Monte Carlo simulation is useful when actual data does not exist or is hard to acquire. Many simulations are conducted for the purposes of predicting the mean or forming a probability distribution. The real estate analyst, often faced with a paucity of data, is tempted to simulate the internal rate of return (IRR) for a project. However, simulation introduces inaccuracies because of Jensen's inequality. Operationally, the problem arises from the curved nature of the IRR function. Simulation uses the concept of the expectation, which is a linear operator. Calculating an expectation for a curved function is a form of linear interpolation that has a built-in error to the extent the straight line between two points does not coincide with the curve. This Demonstration refers to this error as a bias.
Fortunately, the bias is small. But for large sums of money, even an error of few basis points in yield can make a large difference in nominal dollars. More importantly, the amount of the bias grows with the variation. The bias is only zero when there is no variation, a situation rendering simulation unnecessary. Conversely, when variation is great, the bias is also large, making simulation less accurate the more one needs it. The analyst is well advised to consider instead simulating the net present value, which under the right circumstances can be a linear function of cash flow, so that its simulation does not produce misleading conclusions.
This Demonstration uses a stylized set of cash flows in which intertemporal cash flows are fixed and the relationship between initial investment and the net sale proceeds can be reversed. In the equation below the IRR is the root of the equation when .
Jensen's theorem affects concave and convex functions equally (the difference in his conclusion, shown below, is the reversal of the inequality sign). The curve of the IRR changes between convex and concave based on the timing and size of the cash flows, producing an infinite number of error forms matching the infinite number of possible cash flow variations.
J. L. W. V. Jensen, "Sur les fonctions convexes et les inégalités entre les valeurs moyennes," Acta Math., 30, 1906 pp. 175–193.
Jensen's inequality holds that a function is convex in the interval if and only if the following inequality is satisfied for all in and for all with : . A common description of this theorem would be that the function of the expectation is always less than or equal to the expectation of the function. The bias described in this Demonstration is a measure of how these two differ. The bias direction will depend on whether the function is convex or concave (resulting in "less than or equal to…" becoming "equal to or more than…" in the statement above).
R. J. Brown, "Sins of the IRR," The Journal of Real Estate Portfolio Management, 12(2), 2006 pp. 195-200.