Home | Business News | Browse by Publication | M | Management Science

An extension of the internal rate of return to stochastic cash flows.

Article Excerpt
1. Introduction

The internal rate of return (IRR) is a widely used tool for evaluating deterministic cash flow streams, familiar to all students of finance and engineering economics (e.g., Brealey and Myers 1996). When used appropriately, it can be a valuable aid in project acceptance and selection. However, the method is subject to well-known difficulties: a cash flow stream can have multiple conflicting internal rates (both above and below the hurdle rate), or no real-valued internal rate at all and can appear to be inconsistent with net present value calculations. As Rothkopf (1965) and others note, it does not extend well to situations involving uncertainty because different realizations of an uncertain cash flow stream can have different numbers of internal rates or no real-valued internal rate at all. This fact makes calculation of the distribution of the internal rate very difficult (although there are approximations for special cases--see Fairley and Jacoby 1975). But whether one even needs that distribution is an open question--even under risk neutrality, there is no theoretical guidance as to whether one should use, for example, the mean internal rate versus the internal rate of the mean cash flow.

Recently, however, I have devised an approach that circumvents the problem of multiple or nonexisting internal rates (Hazen 2003) for deterministic cash flow streams, while retaining consistency with net present value. Here I explore an extrension of this approach to stochastic cash flow streams. The result is a random rate-of-return measure, here termed the return on mean investment, that is consistent with expected utility of net present value. Even though it reduces to the standard internal rate for deterministic streams, this measure differs from the internal rate for stochastic streams. Nevertheless, it has important conceptual and computational advantages over the internal rate.

We consider here only cash flow streams whose payoffs cannot be hedged by market investments. The latter is treated, for example, in Smith and Nau (1995). The point of view here is the standard one--that net present value, or its expected utility, is the proper way to evaluate nonhedgeable cash flow streams. (1) How ever, the IRR has enduring intuitive appeal and may be a useful supplement to net present value when personal preference or institutional custom dictate. Indeed, a survey of 392 CFOs by Graham and Harvey (2001) found that IRR is employed in capital budgeting with at least as great a frequency as net present value.

In the next section, we review results from Hazen (2003), which are an essential basis for what follows. In [section]3, we introduce the main results and present examples. Section 4 summarizes.

2. Deterministic Cash Flow Streams

The net present value PV(x|r) of a cash flow stream x = ([x.sub.0], [x.sub.1],..., [x.sub.T]) at interest rate r is given by

PV(x|r) = [T.summation over (t = 0)][x.sub.t]/[1 + r).sup.t].

An IRR k for x is any value of r that makes net present value equal to zero. As is well known, for conventional cash flows x that are negative for the first few periods but positive thereafter, a unique proper (k and the investment that generates the cash flow x is worthwhile. This is the fundamental justification for the use of IRR. As is well...