Feasibilitystudyofconstruction investmentrojectsassessment withregardtoriskandrobability ofnpvreaching Andrzej Minasowicz Warsaw University of Technology, Civil Engineering Faculty, Warsaw, PL a.minasowicz@il.w.edu.l Keywords NPV, fuzzy logic, risk the aer resents an analysis of nv and investment risk at the stage of assessment of the strategy and the feasibility study. This analysis, for a secific roject value rofile, allows for secification of robability of occurrence of a given value of cash flows and NPV and for resentation of their fuzziness. INTRODUCTION Decision-making with regard to investment is associated with analysis of risk related to achievement of the basic technical and economic arameters, including the udated value of NPV (Net Present Value). Prior to the decision-making stage, exerienced investors analyse the revenues and exenditures for the investment roject lanned; usually, they have the knowledge sufficient to determine the value of deviations of these lanned [PLN million] 1 9 8 7 6 5 4 3 2 1 1 2 1 2 3 4 5 6 7 8 9 1 11 12 13 14 [PLN million] 3 4 Figure1:Individualcashflows,withouttakingintoaccounttherisk. 1
arameters from the really achieved ones. Such analysis may also be subject to oinion of construction exerts, who may assess the value of such deviation in verbal form. The aer resented discusses the rocedure of determination of robability of reaching the lanned NPV value on the basis of linguistic assessments using fuzzy sets. Assessmentofthe investmentroject Assumtions Any investment roject, analysed in accordance with the NPV method, consists of the aroriate cash flows revenues or exenditures. These flows finally result in the exected NPV value. If we fail to take into account the risks, individual flows may be resented as illustrated by fig. 1. In this case, we disregard the imact of the discount coefficient and we assume that these flows have already been discounted. The recise flow values are rovided in the second column of table 1. (K ) are reresented by negative In fig. 2, variables have the following meanings: Z es extreme essimistic value of revenues, assuming robability =, Z ot extreme otimistic value of revenues, assuming robability =, Z es difference between the most robable value and the extreme es simistic value of revenues, Z ot difference between the extreme otimistic value and the most robable value of revenues, K es extreme essimistic value of exenditures, assuming robability =, K ot extreme otimistic value of revenues, assuming robability =, K es difference between the most robable value and the extreme essi mistic value of exenditures, K ot difference between the extreme otimistic value and the most rob able value of exenditures, z the robability of revenue being equal to Z, t the robability of exenditure being equal to K. K es K es α β K ot z k K Z K ot Z es α Z es β Z ot Figure2:Presentationoftheflowdistributionintriangularform. Z ot S i Activity S 1 Contractual roceedings -1,8 2 Project -2,1 3 Measurement works -1,3 4 Earthworks -1,6 5 Boarding -2,2 6 Concrete works -3,2 7 Steel structures -3,8 8 Water suly system -2,5 9 Ventilation -1,8 1 Power suly works -2,8 11 Sewage system -2 12 year 1 9 13 year 2 9,5 14 year 3 1 NPV = 3,4 Table 1: The assumed cash flows for theroject. values, and revenues (Z ) are reresented by ositive values. In general, exenditures and revenues will be marked as S cash flows. Although the flows calculated within a given investment roject have been resented as the most robable ones, their robability will not be equal to 1. In other words we are not 1% sure that a given flow will assume the value of S, and not a greater or smaller one. This will be influenced by the differences between the exected (most robable) values of cash flows and the real values of these flows that emerge during the roject imlementation. Therefore, the flow can be reasonably resented as a random variable through its robability distribution, secifying the maximum robability, corresonding to flow S and cash flow values, which are characterised by lower robability. We assume that the robability of emergence of value S is the highest. The vertical axis of this chart is robability, while the horizontal axis consists of flow values. Alication of this tye of distribution can be exlained by the fact that at the stage of the feasibility study, we use aroximate data, and alication of more accurate distributions at this stage is not reasonable. The extreme values in the triangular distribution resented are, in fact, characterised by such a low robability hat at the stage of feasibility study, their robability has been assumed to be equal to zero. a. minasowicz feasibility study of construction investment rojects assessment with regard to risk... 1-14 11
The arms of the triangular distribution show how raidly a revenue or an exenditure may change. The character of this change will be determined by roviding tangents of inclination angles α and β. Angle α will determine the risk of increase of exenditure or the risk of decrease of revenue, while angle β will stand for the risk of decrease of exenditure or the risk of increase of revenue. In other words, α a negative risk (associated with a decrease of NPV value), β a ositive risk (associated with an increase of the NPV value). We define risk as the ability, or suscetibility, to changes for individual cash flows. A high risk of an exenditure change means that the most robable value may undergo significant changes uon slight changes of external conditions. On the other hand, a low risk of changes will be tyical for more certain activities, less suscetible to changes. Moreover, a high risk of changes determines the lower robability of emergence of exenditure K, which means we have to take into account substantial changes during the investment roject imlementation in case of a flow characterised by a high α or β. Value S will be obtained from the reliminary assessment of the investment. The robability of the flow assuming value S is Affinity level 1,,9,8,7,6,5,4,3,2,1 robability,7,75,8,85,9,95 1, minimum medium maximum low high Figure3:Presentationofrobabilityasalinguisticvariable. I Activity S (Eks1) (Eks2) (Eks3) 1 Contractual roceedings -1,8 medium medium low 2 Project -2,1 medium low minimum 3 Measurement works -1,3 maximum high high 4 Earthworks -1,6 medium high medium 5 Boarding -2,2 medium medium low 6 Concrete works -3,2 low low low 7 Steel structures -3,8 medium medium low 8 Water suly system -2,5 medium high high 9 Ventilation -1,8 high medium medium 1 Power suly works -2,8 medium high high 11 Sewage system -2 high high medium 12 year 1 9 medium high medium 13 year 2 9,5 medium medium low 14 year 3 1 medium low medium Table2:Abreakdownofexertoinionsoninlinguisticform i Activity S x ΔS es ΔS ot S es S ot ΔS es / S ΔS ot / S S es / S S ot / S 1 Contractual roceedings -1,8,4,83 1,71,69-3,51-1,11,95,38-1,95 -,62 2 Project -2,1,6,8 1,56,94-3,66-1,16,74,45-1,74 -,55 3 Measurement works -1,3,5,92 1,45,73-2,75 -,57 1,12,56-2,12 -,44 4 Earthworks -1,6,5,87 1,54,77-3,14 -,83,96,48-1,96 -,52 5 Boarding -2,2,8,83 1,33 1,7-3,53-1,13,61,48-1,61 -,52 6 Concrete works -3,2,5,8 1,67,83-4,87-2,37,52,26-1,52 -,74 7 Steel structures -3,8,7,83 1,41,99-5,21-2,81,37,26-1,37 -,74 8 Water suly system -2,5,8,88 1,26 1,1-3,76-1,49,5,4-1,5 -,6 9 Ventilation -1,8 1,87 1,15 1,15-2,95 -,65,64,64-1,64 -,36 1 Power suly works -2,8,7,88 1,33,93-4,13-1,87,48,33-1,48 -,67 11 Sewage system -2,6,88 1,42,85-3,42-1,15,71,42-1,71 -,58 12 year 1 9,7,87 1,36,95 7,64 9,95,15,11,85 1,11 13 year 2 9,5,8,83 1,33 1,7 8,17 1,57,14,11,86 1,11 14 year 3 1,9,83 1,26 1,14 8,74 11,14,13,11,87 1,11 Table3:Characteristicvaluesforeachdistribution 12
to be obtained from exerts. To determine α or β, we also need the following ratio: S + S - =x (1) S+ and S- are deviations from values obtained on the basis of reliminary assessments. Information concerning can be rovided by exerts in linguistic form as one of the five exressions: minimum, low, medium, high, maximum. When we obtain such answer, it needs to be transformed into numerical values. For this urose, we aly a resentation of linguistic sace. In our case, it consists of fuzzy sets {minimum; low; medium; high; maximum}. The numerical values of linguistic variable in the case of have been resented in fig. 3. Presented below is information in linguistic form, concerning robability, obtained from three exerts, as an examle is resented in table 2. On the basis of the linguistic sace assumed, numeric values have been determined for robability as established by exerts. In this manner, each of the cash flows has been resented in form of two distributions absolute and relative. On the basis of cash flows, resented as a random variable, NPV has been determined: Figure4:Exemlaryflowsinformofatriangularrobabilitydistribution.,1 1. NPV absolute distribution,9,8,7 3,4;,61,6,5,4,3,2,1 NPV 16,394;, 16,53;, 2 15 1 5 5 1 15 2 Figure5:PresentationofNPVvalueinformofarandomvariabledistribution. (3) (2) Due to the fact that the value of NPV (2) has been resented as a random value, it is ossible to calculate the value of (3) of its emergence, as well as the values of α, β (4). (4) The values of all distributions were determined not on the basis of the entire set of assessment of linguistic variables, but only the values of affinity equal to 1, that is: minimum =.75; low =.8; medium =.85; high =.9; maximum =.95. Therefore, the distribution values constitute acute distributions. In order to gras the fully subjective and imrecise nature of exression of value, a. minasowicz feasibility study of construction investment rojects assessment with regard to risk... 1-14 13
it is necessary to take into account all values of the linguistic variable. Thanks to the fact we have assumed a triangular function of affinity, it is now sufficient to conduct calculations for 3 values: L extreme left of affinity level = ; C central of affinity level = 1; P extreme right of affinity level =. In our case, the linguistic value (L ; C ; P) will be as follows: minimum = (,7 ;,75 ;,8); low = (,75 ;,8 ;,85); medium = (,85 ;,9 ; 95); high = (,85 ;,9 ;,95); maximum = (,9 ;,95 ; 1). In this manner, instead of a single robability value, we have obtained three values of, and thus three robability distributions for a single cash flow (extreme left L, central C, extreme right P). On the basis of flow distributions, resented in this manner, the fuzzy distribution of NPV has been determined. CONCLUSIONS Such resentation of the distribution allows for an objective assessment of distribution of cash flows and the resulting NPV distribution. The levels of fuzziness of the aram- 1. NPV absolute distribution,1,8,6,4,2 2 15 1 5 5 1 15 2 Figure6:AbsolutedistributionofNPVvalues. eters discussed allow for determination of subjectivity and imrecision of assumtion of the basic values, which characterise the individual cash flow distributions and NPV values in linguistic form. The rocedure discussed may be alied in ractice at the stage of strategy assessment and the feasibility study. It allows for determination of robability of emergence of a given value of cash flows and NPV values and for resentation of their fuzziness. REFERENCES extreme right values central values extreme left values Morgan N.G., Henrion M. (199) Uncertainty. A Guide to dealing with uncertainty in quantitative risk and olicy analysis. Cambridge University Press, New York. Piegat, A. (1999) Modelowanie i sterowanie rozmyte. Akademicka Oficyna Wydawnicza EXIT, Warsaw. Minasowicz A. (28) Analiza ryzyka w rojektowaniu rzedsiδwziδcia budowanego, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa. 14