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You are asked to find the present value of a deferred annuity (since the payments start after some time periods). Therefore we will use the present value of a deferred annuity formula

$\\displaystyle P=\\frac{1}{(1+i)^{k-1}}\\frac{C}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)$

where $P$ is the present value, $k$ is the number of time periods until the first cash flow, $C$ is the cash flow per period, $i$ is the interest rate per period, and $n$ is the number of periods.

In our situation we have,

$k=\\var{k}$,

$C=\\var{C}$,

$i=\\frac{\\var{ipa}\\%}{\\var{period[1]}}=\\frac{\\var{ipadec}}{\\var{period[1]}}$, $i=\\var{ipa}\\%=\\var{ipadec}$,

$n=\\var{years}\\times \\var{period[1]}=\\var{n}$, $n=\\var{n}$,

and therefore we have

$\\displaystyle P=\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^{\\var{k}-1}}\\frac{\\var{C}}{\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)}\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)$

Calculating this we find

$\\begin{align}P&\\approx \\var{P}\\\\&=\\$\\var{Prounded}\\quad \\text{(to the nearest cent)}\\end{align}$

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interest rate per annum as a decimal

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Suppose in $\\var{k}$ {period[2]}s you will be given $\\$\\var{C}$ each {period[2]} for $\\var{years}$ years. If the interest rate is $\\var{ipa}\\%$ per annum compounding {period[0]}, what is this cash flow worth {random(\"now\",\"today\",\"at present\")}?

$\\$$ [[0]] (to the nearest cent)

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