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Weekly UpdatesINTRODUCTION
Deterministic discounted cash flow (DCF) analysis is well established as a financial and economic tool for evaluating an investment's feasibility and in capital budgeting. However, writers and practitioners have long acknowledged the presence of uncertainty associated with the main analysis parameters, namely discount rate, cash flows, and investment life span, and the need for tools to establish the risk associated with investments and budgets. To this end, a number of probabilistic extensions to conventional deterministic calculations for present worth (net present value), annual worth, internal rate of return, payback period, and benefit:cost ratio have been proposed as a way of dealing with uncertainty. The extensions permit the trade-off between the magnitude of the return on an investment and the probability of the return, whether the investor is risk averse or a gambler. This article surveys such contributions to the literature.
No comparable survey appears to have been published. The survey ties together, for the first time, many contributions spread over many years and many sources. The survey will be found useful by those engaged in risk management and decisions associated with investments and budgeting having uncertain outcomes. The survey cites 70 references.
The survey centers on discounted cash flow analysis methods that take closed form solutions as far as possible before doing any numerical computation; as such, it excludes wholly numerical techniques such as Monte Carlo simulation (for example, Hertz, 1964; Mercer and Morgan, 1975; Canada and White, 1980; Mosca et al., 2001; Nassar and Al-Mohaisen, 2006). Sensitivity and "what if" analyses, where the deterministic problem is solved for (usually) small changes in the parameters taken one at a time, are also excluded, as are share-trading based approaches and options (for example, Dixit and Pindyck, 1994; Sarkis and Tamarkin, 2005; Alesii, 2006; Block 2007), the use of decision trees, risk-adjusted discount rates, scenario analysis, and break-even analysis (for example, Park, 1997; Canada and White, 1980), and fuzzy sets (for example, Kahraman et al., 2002; Liou and Chen, 2006; Omitaomu and Badiru, 2007). Zinn et al. (1977), among others, comment on some of these alternative approaches. The concentration of the survey on discounted cash flow analysis is not to be taken to imply that these other methods are not useful or not relevant. Rather, the survey emphasis shows the personal preference of the authors. Many people find these other methods very useful and relevant, and they can be related in many ways to this survey's focus.
The survey makes no comment on the debate as to which is a best measure--present worth, annual worth, future worth, internal rate of return, payback period, or benefit:cost ratio--it being reasoned that the measure most appropriate for any given situation would be the one chosen for use. No comment is made on the usage of each of these measures. The article assumes that the reader is familiar with the conventional deterministic versions of these measures as treated in numerous texts.
Generally the assumption is made that probabilistic information is available on the analysis parameters, though obtaining good underlying data represents a considerable obstacle. Probability distributions might be assumed describing the parameters, and the analysis then leads to probability distributions describing the random variables of present worth, annual worth, future worth, internal rate of return, payback period, and benefit:cost ratio. The simpler approach is to do a second-order moment analysis, only working with expected values and variances rather than complete probability distributions.
The obtained probabilistic information on present worth, annual worth, future worth, internal rate of return, payback period, and benefit:cost ratio can then feed into any decision-making regarding investment feasibility and budgeting. As such, present worth, annual worth, future worth, internal rate of return, payback period, and benefit:cost ratio might be used as objective functions or interpreted as constraints to be satisfied in the decision problem.
The structure of the article is in the order of present worth, annual worth and future worth, internal rate of return, payback period, and benefit:cost ratio. Papers that have contributed to probabilistic versions of such methods are cited according to the method. Each method is seen to have attracted the attention of a number of writers. Within each method, the contributions are categorized according to whether the three main parameters--discount rate, cash flows, and investment life span--are treated as being deterministic or probabilistic. Generally it is seen that writers do not allow all three parameters to be probabilistic at the same time because of the intractability of the problem. In the following, a parameter is assumed deterministic unless noted otherwise. The probabilistic version also raises the issue of uncertainty over the timing of the occurrence of cash flows.
The methods can be unified to a certain extent through introducing the notion of feasibility. The definition of feasibility, the probability that an investment is worthwhile, changes slightly between the methods of present worth, annual worth, future worth, internal rate of return, payback period, and benefit:cost ratio, but all can be related to each other. The discussion on feasibility follows the literature survey.
NOTATION AND ASSUMPTIONS
The underlying parameters--discount rate, cash flows, and life span--are assumed to be probabilistic. This leads to the present worth, annual worth, future worth, internal rate of return, payback period, and benefit:cost ratio being random variables.
The probability distributions or probabilistic moments for the discount rate, cash flows, and life span might be based on experience, data, subjective probability estimates (Wagle, 1967), or PERT (program evaluation and review technique) style optimistic, pessimistic, and most likely estimates.
With PERT style thinking, the mean and variance are calculated from
[micro] = 1/6(a + 4m + b) (1)
[[sigma].sup.2] = [[1/6(b-a)].sup.2] (2)
where a, m, and b are optimistic, most likely, and pessimistic estimates (for example, Carmichael, 2006). An appropriate ("two unknowns") probability distribution, such as the normal distribution, can then be fitted to the data.
Generally, discrete time period discounting is adopted by most authors, though some prefer continuous time discounting.
The term risk is used in the sense of Carmichael (2004) to mean the exposure to the chance of occurrences of events adversely or favorably affecting the investment as a consequence of uncertainty.
PRESENT WORTH
Probabilistic Cash Flows
Hillier (1963) appears to have been the first to give an analytical method for determining the probability distribution describing present worth. Hillier's approach treats the periodic cash flows as mutually independent and/or completely correlated random variables. It is assumed that the means and the variances of the cash flows are known.
Let [X.sub.i] be the net cash flow for periods i = 0, 1, 2, ..., n. The present worth for an n-period investment, [PW.sub.n], is given by
[PW.sub.n] = [n.summation over (i=0)][[X.sub.i]/[(1 + r).sup.i]] (3)
where r is the discount rate. [PW.sub.n] will follow a normal distribution where the [X.sub.i] are normally distributed, while the central limit theorem would indicate that the distribution for [PW.sub.n] should, for increasing n, approach that of a normal distribution irrespective of the distributions for [X.sub.i]. See also Hillier (1969) for comment on the normal distribution assumption. Hillier (1963) and Wagle (1967) observe that [PW.sub.n] is the sum of weighted terms, where the weights are the present worth factors, and early cash flows may dominate in determining the distribution shape for [PW.sub.n]. Wagle, however, notes that the [X.sub.i] may themselves be the sum of a number of variates, and hence the [X.sub.i] could be expected to approach being normally distributed.
The expected value and variance of [PW.sub.n] become (Wagle, 1967; Canada and White, 1980):
E[[PW.sub.n]] = [n.summation over (i=0)] E[[X.sub.i]]/[(1 + r).sup.i] (4)
Var[[PW.sub.n]] = [n.summation over (i=0)] Var[[X.sub.i]]/[(1 + r).sup.2i] + 2 [n-1.summation over (i=0)] [n.summation over (j=i+1)] Cov[[X.sub.i], [X.sub.j]]/[(1 + r).sup.i+j] (5)
Alternatively, the variance expression can be written in terms of the correlation coefficient between [X.sub.i] and [X.sub.j] rather than the covariance of [X.sub.i] and [X.sub.j],
Var[[PW.sub.n]] = [n.summation over (i=0] Var[[X.sub.i]]/[(1 + r).sup.2i] + 2 [n-1.summation over (i=0)][n.summation over (j=1+1)] [[rho].sub.ij] [square root of Var[X.sub.i]] [square root of Var[X.sub.j]]/[(1 + r).sup.i+j] (6)
For independent cash flows [X.sub.i],
Var[[PW.sub.n]] = [n.summation over (i=0)] Var[[X.sub.i]]/[(1 + r).sup.2i] (7)
For perfect correlation of the cash flows [X.sub.i],
Var[[PW.sub.n] = ([n.summation over (i=0)] [square root of Var[[X.sub.i]]/[1 + r).sup.i]) (8)
Var[[PW.sub.n]] is smaller for the full assumption of independence compared with the full assumption of correlation. And the assumption of a normal distribution for [PW.sub.n] could be expected to be better for the case of independence compared to the correlated case.
The comment is made by Hillier that data would generally not be available on the covariances or correlations of the cash flows, or if it was available, accurate values could not be anticipated. Nevertheless, suggestions for obtaining estimates for correlation coefficients between cash flows in different periods have been advanced (see, for example, Hillier, 1969; Kim and Elsaid, 1988; and Kim et al., 1999).
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