Project Net Present Value estimation under uncertainty
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Abstract
The paper contains a description of a possible modification of the original Net Present Value which allows one to evaluate projects under uncertainty with unknown probabilities (understood mainly as frequencies). Cash flows are usually uncertain since both incomes and expenditure related to the project concern the future. Additionally, probabilities of particular scenarios may be unknown due to many factors (e.g. the diversity of definitions for probability, lack of historical data, lack of sufficient knowledge about possible states of nature). The novel approach is based on a hybrid of Hurwicz and Bayes decision rules and is supported by a sensitivity analysis. The new method applies scenario planning and takes into account the decision maker’s attitude towards a given decision problem (measured by coefficients of pessimism and optimism). The procedure can be used even in the case of asymmetric distributions of net cash flows at particular periods since it considers the frequency of each value. The modification of the Net Present Value may support any uncertain multiperiod economic decision.
Keywords
Project Net Present Value Decision maker’s nature Uncertainty Unknown probabilities (frequencies) Decision making Sensitivity analysis1 Introduction
Projects may be evaluated and compared according to many statistic and dynamic methods of investment profitability assessment. One of them is the Net Present Value (NPV), which was formalized and popularized by Fisher (1907). This measure is computed on the basis of all foreseen and discounted revenues and costs over the lifetime of the project. The traditional version of NPV treats future cash flows as certain (deterministic) values. Nevertheless, many contributions devoted to the NPV estimation are based on the assumption that those data are uncertain, which is totally justifiable since both incomes and expenditure related to the project concern the future (except for the cash flow at moment 0). There are several different methods designed for taking uncertainty in such calculations into account, for instance: (1) to increase the discount rate, (2) to apply sensitivity analysis, (3) to compare pessimistic and optimistic cash flows or, (4) to estimate the expected cash flows by means of scenario planning and the probability distribution.
Note that the last procedure may be applied on condition that the decision maker knows or is able to estimate the likelihood of particular scenarios (states of nature, events). Meanwhile, it is sometimes quite complicated to compute probabilities due to the existence of many discrepant definitions of probability (Carnap 1950; Frechet 1938; Hau et al. 2009; Knight 1921; Kolmogorov 1933; Piegat 2010; Popper 1959; van Lambalgen 1996; von Mises 1949, 1957), the lack of historical data in the case of totally new decisions and events (Guo 2011; Guo and Ma 2014; GasparsWieloch 2015c, d, 2016a, b, 2017a, b), the lack of sufficient knowledge about particular states or the fact that the set of possible scenarios forecasted by experts in the scenario planning stage does not satisfy the probability axioms—the sum of state probabilities should be equal to 1, the whole sample space must be precisely defined (see, Kolmogorov 1933). Furthermore, Finetti (1975) argues that objective probabilities do not exist: “No matter how much information you have, there is no scientific method to assign a probability to an event”, there are only subjective probabilities—different for particular decision makers. However, according to Caplan (2001), people may be even unable to declare subjective probabilities—they just implicitly set probabilities when acting. Additionally, according to von Mises (1949), the theory of probability can never lead to a definite statement concerning a single event—the probability of a single event cannot be presented numerically. Roszkowska and Wachowicz (2015) also point out that people prefer ordinal measures to cardinal ones.
Despite the fact that the probability, in some circumstances, may be difficult to estimate, supporters of the theory of economics stress that some probabilitylike quantities can be often estimated and applied. Hence, in many cases, except for the aleatory uncertainty which is not reducible due to its random nature (Tannert et al. 2007; Zio and Pedroni 2013), uncertainty may be measured and quantified somehow (Piasecki 2016). Therefore, we would like to propose a new approach which can be used for project Net Present Value estimation under uncertainty with unknown objective probabilities. The procedure described in the paper will allow the decision maker (DM) to use scenario planning and to take into account his or her risk aversion measured by coefficients of optimism and pessimism. We assume that the novel procedure is designed for the selection of projects performed only once (see oneshot decisions, GasparsWieloch 2015a, 2016a, b, 2017a, b, c; Kofler and Zweifel 1993), since the choice of a project from the same set of potential projects in the future requires rescenario planning and a new declaration of the level of optimism.
The paper is organized as follows. Section 2 deals with the main features of the traditionally understood NPV method. Section 3 concerns the NPV technique with uncertain parameters. Section 4 describes the problem in the context of uncertainty with unknown frequencies. Section 5 presents a decision rule that may be used as a tool in evaluating NPV under uncertainty. Section 6 provides a case study. Conclusions are gathered in the last part.
2 Net Present Value
The Net Present Value or Net Present Worth (Lin and Nagalingam 2000; Berk et al. 2015) is defined as the sum of the present values of incoming (benefits) and outgoing (costs) cash flows over a period of time. NPV can be described as the difference between the sums of discounted cash inflows and cash outflows. According to the investment profitability assessment method based on NPV, a cash flow today is more valuable than an identical cash flow in the future because a present flow can be invested immediately and begin earning returns, while a future flow cannot (Berk et al. 2015).
Equation (1) allows one to compute the sum of the discounted net cash flows. Equation (2) enables one to set the difference between the present value of benefits and the present value of costs. Formula (3) is a simplified version of Eq. (2) and it may be applied to situations with only one expense at the beginning of the project. The construction of NPV is based on a simplifying assumption that the net cash received or paid occurs in a single transaction on the last day of each period (https://www.boundless.com/finance).
NPV can also be written in a continuous variation (Buser 1986; Grubbström 1967), but in this paper we investigate only the discrete case. The original version of NPV applies a constant discount rate, which is suitable for shortterm projects. The use of a variable discount rate is desirable when appreciating longterm investments (Fabozzi and Fong 1994; Piasecki and RonkaChmielowiec 2011). Here, we focus on constant discount rates. Due to the fact that the decision maker can reinvest particular cash flows, the true NPV may be higher. Therefore, the Modified Net Present Value has been proposed (Chandra 2009; Filho et al. 2012; McClure and Girma 2004). In this contribution we concentrate on NPV, but the reinvestment factor can be easily introduced.
3 Net Present Value under uncertainty

under certainty (DMC) where parameters are deterministic,

under risk (DMR) where possible scenarios and their likelihood are known,

with partial information (DMPI) where possible states of nature are known, but their probability is known incompletely: the DM only knows the order of scenarios or the intervals with possible probabilities for each scenario,

under complete uncertainty (DMCU) where the scenarios are known, but the probability of their occurrence is not, or

under total ignorance (DMTI) where the DM is not able to define possible events.
In this paper we consider both the epistemic and aleatory uncertainty, which leads us to a conclusion that the likelihood of the occurrence of particular states of nature cannot be estimated in an accurate way. Additionally, owing to the fact that the contribution concerns oneshot decisions only (the selected project is performed only once, i.e. only one state of nature has the chance to occur within a given period), we refer in a sense to the Austrian approach where the probability understood as frequency cannot be computed for a single event. The theory of economics is also partially applied in this research since unknown probabilities are going to be replaced by some probabilitylike quantities.
Uncertainty in NPV has been already taken into account in many ways. For instance, one can increase the discount rate (Method I). Nevertheless, some researchers state that it is not a reasonable approximation since the increased discount rate reduces the impact of potential losses below their true financial cost. Actually, the increased discount rate reduces the impact of potential losses, but the use of a higher discount rate signifies that the DM intends to earn more and that he/she is more willing to risk (lose) his/her money—usually more profitable investments are riskier. On the other hand, if the DM applies a lower discount rate, that means that he/she intends to earn less and that he/she does not accept the possibility of the occurrence of significant losses—usually less profitable investments are less risky. Thus, in our opinion, the level of the discount rate reflects the DM’s nature (riskprone behavior, riskaverse behavior), not the degree of uncertainty.
Another way consists in compounding the risk premium with the risk free rate (Method II). However, as a result, future cash flows are discounted by both rates, which entails an extremely low NPV.
The sensitivity analysis may be also applied to NPV (Method III), which enables one to check how NPV varies depending on the level of particular cash flows. Furthermore, it gives the possibility to set the interval for a cash flow at a given moment, within which NPV remains positive or higher than the NPV of another project.
When the DM knows possible scenarios at particular moments, it is recommended to refer to scenario planning (Pomerol 2001; Van der Heijden 1996). One can, for example, compare pessimistic and optimistic cash flows (Method IV), and, if the likelihood of each state of nature is estimated, the expected NPV may be calculated. The latter method (rNPV: riskadjusted Net Present Value) consists of the following steps: (1) computing the expected net cash flow \(E({ NCF}_{t})\) and standard deviations \(({ SD}_{t})\) at particular moments, (2) calculating the expected NPV, i.e. \(E({ NPV})\), and the mean standard deviation (MSD) for the project, (3) computing the coefficient of variation CV (the quotient of MSD and E(NPV))—the lower CV is, the less risky the project is (Method V).
When probabilities are not known, but the mean net cash flows \((MNCF_{t})\), standard deviations \((SD_{t})\) and correlations between sequences of cash flows from different periods (\(\rho _{t,s}\)) are given, \(MNCF_{t}\)values are treated as \(E(NCF_{t})\) and the whole standard deviation is computed on the basis of the partial standard deviations and the aforementioned correlations (Method VI).
The NPV estimation under uncertainty may be also supported by fuzzy numbers and interval arithmetic (Chiu and Park 1994; Gutiérrez 1989; Filho et al. 2012) (Method VII).
The above list of procedures designed for NPV calculation under uncertainty is not exhaustive, but allows one to be aware of the impressive variety of the existing approaches. In Sect. 4 we will concentrate on NPV in the context of scenario planning, risk aversion and uncertainty with unknown frequencies.
4 Scenario planning, risk aversion and NPV under uncertainty
Now, let us analyze the case of NPV estimation when the DM knows possible scenarios at particular moments and net cash flows connected with them, but the likelihood, mainly understood as frequency, is not known. Additionally, we assume that the DM is not able to define particular subjective probabilities exactly since his knowledge about future scenarios is not sufficient—the assessed projects are new. The result of the choice made under uncertainty depends on two factors: which decision (project, investment) is selected (internal factor) and which scenarios will occur in the future (external factor).
The uncertain cash flow matrix for a set of projects.
Source: Prepared by the author
Periods\(\backslash \)investments  \(\hbox {I}_{1}\)  \(\hbox {I}_{{\mathrm{j}}}\)  \(\hbox {I}_{{\mathrm{n}}}\) 

0  \(\hbox {NCF}_{0,1}\)  \(\hbox {NCF}_{0,{\mathrm{j}}}\)  \(\hbox {NCF}_{0,{\mathrm{n}}}\) 
1  \(\hbox {NCF}^{1}_{1,1}\)  \(\hbox {NCF}^{1}_{1,{\mathrm{j}}}\)  \(\hbox {NCF}^{1}_{1,{\mathrm{n}}}\) 
\(\hbox {NCF}^{{\mathrm{k}}(1,1)}_{1,1}\)  \(\hbox {NCF}^{{\mathrm{k}}(1,{\mathrm{j}})}_{1,{\mathrm{j}}}\)  \(\hbox {NCF}^{{\mathrm{k}}(1,{\mathrm{n}})}_{1,{\mathrm{n}}}\)  
\(\hbox {NCF}^{{\mathrm{m}}(1,1)}_{1,1}\)  \(\hbox {NCF}^{{\mathrm{m}}(1,{\mathrm{j}})}_{1,{\mathrm{j}}}\)  \(\hbox {NCF}^{{\mathrm{m}}(1,{\mathrm{n}})}_{1,{\mathrm{n}}}\)  
t  \(\hbox {NCF}^{1}_{{\mathrm{t}},1}\)  \(\hbox {NCF}^{1}_{{\mathrm{t,j}}}\)  \(\hbox {NCF}^{1}_{{\mathrm{t,n}}}\) 
\(\hbox {NCF}^{{\mathrm{k(t,1)}}}_{{\mathrm{t,1}}}\)  \(\hbox {NCF}^{{\mathrm{k(t,j)}}}_{{\mathrm{t,j}}}\)  \(\hbox {NCF}^{{\mathrm{k(t,n)}}}_{{\mathrm{t,n}}}\)  
\(\hbox {NCF}^{{\mathrm{m(t,1)}}}_{{\mathrm{t,1}}}\)  \(\hbox {NCF}^{{\mathrm{m(t,j)}}}_{{\mathrm{t,j}}}\)  \(\hbox {NCF}^{{\mathrm{m(t,n)}}}_{{\mathrm{t,n}}}\)  
T  \(\hbox {NCF}^{1}_{{\mathrm{T,1}}}\)  \(\hbox {NCF}^{1}_{{\mathrm{T,j}}}\)  \(\hbox {NCF}^{1}_{{\mathrm{T,n}}}\) 
\(\hbox {NCF}^{{\mathrm{k(T,1)}}}_{{\mathrm{T,1}}}\)  \(\hbox {NCF}^{{\mathrm{k(T,j)}}}_{{\mathrm{T,j}}}\)  \(\hbox {NCF}^{{\mathrm{k(T,n)}}}_{{\mathrm{T,n}}}\)  
\(\hbox {NCF}^{{\mathrm{m(T,1)}}}_{{\mathrm{T,1}}}\)  \(\hbox {NCF}^{{\mathrm{m(T,j)}}}_{{\mathrm{T,j}}}\)  \(\hbox {NCF}^{{\mathrm{m(T,n)}}}_{{\mathrm{T,n}}}\) 
Given such data, one could calculate the expected NPV on the basis of Method VI (see Sect. 3). Nevertheless, this time, we assume that the number of scenarios for a given period may be different for particular investments. Additionally, we will attempt to take into account the DM’s nature measured by the coefficients of pessimism (\(\alpha \)) and optimism (\(\beta \)). These parameters belong to interval [0, 1] and satisfy the condition \(\alpha +\beta =1\). \(\alpha \)(\(\beta \)) tends to 0 (1) for extreme optimists (riskprone behavior) and is close to 1 (0) for extreme pessimists (riskaverse behavior). Coefficients of pessimism and optimism have been already used in the decision making process, for instance in Hurwicz (1952) and Perez et al. (2015). In this contribution they will allow us to generate some probabilitylike quantities. Note that those quantities will not be directly estimated by the decision makers, but will be calculated on the basis of their level of optimism/pessimism. Hence, coefficients \(\alpha \) and \(\beta \) are the initial parameters and probabilitylike quantities constitute the secondary parameters.
There are numerous decision rules devoted to decision making under uncertainty (see, for example, Basili 2006; Basili et al. 2008; Basili and Chateauneuf 2011; Beauchene 2015; Chassein and Goerigk 2016; Ellsberg 2001; Etner et al. 2012; GasparsWieloch 2012, 2013, 2014a, b, c, d, 2015a, b, c, d, e, 2016a, b, 2017a, b, c, d; Ghirardato et al. 2004; Gilboa 2009; Gilboa and Schmeidler 1989; Halpern and Leung 2014; Hayashi 2008; Hurwicz 1952; Ioan and Ioan 2011; Marinacci 2002; Nakamura1996; Perez et al. 2015; Piasecki 1990; Savage 1961; Schmeidler 1986; Tversky and Kahneman 1992; Wald 1950), but some of them are based on the probability calculus or do notconsider the DM’s nature—thus, they cannot be applied to the aforementioned problem.
In Sect. 5 we are going to present the H+B decision rule for NPVU (Net Present Value under uncertainty). The original version of that procedure, i.e. the H+B rule (a hybrid of Hurwicz and Bayes rules), is described in GasparsWieloch (2014a, 2015b, c, 2016b). It is designed for decision making under uncertainty with unknown probabilities. Thanks to parameters \(\alpha \) and \(\beta \), the H+B rule enables one to take into consideration the DM’s attitude towards a given problem The procedure is devoted to searching optimal pure (not mixed) strategy and to solving oneshot decision problems (Guo 2011). A pure strategy, as opposed to a mixed strategy, allows the DM to select and perform only one accessible alternative (GasparsWieloch 2015e). Oneshot decision problems are connected with decisions performed only once.
Of course, one should wonder why we do not intend to apply the original Hurwicz decision rule, which also uses \(\alpha \) and \(\beta \) and usually leads to sensible results. The disadvantage of that approach is related to the following factor. In some cases the Hurwicz rule provides answers which are contradictory to the logic and do not reflect the decision maker’s preferences. Such a phenomenon stems from the fact that the Hurwicz criterion takes extreme payoffs into consideration only—transitional values connected with a given decision are ignored. Additionally, the above rule does not examine the frequency of relatively high and low payoffs belonging to the set of all profits assigned to particular alternatives (see, GasparsWieloch 2012, 2014a, c, 2016b). In the H+B rule, in contrast to the Hurwicz approach, all outcomes have an influence (though not the same) on the value of the final measure. Hence, the H+B approach recommends logic rankings for both symmetric and asymmetric distributions of payoffs. And that is why it may be useful in the evaluation (choice) of investment projects on the basis of NPV under uncertainty. Note that the original H+B decision rule needs to be adjusted to that goal (i.e. NPV estimation) since, this time, payoffs connected with particular projects come from different periods.
5 The H+B decision rule for NPV under uncertainty (NPVU)
 1.
Define n(the number of projects) and generate the cash flow matrix for the whole set of projects, see Table 1.
 2.
Determine \(\alpha \) and \(\beta \) for a given problem. If \({\alpha \in [0,0.5)}\), then \({\alpha =\alpha _o ,\beta =\beta _o}\) (\(\alpha _{o}\) and \(\beta _{o}\) are the optimist’s coefficients). If \({\alpha \in (0.5,1]}\), then \({\alpha =\alpha _p ,\beta =\beta _p}\) (\(\alpha _{p}\) and \(\beta _{p}\) are the pessimist’s coefficients).
 3.
Find a nonincreasing sequence of net cash flows \(Sq_{t,j}^ =(a_{t,j}^1 ,...,a_{t,j}^s ,...,a_{t,j}^{m(t,j)} )\) for each project \(I_{j}\) and for each period \(t=1,{\ldots },T\), where \(a_{t,j}^s \ge a_{t,j}^{s+1} \) (\(s=1,{\ldots },m(t,j)1\)), m(t, j)—number of terms in the sequence, s—number of the term in the sequence.
 4.Calculate, for each project and each period, index \(hb_{{t,j}} (hb_{{t,j}}^p \), \(hb_{{t,j}}^o \) or \(hb_{{t,j}}^{0.5} \) depending on parameter \(\alpha \)). If \({\alpha \in (0.5,1]}\), calculate \(hb_{{t,j}}^p \) (index for pessimists) according to Eq. (4). If \({\alpha \in [0,0.5)}\), compute \(hb_{{t,j}}^o \) (index for optimists) following formula (5). If \(\alpha \) = 0.5, calculate \(hb_{{t,j}}^{05} \) using Eq. (6), where \(b_{t,j }\) denotes the Bayes criterion, i.e. the average of all net cash flows at a given period.$$\begin{aligned} hb_{{t,j}}^p= & {} \frac{\alpha _{p} \cdot a_{{t,j}}^{m(t,j)} +\beta _{p} \cdot \sum \nolimits _{s=1}^{m(t,j)1} {a_{t,j}^s } }{\alpha _{p} +(m(t,j)1)\cdot \beta _{p} } \end{aligned}$$(4)$$\begin{aligned} hb_{{t,j}}^o= & {} \frac{\alpha _{o}^ \cdot \sum \nolimits _{s=2}^{{m(t,j)}} {a_{{t,j}}^s } +\beta _{o}^ \cdot a_{{t,j}}^1 }{(m(t,j)1)\cdot \alpha _{o} +\beta _{o} } \end{aligned}$$(5)The denominators in Eqs. (4)–(5) are introduced so that the final value of particular indices belongs to interval \([a_{{t,j}}^{m(t,j)} ,a_{t,j}^1 ]\). If, for a given project, there is no net cash flow at period t, then \(hb_{{t,j}}=0\). In the case of period \(t= 0\), \(hb_{0,j}=NCF_{0,j}\).$$\begin{aligned} hb_{{t,j}}^{0.5}= & {} hb_{{t,j}}^p =hb_{{t,j}}^o =b_{{t,j}} =\frac{1}{m(t,j)}\cdot \sum \limits _{k(t,j)=1}^{m(t,j)} {NCF_{{t,j}}^{k(t,j)}} \end{aligned}$$(6)
 5.Compute, for each project, \({\alpha NPV_{j}}\) (i.e. the Net Present Value considering the DM’s nature) on the basis of formula (7).where r still denotes the discount rate, i.e. the rate of return that could be earned from an investment in the financial markets with similar risk.$$\begin{aligned} \alpha { NPV}_j =\sum _{t=0}^T {\frac{{ hb}_{t,j} }{\left( {1+r} \right) ^{t}}} , \end{aligned}$$(7)
 6.
 7.
If one or more projects satisfy Eq. (8) and its (their) \({\alpha NPV_{j}}\) value is not positive, reject all projects or decrease the level of the discount rate, if it is justifiable, and go to step 5. Otherwise, go to step 8.
 8.If more than one project satisfies Eq. (8) and their \({\alpha NPV_{j}}\)value is positive, calculate the mean standard deviation (nMSD) only for projects \(I^{*}_{j}\) following condition (9). Pessimists should choose the project with the lowest mean standard deviation. Moderate DMs and optimists may select a project \(I^{*}_{j }\)with more diffused cash flows (i.e. with a higher MSD).In the last part of Sect. 5 we would like to explain equations given in step 4 of the algorithm presented above in detail.$$\begin{aligned} \overline{{ MSD}_j } =\frac{1}{T}\sqrt{\sum _{t=1}^T {\left( {\frac{1}{m(t,j)}\sum _{s=1}^{m(t,j)} {\left( {a^{s}_{i,j} \overline{a_{t,j} } } \right) ^{2}} } \right) } } \end{aligned}$$(9)
It is worth emphasizing that in the procedure proposed in GasparsWieloch (2014a) the index value depends on the number of states of nature, which is not the case of Hurwicz rule. For pessimists, when the number of scenarios increases, the importance of payoff \(a_{j,min }\) in the index decreases and the significance of the remaining profits increases. On the other hand, for optimists, the importance of payoff \(a_{j,max }\) decreases and the significance of the remaining profits increases. Hence, again, we can observe the impact of Bayes rule in the analyzed hybrid, because the chance of the occurrence of a given event decreases along with the growth in the number of states of nature (GasparsWieloch 2016b).
Note that the sensitivity analysis may effectively support the H+B rule for NPVU. If possible, it is recommended to generate project rankings for different values of the discount rate and the coefficient of optimism. Given such a specification, the decision maker can make his or her final decision more consciously (see Method III in Sect. 3).
As it can be noticed, the H+B rule for NPVU contains an additional step (in comparison to the original H+B), i.e. the computation of the mean standard deviation. That element is characteristic of Method V (Sect. 3), but this time, we apply that measure only to the multiple solutions case.
In the described algorithm (H+B for NPVU) coefficients of pessimism/optimism are supposed to be constant, i.e. the same for each period. However, the DM may state that each period could be influenced by different factors, negative and positive, and that the level of those parameters should vary. The procedure may be easily extended and provides an opportunity to declare different values for \(\alpha \) and \(\beta \), i.e. \(\alpha _{1}\), \(\alpha _{2}\), ..., \(\alpha _{t}\), ..., \(\alpha _{T }\) and \(\beta _{1}\), \(\beta _{ 2}\), ..., \(\beta _{ t}\), ..., \(\beta _{ T }\)(step 2) and to apply different formulas (Eqs. 4, 5 or 6, step 4) for each period depending on the level of the aforementioned coefficients. Note that all coefficients must be given before the choice of the project.
6 Case study
The cash flow matrix for a set of projects (Example)—in thousand dollars.
Source: prepared by the author
Periods\(\backslash \)investments  \(\hbox {I}_{1}\)  \(\hbox {I}_{2}\)  \(\hbox {I}_{3}\) 

0  \(\) 50  \(\) 0  \(\) 120 
1  \(\) 30  0  \(\) 50 
\(\) 20  \(\) 60  
\(\) 40  \(\) 80  
0  
2  100  200  0 
130  170  
150  160  
140  80  
3  200  0  300 
50  50  250  
0  70  500  
150 
 1.
\(n=3\), the cash flow matrix is given in Table 2.
 2.
\({\alpha =\alpha _{p}=0.7}\), \({\beta =\beta _{p}=0.3}\) (here the DM declares constant parameters for each period).
 3.The nonincreasing sequences are as follows:

for \(\hbox {I}_{1}\): \(Sq_{1,1} =(0,20,30,40)\), \(Sq_{2,1} =(150,140,130,100)\), \(Sq_{3,1} =(200,50,0)\),

for \(\hbox {I}_{2}\): \(Sq_{2,2} =(200,170,160,80)\), \(Sq_{3,2} =(150,70,50,0)\),

for \(\hbox {I}_{3}\): \(Sq_{1,3} =(50,60,80)\), \(Sq_{3,3} =(500,300,250)\).

 4.Indices \(hb_{t,j}^p \) are computed in the following way (all the indices are presented in Table 3):$$\begin{aligned} hb_{1,1}^p= & {} \frac{0.7\cdot (40)+0.3\cdot (3020+0)}{0.7+3\cdot 0.3}=26.875,\\ hb_{2,1}^p= & {} \frac{0.7\cdot 100+0.3\cdot (150+140+130)}{0.7+3\cdot 0.3}=122.5,\\ hb_{3,1}^p= & {} \frac{0.7\cdot 0+0.3\cdot (200+50)}{0.7+2\cdot 0.3}=57.692. \end{aligned}$$Table 3
Indices \(hb_{t,j}^p \) (in thousand dollars).
Source: prepared by the author
Periods\({\backslash }\)investments
\(\hbox {I}_{1}\)
\(\hbox {I}_{2}\)
\(\hbox {I}_{3}\)
0
\(\) 50.000
\(\) 90.000
\(\) 120.000
1
\(\) 26.875
0.000
\(\) 68.462
2
122.500
134.375
0.000
3
57.692
50.625
334.615
 5.We assume that \(r=9\%\). Measure \({\alpha NPV_{j}}\) is computed below for each project. Table 4 presents the values of that measure for \(r\in [0.05,0.15]\).$$\begin{aligned}&\alpha { NPV}_1 =50\frac{26.875}{1.09}+\frac{122.5}{(1.09)^{2}}+\frac{57.692}{(1.09)^{3}}=73.0,\nonumber \\&\quad \alpha { NPV}_2 =62.19, \quad \alpha { NPV}_3 =75.58. \end{aligned}$$Table 4
Measures \({\alpha NPV_{j}}\) (in thousand dollars) for different discount rates (\({\alpha =0.7}\)).
Source: prepared by the author
Discount rate\(\backslash \)investments
\(\hbox {I}_{1}\)
\(\hbox {I}_{2}\)
\(\hbox {I}_{3}\)
0.15
57.19
44.89
40.48
0.14
59.63
47.57
45.80
0.13
62.14
50.32
51.32
0.12
64.72
53.16
57.05
0.11
67.40
56.08
62.99
0.10
70.15
59.09
69.16
0.09
73.00
62.19
75.58
0.08
75.94
65.39
82.24
0.07
78.97
68.69
89.16
0.06
82.11
72.10
96.36
0.05
85.35
75.61
103.85
Table 5Measures \({\alpha { NPV}_{j}}\) (in thousand dollars) for different values of \({\alpha (r=9\%)}\).
Source: prepared by the author
Coefficient of pessimism\(\backslash \)investments
\(\hbox {I}_{1}\)
\(\hbox {I}_{2}\)
\(\hbox {I}_{3}\)
0.60
89.34
77.91
84.46
0.65
81.54
70.51
80.18
0.66
79.89
68.93
79.29
0.67
78.21
67.30
78.38
0.68
76.51
65.64
77.46
0.69
74.77
63.94
76.53
0.70
73.00
62.19
75.58
0.71
71.19
60.40
74.61
0.72
69.35
58.57
73.63
0.73
67.48
56.68
72.64
0.74
65.57
54.75
71.63
0.75
63.62
52.76
70.60
0.80
53.25
41.99
65.21
 6.
\(I^{*}_{j}=I_{3}\). The DM can select project \(I_{3}({\alpha NPV_{3}}>0)\).
7 Conclusions
In this contribution we have described a possible modification of the original NPV in order to evaluate projects, and choose the best one, under uncertainty with unknown probabilities (probabilities are not treated as initial parameters of the decision problem). The proposed method allows one to apply scenario planning and to take into account the decision maker’s attitude towards a given problem (measured by coefficients of pessimism and optimism). The new procedure can be used even in the case of asymmetric distributions of net cash flows at particular periods. The novel method does not require the estimation of probabilities, which is extremely desirable especially in the case of totally new decisions (projects) and for passive decision makers who do not intend to analyze each scenario, period and value very meticulously. Coefficients of optimism and pessimism are used to generate some probabilitylike quantities, which coincides with the theory of economics, according to which for the majority of uncertain problems, unknown objective or subjective probabilities may be replaced by other measures in order to quantify uncertainty. The approach presented in the paper has been called H+B rule for NPVU since it is partially based on the original version of H+B rule, which constitutes a hybrid of the Hurwicz and Bayes decision rules and which is designed for oneperiod decision problems. The modified version of H+B rule enables one to consider multiperiod scenariobased decision problems.
We are aware of the fact that the suggested method may lead only to partially rational decisions since, due to some unknown factors concerning the future, decision makers possess only “bounded rationality” and have to make decisions by “satisficing” or choosing what might not be optimal, but will make them happy enough (Frish and Baron 2006; Simon 1957, 1991).
It is worth stressing that the H+B rule for NPVU may be applied rather on the assumption that the cash flow matrix is estimated by experts instead of the decision maker, since the values are objective^{2} and DM’s attitude towards a given problem is considered by means of coefficients of pessimism/optimism. If the aforementioned matrix is estimated by the DM, there is a risk that his/her attitude will be considered twice: in the cash flow matrix and in the coefficients of pessimism/optimism, which may distort real DM’s preferences (GasparsWieloch 2015d).
Note that, as in the case of the Hurwicz rule, the suggested decision rule is a subjective procedure since the coefficient of optimism/pessimism is estimated subjectively.
As part of the concluding remarks, we ought to discuss the problem of group (collaborative) decision making because investment decisions are usually made collectively. Thus, particularly in corporate reality, the procedure described in the contribution cannot be directly used in the decision making process. Nevertheless, it can hold a consultative (advisory) function as the final collective decision may constitute, for instance, the most frequent response from among the results obtained separately for each decision maker on the basis of that decision rule. Additionally, using the H+B rule for NPVU by particular members of a group facilitates the final collective choice of the best project since the original set of potential projects has the chance to be significantly reduced.
Another aspect, which needs to be mentioned in the paper, is related to the multistage character of the decision making process in the case of project selection. On the face of it, we could state that the H+B rule for NPVU is designed only for onestage decisions. However, it can be easily used in multistage group decisions and combined with the Delphi method, where it is believed that during consecutive rounds the range of the answers will decrease and the group will converge towards the “correct” answer. In the case of multistage individual decisions the use of the H+B rule for NPVU could be also advantageous. First stages can, for example, concern situations where the investor is not able to determine a precise degree of optimism and, instead, he/she declares this parameter as an interval. Further stages concern a less “uncertain” uncertainty. Hence, the parameter can be estimated more precisely.
Owing to the fact that the project selection is a task requiring high responsibility, entailing expenditure and timeconsuming execution, we would like to emphasize that the H+B rule for NPVU cannot be applied rashly. We strongly recommend that decision makers support the procedure with the sensitivity analysis. Thus, before making the final decision, it is desirable to check how the NPVs and the project rankings change after a slight modification of the level of the optimism coefficient.
The H+B rule for NPVU can be easily adjusted to further nondeterministic applications in areas such as the estimation of NPV with variable discount rates (for longterm investments), the NPV estimation with reinvested cash flows (Chandra 2009) and NPV assessment for maketoorder and maketostock manufacturing systems (Naim et al. 2007).
Footnotes
 1.
The converse process in DCF analysis consists in taking a sequence of cash flows and a price as input and inferring a discount rate as output, see Internal Rate Return (IRR). IRR or other efficiency measures are used as a complement to NPV because the latter tool does not provide an overall picture of the gain or loss of executing a certain project. Meanwhile IRR allows one to see a percentage gain relative to the investments in the project (https://en.wikipedia.org/wiki/Discounted_cash_flow, https://www.boundless.com/finance/).
 2.
The more objective nature of scenario planning in the case of experts results from the fact that experts are better prepared to perform particular steps of that process. They are able (thanks to various methods such as brainstorming, 80:20 rule, Important Uncertainties Matrix) to determine the most important factors (variables) that will decide the nature of the future environment. They have sufficient skill to assess the impact of a given factor on the remaining ones. And finally, they can define the set of possible scenarios.
Notes
Acknowledgements
This research is financed by the National Science Center in Poland (Project Registration Number: 2014/15/D/HS4/00771).
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