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Homerate of return
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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 trialanderror, determine if such an $r$ exists. If $r$ exists, and if $P_{i}$ are all nonnegative, then by Descartes’ rule of signs, $r$ is always unique and $r>1$.
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