Fuzzy Decision Analysis for Alternative Selection Using a Fuzzy Annual Worth Criterion

Very often, in industry, discounted cash flow techniques are applied for analyzing and selecting investment alternatives under consideration. These techniques are usually based on the data under certainty or risk. In reality, however, the decision makers are often facing the situation of vague cash flows and discount rates, or even uncertain durations, when evaluating and selecting potential investments. Fuzzy set theory has the capability of capturing vague data and allows mathematical operations. This article proposes a fuzzy equivalent uniform annual worth (fuzzy EUAW) method to assist practitioners in evaluating investment alternatives utilizing the theory of fuzzy sets. Triangular fuzzy numbers (TFNs) are used throughout the analysis to represent the uncertain cash flows and discount rates. Further, fuzzy capital recovery factors and fuzzy sinking fund factor are derived. Using these two factors, the fuzzy equivalent annual worth of each investment alternative can be found. By ranking these fuzzy numbers with the integral value, the optimal investment alternative is selected. A numerical example is provided to illustrate the results of the alternative selection.     

A Comparison of Fuzzy Set and Probabilistic Paradigms for Ranking Vague Economic Investment Information Using a Present Worth Criterion

This article introduces an approach for comparing the fuzzy set and probabilistic paradigms for ranking vague economic investment information when a present worth criterion is used. Comparisons are made between the two paradigms by forming a ratio of fuzzy ranking value to probabilistic ranking value for a set of tested economic decision conditions (vague cash flows and interest rates). A multivariable analysis of the comparison ratios indicates the dominance or spread required in the expected present worths of competing alternatives to guarantee consistent decision ranking among the paradigms. Results of the research suggest that probabilistic methods will not rank vague economic investment information the same as fuzzy set methods. As a result, decision makers who use probabilistic methods to model vagueness in human thought (the condition for which fuzzy theory was developed) may obtain rankings inconsistent with those obtained via the fuzzy models. In such cases, if decisions are based on project rankings, then different choices or recommendations are likely.     

Fuzzy Present Value Analysis Model For Evaluating Information System Projects

In this article, the economic evaluation of information system projects using present value is analyzed based on triangular fuzzy numbers. Information system projects usually have numerous uncertainties and several conditions of risk that make their economic evaluation a challenging task. Each year, several information system projects are cancelled before completion as a result of budget overruns at a cost of several billions of dollars to industry. Although engineering economic analysis offers tools and techniques for evaluating risky projects, the tools are not enough to place information system projects on a safe budget/selection track. There is a need for an integrative economic analysis model that will account for the uncertainties in estimating project costs, benefits, and useful lives of uncertain and risky projects. In this study, we propose an approximate method of computing project present value using the concept of fuzzy modeling with special reference to information system projects. This proposed model has the potential of enhancing the project selection process by capturing a better economic picture of the project alternatives. The proposed methodology can also be used for other real-life projects with high degree of uncertainty and risk.     

ALTERNATIVE MODELS FOR FINDING CUMULATIVE PAID CAPITAL AND REMAINING BALANCE FOR A UNIFORM SERIES OF CASH FLOWS

The purpose of this paper is to propose models for finding the cumulative amount of paid capital up to a certain point in time for a discrete uniform series of cash flows, and also the unpaid capital amount of a similar séries of cash flows. An exponential model for finding the present worth for a séries of cash flows that is increasing exponentially is used to find the sum of paid capital. The second model that is used for finding the unpaid capital is developed based on the first model. The major advantages of the first model over the existing models is that it provides a direct tool for finding the cumulative amount of paid capital or unrecovered investment. The proposed models are easy to use since they are independent of finding the amount of the cash flow A. The cumulative paid interest up to a certain point in time can be found by subtracting the total repaid capital from the amount of total paid money up to that point.     

A review of: “PROJECT MANAGEMENT IN MANUFACTURING AND HIGH TECHNOLOGY OPERATIONS” by Adedeji B. Badiru,John Wiley and Sons,Inc., New York, 1988, xxii + 319 pp., ISBN 0471-62892-1, List $44.95.

The conventional approach to considering working capital cash flows in capital budgeting is to omit them or include some ad hoc figures at the initiation and termination of the project. The authors argue for an endogenous system of estimating relevant working capital cash flows on a periodic basis. Otherwise, the present value of working capital cash flows is biased against the project's acceptance. Examples of calculating working capital cash flows as related to changes in annual sales are presented for three time patterns of sales and contrasted to the conventional method. An empirical study of the linear relationship of net working capital and sales revenue of 770 companies is reported, and an alternative cash flow model is offered thai includes working capilal cash flows.     

Adjustments within discount rates to cater for uncertainty—Guidelines

Deterministic discounted cash flow (DCF) analysis is a well-accepted technique in engineering appraisals. Common practice is to incorporate all uncertainty influences within a single variable—namely, the discount rate—which also represents the time value of money. Commentary already exists in the literature that such a practice is expedient but not rational and has shortcomings. This article examines the error involved in this practice and provides guidelines and precautions for using blanket or constant discount rates in dealing with uncertainties. It shows the adjustments necessary for any given investment scenario. This is done through establishing equivalence of the expected utility of deterministic and probabilistic present worth, allowing a rate adjustment to be calculated. Numerical studies look at the relationship or trends of this rate adjustment to the key analysis variables. Generally, it is found that the rate adjustment should be decreased as the timing of a cash flow's occurrence increases, increased as the variance of the cash flow increases, kept almost as an additive constant as the base rate increases, and increased as the investor's level of risk aversion increases. The article provides practitioner-friendly usable guidelines for adjusting rates, something that is unavailable elsewhere in the literature.     

Risk Allowance in Original Capital Investments*

In engineering economy studies, the total risk capital is often not the original capital investment. If a firm remains profitable in the future, a portion of a completely unsuccessful investment can be recovered (1) through income-tax saving as a result of the depreciation cash flows, and (2) through possible reuse of the idle depreciating facilities.

To allow for income-tax savings, the authors propose that the present worth of the guaranteed depreciation cash flow be discounted at the cost of capital and subtracted from the total initial investment to give a better measure of the risk capital. The operating profit, depreciation-free net income, can then be treated in an appropriate fashion using probabilities or a higher discount rate to account for future uncertainties in forecasting market volume, price, manufacturing costs, etc. The application of this principle has been illustrated through a number of ex amples. The results indicate the value of distinguishing between the depreciation and operational cash flow in evaluating high-risk projects in which the yield criterion is used and in mutually exclusive evaluations in which capital investment and depreciation life vary.

A further reduction in original risk capital investment may be justified if the investment still has alternate use value should the project fail; that is, in addition to the depreciation tax credit from an idle piece of equipment. The application of this principle to a mutually exclusive decision involving a grass roots plant versus a plant located as a part of an integrated facility is illustrated. Interestingly, while most decision-makers tend to be conservative with regard to reducing risk capital, ignoring the reuse potential is inconsistent in this situation as it will tend to favor the investment with the greater risk, i.e., the grass roots location.     

A NOTE ON BIAS IN CAPITAL BUDGETING INTRODUCED BY STOCHASTIC LIFE

This note extends work by Young and Contreras and by Rosenthal on the present worth of cash flows under uncertain timing. In capital budgeting analysis, the use of the expected life of a project instead of the life distribution of the project biases the estimate of its expected net present value. In most situations the bias results in an overestimate of the expected net present value of the project. When the exact life distribution is unknown, the bias can be approximated by Taylor series expansion. The sensitivity of the bias to the discount rate, to cash flow patterns, and to income taxes is also investigated.     

Nominal and Real Required Returns in Present Value Analysis

Present value analysis is complicated by uncertainty about future inflation. If it so happened that real cash flows did not depend on inflation and real required returns were close to zero, then inflation and required returns could be omitted from present value analysis. Unfortunately, they cannot. This paper shows that real cash flows are typically sensitive to inflation and that, outside of the 1950s and 1960s, real interest rates, have not been very stable or close to zero. There have consequently been many years when a neglect of inflation and interest rates would have caused substantial present-value errors. It is also argued that interest rates on Treasury and Corporate AAA bonds are only appropriate for virtually risk-free projects. An engineering project normally commands a substantial risk premium and the neglect of such premia can cause a large overestimate of a project's present value.     

Probability Distribution of the Project Payback Period Using the Equivalent Cash Flow Decomposition

For stochastic cash flows, probabilistic approaches to determine a complete distribution of payback period are very limited. The payback analysis based on the net present value (NPV) has several advantages. For annual cash flows, however, the NPV-based method does not provide a complete payback distribution. This article proposes a new technique, the equivalent cash flow decomposition (ECFD), which converts an annual cash flow into an equivalent subannual cash flow at a desired level of precision. The ECFD technique can be used in conjunction with any probabilistic cash flow technique. This article demonstrates that the ECFD technique overcomes the discontinuity limitation of the conventional NPV-based payback period method and generates a complete distribution of the payback period of annual cash flows. Examples indicate that the proposed method is robust with the accuracy comparable to Monte Carlo simulation.     

Duration Measures for Corporate Project Valuation

Sensitivity analysis is a very common exercise performed with the forecasting of project cash flows. In this article, a duration-type measure is generated that provides a single number for the assessment of project cash flows relative to changes in the discount rate (or adjusted for changes in a particular cash flow model parameter). The calculation is no more difficult than the duration measures that already exist for bonds. Yet, the calculation provides valuable insight that many times is lost when performing sensitivity analysis. Further, at a minimum, the measure provides a gauge for the consequences of misspecifiying the discount rate for a project.     

Evaluation of a mining project under the joint effect of commodity price and exchange rate uncertainties using real options valuation

Cash flows generated by mining projects tend to be volatile and are extensively influenced by exogenous variables, notably commodity prices and exchange rates. The traditional discounted cash flow (DCF) method, which is normally used for economic feasibility studies and mining project evaluations, presents inconsistencies because the method fails to adequately address uncertainties and operational flexibilities and often ignores certain specific market conditions. Numerous studies have been carried out for mining project evaluations using the real options valuation (ROV) technique for assessing commodity price uncertainty, but there is no research on the combined effects of price and exchange rate uncertainties. Therefore, in order to assess the economic viability of a mining project more accurately, the commodity price and its inherent volatility, the exchange rate and its inherent volatility, and the correlation parameters between them have been incorporated into the model and used in the evaluation process. One of the interesting findings revealed in the study is that project values are overestimated if only commodity price uncertainty is considered in evaluating the project value instead of the joint effect of commodity price and exchange rate uncertainties. This new ROV technique will explore the opportunity to utilize an alternative methodology for approximating project values and to identify valuation opportunities to enhance economic gains or to mitigate economic losses, where the DCF valuation method does not.     

Economics of Engineering: A New Fabric from Some Old Threads

Three axioms are used to present the basic fundamental ideas of engineering economics. These axioms are then used to derive a generalization of present value that includes both a local and global perspective. The relationship between the local and global concepts is derived and then used to solve cash flow problems that involve flows placed arbitrarily in time.     

A Review of: “STATISTICS OF EXTREMES” By Emil J. Gumbel Columbia University Press New York, 1958 375 pp. $15.00

Standard procedures for evaluating future cash flows are to find an appropriate discount rate consistent with the cash flow's risk and then to derive a present value. While discounted cash flows seem appropriate for many instances, finding appropriate discount rates is often difficult, or discount rates may not exist when the risk is actually a function of a decision that requires the cash-flow valuation. We consider two approaches that have been suggested to alleviate this problem: the capital asset pricing model (CAPM) and the risk-neutral pricing arguments from option theory. We discuss the assumptions inherent in these models and show the results on the well-known news vendor model. Our option pricing results correspond to Singhal's [17] results using CAPM and a different valuation procedure for the option pricing model. We, however, derive a simpler expression that clearly illustrates differences from the standard form ignoring risk.     

Valuation of projects with minimum revenue guarantees: A Gaussian copula–based simulation approach

This technical note presents a numerical simulation technique to perform valuations of infrastructure projects with minimum revenue guarantees (MRG). It is assumed that the project cash flows—in the absence of the MRG—can be described in a probabilistic fashion by means of a very general multivariate distribution function. Then, the Gaussian copula (a numerical algorithm to generate vectors according to a prespecified probabilistic characterization) is used in combination with the MRG condition to generate a set of plausible cash flow vectors. These vectors form the basis of a Monte Carlo simulation that offers two important advantages: it is easy to implement and it makes no restrictive assumptions regarding the evolution of the cash flows over time. Thus, one can estimate the distribution of a broad set of metrics (net present value, internal rate of return, payback periods, etc.). Additionally, the method does not have any of the typical limitations of real options–based approaches, namely, cash flows that follow a Brownian motion or some specific diffusion process or whose volatility needs to be constant. The usefulness of the proposed approach is demonstrated with a simple example.     

A multi-item fuzzy economic production quantity problem with a finite production rate

Inventory Management and production planning are essential tasks for every company in the industry. Therefore, the development of a large set of Economic Order Quantity (EOQ) models is needed. In this paper, a fuzzy multi-item Economic Production Quantity (EPQ) model is developed. This paper contributes to the state-of-the-art with a theoretical study of a problem, where a company has to decide the size of some production batches under uncertain cycle times. The uncertainty will be handled with triangular fuzzy numbers and an analytical solution will be found to the optimization problem.     

On delineating supply chain cash flow under collectionrisk

The credit terms for accounts receivable (AR) offered by sellers to buyers not only create a time lag between supply chain physical flow and cash flow, but also increase the collection risk contributed by late collection and default. Previous studies describing the relationship between the two major supply chain flows did not consider the collection risk, which poses a serious challenge to companies with limited cash resources when seeking growth opportunities in sales. This study first delineates the relationship between the two flows during a growth period without imposing any constraints. A stochastic optimization model is then developed to observe the managerial implications of cash flow risk under tight cash constraints.     

Technical Note: Formulas for Perpetual Gradients

Virtually all engineering economy texts include the formula to find the present worth of a perpetual annuity, but only two seem to include the similar formula for a perpetual arithmetic gradient. This note adds a new formula for the present worth of a perpetual geometric gradient. Though no cash flow stream really lasts forever, there are many infrastructure projects whose lives are long enough to be economically “forever.” In addition, many endowment problems would be more realistic if they assumed a perpetual geometric gradient. Moreover, the simplicity of the formula may help practitioners and students develop a feel for the economics of long-lived gradients.     

MODELLING REGULATORY DISTORTIONS WITH REAL OPTIONS

The introduction of uncertainty can make a significant difference in the valuation of a project. This manifests itself, inter alia, in the regulatory constraints that can affect the valuations of the firm's investment which, in turn has an adverse impact on consumers' welfare. In particular, the inability to exercise any or all of the delay, abandon, start/stop, and time-to-build options has an economic and social cost. With this view in mind, we specify and estimate a model where regulatory constraints impact on the firm's cash flow and on investment valuation with real options methods.

This paper uses real options analysis to address issues of regulation that have not been previously quantified. We show that regulatory constraints on cash flow have an impact on investment valuations in the telecommunications industry. Specifically, a model is developed to estimate the cost of regulation for broadband services. We show that the cash flow constraints and the inability to delay and abandon has a significant cost. Because some costs are not recognized in a static view of the world, this failure to recognize the operation and implications of non-flexibility by regulators (which can be modeled by real options methods) will lead to a reduction in company valuations which in turn will lead to a reduction in economics welfare.     

DERIVATION OF AN INTERTEMPORAL CORRELATION COEFFICIENT MODEL BASED ON CASH FLOW COMPONENTS AND PROBABILISTIC EVALUATION OF A PROJECT'S NPV

If the first two moments (mean and variance) of the net present value (NPV) are known, various probabilistic information of possible NPV can be derived. However, in general, it is practically impossible to estimate the variance of lengthy investment projects due to difficulties in estimating all the intertemporal correlation coefficients between cash flows of two different periods. In this paper, we derive an estimation model for the intertemporal coefficients based on cash flow components and show how the model, under a certain assumption, can be used for estimating and deriving probabilistic information.