Question

You have just won a lottery.  Your prize is an annuity with a payment of $50,000 at the beginning of each year, starting on 1/1/2020, for 10 years.

(a)        What is the present value of your prize as of 1/1/2020? Use an annual interest rate of 4%.

You decide to donate your prize to fund an annual scholarship to a university.  The annual scholarship is for $x, given on 5/1 of each year, starting on 5/1/2020, and is supposed to last forever.  This scholarship has the same present value as your lottery prize as of 1/1/2020, at an annual interest rate of 4%.

(b)       Calculate $x.

(c)        Assume that the calculations in parts (a) and (b) are repeated at an annual interest rate of 5%.  Withoutdoing the actual calculations, do you expect $x to be higher than, the same as, or lower than your answer in part (b)? Explain.

Answer 1

a)Annual Payment (A)=$50,000 , n=number of years =10,r=interest rate =0.04

The problem belongs to Present Value of Annuity.

The formula for annuity is PVAF=A[((1+r)^n-1))/(r(1+r)^n)]. Use PVIFA table and look for 4% on row and 10 years on column we get 8.1109.

hence substitute the values into the formula.

=50000*PVIFA(0.04,10)

=50000*8.1109

=$4,05,545

b)Annual Scholarship Amount

since n mentioned is forever . the formula to use is of perpetual bond in which there is only interest payment and no maturity years .

hence

P=A/r

4,05,545=A/0.04

=$16221.8

c)The annual interest rate is now increased from 4 to 5%, $x value will be higher than in part (b). Since interest rate is now increased from 4 to 5%, the scholarship amount is increased by a slight amount than in part (b).