Question 1177633
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Future Value  of a single investment

A principal value, *[tex \Large PV], invested at a nominal rate of *[tex \Large r] per annum expressed as a decimal, compounded *[tex \Large n] times per year for *[tex \Large t] years has a future value, *[tex \Large FV], of


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ FV\ =\ PV\(1\,+\,\frac{r}{n}\)^{nt}]


So, if you consider *[tex \Large PV\ =\ 1], then *[tex \Large (1\,+\,\frac{r}{n}\)^{nt}] is the factor by which *[tex \Large PV] is multiplied.  Then *[tex \Large (1\,+\,\frac{r}{n}\)^{nt}\ -\ 1], multiplied by 100 to convert to percent, becomes the APY.


*[tex \LARGE\ \ \ \ \ \ \ \ \ \ APY\ =\ \[\(1\,+\,\frac{r}{n}\)^{nt}\ -\ 1\]\ \times\ 100]


For continuous compounding, the formula is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ FV\ =\ Pe^{rt}]


Where *[tex \Large P] is the principal invested, *[tex \Large e] is the base of the natural logarithms, *[tex \Large r] is the nominal rate expressed as a decimal, and *[tex \Large t] is the number of years of the investment.


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ APY\ =\ \(e^{rt}\ -\ 1\)\ \times\ 100]


All you have to do is plug in the numbers and do the arithmetic.

																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
*[illustration darwinfish.jpg]

From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> 
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