Question

If Joey purchased a $100,000 house with a 20 percent down payment and borrowed the rest on a 30-year mortgage at 4% interest, what would his monthly payment be?

Answer 1

Since Joey put a 20 percent down payment on the house, he only needed to take out a mortgage of $80,000 to purchase the house.  He will pay equal monthly payments  over a 30 year period. The formula to be used to calculate the monthly payment is the formula for the present value of an annuity.

The formula for the present value of an annuity is,

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

Where $P$ is the present value of the mortgage, i.e the amount that Joey gets from the bank, $R$ is the monthly payment, $i =\frac{0.04}{12}=0.0033$ because the interest is calculated monthly, $n= 12\times 30 = 360$ because the payment will be made for 360 months.

The next step is to solve for $R$ in the formula for the present value of annuity. Solving for $R$ we get

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

Substituting the values for $n$ and $i$ in the equation to evaluate the value for $R$.

$R=\frac{0.033\times 80,000}{[1-(1+0.0033)^{-360}]} \\R=381.93$.

The monthly payment for the $80,000 is $381.93.