# rate of return

Suppose you invest $P$ at time $0$ and receive payments $P_{1},\ldots,P_{n}$ at times $t_{1},\ldots,t_{n}$ corresponding to interest rates (evaluated from $0$) $r_{1},\ldots,r_{n}$. The net present value of this investment is

 $NPV=-P+\frac{P_{1}}{1+r_{1}}+\frac{P_{2}}{1+r_{2}}+\cdots+\frac{P_{n}}{1+r_{n}}.$

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:

 $P=\frac{P_{1}}{(1+r)^{t_{1}}}+\frac{P_{2}}{(1+r)^{t_{2}}}+\cdots+\frac{P_{n}}{% (1+r)^{t_{n}}}.$

Remarks.

• We typically assume that  $t_{1}\leq t_{2}\leq\cdots\leq t_{n}$,  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 $r>-1$.

Title rate of return RateOfReturn 2013-03-22 16:41:05 2013-03-22 16:41:05 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 91B28