On Evaluation of Risky Investment Projects. Investment Certainty Equivalence
OOn Evaluation of Risky Investment Pro jects.Investment Certainty Equivalence
Andrey Leonidov ∗ a,b , Ilya Tipunin † a , and EkaterinaSerebryannikova ‡ a,baP.N. Lebedev Physical Institute, 53, Leninsky prospect, Moscow, Russia, 119333 b Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, MoscowRegion, Russia, 141701
May 26, 2020
Abstract
The purpose of the study is to propose a methodology for evalua-tion and ranking of risky investment projects. An investment certaintyequivalence approach dual to the conventional separation of risklessand risky contributions based on cash flow certainty equivalence is in-troduced. Proposed ranking of investment projects is based on gaug-ing them with the Omega measure, which is defined as the ratio ofchances to obtain profit/return greater than some critical (minimalacceptable) profitability over the chances to obtain the profit/returnless than the critical one. Detailed consideration of alternative risklessinvestment is presented. Various performance measures characterizinginvestment projects with a special focus on the role of reinvestmentare discussed. Relation between the proposed methodology and theconventional approach based on utilization of risk-adjusted discount ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ q -f i n . R M ] M a y ate (RADR) is discussed. Findings are supported with an illustrativeexample. The methodology proposed can be used to rank projects ofdifferent nature, scale and lifespan. In contrast to the conventionalRADR approach for investment project evaluation, in the proposedmethod a risk profile of a specific project is explicitly analyzed interms of appropriate performance measure distribution. No ad-hocassumption about suitable risk-premium is made. Keywords:
Investment appraisal; Ranking of investment projects;Certainty equivalence; Riskless alternative; Omega measure.
Evaluation and ranking of investment projects is one of the most importantproblems in corporate finance (Brealey et al., 2012). At the heart of thisproblem is a necessity of formulating investment goal taking into account thevariability of the project outcomes.A risky investment project is naturally defined as a project that couldbring return exceeding that of the alternative riskless investment generatingthe same cash flow pattern. A risk premium corresponds to return above theriskless one that, however, is not guaranteed, i.e. is uncertain. Therefore thepossible outcomes for risk premium are specified in terms of a probabilitydistribution that provides a quantitative description of its variability. Adecision to go for a project is thus contingent on the estimate of chancesof achieving investor’s profit benchmarks that depend on the shape of therisk premium probability distribution. To our knowledge, this idea was firstformulated in (Fishburn, 1977).Let us note that historically the mean return and its standard deviationwere used as project ”coordinates” in the risk-return space (Markowitz, 1959;Bouchaud and Potters, 2003). This works fine for symmetric return distri-butions which shape is close to that of the Gaussian (normal) distribution.However, it was soon realized that a much better characterization of risk-return space is achieved by replacing mean return and its standard deviationby the required return and some asymmetric risk measure taking into ac-2ount that the notion of risk is naturally related only to returns smaller thanthe required one (Bouchaud and Potters, 2003; Caporin et al.; Krokhmalet al., 2013; Laughhunn et al., 1980; Miller and Leiblein, 1996; Nawrocki,1999; Sortino and Satchell, 2001). For recent applications of this method toevaluation of complex investment projects see e.g. (Dimitrakopoulos et al.,2007; Leite and Dimitrakopoulos, 2007).Assessment of the quality of the project under consideration proceedsthrough two basic steps: • specification of the acceptable risky profit range, in particular - of thelowest acceptable profit in terms of a risk premium defined with respectto an appropriate riskless benchmark, • quantification of chances for the risky profit to miss the acceptablerange (risk) as following from the analysis of variability of risk premiumcharacterized by the shape of its probability distribution.A project is thus characterized by a given (chosen by the investor) hurdlerisk premium and uncertainty associated with chances of its realization. Thisalso lays a basis for ranking several projects: this is done by ranging, for agiven threshold risk premium, values of some suitably defined measure ofmissing the target risk premium.A widely used approach to this problem is to use averaged cash flowsin combination with a risk-adjusted discount rate (RADR) (Brealey et al.,2012). It is, however, known that the RADR approach is not universal andis applicable only to investment projects possessing specific characteristics(Robichek and Myers, 1966; Myers and Turnbull, 1977; Fama, 1977), see also(Hull, 1986). In particular, in (Robichek and Myers, 1966) it was shownthat using RADR one implicitly fixes very special structure of the investor’spreferences. This follows from comparison of the RADR estimate and theexpression obtained by using the certainty equivalence approach. An inter-esting recent development along these lines was described in (Espinoza andMorris, 2013; Espinoza, 2014).In the present paper we propose a method of separation of riskless andrisky contributions of investment project cash flows based on constructing3 replicating riskless investment for each possible cash flow realization. Weterm the corresponding principle Investment Certainty Equivalence. This is,in particular, to stress that an important question of reinvestment shouldbe treated separately, see e.g. a recent discussion in (Cheremushkin, 2012).With respect to interrelation between the RADR and certainty equivalenceapproaches it is shown that fixing some risk-adjusted discount rate implies acertain separation of riskless and risky contributions to the cash flows. As tothe quantitative assessment of risk/return profile we will use a particularlyinteresting asymmetric risk/return measure, the so-called Ω, considered in(Kazemi et al., 2004; Keating and Shadwick, 2002; Bertrand and luc Prigent,2011).The main objective of the paper is to describe a new methodology ofinvestment projects ranking. Therefore detailed description of sources ofrisks and of methodologies of their modelling as well as a detailed analysis ofmechanisms underlying projects cash flows are out of the scope of the currentstudy. However, some comments on classification of risk factors can be made.All the risks that influence efficiency of investment projects can be dividedinto two classes: cash flow risks and alternative investment/reinvestmentrisks. The first class contains such risk factors as variability of macroeco-nomic indicators, market prices or operational risks. Such risks exert directinfluence on project’s cash flows. The second class contains risks related tovariability of reinvestment rate. As this paper analyses riskless alternativeonly, the methodology of accounting for risks of the second class is out ofthe scope of the present analysis and consitutes an interesting direction forfuture studies.The paper is organized as follows.In the Section 2 we detail a procedure of evaluation and ranking of riskyprojects. In paragraph 2.1 we outline a general description of the key ingre-dients of an investment project as well as quantitative characteristics used inits assessment. In paragraph 2.2 we give a schematic outline of the evaluationprocedure considered in the the present paper. In paragraph 2.3 we constructa riskless portfolio replicating a given cash flow stream. In paragraph 2.4 wedescribe key quantitative characteristics used in evaluation of an investment4roject. In paragraph 2.5 we discuss risk premium measures that arise inthe approach under consideration. In paragraph 2.6 we describe a quantita-tive criterion suggested to make an investment-related decision based on therisk profile of an investment project. In paragraph 2.7 we derive formulaefor the critical/threshold values for the key quantitative characteristics of aninvestment project.In the Section 3 we discuss some quantitative aspects of comparison be-tween the conventional RADR approach and the approach discussed in thepresent paper. In paragraph 3.1 we outline the conventional RADR method-ology. In paragraph 3.2 we discuss a conventional certain equivalence ap-proach. In paragraph 3.3 we discuss the relation between the RADR ap-proach and the one suggested in the present paper.In the Section 4 we compare the results of ranking of two model invest-ment projects using RADR and suggested approaches correspondingly.In the Section 5 the example of ranking of real industrial projects isprovided.In the Section 6 we present our conclusions. In this paragraph we focus on the description of projects with the simpleststructure of cash flows with the single negative contribution corresponding toan initial investment. The generalization for the case of arbitrary structureof cash flows is described in Paragraph 2.3.In what follows we assume that a description of an investment project ofduration T to be assessed is available in the following form. A project involvestwo major interrelated processes, those of investment and reinvestment: • Investment
Investment process corresponds to a transformation of an initial in-vestment outlay I into one of the possible realizations { F ( i ) } of the5ash flow stream generated by the project I (cid:55)−→ F (1) = ( F (1)1 , · · · , F (1) T ) , · · · , F ( N ) = ( F ( N )1 , · · · , F ( N ) T ) , (1)or, in condensed notation, I (cid:55)−→ investment { F ( i ) } , i = 1 , · · · , N, (2)where cash flow streams F ( i ) = ( F ( i )1 , · · · , F ( i ) T ) correspond to N pos-sible outcomes that are, e.g., generated through Monte-Carlo simula-tion or scenario analysis. The uncertain nature of the project outcomemakes it natural to use a probabilistic description where, generically,a project is fully characterized by some multinomial probability distri-bution P F ( F , · · · , F T ): I (cid:55)−→ investment P F ( F , · · · , F T ) (3) • Reinvestment
Reinvestment process corresponds to transformation of cash flow streams { F ( i ) } into the set of terminal cash flows at maturity T { F tot( i ) T } throughreinvesting the components of { F ( i ) } into the same project or someother riskless/risky projects (reinvestment) providing (uncertain) for-ward profitability rates K f = ( K f → T , K f → T , · · · , K fT − → T ) describedby a probability distribution P K f ( K f ) such that for each intermediatecash flow we have F ( i ) t (1 + K ft → T ) = F t ot ( i ) T ≡ FV( F ( i ) t | K ft → T ) , (4) In general the partial cash flows F t can be both positive (profits) and negative (ad-ditional investments). Of course, reinvestment process operates only with intermediatepositive cash flows. F ( i ) t | K ft → T ) denotes the future value at time T of the partialcash flow F ( i ) t corresponding to the partial rate K ft → T . Therefore, foreach realization of the cash flow stream we have { F ( i ) } K f (cid:55)−→ reinvestment { FV( F ( i ) t | K ft → T ) } (5)or, in condensed notation, F K f (cid:55)−→ reinvestment FV( F | K f ) (6)Within probabilistic description the reinvestment process generates afinal probability distribution of terminal cash flow P F tot T ( F t otT ): P F ( F , · · · , F T ) P Kf ( K f ) (cid:55)−→ reinvestment P F tot T ( F t otT ) (7)More explicitly, P F tot T ( F t otT ) = (cid:90) d K f d F P K f ( K f ) P F ( F ) δ (cid:0) F t otT − FV( F | K f ) (cid:1) (8)The full description of an investment project can therefore be summarizedby the following superposition of investment and reinvestment processes: I (cid:55)−→ investment { F ( i ) } K f (cid:55)−→ reinvestment { F ( i )t otT = FV( F ( i ) | K f ) } (9)or, in probabilistic terms, as I (cid:55)−→ investment P F ( F ) P ( K f ) (cid:55)−→ reinvestment P F tot T ( F t otT = FV( F | K f )) (10)From the distribution of terminal cash flow P F tot T ( F t otT ) one can calculatethe distributions of the project profit Π T = F t otT − I P Π (Π T ) = (cid:90) dF tot T δ (Π T − F t otT + I ) P F tot T ( F t otT ) = P F tot T (Π T + I ) (11)and/or its return M T = ( F t otT − I ) /I (or, in the annualized form, M T ≡
71 + µ ) T − P M ( M T ) = (cid:90) dF tot T δ ( M T − F t otT − I I ) P F tot T ( F t otT ) = I P F tot T ( I ( M T − . (12)Although some details of the full probabilistic description like smoothnessof the intermediate incomes can be of interest for evaluating the quality of aninvestment project, the consideration is usually restricted to analyzing theproperties of the distributions P Π (Π T ) and/or P M ( M T ) that, as describedabove, are fully determined, see (11) and (12), by the distribution of theterminal cash flow P F tot T ( F t otT ). In this section we provide a logical outline of the proposed method of invest-ment evaluation against riskless alternative and define a notion of investmentcertainty equivalence. For convenience we break the procedure into stages asfollows:1. As described in the previous paragraph, a risky investment projectwith duration T can generically be described as a superposition of twoprocesses: • transformation of initial investment I into a cash flow stream F = ( F , · · · , F T ) realizing possible outcomes of investing I intothe particular project under consideration (investment) I (cid:55)−→ investment F ; (13) • transformation of F into a terminal cash flow at the end of theproject F tot T through reinvesting F into the same or some otherriskless or risky projects (reinvestment): F (cid:55)−→ reinvestment F tot T . (14)8. An investment certainty equivalent is defined as a riskless investment˜ I generating the same cash flow pattern F = ( F , · · · , F T ):˜ I (cid:55)−→ riskless investment F . (15)The difference ˜ I − I between the investment certainty equivalent andthe project investment outlay quantifies the investment risk premium.The notion of investment certainty equivalence is dual to the conven-tional certainty equivalence related to the riskless investment of theoriginal investment outlay I producing a modified cash flow pattern,see discussion in the paragraph 3 below.3. Generically the outcomes of both investment and reinvestment are un-certain and, therefore, both processes are risky. A risk premium (a gapbetween risky and riskless return/profit) does thus include two differentcontributions so that generically there exist two different risk premiumscorresponding to uncertainties in investment and reinvestment.4. The present study is mainly focused at investment risk premium andassumes riskless alternative investment and riskless reinvestment.
5. For investment project to be attractive it should with acceptable prob-ability bring profit exceeding the minimally acceptable one specified bythe investor (Fishburn, 1977).6. The risk related to risk premiums is quantified by analyzing their dis-tributions and evaluating the corresponding quantities characterizingthe risk/return profile of the investment project. In what follows weshall restrict our consideration to the parameter Ω (defined below inParagraph 2.6 expression (38)) characterizing risk/return relation.7. The projects are then accepted/ranged according to the investor risk/returnpreferences. Namely, the projects are ranked in decreasing order in Ω,the projects with highest values of Ω being the best. For a discussion of some features of risky alternative investment see paragraph 3. .3 Investment project: alternative riskless investment The suggested method of evaluation and ranking of investment projects isbased on gauging investment I generating the set of expected cash flow tra-jectories against the riskless alternatives generating the same set of cash flowtrajectories and characterized by the riskless yield curve R = ( R , · · · , R T )or, equivalently, its annualized counterpart r = ( r , · · · , r T ) and the risk-less reinvestment forward rates R f = ( R f , · · · , R fT ) and r f = ( r f , · · · , r fT )which can be calculated from the riskless yield curve. The exact procedureis described below.In the case of riskless reinvestment the expression (9) describing invest-ment and reinvestment processes takes the following form: I (cid:55)−→ investment { F ( i ) } R f (cid:55)−→ reinvestment { F ( i )t otT = FV( F ( i ) | R f ) } (16)or, in probabilistic terms (cf. equation (10)), I (cid:55)−→ investment P F ( F ) R f (cid:55)−→ reinvestment P F tot T ( F t otT = FV( F | R f )) (17)In general a set of cash flows F contains both positive F + and negative F − contributions corresponding to profits and additional future investmentoutlays correspondingly: F = F + − F − , (18)where F + t = max( F t , , F − t = max( − F t ,
0) (19)Constructing a proper treatment of negative contributions to cash flow havingnatural interpretation of future additional investments is a subtle issue . Inthe considered case of riskless alternative investment/reinvestment universethe procedure is, however straightforward. The future investments can be For an early discussion see e.g. (Beegles, 1978; Booth, 1982; Miles and Choi, 1979). F − | R ) = T (cid:88) t =1 F − t R t (20)Such an additional investment can be arranged by buying a portfolio ofcouponless riskless bonds with payments replicating future investments F − t at the appropriate time horizons. The portfolio consists from partial invest-ments ( I (1)0 , · · · , I ( T )0 ) such that I ( t )0 (1 + R t ) = F − t → (cid:88) t I ( t )0 = T (cid:88) t =1 F − t R t (21)so that for the particular cash flow pattern under consideration the initialinvestment outlay should include the additional investment (20), and, there-fore, the investment pattern in (16) is for this realization replaced by I t ot ≡ I + PV( F − | R ) (cid:55)−→ investment F + (22)Let us now turn to an explicit description of the riskless reinvestmentpattern and note that the cash flow pattern F + can be arranged by the risklessinvestment ˜ I through investing into a portfolio of bonds ( B , · · · , B T ) suchthat B t (1 + R t ) = F + t , ˜ I = T (cid:88) t =1 B t = PV( F + | R ) (23)and assume that at t = 0 we fix a forward contract for buying at time t at theprice F + t the bond maturing at T thus fixing the corresponding rate R fT − t .This leads to the cash flow at T equalling F + t (1+ R fT − t ) = B t (1+ r t ) t (1+ R fT − t ).On the other hand the same cash flow can be fixed by buying at t = 0 thebond maturing at T so that B (1 + r T ) T = B t (1 + r t ) t (1 + R fT − t ). The tworiskless portfolios giving the same profit should have equal initial investmentsat t = 0, i.e. B t = B . We obtain therefore(1 + r T ) T = (1 + r t ) t (1 + R fT − t ) , (24)11hus fixing the forward rate in question R fT − t = (1 + r T ) T (1 + r t ) t − . (25)The vector of positive cash flows F + has the same riskless present value asthe riskless cash flow (cid:80) Tt =1 F + t (1 + R fT − t ) and thus we have indeed fixed theforward rate curve determining the forward valueFV( F + | R f ) = T (cid:88) i =1 F + i (1 + R fT − i ) , (26)so that the complete description of some particular outcome of an investmentproject taking into account the necessity of additional investment outlays canbe described as I t ot ≡ I + PV( F − | R ) (cid:55)−→ investment F + R f (cid:55)−→ reinvestment F t otT = FV( F + | R f ) (27) The profitability of an investment project on each cash flow trajectory can becharacterized in several ways. The list of the corresponding characteristicsincludes, in particular, • the net terminal profit Π T ( F );Π T = FV( F + | R f ) − I − PV( F − | R ) ≡ FV( F + | R f ) − I tot0 (28)12 the terminal return M T and its annualized version µ M T = (1 + µ ) T = FV( F + | R f ) I tot0 ; (29) • the net present valueNPV( F ) = ˜ I − (cid:0) I + PV( F − | R ) (cid:1) ≡ PV( F + | R ) − I tot0 ≡ PV( F | R ) − I (30) • the profitability index PI = NPV( F ) I tot0 . (31)Let us stress that evaluation of the risk/return profile of an investment projectdoes depend on the target characteristics chosen by an investor. An amount of risk premium collected by an investor on the particular trajec-tory described in (27) can be quantified by comparing the risky investment(27) with its riskless alternative ˜ I . In the case under consideration the twoinvestments to compare are I tot ≡ I + PV( F − | R ) (cid:55)−→ investment F + (32)˜ I ≡ PV( F + | R ) R (cid:55)−→ investment F + (33) The meaning of µ if close to that of MIRR( k, d ) defined by(1 + MIRR( k, d )) T = (cid:80) Tt =1 F + t (1 + k ) T − t (cid:80) Tt =0 F − t (1 + d ) − t . Let us stress the above-defined MIRR assumes the flat term structure structure of boththe reinvestment and financing rate curves (see also(Kierulff, 2008)). I + PV( F − | R ) < PV( F + | R ) (34)i.e. (see (30)) if NPV( F | R ) > NPV is thus simply equal to NPV( F | R ).Let us now find an explicit expression for the risk premium ∆ M for theterminal return M ≡ R T + ∆ M . This follows directly from1 + R T + ∆ M = FV( F + | R f ) I + PV( F − | R )1 + R T = FV( F + | R f )PV( F + | R ) (36)so that ∆ M = (1 + R T ) ∆ NPV I ( tot )0 ≡ (1 + R T )PI( F | R ) (37) For the riskless investments the risk premium is, obviously, absent, ∆
NPV =∆ M = 0. For risky cash flows the contributions F t , t ≥ P (∆ NPV ) or P (∆ M ) corresponding to the set ofpossible cash flow trajectories. The investor’s evaluation of the project shouldtake this into account. A natural way of dealing with the uncertainty of therisk premium is to fix a hurdle risk premium ∆ NPV or ∆ M and quantify riskby analyzing the chances of the project risk premium being below this target.A natural way of weighting risks against gains is to use the ratio Ω (Keatingand Shadwick, 2002; Kazemi et al., 2004) with some threshold risk premiumscale Λ separating desirable and undesirable risk premium outcomes:Ω = (cid:82) ∞ Λ dx ( x − Λ) P ( x ) (cid:82) Λ −∞ dx (Λ − x ) P ( x ) ≡ Call(Λ)Put(Λ) (38)14he second equality in (38) reflects the fact that the ratio Ω has, for the riskpremium ∆
NPV , a natural interpretation in terms of the ratio of prices ofthe so-called Bachelier (Bouchaud and Potters, 2003) call and put options,see (Kazemi et al., 2004). Let us stress that these prices are different fromthe commonly considered Black-Scholes ones, see a detailed discussion in(Bouchaud and Potters, 2003). In this case equation (38) Call(Λ) is a priceof an European option on buying the risk premium at a price Λ while Put(Λ)is that of a European option on selling it for the same price. The final decisionon the project does thus depend on whether the value of Ω associated withthe required risk premium characteristics ∆
NPV or ∆ M is acceptable in termsof the investor’s risk/return considerations.The key property of Ω is that it is a monotonously decreasing functionof the threshold risk premium. This allows to establish a simple criterion foradmissible risk limiting the corresponding risk premium:Λ : Ω(Λ) ≥ > < In the general case the threshold NPV ∗ for NPV(or equivalently to ∆ NPV )provides the scale separating the distribution in P (NPV) (or P (∆ NPV )) intodomains corresponding to gains/losses. The value of NPV ∗ can be fixedin different ways. It can be fixed by management based on the minimalacceptable gain Π ∗ , this directly determines NPV ∗ . Alternatively, it can becalculated from the value of minimally acceptable profitability of the project.Let us assume that one specifies the risk premium ∆ ∗ µ . This means that µ should exceed r T + ∆ ∗ µ . Defining µ ∗ = r T + ∆ ∗ µ we get, for a given cash15ow trajectory, an acceptance criterion µ > µ ∗ . (40)The value NPV ∗ of the threshold risky NPV corresponding to µ ∗ shouldsatisfy − I + PV( F | r ) = NPV ∗ ⇐⇒ µ = µ ∗ . (41)Based on (41) we get (for details see Appendix) µ ∗ = r T + ∆ ∗ µ , (42)NPV ∗ = (cid:32) (1 + r T + ∆ ∗ µ ) T (1 + r T ) T − (cid:33) (cid:32) I + T (cid:88) t =1 I ( t )0 (cid:33) . (43) Let us now turn to the comparison with the widespread methodology ofan investment project valuation - the risk-adjusted discount rate formalism(RADR) and apply the same approach as in previous paragraphs to its de-scription. For simplicity in this paragraph we will consider only the case ofthe simplest canonical cash flow (i.e. only positive cash flows at t ≥
1) andassume the flat term structure of the riskless rate.
The standard algorithm of valuation within RADR approach includes thefollowing three stages:1. The ensemble of cash flow trajectories is characterized by the vector ofaverages (cid:104) F (cid:105) = ( (cid:104) F (cid:105) , . . . , (cid:104) F T (cid:105) ). In the case when the ensemble of F consists of N realisations { F ( i ) } this vector is generated by ”vertical”16veraging: I (cid:55)−→ F (1) = ( F (1)1 , · · · , F (1) T ) · · · F ( N ) = ( F ( N )1 , · · · , F ( N ) T ) (44) ⇓(cid:104) F (cid:105) = ( (cid:104) F (cid:105) , · · · , (cid:104) F T (cid:105) ) (45)2. The mean cash flows (cid:104) F t (cid:105) , t = 1 , . . . , T, are discounted with the effec-tive (risk-adjusted) rate k = r + ∆ r .3. The initial investments I are compared with the sum of discountedvalues of (cid:104) F t (cid:105) . The investor goes into a project if I < T (cid:88) t =1 (cid:104) F t (cid:105) (1 + k ) t . (46) Let us provide the explanation of the idea underlying this valuation methodin terms of approach presented in the previous paragraphs. Let us assumethat there exists a possibility of investment with some rate k . In such a caseto obtain the cash flow F t in the period t one should make an initial outlay I ( t ) I determined by the following expression: I ( t ) I (1 + k ) t = F t . (47)The total initial outlay I I guaranteeing the the cash flow stream F = ( F , . . . , F T )is therefore I I = T (cid:88) t =1 I ( t ) I ≡ T (cid:88) t =1 F t (1 + k ) t ≡ PV( F | k ) (48)Within the RADR approach one assumes that there is a possibility to17ake investments with (risky) rate k replicating the mean cash flows {(cid:104) F t (cid:105)} : I t ( (cid:104) F t (cid:105) , k ) · (1 + k ) t = (cid:104) F t (cid:105) ⇒ I t ( (cid:104) F t (cid:105) , k ) = (cid:104) F t (cid:105) (1 + k ) t . (49)However, as the rate k is risky, the income from these partial investments isnot guaranteed. The risk-free income that can be obtained from the partialinvestment outlay I ( t ) I ( (cid:104) F t (cid:105)| k ) is determined by the risk-free rate r : I ( t ) I ( (cid:104) F t (cid:105)| k ) r (cid:55)−→ (cid:18) r k (cid:19) t (cid:104) F t (cid:105) ≡ α ( r, k ) t (cid:104) F t (cid:105) . (50)This is the so-called certainty equivalent of the mean cash flow (cid:104) F t (cid:105) . There-fore, the mean cash flows (cid:104) F t (cid:105) can be represented as a composition of risklessand risky contributions: (cid:104) F t (cid:105) = α ( r, k ) t (cid:104) F t (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) riskless + (1 − α ( r, k ) t ) (cid:104) F t (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) risky . (51)The distinguishing feature of the certainty equivalent is in the fact that itsriskless present value is equal to the risky present value of the underlyingmean cash flow: PV( (cid:104) F t (cid:105)| k ) = PV( α ( r, k ) t (cid:104) F t (cid:105)| r ) (52)Thus, the acceptance criterion in the RADR formalism as expressed in termsof the certainty equivalents is expressed as follows: − I + T (cid:88) t =1 PV( α ( r, k ) t (cid:104) F t (cid:105)| r ) > , (53)or, equivalently, − I + T (cid:88) t =1 PV( (cid:104) F t (cid:105)| r ) > T (cid:88) t =1 PV((1 − α ( r, k ) t ) (cid:104) F t (cid:105)| r ) . (54)This means that the risk premium from the project should exceedthe outlayof the riskless investment guaranteeing risky part of the cash flows stream.18et us note that (cid:104) NPV( F | r ) (cid:105) ≡ − I + T (cid:88) t =1 PV( (cid:104) F t (cid:105)| r ) (55)and introduce the following notationΛ RADR = T (cid:88) t =1 PV((1 − α ( r, k ) t ) (cid:104) F t (cid:105)| r ) . (56)In such a case, the criterion (54) takes the form (cid:104) NPV( F | r ) (cid:105) > Λ RADR , (57)which is equivalent to Ω(Λ RADR ) > . (58)Thus, the RADR evaluation imposes the exact value for the risk/profitseparation scale Λ = Λ RADR . Both Λ
RADR and NPV ∗ (introduced in the paragraph 2.7) have a naturalinterpretation of critical values for NPN( F | r ), however there are a principaldifferences in their meanings. • The investment acceptance criterion
N P V ( F | r ) > N P V ∗ refers to each trajectory separately, whereas the RADR criterion (cid:104) N P V ( F | r ) (cid:105) > Λ RADR operates with mean values only. • The criterion
N P V ( F | r ) > N P V ∗ is equivalent to the investment ac-19eptance criterion in the form µ > µ ∗ .In the case RADR approach one can write out the following chain ofequivalent expressions :NPV( (cid:104) F (cid:105)| k ) > ⇐⇒ MIRR( (cid:104) F (cid:105)| k ) > k ⇐⇒ (cid:104) NPV( F | r ) (cid:105) > Λ RADR (59) • From the above equivalence it immediately follows that the RADRcriterion implicitly fixes the reinvestment rate k , albeit for the meancash flows only. • In case of RADR there is only one choice for the minimal acceptableΩ level: Ω(Λ
RADR ) >
1, as the comparison is made in terms of means.In general, Ω(NPV ∗ ) > Ω ∗ , where Ω ∗ is equal to one only for the riskneutral investor and is less or greater than one for risk averse and riskseeking investors respectively. • The approach presented in the previous section is also capable of ac-counting for sensitivity of the risk measure Ω to small changes in thevalue of the critical threshold (e.g. NPV ∗ ) which can be very usefulfor project’s ranking. Such an analysis allows to choose the projectwith a ”more stable/less sensitive” Ω-ratio thus corresponding to abetter risk profile. In contrast, in the RADR approach the thresholdΛ RADR is fixed and there is no possibility to take into account the exactrisk-profile of the project.
Let us consider two investment projects with the structure of cash flowsshown in Table 1, where the cash flow F + at time 1 is a random quantitywhile negative cash flows at times 0 and 2 are fixed. The projects differ in Where, in general, (1 + MIRR( F | k, k )) T = (cid:80) Tt =1 F + t (1+ k ) t I + (cid:80) Tt =1 F − t (1+ k ) t . If only one investment cashflow I exists (1 + MIRR( F | k )) T = (cid:80) Tt =1 F + t (1+ k ) t I F + , the right-skewed for the first one, see Fig. 1(Right-Skewed) and left-skewed for the second, see Fig. 1(Left-Skewed). The char-acteristics of these distributions are shown in Table 2. Let us note that thesecond ”Left-Skewed” project has larger mean and median values.Time 0 1 2Cash Flow -200 F + -100Table 1: Cash flow structureFigure 1: Distributions of F + for the two projects. Dashed vertical line showsthe position of mean values 21ight-Skewed Left-SkewedMean 350 355Median 334 370Std. Dev. 40 40Skewness 2.7 -2.8Table 2: Characteristics of F + distributions for the two projects.Let us assume for simplicity the flat riskless discounting and forwardingrates of 5% and that both projects are correlated with the market with thecorrelation coefficient ρ .1. RADR evaluation
As standard deviations of cash flows in the two projects are the same,according to CAPM their β coefficients are also equal ( β = ρσσ m ), where σ is a standard deviation of the projects return and σ m - that of thereturn of the market portfolio. Thus within the RADR framework thediscounting rate for the two projects is the same, r RADR = r f + β ( r m − r f ) , where r f - is the risk-free rate and r m - the return of the market portfolio.We haveNPV( (cid:104) F (cid:105)| r RADR ) = −
200 + (cid:104) F + (cid:105) r RADR − r RADR ) , where (cid:104) F + (cid:105) is the average of F + (the value is provided in Table 2).Let us note that in the RADR/CAPM approach the ”Left-Skewed” isbetter than the ”Right-Skewed” one for any β simply because of theranking of the average values of positive cash flow.In Table 3 we compare characteritistics of both projects. We see thatwith r RADR = 15% ( β ( r m − r f ) = 10%) the ”Left-Skewed” project isalways better than the ”Right-Skewed” one.22riterion Right-Skewed Left-SkewedNPV( (cid:104) F (cid:105)| (cid:104) F (cid:105)| , (cid:104) F (cid:105)| , r RADR = 15%2.
Evaluation in the new approach
Let investor’s preferences be characterized by the desired risk premiumof ∆ = 10% (e.g. ∆ = β ( r m − r f )), i.e. critical µ equal to µ ∗ = 15%.As negative cash flows and bond rate are fixed one can reconstruct thecritical value of NPV (NPV ∗ ). The corresponding values are shown inTable 4.The histograms of the NPV( F | r ) distributions of the projects are shownin Fig.2, where the vertical line shows the hurdle scale NPV ∗ . Thecharacteristics of the NPV distribution are shown in Table 5. Thedistributions of µ are shown in Fig. 3 and the corresponding distributioncharacteristics in Table 6.Criterion hurdle scale µ ∗ ∗ F | r ) in the two projects. Vertical line showsthe position of NPV ∗ Figure 3: Distributions of µ in the two projects. Vertical line shows theposition of µ ∗ µ distributions for the two projectsKnowing the critical values of NPV ∗ ( µ ∗ ) one can calculate the value ofΩ NPV (Ω µ ) for the two projects under consideration. The correspondingvalues are shown in Table 7.Project Ω NPV Ω µ Right-Skewed 0.4 25.7Left-Skewed 0.3 3.1Table 7: The values of Ω for the two projectsTherefore, with ∆ of 10% one should prefer the Right-Skewed project.25
Example of ranking of real industrial pro jects
Let us consider an example of ranking two real industrial projects relatedto production of chemical fertilisers. Both projects were described in theform of excel table calculating project characteristics (e.g. project’s NPVor µ ) from some inputs (e.g. price and macroeconomic indicators dynamicforecasts, plants characteristics, transportation tariffs forecasts etc.).At the first stage we simulate 1000 Monte-Carlo scenarios for the models’inputs. After that we calculate different projects’ characteristics, such asNPV and µ , in each Monte-Carlo scenario. As a result of these proceduredistributions of the projects under consideration were obtained. Histogramsof these distributions are shown in Fig. 4.26igure 4: Distributions of µ of two real projectsAs it was described in the previous sections, the procedure of projectsranking consists of three steps. The first is to specify hurdle rate, the secondis to calculate Ω using the chosen value and the third is to rank projects indecreasing order in Ω.In Fig. 5 we show how values of Ω change with hurdle rate µ ∗ . From thisplot it follows that investors with different hurdle rates µ ∗ may have differentprojects ranking. For example, with µ ∗ = 5% the Project A is preferable,however, with µ ∗ = 7% an investor would choose the Project B.27igure 5: Distributions of µ of two real projects The present study addresses one of the most important and, at the sametime, controversial problems in corporate finance – evaluation and ranking ofinvestment projects. The current industry standard is based on using for thispurpose the risk-adjusted discount rate (RADR). It is however well knownfor quite a long time that the RADR methodology is plagued with seriouslimitations, e.g.: • it assumes some specific investor preferences that might not reflect thereal ones; • all the information on risks, i.e. on probabilistic description of pre-mium, its moments, nature of its tail, etc. is compressed into onenumber, the risk premium, with no clear methodology of translatingproject-specific risks into this number,etc. This absence of clear-cut methodology makes it very difficult to compareprojects with different timespan, from different industries, etc. An invest-ment certainty equivalence approach proposed in the present paper allows28o perform en explicit separation of risky and riskless contributions to eachpossible realization of cash flows characterizing each particular investmentproject thus making it possible to apply modern criteria of evaluating andranking of investment projects based on the corresponding exact distributionof risk premium. Detailed properties of these distribution are determined bysuch risk factors as sovereign, industry-specific or project-specific ones.The approach makes it possible to • compare investment projects from different industries through an as-sessment of differences in variability patterns of historical premiums ofprojects in these industries; • use exact accounting for different risk sources resulting in fully rationalrisk-adjustment selection; • describe investor’s risk-return preferences using only one parameter –the hurdle rate, i.e. the premium scale that for a given investor marksthe range of acceptable/non-acceptable risk premiums.The proposed methodology can be very helpful in organising a systematicprocedure of evaluation and ranking of investment projects in large firmsin which hundreds of investment projects with widely different timespans,economic significance and risk profiles are simultaneously considered.Let us stress once again that in the present paper we discuss only thecase of the riskless alternative investment. There remains a very importantquestion of how to account for reinvestment risks. We plan to return to thisquestion in future. Acknowledgements
We are very grateful to A. Landia, V. Kalensky, A. Botkin, A. Djotyanand I. Nikola for numerous discussions that were crucial for shaping ourunderstanding of the subject of the present paper.29 eferences
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Appendix
Let us define the minimally acceptable terminal rate µ ∗ = r T + ∆ ∗ µ . Let usshow, how to get the corresponding critical values for NPV. From(1 + R fT − t ) = (1 + r T ) T (1 + r t ) t , (60) B t (1 + r t ) t = F + t , (61)we get(1 + µ ) T = (cid:80) Tt =0 F + t (1 + R fT − t ) I + (cid:80) Tt =1 I ( t )0 = (cid:80) Tt =0 B t (1 + r t ) t (1 + R fT − t ) I + (cid:80) Tt =1 I ( t )0 = (62) (cid:80) Tt =1 B t (1 + r t ) t (1+ r T ) T (1+ r t ) t I + (cid:80) Tt =1 I ( t )0 = (1 + r T ) T (cid:80) Tt =1 B t I + (cid:80) Tt =1 I ( t )0 . (63)In addition, we have (cid:80) Tt =1 B t = NPV( F | r ) + I + (cid:80) Tt =1 I ( t )0 and, therefore,(1 + µ ) T = (1 + r T ) T (cid:32) NPV( F | r ) I + (cid:80) Tt =1 I ( t )0 + 1 (cid:33) (64)32rom (64) we finally get the relation between NPV ∗ and µ ∗ (and, therefore,∆ ∗ µ ) (1 + µ ∗ ) T = (1 + r T ) T (cid:32) NPV ∗ I + (cid:80) Tt =1 I ( t )0 + 1 (cid:33) , (65)i.e. NPV ∗ = (cid:32) (1 + r T + ∆ ∗ µ ) T (1 + r T ) T − (cid:33) (cid:32) I + T (cid:88) t =1 I ( t )0 (cid:33) ..