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Suppose you borrow $\\$\\var{P}$to buy a house. The term of the loan is$\\var{years}$years, and you will need to make {period} repayments on the loan at the beginning of each {period}. Interest is$\\var{ipa}\\%$per annum compounding {period}. Calculate the size of your monthly repayment. \n \n$\\ [] (to the nearest cent)

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You are asked to find the cash flow amount of an annuity due (since the payments are at the beginning of each period) and we are given the present value of the annuity. Therefore we will use the present value of an annuity due formula

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

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where $P$ is the present value, $C$ 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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$P=\\var{P}$,

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

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$n=\\var{years}\\times \\var{period}=\\var{n}$,

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and therefore we have

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

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which we need to rearrange to solve for $C$.

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We want to get $C$ by itself. We start by multiplying both sides by $\\left(\\simplify[unitDenominator]{{ipadec}/{period}}\\right)$ (to remove the division of $C$ by it on the right hand side)

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

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Next we will divide both sides by $\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period}}\\right)^\\var{n}}\\right)\\left(1+\\simplify[unitDenominator]{{ipadec}/{period}}\\right)$ (to remove the multiplication by it on $C$)

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

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Calculating this we find

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

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