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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.)

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Jen
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  • The first year you can give the scholarship the capital will almost certainly be a little more than the minimum needed to maintain itself. The extra can be used for part (b). – Peter Sep 03 '20 at 00:22

1 Answers1

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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.

swordlordswamplord
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