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}\\%$per annum compounding {period[0]}, what is this cash flow worth {random(\"now\",\"today\",\"at present\")}? \n \n$\\ [[0]] (to the nearest cent)

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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

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

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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.

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In our situation we have,

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$k=\\var{k}$,

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$C=\\var{C}$,

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

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

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and therefore we have

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

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Calculating this we find

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