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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[0]} repayments on the loan at the beginning of each {period[2]}. Interest is $\\var{ipa}\\%$ per annum compounding {period[0]}.

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You are asked to find the repayment 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

$\\displaystyle P=R(1+i)[ \\frac{1-(1+i)^{-n}}{i} ]$

where $P$ is the present value, $R$ is the repayment per period, $i$ is the interest rate per period, and $n$ is the number of periods.

In our situation we have,

$P=\\var{P}$,

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

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

and therefore we have

$\\displaystyle \\var{P}=R (1+\\frac{\\var{ipadec}}{\\var{period[1]}})[\\frac{1-(1+\\frac{\\var{ipadec}}{\\var{period[1]}})^{-\\var{n}}}{\\frac{\\var{ipadec}}{\\var{period[1]}}}]$

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

Calculating this we find

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

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Which formula should you use to calculate the size of your monthly repayment?

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Calculate the size of your monthly repayment.

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

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The formula for calculating the present value ($P$) of an annuity due is:

$P=R(1+i)\\left[\\frac{1-(1+i)^{-n}}{i}\\right]$

where $R$ represents the value of each repayment, $i$ represents the interest rate per compounding period and $n$ represents the number of repayments.

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