عدم اطمینان و آنالیز حساسیت های جهانی در ارزیابی پروژه های سرمایه گذاری

This paper discusses the use of global sensitivity analysis (SA) techniques in investment decisions. Global SA complements and improves uncertainty analysis (UA) providing the analyst/decision- maker with information on how uncertainty is apportioned by the uncertain factors. In this work, we introduce global SA in the investment project evaluation realm. We then need to deal with two aspects: (1) the identification of the appropriate global SA method to be used and (2) the interpretation of the results from the decision maker point of view. For task 1, we compare the performance of two family of techniques: non-parametric and variance decomposition based. For task 2, we explore the determination of the cash flow global importance (GI) for valuation criteria utilized in investment project evaluation. For the net present value (NPV), we show that it is possible to derive an analytical expression of the cash flow GI, which is the same for all the techniques. This knowledge enables us to: (1) offer a direct way to compute cash flow GI; (2) illustrate the practical impact of global SA on the information collection process. For the internal rate of return (IRR), we show that the same conclusions cannot be driven. In particular, (a) one has to utilize a numerical approach for the computation of the cash flow influence, since an analytical expression cannot be found and (b) different techniques can produce different ranking. These observations are illustrated by means of the application to a model utilized in the energy sector for the evaluation of projects under survival risk. The quantitative comparison of cash flow ranking with respect to the NPV and IRR concludes the paper, illustrating that information obtained from the SA of the NPV cannot be transferred to the IRR.

This paper introduces the use of global sensitivity analysis (SA) techniques in investment valuation. When firms deal with investment projects, many factors are uncertain. Uncertainty analysis (UA) is performed as part of the decision-making (DM) process to enable the decision maker (DMr) to understand the degree of confidence in the decision (Apostolakis, 1995), and to assess the project risk (Winston, 1998; Bodie and Kane, 2001; Helton, 1993; Borgonovo et al., 2003; Hofer, 1999). Dedicated subroutines are nowadays included in the most diffuse business software (Excel or Lotus) or in dedicated software packages (Winston, 1998). UA results alone, however, do not provide information on how uncertainty is apportioned by uncertainty in the input factors, and, therefore, on which factors to devote data collection resources so as to reduce uncertainty most effectively/rapidly (Saltelli, 1999). This is, however, the information one obtains from global SA. Several SA methods have been recently developed in the literature, outside the investment project valuation realm (Archer et al., 1997; Borgonovo 2001; Campolongo and Saltelli, 1997; Downing et al., 1985; Hamby, 1994; Hamby and Tarantola, 1999; Helton, 1993; Hofer, 1999; Homma and Saltelli, 1996; McKay, 1996; Saltelli, 1997 and Saltelli, 1999; Saltelli and Bolado, 1998; Saltelli and Marivoet, 1990; Saltelli et al., 1999 and Saltelli et al., 2000; Sobol’, 1967, Sobol’, 1990, Sobol’, 1993 and Sobol’, 2001; Saltelli and Sobol’, 1995). It is the purpose of this paper to illustrate the utilization and meaning of global SA in the uncertainty management of investment project evaluation. We undertake the analysis in two steps. The first step is the identification of the appropriate techniques to be applied to discounted cash flow (DCF) valuation models (Borgonovo and Peccati, 2005; Bodie and Kane, 2001; Beccacece et al., 2000; Koltai and Terlaky, 2000; Taggart, 1996). The second step is the analysis of the application of cash flow GI and of its role in the DM process. For the first step, we examine the following global SA techniques: Sobol’ global sensitivity indices [Sr(xi)] 1 ( Table 1) ( Sobol’, 1993 and Sobol’, 2001; Saltelli and Sobol’, 1995; Saltelli et al., 1999), the Pearson correlation coefficients (PEAR) and the standardized regression coefficients (SRC) ( Saltelli and Marivoet, 1990). Sr(xi) belong to the family of variance decomposition-based (VDB) techniques and estimate the GI of a parameter by means of the complete decomposition of the model variance. PEAR and SRC are non-parametric (NP) global SA techniques and compute the GI by means of a regression of the output on the uncertain parameters. Table 1. Acronyms used in this work Acronyms Name UA Uncertainty Analysis SA Sensitivity Analysis VDB Variance Decomposition Based DM Decision-Making DMr Decision-Maker NP Non-Parametric GI Global Importance PEAR Pearson correlation coefficient SRC Standardized regression coefficient Sr(xi) Sobol’ Global Sensitivity Indices of order r ST(xi) Sobol’ total indices FAST Fourier Amplitude Sensitivity Test Table options We show that, if the DMr selects as a valuation criterion a net present value (NPV) or one of its generalized forms, then the GI of cash flows: (a) can be computed analytically; (b) coincides with the fraction of the NPV variance associated with the cash flow; is equivalently estimated by all the techniques; (c) depends only on the cash flow standard deviation and not on the type of cash flow distribution; (d) has a straightforward interpretation in terms of uncertainty management. Thanks to these properties, it is then possible to study the relationship between GI, timing and uncertainty of a cash flow. We show that, if the DMr selects an internal rate of return IRR to value the investment, an analytical approach is not feasible. As a consequence, not all the techniques can be equivalently used to estimate cash flow GI. In particular, NP techniques should not be utilized since their ability to correctly estimate GI declines as the model becomes non-linear. For the second step, we show that these results have a direct impact on the information collection process. To do so, we illustrate the global SA of a sample model utilized in the energy sector for the evaluation of projects under survival risk (Beccacece et al., 2000). The model estimates three investment criteria: NPV, value at any time t(Vt), and IRR. For the project NPV and Vt the computation of the cash flow GI is direct thanks to the analytical results mentioned above. In particular, it is enough that the DMr has assesses the cash flow standard deviation, which is a direct output of a standard UA. We show that collecting information on the cash flows associated to the highest values of GI provides the most effective way to reduce the valuation criterion variance. We also illustrate that the DMr has immediate information on the amount of the decrease. For the project IRR estimated via the same model, we compute the cash flow GI numerically, comparing the performance of ST(xi), PEAR and SRC. We show that if one relied upon PEAR and SRC to rank cash flows based on their GI, then incorrect conclusions could be drawn. We then compare the cash flow ranking w.r.t. the NPV and to the IRR, obtaining quantitative information through Savage Scores ( Borgonovo et al., 2003; Campolongo and Saltelli, 1997). The little agreement between the ranking shows that a cash flow does not necessarily influence uncertainty in the investment NPV and IRR in the same way. As a consequence, global SA results for the NPV cannot be directly transferred to the IRR: if one collects information on a cash flow with a high GI w.r.t. the investment IRR, one would not automatically reduce uncertainty in the NPV effectively and vice versa. In Section 2, we present the principles of global SA and the techniques used in this paper. In Section 3, we discuss the global SA of DCF valuation models. Section 4 presents the application of the results and techniques of Section 3 to a project evaluation model proposed by Beccacece et al. (2000) and in use in the energy sector. Conclusions are offered in Section 5.

We have discussed the use of global SA techniques in the evaluation of investment projects. Global SA is a set of techniques that have been recently developed in the literature (Saltelli, 1999) to complement Uncertainty Analysis (UA), providing information on how uncertainty in the model output is generated by uncertainty in the input factors. Nowadays, most of the standard software used in industrial decision-making is equipped with Monte Carlo subroutines that enable the DMr to perform uncertainty or scenario analysis. By means of UA, the DMr can quantify His/Her uncertainty in the project and assess the likelihood of favorable and adverse scenarios. If in addition to UA a global SA is performed, then the DMr is able to derive quantitative information on what are the factors driving uncertainty. These parameters are the ones deserving better attention in the information gathering and data collection processes, in order to reduce uncertainty in the fastest way. It has been the purpose of this work to introduce global SA in the investment project evaluation realm. To do so, we have examined the global SA of the most used valuation criteria. For the NPV valuation criterion we have seen that • The GI of the cash flows [GINPV(xi)] can be obtained analytically. • GINPV(xi) can be seen as the product of three effects: the cash flow uncertainty, the probability of receiving the cash flow and the discount factor. • The ranking of the cash flows does not necessarily follow their timing, but uncertainty and probability effects can drive the results. • GINPV(xi ) is independent of the type of cash flow epistemic distribution, but dependent only on View the MathML sourceσi2. • GINPV(xi) can be equivalently estimated using variance decomposition-based techniques, and NP techniques (SRC and PEAR). These results imply that, once the DMr has estimated His/Her uncertainty in the investment cash flows, then their GI can be found directly, without further calculations, since an analytical expression is available. More in detail, it is enough that the DMr assesses a standard deviation of the cash flows, and their GI is directly found from Eq. (11). The second advantage of the above results is that they rule out any problem related to the numerical computation of cash flow GI. Different results, however, have been obtained for the IRR valuation criterion. We have seen that: • GIIRR(xi) must be computed numerically, since an analytical expression for GIIRR(xi) is not achievable, in general. • Due to the non-linearity of the model, PEAR, SRC and VDB techniques produce different cash flow ranking. • VDB techniques produce the most reliable estimates since PEAR and SRC fail in assessing the importance of interaction terms in the case of non-linear models. The fact that the global SA of the IRR can be performed only numerically, brings into the picture the limitations connected with the computational aspects discussed in Section 2 (Table 2). We have illustrated the previous results numerically, through the global SA of a model developed for the evaluation of projects under serious survival risk proposed by (Beccacece, et al. 2000) and in use in the energy sector (Beccacece, et al., 2000). We have discussed the cash flow GI in two cases, a base case and a case of uniform increase in uncertainty in the cash flows. The application of the general NPV results listed above has enabled us to: —derive the cash flow GI analytically; —quantify the reduction in NPV variance following a reduction in cash flow variance. We have discussed how the global SA results can be utilized in the information collection process in order to manage uncertainty effectively. More precisely, we have seen that reducing uncertainty in the first cash flow of the revenue period (r5) would lead to the fastest reduction in the NPV variance. The analytical approach has been utilized for the global SA of the value of the project as a function of time. Similar results w.r.t. those of the NPV have been obtained. We have then performed the global SA of the project IRR. We have resorted to a numerical approach utilizing the extended FAST method to obtain quantitative estimates for GIIRR(xi) and we have also compared the results to the estimates of GIIRR(xi) obtained by utilizing the PEAR and SRC methods. Results confirmed the foreseen poor performance of PEAR and SRC, due to the non-linear dependence of the IRR on the cash flows. We have then compared the cash flow ranking w.r.t. the NPV to the respective IRR ranking. Results have shown little agreement. In particular, the parameter influencing uncertainty in the IRR the most (r5) is not the one influencing the NPV the most (γ1). Thus, if a DMr collects information on γ1, then He/She would reduce the IRR variance effectively and not the NPV one. This result shows that information obtained from the SA of the NPV cannot be transferred to the SA of the IRR. Thus, also global SA results state the non-equivalence of the IRR and NPV valuation criteria.