Question

Tom and Sally are purchasing a home. They wish to save money for 8 years and purchase a house that has a value of $210,000 with cash. They deposit money into an account paying 10.2% interest.Step 1 of 2 : How much do they need to deposit each month in order to make the purchase?Step 2 of 2 : How much money did they deposit into the account in all?

Answer 1

We have to calculate the monthly deposit in order to achieve a certain amount after certain number of years.

The amount is P = $210,000 and the number of years is n = 8 years.

Each deposit will increase a capital that will be compounding the interest, which has a annual rate of r = 10.2%.

As the deposits are made monthly (and we assume this is also the compounding period), we have a number of subperiods m = 12 subperiods per year.

We can use the annuity formula to find the monthly payment M so as to have a future value of FV = 210000:

$M=\frac{FV*\frac{r}{m}}{(1+\frac{r}{m})^{n*m}-1}$

We can replace with the values and calculate the amount as:

$\begin{gathered} M=\frac{210000*\frac{0.102}{12}}{(1+\frac{0.102}{12})^{8*12}-1} \\ M=\frac{210000*0.0085}{(1.0085)^{96}-1} \\ M\approx\frac{1785}{2.25365-1} \\ M\approx\frac{1785}{1.25365} \\ M\approx1423.84 \end{gathered}$

2) We have to calculate how much they deposit into the account.

As they deposit monthly during 8 years, the amount of deposits will be 12*8 = 96 deposits.

We can then multiply by the deposit amount and obtain:

$\begin{gathered} TP=M*(n*m) \\ TP=1423.84*96 \\ TP=136688.64 \end{gathered}$

Answer:

1) Monthly payment = $1,423.84

2) $136,688.64