Question

            6.         Lara is planning for retirement. She believes she will need $7,500 a month (payable at the end of each month) for approximately 20 years after retirement. She has 25 years remaining until her retirement starts, and during this time she will deposit an annual amount into her company’s retirement plan at the beginning of each year. Assuming her investment will earn 12% (nominal APR), compounded monthly, throughout both time periods, how much will Lara need to save each year before retirement? (5 pts)

Answer 1

Calculation of amount needed at retirement:

Monthly Payment = $7,500
Time Period = 20 years or 240 months

Annual Interest Rate = 12.00%
Monthly Interest Rate = 1.00%

Amount needed = $7,500/1.01 + $7,500/1.01^2 + … + $7,500/1.01^239 + $7,500/1.01^240
Amount needed = $7,500 * (1 - (1/1.01)^240) / 0.01
Amount needed = $7,500 * 90.819416
Amount needed = $681,145.62

Calculation of annual deposit:

Effective Annual Rate = (1 + Monthly Interest Rate)^12 - 1
Effective Annual Rate = (1 + 0.01)^12 - 1
Effective Annual Rate = 1.126825 - 1
Effective Annual Rate = 0.126825 or 12.6825%

Time period = 25 years

Let annual deposit be $x

$x*1.126825^25 + $x*1.126825^24 + … + $x*1.126825^2 + $x*1.126825 = $681,145.62
$x * 1.126825 * (1.126825^25 - 1) / 0.126825 = $681,145.62
$x * 166.933165 = $681,145.62
$x = $4,080.35

Annual Deposit = $4,080.35

Therefore, Lara needs to save $4,080.35 each year before retirement.