Suppose 40,000 was invested on January 1, 1980 at an annual effective interest rate of 7% in order to provide an annual (calendar-year) scholarship of $5,000 each year forever, the scholarships paid out each January 1.
(a) In what year can the first $5,000 scholarship be made?
(I answered this one and got it correct.)
(b) What smaller scholarship can be awarded the year prior to the first $5,000 scholarship?
(This one, I have trouble with, how do I find the payment without having to disrupt the 5,000 payments? This is the equation I came up with and had adjusted with an equivalent discount rate, nothing seems to work, answer is 2,109.79.)
First part is good. Since we are dealing with annuities-due you need to divide the contribution of $5000 / d where d = i/1+i = .0654
Now, to find the smaller payment 1 year prior to the first $5000 the way I did it was,
FV = 40,000 (1.07)^9 = $73,538.37
Now, we know in order for the scholarship to be available, the funds need to increase up to $76,428.57.
So at year 9, which is .57 years away before the funds reach maturity is
76,428.57 - 73,538.37 = 2,890.20
Now, that value represents the .57 or in this case, we want to be exact so, .56976171 years left until maturity which means that at the current moment
C - C(.56976171) = 5000 - 2890.20 = $2,109.80 has already been earned and can be paid on January 1, 1989.