rate of return

Suppose you invest $P$ at time $0$ and receive payments $P_1,\mathrm{\dots },P_n$ at times $t_1,\mathrm{\dots },t_n$ corresponding to interest rates (evaluated from $0$) $r_1,\mathrm{\dots },r_n$. The net present value of this investment is

$$NPV=-P+\frac{P_1}{1+r_1}+\frac{P_2}{1+r_2}+\mathrm{\cdots }+\frac{P_n}{1+r_n}.$$

The rate of return $r$ of this investment is a compound interest rate, compounded at every unit time period, such that the net present value of the investment is $0$. In other words, if $r$, as a real number, exists, it satisfies the following equation:

$$P=\frac{P_1}{{(1+r)}^{t_1}}+\frac{P_2}{{(1+r)}^{t_2}}+\mathrm{\cdots }+\frac{P_n}{{(1+r)}^{t_n}}.$$

Remarks.

• We typically assume that  $t_1\le t_2\le \mathrm{\cdots }\le t_n$,  and, in most situations, that they are integers, so that the equation is a polynomial equation.

• However, there is no guarantee that $r$ exists, and if it exists, that it is unique.

• Nevertheless, one can usually, by trial-and-error, determine if such an $r$ exists. If $r$ exists, and if $P_i$ are all non-negative, then by Descartes’ rule of signs (http://planetmath.org/DescartesRuleOfSigns), $r$ is always unique and $r>-1$.