Question

4. (3pts) Deposits of $500 are placed into a fund at the beginning of each year for the next 20 years in order to provide an annual scholarship forever, where the constant annual payments start at the end of 30th year. Find the amount of each payment with annual effective rate of 4%. (Answer: $881.58)

Answer 1

We can calculate the Future value (FV) of these annual deposits by using interest rate of 4% (Future value of annuity due formula as the deposits are at the beginning of each year)

FV = PMT*(1+i) *{(1+i) ^n−1} / i

Where FV =?

PMT = Annual deposit = $500

n = N = number of payments = 20 (year)

i = I/Y = interest rate per year = 4%

Therefore,

FV = $500*(1+4%) *{(1+4%) ^20−1} / 4%

FV = $15,484.60

This account will have $15,484.60 after 20 years but the scholarship will start from the end of 30^th year. Therefore we need to know the future value of the above amount at the end of year 29 (9 years left after deposit) which we can calculate in following manner -

FV = PV * (1+i) ^n

Where, FV is the future value at the end of year 29 =?

Present Value (PV) is future value at the end of year 20 = $15,484.60

i = I/Y = interest rate per year = 4%

And n is time period = 9 years

Therefore,

FV = $15,484.60* (1+4%) ^9

= $22,039.42

Now we can use perpetuity method which pays equal amount of payment with periodic interval for indefinite time.

Formula for calculation of Present value of perpetuity is

Present Value of a perpetuity = C/ r

Where,

The present value of a perpetuity at the time (t=0 at 29^th year) = $22,039.42 (payment will start after one year)

Annual payments C =?

Interest rate r = 4% per year

Therefore,

$22,039.42 = C/4%

Or C = $22,039.42 * 4% = $881.58