Mr. Strupp expects to retire in 12 years. Beginning one month after his retirement, he would like to receive $500 per month for twenty years. How much must he deposit into a fund today to be able to do so if the rate of interest on the deposit is 12% compounded monthly?
t = 20yrs * 12mo/yr = 240mo.
Pt = $500/mo * 240mo = $120,000.
Pt = Po(1+r)^n = $120,000.
r = 12%/12mo = 1%/mo = 0.01 = Monthly
% rate expressed as a decimal.
n = 1 comp/mo. * 240mo = 240 comp. periods.
Po(1.01)^240 = 120,000,
Po = 120,000 / (1.01)^240 = $11,016.70.
To calculate the amount Mr. Strupp must deposit today, we can use the formula for the present value of an annuity.
The formula for the present value of an annuity is:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PV is the present value
PMT is the payment per period
r is the interest rate per period
n is the total number of periods
In this case, Mr. Strupp wants to receive $500 per month for 20 years, and the interest rate is 12% compounded monthly. We need to convert the interest rate to a monthly rate by dividing it by 12, and multiply the number of years by 12 to get the total number of months.
PMT = $500
r = 12% / 12 = 0.01
n = 20 years * 12 months/year = 240 months
Now we can plug these values into the formula and calculate the present value:
PV = $500 * [(1 - (1 + 0.01)^(-240)) / 0.01]
Using a calculator or spreadsheet, calculate the value inside the brackets first [(1 - (1 + 0.01)^(-240)) / 0.01], and then multiply it by $500. The result will be the amount Mr. Strupp must deposit into the fund today to be able to receive $500 per month for 20 years at a 12% interest rate compounded monthly.