A Probabilistic Internal Rate of Return: Theory and Illustration
The purpose of this paper is to provide a theoretical background on the internal rate of return (IRR), on the probabilistic IRR, and to present an illustration based upon both a Taylor series expansion and a Monte Carlo simulation. It is shown that Monte Carlo simulation results in a more precise outcome as compared to the theoretical expectations from a Taylor series expansion. This precision is more than twice in terms of the standard deviations of the IRR, and around six times more in terms of the standard errors of the IRR. Second, the distributions of the internal rate of return follow approximately a normal distribution, and this allows a sound basis for project appraisal and risk management. Third, the grand means of the internal rates of returns for all four cases considered are statistically insignificantly different from each other, as expected, and they are statistically insignificantly different from the average internal rate of return, obtained by discounting the mean amounts of the cash flows. Fifth, the standard deviations and the standard errors of the IRR are directly proportional to the assumed standard deviations of the cash flows. The purpose of this paper is to demonstrate the benefits of undertaking Monte Carlo simulation in the estimation of the probability distribution of the internal rate of return (IRR) of a project for which future cash flows are stochastic but independent. Illustrative examples are provided for four cases. The first is when future cash flows follow a uniform distribution, which is a usual assumption in the literature (Sarper et al., 2010). The second is when future cash flows follow a normal distribution with the same mean and variance as the above uniform distribution. The third is when future cash flows follow a normal distribution with the same mean as above but with double the variance. The fourth is when future cash flows follow a normal distribution with the same mean as above but with half the variance. Assuming normal distributions conforms also to the literature (Hillier, 1963; Fairley & Jacoby, 1975; Kim & Reinschmidt, 2012). In all cases the initial cash outlay is fixed in value. Ten thousand simulation runs are produced for each case, and these are repeated a hundred times. It is shown that Monte Carlo simulation results in a more precise outcome, i.e. provides a lower variance for the distribution of the internal rate of return relative to the theoretical expectations from a Taylor series expansion. This precision is more than twice in terms of standard deviations and more than four times in terms of variance. In turn, the precisions of the standard errors of the IRR are around six times more precise in terms of standard deviations, and around thirty six times more precise in terms of variances. Second, the distributions of the internal rate of return follow approximately a normal distribution, together with other statistics from the simulations, and this provides for a sound basis for project appraisal and risk management. Third, the grand means of the internal rates of returns, for all four cases considered, are statistically insignificantly different from each other, as expected. Fourth, the grand means are statistically insignificantly different from the average internal rate of return, obtained by discounting the means of the cash flows. Fifth, the variances of the IRR are directly proportional to the assumed variances of the future cash flows. Finally, the squares of the standard errors of the IRR are directly proportional to the assumed variances of the future cash flows.