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.

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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.

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

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Nevertheless, one can usually, by trialanderror, determine if such an $r$ exists. If $r$ exists, and if ${P}_{i}$ are all nonnegative, then by Descartes’ rule of signs (http://planetmath.org/DescartesRuleOfSigns), $r$ is always unique and $r>1$.
Title  rate of return 

Canonical name  RateOfReturn 
Date of creation  20130322 16:41:05 
Last modified on  20130322 16:41:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 91B28 