis- tic ADT estimates of 10, 20, and 34 thousand, respect- ively, the value of the project with an option to defer was calculated to be $25.5 million. Thus, they con- cluded that the option to defer the project five years to acquire better information would have added $111.8 million ($25.5 � (�$86.3)) to the value of the project. Thus, the revised financial analysis would have indicated that the project could have proceeded if such an option was included in the contract. Ironically, as the project was delayed nearly four years, it appears that the forced delay was either not used by the developer to acquire better/additional information or the option was not as valuable as origin- ally estimated.

For discussion purposes, it is important to under- stand how the option to defer construction was calcu- lated. First, the NPV for initial ADTs of 10, 20, and 34 thousand were calculated without including the initial

Table 1. Toll schedule from Garvin and Cheah (2004). Years 3–5 6–7 8–10 11–13 14–16 17–42 Toll $2.0 $2.25 $2.5 $2.65 $2.85 $3.00

266 D. ESPINOZA ET AL.

CAPEX (i.e., $279 million), which was considered, using real options lingo, as the exercise price (i.e., the pre- sent value of the CAPEX representing the lower-bound target net revenues that the developer would need to attain to execute the project). These values can be easily calculated using the cashflow model in Table 2 and setting the initial CAPEX to zero. The calculated values for the pessimistic, expected, and optimistic ini- tial ridership are $37.0, $138.7, and $281.1million, respectively (see shaded boxes in Figure 7a). Furthermore, Garvin and Cheah (2004) assumed a binomial model to describe the evolution of the initial ADT with time from 20,000 at t ¼ 0, to either a high of 34,000 or a low of 10,000 at t ¼ 5. These values were used to calculate the risk neutral probability (p) as 61% for the optimistic ADT and 39% for the pessimis- tic ADT (Figure 7a). Noting that the present value of the CAPEX discounted at 15.6% is $225.1 million, the corresponding NPV for the optimistic/pessimistic ADTs including the initial CAPEX is calculated to be $56 mil- lion (i.e., $281.1 � $225.1) and �$188.1 million (i.e., $37.0 � $225.1), respectively. Because the NPV for the pessimistic ADT is -$188.1 million, project develop- ment for this condition is not attractive and conse- quently would not proceed (i.e., the option would not be exercised). Thus, consistent with ROV, the NPV for the pessimistic ADT is set to zero. The expected NPV in Year 5 is then calculated as $34.2 million (i.e.,

0.61 � 56 þ 0.39 � 0). Hence, discounting this value using the nominal risk-free rate (i.e., 6%), the NPV at t ¼ 0 is $25.5 million (i.e., the project with an option to defer would be a go).

This analysis as presented has several drawbacks. First, the 15.6% discount rate was selected based on loose correlation of the project with stock market returns, estimations of the project’s long-term per- formance, and ill-defined correlations between the overall regional economic performance and expected daily traffic levels. Further, it assumes no toll price elasticity in traffic demand and the analysis is incom- patible with option pricing, as the NPV in Year 5 was calculated using a discount rate of 15.6% instead of the risk-free rate. Finally, the analysis implicitly assumes that the option to delay had no cost.

For this example, the only source of revenues is total ridership tolls, that is, the number of vehicles that will ultimately use the toll road times the toll paid per vehicle. Postponing construction to improve the traffic study may have reduced some uncertainty regarding the daily ridership forecast but it would not have eliminated it. A thorough analysis of traffic pat- terns together with the demand for alternative roads would certainly help reduce uncertainty but a more effective strategy would be to undertake a fraction of the project (e.g., fewer lanes, shorter distance) as a pilot and measure actual ridership.

Table 2. Cashflows replica (in millions of $unless noted) from Garvin and Cheah (2004). Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .. 42

ADT(�1000) 20.0 22.8 25.9 29.6 33.8 38.5 41.2 44.9 47.2 50.5 54.0 57.8 61.8 66.2 70.8 75.8 81.1 86.7 .. 384.3 Toll/vehicle ($) 2.00 2.00 2.00 2.25 2.25 2.50 2.50 2.50 2.65 2.65 2.65 2.85 2.85 2.85 3.00 3.00 3.00 3.00 .. 3.00 gross revenue 0 0 14.6 16.6 19.0 24.3 27.7 35.1 37.6 40.2 45.6 48.8 52.2 60.1 64.3 68.8 77.5 82.9 88.8 95.0 .. 420.7 OPEX 0 0 9.0 9.5 9.9 10.4 10.9 11.5 12.1 12.7 13.3 14.0 14.7 15.4 16.2 17.0 17.8 18.7 19.6 20.6 .. 60.3 CAPEX 139.5 139.5 0 0 0 0 0 0 0 3 0 1.7 0 0 0 9.4 0 0 0 10.4 .. 0 Net Revenue �139.5 �139.5 5.6 7.2 9.1 13.9 16.8 23.7 25.5 24.6 32.3 33.2 37.6 44.7 48.2 42.5 59.7 64.2 69.1 63.9 .. 360.4

Figure 7. Binomial representation of initial traffic risk.

CONSTRUCTION MANAGEMENT AND ECONOMICS 267

As initially forecasted in Table 2, the consortium considered that an appropriate nominal return for this investment calculated using CAPM would be 15.6%. Assuming a 3% inflation rate consistent with U.S. infla- tion data from 1990s, investors in this project expected to receive $2.7 billion (in 1992 dollars) for an initial $279 million investment (i.e., a ten-fold return over the contract period) and an initial ADT of 34,000. If the initial ADT of 20,000 along with the expected growth rates had materialized, investors would have received $1.7 billion for their initial $279 million investment (i.e., a six-fold return). One might ask if this is a commensurate return for the risk associated with a private sector investment. Indeed, in the next section we apply DNPV in part to review how propor- tional these demanded returns were to invest- ment risks.

DNPV valuation

The toll road was considered an essential infrastruc- ture needed to encourage development and improve

commuting in the Washington, DC metropolitan area.8

The analysis herein is not aimed to rework the expected traffic scenarios nor the proposed traffic growth rates but rather to analyze and evaluate the risks and the variability associated with the assump- tions made at the time to define the proposed road ADT scenarios. However, before the DNPV analysis is performed, a discussion of some of the assumptions made by Gavin and Cheah (2004) is provided in Table 3 along with modifications proposed in an effort to provide a better decision-making tool using data avail- able at the time without the benefit of hindsight. The resulting cashflow analysis is shown in Table 4. The calculated NPV for the revised parameters for the expected initial ADT (i.e., 20,000) was estimated to be -$80.3 million, similar to the original -$86.4 million estimated. Using this standard cashflow analysis, risks are identified, and corresponding annual cost of risks can be estimated (Espinoza 2014, Espinoza and Rojo 2015). For this example, the main risk sources identi- fied are the initial ADT estimate and annual ADT variability.

Table 3. Financial model commentaries and observations. Garvin and Cheah (2004) DNPV analysis

Cash flows It is unclear if the analysis uses nominal or real (net of inflation) cash flows.

DNPV analysis is performed using real cash flows.

Inflation The analysis ignores inflation. This is inconsistent with the discount rate (15.6%) obtained using a nominal risk-free rate of 6% instead of a real risk-free rate.

Inflation expectation in the 90’s was on the order of 3%. This inflation value is used to correct nominal rates used in the financial analysis.

discount rates Because CAPM’s ability to predict stock returns is not supported by evidence, its extension to infrastructure investments is questionable. At best, the discount rate used (i.e., 15.6%) represents the investors’ desired nominal return not project risk.

DNPV discounts cash flows using risk-free rates. A real risk-free rate of 3% (i.e., 6%-3%) is used to be consistent with projected real cash flows. The nominal risk-free rate of 6% is consistent with the return of long-term U.S. treasuries in the early 1990s.

Traffic demand growth rate The assumed traffic growth (14% for the first 6 years followed by 7%) ends up with an expected daily traffic volume at the end of the concession of 384,240 when the ADT for an 8- lane highway ranges from 90,000 to 200,000 (Highway Capacity Manual, 6th Edition, 2018).

The growth rates are left unchanged. The expected ADT is limited to a more realistic maximum of 145,000 (the average of an 8-lane range). For the year-to-year traffic variability, it was assumed that the ADT follow a normal distribution. The statistical parameters were estimated assuming the mean would be between 0.75 and 1.25 of the forecasted value with a 50% confidence level. The corresponding standard deviation was estimated equal to 37% of the forecasted ADT.

Initial average daily traffic (ADT) demand

Assumed a binomial distribution with an average, pessimistic and optimistic ADT of 20, 10, and 34 thousand, respectively. The high and low demands were assumed to take place in Year 5. The probability of a low ADT was calculated to be 39%.

The estimated pessimistic ADT (10,000) and optimistic ADDT (34,000) with probability of occurrence of 39% and 61% are left unchanged. These values are used to calculate the risk profile for the initial ADT.

OPEX The analysis assumes a 5% annual increase irrespective of traffic volume. Since wear and tear are linked to traffic, OPEX would be more realistic if correlated to this parameter.

The analysis assumes that OPEX is directly correlated to traffic demand.

Tolls It is unclear if tolls are in nominal or real dollars. The toll schedule at the end of the first 15 years somewhat matches the expected rate of inflation. However, the remaining 25 years, tolls remain unchanged.

Tolls are assumed real (net of inflation); thus, the same toll schedule is used. This implies that the initial schedule (i.e., $2.00) is used to attract users and then tolls are gradually increased to $3.00 to generate more sustainable revenues.

268 D. ESPINOZA ET AL.

For comparison purposes, the cost of risk associated with the initial ADT was estimated using the risk pro- file estimated by Garvin and Cheah (2004), which assumed a binomial distribution. That is, the pessimis- tic ADT (10,000) and optimistic ADT (34,000) had a probability of occurrence of 39% and 61%, respect- ively (Figure 7b). This resulted in a normalized cost of risk (H) equal to 23.2% of the expected total future ADT demand. Because future revenues are linked to the initial traffic demand, it follows that this is a risk that would manifest in the first year of operation when the initial traffic volume is materialized. Hence, the initial ADT risk would equal to sum of all the expected ADT, that is:

RADT ¼ H X40 t¼1

365 � Tolli � ADTt 1 þ rð Þt

(4)

For the year-to-year traffic variability, the forecasted ADT was assumed to fit a normal distribution and the cost of risk associated with an attaining a lower-than- forecasted average ADT was estimated using the expression for a put option for a normal distribution derived by Charlin and Cifuentes (2019) setting the strike price equal to the forecasted average ADT. The put option (i.e., expected value of the downside) was estimated as H ¼ 14.8% of the forecasted average ADT (i.e., the annual cost of risk is equal to 14.8% of the annual revenue).

The calculated cost of risk for these two sources (i.e., annual ADT demand and initial ADT accuracy) are summarized in Table 5. As shown, the calculated net

revenues from the cashflow analysis in Table 4 is also included in the first row of Table 5. The present value (discounted at the real risk-free rate of 3%) of the cost of risks for annual ADT growth rate and initial ADT are $423.6 and $277.9 million, respectively. The project DNPV is estimated to be $95.2 million (higher than the calculated NPV). The DNPV > 0 analysis indicates that the project should be executed. The equivalent real RADR is 8.0% (i.e., the discount rate that applied to the net revenues calculated in Table 4 would result in an NPV of $95.2 million). This corresponds to a nominal RADR of 11.0%, lower than the 15.6% origin- ally considered by the project developers.

This explicit identification of the cost of risk allows stakeholders to better understand the risk sources and their impact on project value. For this example, where only two risk sources were considered, if deferring the project by five years were to result in having 100% certainty of the initial traffic demand, the cost of risk associated with initial traffic demand would be nil and the project DNPV at t ¼ 5 would be $518.8 million (i.e., $95.2 þ $423.6). The project DNPV at t ¼ 0 would thus be $447.4 million. It is noted, however, that deferring construction will likely be insufficient to eliminate the uncertainty surrounding the initial ADT. A more realis- tic assumption would be that deferring the project would result in a reduction of the downside because of having better/more traffic data. Furthermore, if all that is needed to resolve traffic uncertainty is to defer projects for a few years, public entities would be bet- ter off by owning the initial ADT risk and negotiating

Table 5. Decoupled net present value analysis – additional risks included (in millions of $unless noted). Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .. 42

Net Revenue �139.5 �139.5 5.6 6.4 7.3 11.0 12.5 17.8 19.1 17.4 24.4 24.4 27.9 34.1 36.5 29.7 45.7 48.9 52.3 45.5 .. 93.5 Risks .. 1-Annual ADT 2.2 2.5 2.8 3.6 4.1 5.2 5.5 5.9 6.7 7.2 7.7 8.9 9.5 10.2 11.4 12.2 13.1 14.0 .. 23.4 2-Initial ADT 463 3-CAPEX 14.0 14.0 4-OPEX 0.23 0.26 0.29 0.33 0.38 0.43 0.46 0.50 0.53 0.57 0.61 0.65 0.70 0.74 0.80 0.85 0.91 0.98 .. 1.63 5-Revenue loss 0.20 0.23 0.26 0.34 0.38 0.49 0.52 0.56 0.63 0.67 0.72 0.83 0.89 0.95 1.07 1.15 1.23 1.31 .. 2.19 6-Repairs 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 .. 1.12 Decoupled CF1 �139.5 �139.5 �459 3.9 4.5 7.4 8.4 12.6 13.5 11.5 17.7 17.2 20.2 25.2 27.0 19.5 34.2 36.6 39.2 31.5 .. 27.1 Decoupled CF2 �153.5 �153.5 �459 3.7 4.2 7.1 8.1 12.2 13.0 11.0 17.1 16.6 19.6 24.6 26.3 18.7 33.4 35.8 38.3 30.5 68.5 Decoupled CF3 �153.5 �153.5 �462 1.0 1.4 4.2 5.1 9.0 9.8 7.6 13.6 13.1 15.9 20.7 22.3 14.6 29.0 31.2 33.6 25.7 .. 61.8 1. Calculated considering risks 1 and 2 (i.e., annual ADT and Initial ADT). DNPV ¼ $95.2 million. 2. Calculated considering risks 1 through 4. DNPV ¼ $48.3 million. 3. Calculated considering all six identified risks. DNPV ¼ �$2.1 million.

Table 4. Modified cashflows (in millions of $unless noted). Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .. 42

ADT(�1000) 20.0 22.8 26.0 29.6 33.8 38.5 41.2 44.1 47.2 50.5 54.0 57.8 61.8 66.2 70.8 75.8 81.1 86.7 .. 145.0 Toll/Vehicle ($) 2.00 2.00 2.00 2.25 2.25 2.50 2.50 2.50 2.65 2.65 2.65 2.85 2.85 2.85 3.00 3.00 3.00 3.00 .. 3.00 gross revenue 0 0 14.6 16.6 19.0 24.3 27.7 35.1 37.6 40.2 45.6 48.8 52.2 60.1 64.3 68.8 77.5 82.9 88.8 95.0 .. 158.8 OPEX 0 0 9.0 10.3 11.7 13.3 15.2 17.3 18.5 19.8 21.2 22.7 24.3 26.0 27.8 29.8 31.9 34.1 36.5 39.0 .. 65.3 CAPEX 139.5 139.5 0 0 0 0 0 0 0 3 0 1.7 0 0 0 9.4 0 0 0 10.4 .. 0 Net Revenue �139.5 �139.5 5.6 6.4 7.3 11.0 12.5 17.8 19.1 17.4 24.4 24.4 27.9 34.1 36.5 29.7 45.7 48.9 52.3 45.5 .. 93.5

CONSTRUCTION MANAGEMENT AND ECONOMICS 269

a lower return on investment for the private investors, as their risk would be lowered.

In general, developers would not be comfortable taking on initial ADT risk as this parameter depends on many factors that are not within their control (e.g., regional economic growth, future construction of alternative routes). Moreover, since the risk associated with initial ADT is linked to a potential excess capacity that may not be needed for several years or even dec- ades, rather than providing minimum ridership guar- antees to attract private investment, it may be more advisable if public entities require developers to adopt a phased approach to better forecast future traffic demand and reduce initial ADT risk, even if such an approach would result in higher CAPEX. Better yet, public entities could require developers to provide separate pricing for building the entire infrastructure at once or using a phased approach. An increase in CAPEX would represent the cost to acquire the option to build the road in phases.

If the main source of uncertainty was the initial ADT, then scaling the project down to reduce the ini- tial investment and collect actual traffic information would have been an appropriate response to address low traffic demand risk. Furthermore, if minimum guarantees or similar mechanisms aimed at reducing low initial ridership risk had been provided to attract private investment, public entities should not afford investors large risk premiums (i.e., high RADRs) that compensate them disproportionately for a risk that has largely been mitigated. The touted benefits of PPPs for public entities include attracting upfront pri- vate capital and taking advantage of the private sec- tor’s construction and operational expertise to render a public service at a lower total price. Therefore, over-

compensating private investors for initial ADT uncer- tainty, especially if developers are not bearing the low ridership risk, is not a good use of the PPP contract- ing structure.

To further illustrate the ability of DNPV to deal with different risks, other risks (e.g., construction risks, oper- ational risks) associated with large infrastructure proj- ects were considered in this analysis to allow a better comparison to the original analysis. To simplify the discussion, it is assumed that CAPEX overrun risk (Risk 3) is represented by 8.6% of the initial estimate (Flybvjerg et al. 2002) whereas annual OPEX is assumed to vary þ/� 15%, with this variation repre- sented by a triangular distribution. The OPEX cost of risk (Risk 4) is estimated to be 2.5% of the annual expected cost. The estimated DNVP considering Risks 1–4 is calculated to be 48.3 million (Table 5). The back calculated real RADR is 8.8% (i.e., 11.9% nominal). This value can be viewed as the investment risk signature that can be used to convey the riskiness of the investment.

Although not part of the original investment ana- lysis, to illustrate how physical risks can be incorpo- rated within the DNPV framework, the financial impact associated with climate change is also included using hypothetical values of probability of occurrence and impact. To that end, it is assumed that climate change could affect revenue collection due to a temporary shutdown (e.g., road collapse that would result in an extended road closure for repairs and therefore reduced toll collection). Let’s further assume that wea- ther data indicates that the annual probability (k) of an extreme rainfall event is equal to 2 in 100 years (or 2% per year) and the corresponding engineering esti- mates indicates that such an event could result in the

Figure 8. Evaluation of hypothetical downside climate change risk.

270 D. ESPINOZA ET AL.

loss of 35% of toll revenue due to an extended road closure associated with repair work. Under these assumptions, the cost of risk is estimated to be 1.4% of the annual revenues (Figure 8). The cost of risk for revenue losses associated to this event for Years 3–42 are included in Table 5 (Risk 5). Let’s further assume that such an event would generate infrastructure dam- ages with an estimated repair cost equal to 20% of the initial CAPEX (Risk 6). This would result in an annual cost of risk of $1.1 million (i.e., 2% � 0.2 � $279). The estimated DNPV considering these two additional risks is �$2.1 million. Note that for the parameters selected the total cost of risk due to climate change is $50.3 million (i.e., $26.0 million associated with revenue losses and $24.3 million related to infrastructure repairs). The cost of risk associated with climate change could then be used to evaluate options to protect future revenues as well as infrastructure repairs (e.g., buy insurance protection, implement resilient features to the design that would reduce the potential for a temporary shutdown).

Summary and conclusions

The main purpose of this paper was to establish a connection between the decoupled net present value (DNPV) methodology and prospect theory. As demon- strated, DNPV combines multiple concepts of prospect theory and the certainty equivalent method (CEM).9

The main feature borrowed from CEM is separation of the time value of money (represented by the risk-free rate) and risk, whereas the main features borrowed from prospect theory are investors’ loss aversion and asymmetric attitudes towards gains versus loses. DNPV accurately captures this asymmetry with its cost-of-risk concept. Furthermore, DNPV is shown to be compat- ible with financial derivative concepts to calculate the cost of risk, as the potential downside depends on (i.e., is derived from) the cashflow profile and thus is endogenous to the project. In other words, using DNPV the cost of risk is project dependent and inde- pendent of investors’ risk preferences. Although not necessary to successful application of DNPV, this paper also illustrates how an equivalent discount rate can be calculated for different risk profiles. The equivalent dis- count rate can therefore be viewed as risk signature that summarizes the project risk in a single calculated parameter, which is in stark contrast with current NPV- based practices whereby infrastructure investment risk is exogenously calculated by CAPM or similar means independent of the project’s risk characteristics. Once project risks have been identified and valued using

DNPV, risk preferences and investors’ understanding of these risks can be accounted for to more effectively manage (i.e., avoid, reduce, mitigate transfer, or retain) them. Connecting prospect theory and derivatives establishes a link between market and non-market risks, allowing users to quantify risk regardless of its source in the same manner.

Investing at any stage can be viewed as an option to be exercised if the present value of the estimated future cashflows discounted at the risk-free rate is greater than, or equal to, the cost of risk. Managerial flexibility (e.g., options to expand, delay, contract, or switch) can be easily accommodated as a suite of add- itional DNPVs. For any option to add value to the overall investment, the additional DNPV must be posi- tive. This is in stark contrast to the expanded NPV con- cept promoted to calculate real options, which assumes that all options are positive and therefore valuable, implicitly neglecting the cost to acquire an option.

A case study from the literature was re-evaluated to illustrate the power and relative simplicity of the DNPV method. The case study discusses how DNPV quantifies risks, which in turn could be used to struc- ture elaborated contractual arrangements including public-private partnerships with transparent risk alloca- tions among different partners. The case study was extended to illustrate how climate change risk may affect the estimated DNPV and how the cost of risk associated with climate change may be used to argue in favor of investing in resilience.

In summary, the main features of the DNPV meth- odology are: (i) separation of risk and the time value of money; (ii) accounting for the time value of money using the risk-free rate; (iii) defining market and non- market cost of risk as the expected negative deviation from the expected cashflow (i.e., the potential loss), so that the higher the potential loss, the higher the potential cost of risk; (iv) considering project risks as costs to the project and calculating them as synthetic insurance premiums designed to protect against a potential shortfall in expected cashflows; (v) defining “riskless” cashflows as the expected cashflows minus the synthetic insurance premiums (i.e., the cost of risk); (vi) discounting the resulting riskless cashflows using risk-free rates; (vii) equating investors to insur- ance/hedge providers of the project’s forecasted cash- flows; and (viii) compensating investors with the estimated cost of risk for their assumed project risks. Most powerfully, DNPV assigns risk where it belongs, to the variability in cashflows rather than project returns. This feature provides for a more equitable

CONSTRUCTION MANAGEMENT AND ECONOMICS 271

allocation of risk and reward amongst project stakeholders.

A key challenge and limitation in the application of DNPV is that accuracy of a cost of risk calculation is directly related to the cashflow distribution profile accuracy. Depending on the type of risks involved, accurate calculation of the cost of risks can have vary- ing difficulties and effort requirements and may suffer from data scarcity. Future research should be devoted to collect, compile and process relevant industry-wide, risk-specific databases that can be used to construct PDFs of revenues/expenditures such that better and more transparent quantification of the cost of risk can be performed. Data analytics may be used to improve the accuracy of such PDFs. The reward of this added effort will be a better understanding of investment risks and the effects that different risk management strategies may have on investments.

Disclosure statement

No potential conflict of interest is reported by the authors.

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplemen- tary materials.

Notes

1. A discount factor applied to negative cash flows would have the opposite effect (i.e., it would reduce the liability), hence it would increase the project NPV.

2. The risk factor depends on the investor’s degree of risk aversion. For risk-neutral investors, k ¼ 1 whereas for individuals that cannot tolerate any degree of uncertainty k¼0.

3. In reality, if the option is fairly priced such that there is no risk-free arbitrage opportunity, then the value of the option minus the price paid for it should be zero.

4. The downside for expenditures is opposite to that of revenues, that is, the downside is to the right of the expected value and cost of risk is added to the expenditures expected value.

5. Higher levels of risk neutrality can be defined in a similar fashion.

6. Readers familiar with derivatives can relate Line 1 to a call option and also relate DNPV with real options; however, despite the equivalence, the consistent treatment of the downside risk in the DNPV method to obtain certainty equivalents for the cash flow renders it superior to real options.

7. Put options are financial instruments that work as insurance products that protect stock owners against a potential drop below a specified value.

8. Although not used in the analysis, a review of the census data indicates that the region indeed grew between 1990 and 2010 from 86,000 to 315,500 which equated to an annual population growth rate of 6.5%. This rate was consistent with the assumed long-term traffic growth of 7% assumed the original traffic projections in Table 2.

9. The main difference of the proposed cost of risk with those presented elsewhere for market risk (e.g., Fama, 1977) is that in the proposed DNPV procedure the cost of risk is derived from information of the asset under consideration and not from its correlation with the overall market.

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Appendix A

The capital markets are a reliable social laboratory where the cost of risk concept for the average loss-averse investor can be tested. The insurance concept to define risk as the loss of value of individually traded assets was introduced by Galai (1977) to describe financial options (i.e., puts and calls). At-the-money future put options represent the average value of an insurance premium (i.e., the cost of risk) that the market assigns to uncertain (i.e., fluctuating) asset prices to protect investments from falling below expected future value. To facilitate the subsequent DNPV discussion, the fol- lowing three propositions are introduced.

Proposition I: Average investors10 in options of traded securities (i.e., puts and calls) are Risk Neutral Level 0 (or simply, risk neutral), that is, the expected return of investing in derivatives is the risk-free rate.

Proposition II: Assuming that ~RtþDt is the future value at time tþDt of the cost of risk of a traded security represent- ing the risk that the StþDt � XtþDt where StþDt is the expected cashflow of the security and XtþDt ¼ Xtð1 þ rDtÞ is the future value of the exercise price and ~Rtþt¼tþDt~Rtð1 þ rDtÞ; then average loss-averse investors in publicly traded assets are Risk Neutral Level I, that is, the average return of a stream of cashflows represented by (~St�~Rt) is the risk-free rate. In other words, the expected cashflows minus the cost of risk should be, on aver- age, riskless.

Proposition III: An investment is considered suitable if the expected cashflows discounted at the risk-free rate are greater than the cost of risk (i.e., DNPV � 0).

DPNV ¼ ~StþDt�~XtþDt 1 þ rDtð Þ �

~Rt � 0 (A.1)

The demonstration of Propositions I and II follows from the no-arbitrage argument which requires that expected changes in a call option (c) must return the risk-free rate (Shackleton and Sødal, 2005), that is:

E dc½ � ¼ rc dt (A.2) where E[.] is the expected value operator. Equation (A.2) is written in discrete form as:

ctþDt � ct ¼ rctDt (A.3) Using the put-call parity relationship for European

options at time t:

pt þ St e�dDt ¼ ct þ XtþDte�rDt (A.4) where pt is the put option and d is the yield. Setting the exercise price at time t equal to the current value of the traded asset (i.e., Xt ¼ St), the future value of the exercise

price (XtþDt) can be calculated as St exp[(r – d)Dt]. This value represents a reference future value for the status quo. Replacing this value in Equation (A.4), a unique relationship between at-the-money call and put options are obtained (i.e., pt ¼ ct). Hence, using the call option definition to describe its value at time tþDt (i.e., ctþDt ¼ ~StþDt � ~XtþDt), and replacing pt ¼ ct on the left-hand side, Equation (A.2) can be written as:

~StþDt�pt 1 þ rDtð Þ ¼ ~XtþDt ¼ ~Xt 1 þ rDtð Þ (A.5) Since put options can be interpreted as the expected

downside times the probability of occurrence (Carmichael et al., 2011), for the particular case of at-the-money put options, pt is essentially equivalent to the present worth of the cost of risk (an insurance that protects the potential downside) discounted at the risk-free rate. Thus, Equation (A.5) establishes that the return of uncertain future cash- flows minus their cost of risk ~RtþDt ¼ ptð1 þ rDtÞ is equal to the risk-free rate. This demonstration completes the proof of Proposition II.

It follows from the definition of risk neutrality level dis- cussed above that investors in traded assets with well- defined PDFs behave as Risk Neutral Level I investors (i.e., the expected value of the downside constitutes a fair esti- mate of the cost of risk). Furthermore, using the cost of risk definition, Equation (A.5) is rewritten in today’s currency as:

~StþDt�~XtþDt�~RtþDt 1 þ rDtð Þ ¼

~StþDt 1 þ rDtð Þ �

~Xt � pt ¼ 0 (A.6)

Then, for each time period, the left-hand side of Equation (A.6) is equivalent to the net cashflow of Equation (2). Since Risk Neutral Level I individuals invest if DNPV ¼ 0, then such investors would certainly invest if DNPV > 0, thus complet- ing the proof of Proposition III. The DNPV � 0 investing rule is equivalent to recommending investing when the present value of the expected future cashflows (discounted using the risk-free rate) is greater than or equal to the present value of the cost of risks. The cost of risk provides a measure of the potential variation between the actual and the expected cashflows. Thus, calculating the cost of risk as the expected downside and subtracting it from expected future cashflows results in expected riskless cashflows that can be discounted using the risk-free rate.

It is noted that, for traded securities, Equation (A.6) is exact as Dt ! 0 (i.e., the asset is continuously hedged with at-the-money put options). For discrete time intervals, Equation (A.6) is valid on average, that is, the portfolio return is not always the risk-free rate but is so on average over multiple trading periods. Hence, the return of a port- folio comprised of traded securities minus their correspond- ing at-the-money put options at any time may be higher/ lower than the risk-free rate, but on average, the return should be the risk-free rate. Hence, the value of a real pro- ject with market and non-market risks can be calculated using Equation (A.6).

274 D. ESPINOZA ET AL.