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...methods are subject to sensitivity analysis to determine the effects of changes in forecasting parameters. Generally, this is performed by assessing various scenarios for the parameters and may even lead to probability distribution for the project's profitability. The process is tedious and has the potential to provide analysis that is equally complex to interpret.
Ultimately, the goal of sensitivity analysis is to provide a measure of how susceptible the forecasted cash flows are to a change in the environment in which the forecasted cash flows have been generated. When analyzing a bond portfolio, duration measures provide the equivalent of sensitivity analysis relative to interest rate changes. The purpose of this article is to generate an equivalent duration measure for project cash flows. Such a measure can be simple to calculate (depending on the cash flow structure or the modeling of the cash flows) and captures the intuition of project sensitivity analysis in a single number.
The current literature takes duration analysis beyond bonds and analyzes the duration of equities (e.g., Havert, McLaughlin, and Taggart, 1998). This investigation takes duration analysis to the project level and can in theory provide a duration measure for the cash flows of the entire firm or for the cash flows to equity holders. The primary goal is to provide a duration measure relative to the discount rate (keeping cash flows static) because the discount rate is very difficult to assess for a project even when cash flows are relatively certain. However, duration measures can be calculated based on a cash flow model parameter (e.g., the sales growth rate) or based on the discount rate, which can in turn also affect a cash flow model parameter.
In the first section of the article, a duration measure for project valuation is derived and discussed. An associated duration measure specific to a project input parameter is derived in the second section. The third section concludes the article. In addition, there is an appendix containing VBA code for Excel-type functions to measure project duration and project convexity.
THE DURATION MEASURE FOR PROJECT CASH FLOWS
Macaulay's (1938) bond duration, D, is the negative of the first derivative of the bond price multiplied by (1 + discount rate) and divided by the bond price. More specifically, let the discount rate be symbolized by k and let the bond price be symbolized by P.
D = -[[DELTA]P/[DELTA]k] * [(1 + k)/P] (1)
The duration is then used to find an approximate percentage change in the bond price given a change in the discount rate (see Bodie, Kane, and Marcus, 2004).
[DELTA]P/P = 1 D * [[DELTA]k/(1 + k)] (2)
One can perform a similar calculation based on modified duration, [D.sub.M], which is Macaulay's duration divided by (1 + k).
The magnitude of the duration measure indicates the susceptibility of the bond price to an interest rate change. A larger duration indicates a greater percentage change in bond price should interest rates move. To mitigate the risk of interest rate fluctuations, securities of varying duration can be combined to produce an optimal or target portfolio duration. Consequently, duration is a simple metric that provides valuable insight in regard to bond portfolios. This is the same intuition desired for evaluating project cash flows in regard to sensitivity analysis.
To create a "project duration" measure, [D.sub.P], it is simply a matter of following the formula provided in Eq. (1). Project cash flows ([CF.sub.i]), positive or negative, are evaluated on a periodic basis with a terminal value ([TERM.sub.N]) used to value all cash flows beyond period N. Notice the project valuation, V, resembles a bond valuation without fixed coupons.
V = [N.summation over (i=0)][[CF.sub.i]/[(1 + k).sup.i]] + [TERM.sub.n]/[(1 + k).sup.N] (3)
In practice, the terminal value is usually based on the Nth cash flow, [CF.sub.N] x...
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