Question

How much should Sean's dad invest into a savings account today, to be able to pay for Sean's rent for the next five years if rent is $850 payable at the beginning of each month? The savings account earns 2.50% compounded monthly.____________________Round to the nearest cent

Answer 1

We are asked to determine the present value of an annuity that is compounded monthly and needs to be paid at the beginning of the time period. To do that we will use the following formula:

`PV_{\text{due}}=PMT((1-(1+i)^(-n))/(i))(1+i)`

Where:

`\begin{gathered} \text{PMT}=\text{ monthly payments} \\ i=interest\text{ rate in decimal form} \\ n=nu\text{mber of periods} \end{gathered}`

Now, the PMT is $850 since this is the required amount for each period. The interest rate in decimal form is:

`(2.50)/(100)=0.025`

The number of periods "n" is the number of months is 5 years, since there are 12 months per year, then in five years we have:

`n=5\text{year}(12month)/(1year)=60months`

Now we replace the values in the formula:

`PV_{\text{due}}=(850)((1-(1+0.025)^(-60))/(0.025))(1+0.025)`

Now we solve the operations and we get:

`PV_{\text{due}}=(850)((1-0.23)/(0.025))(1.025)`

Solving the operations:

`PV_{\text{due}}=26929.17`

Therefore, the investment must be $26929.17.