Question

Find the present value of an ordinary annuity with deposits of $5,618 semiannually for 4 years at 10.0% compounded semiannually.

What is the present value?
$
(Round to the nearest cent.)

Answer 1

`~~~~~~~~~~~~\underset{\textit{payments at the end of the period}}{\textit{Future Value of an ordinary annuity}}\\ \\\\ A=pymnt\left[ \cfrac{\left( 1+(r)/(n) \right)^(nt)-1}{(r)/(n)} \right]`

`\begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\\ pymnt=\textit{periodic payments}\dotfill &\$5618\\ r=rate\to 10\%\to (10)/(100)\dotfill &0.10\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus twice} \end{array}\dotfill &2\\ t=years\dotfill &4 \end{cases}`

`A=5618\left[ \cfrac{\left( 1+(0.10)/(2) \right)^(2\cdot 4)-1}{(0.10)/(2)} \right]\implies A=5618\left( \cfrac{1.05^8~~ - ~~1}{0.05} \right) \\\\[-0.35em] ~\dotfill\\\\ ~\hfill A\approx 53646.89~\hfill`