A church congregation has raised ​$37,625 for future outreach work. If the money is invested in a fund paying​ 7% compounded​ quarterly, what annual payment can be made for ten years from the fund if the first payment is to be made four years from the date of investment in the​ fund?

Question

Question content area bottom
Part 1
The size of the payment is ​$
  
enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

Answer 1

To calculate the annual payment that can be made for ten years:

1. Calculate the future value of the investment after 4 years:
PV = $37,625
r = 0.07 (7% annual interest rate)
n = 4 (compounded quarterly)

FV = PV * (1 + r/n)^(n*t)
FV = $37,625 * (1 + 0.07/4)^(4*4)
FV = $37,625 * (1 + 0.0175)^16
FV = $37,625 * (1.0175)^16
FV = $37,625 * 1.300628
FV = $48,970.98

2. Calculate the annuity payment using the future value as the present value for an ordinary annuity formula:
PV = $48,970.98
r = 0.07
n = 4
t = 6 (10 years - 4 years)

Annuity Payment = PV * (r/n) / [(1 - (1 + r/n)^(-n*t))]
Annuity Payment = $48,970.98 * (0.07/4) / [1 - (1 + 0.07/4)^(-4*6)]
Annuity Payment = $48,970.98 * 0.0175 / [1 - (1.0175)^(-24)]
Annuity Payment = $48,970.98 * 0.0175 / [1 - 0.606101]
Annuity Payment = $48,970.98 * 0.0175 / 0.393899
Annuity Payment = $855.20

Therefore, the annual payment that can be made for ten years from the fund is $855.20.