A scholarship fund is set up with a lump sum payment of $A today.The fund will make a payment at the end of every year.The payments are: $60,$140,$220,…..If j = 8%,what is A?

Question

Quizplus · Accepted Answer

This is a case of an increasing annuity. The payments increase by $80 each year ($60, $140, $220, etc.), starting from $60. To find the present value (PV) of this annuity, we can use the formula for the present value of an increasing annuity: $PV = P \times \frac{1 - (1 + i)^{-n}}{i} + G \times \left( \frac{1 - (1 + i)^{-n}}{i} - \frac{n \times (1 + i)^{-n}}{i} \right)$, where $P$ is the first payment, $G$ is the increase in payment per period, $i$ is the interest rate per period, and $n$ is the number of periods. However, since the payments increase indefinitely, this becomes a perpetuity.For a perpetuity, the formula simplifies since $n$ approaches infinity, and the term $(1 + i)^{-n}$ approaches 0. The simplified formula for an increasing perpetuity is $PV = \frac{P}{i} + \frac{G}{i^2}$.Given that $P = $60$, $G = $80$, and $i = 8\% = 0.08$, we can calculate $A$ as follows:$$A = \frac{60}{0.08} + \frac{80}{(0.08)^2} = 750 + 12,500 = $13,250.$$Therefore, the correct answer is $13,250.

A Scholarship Fund Is Set Up with a Lump Sum