Solving for compound interest and annuities

Question

Sandra’s grandmother set up an investment portfolio when she was 20. She invested $200 every year and earned an average annual interest rate of 8.8%, compounded annually. When she was 75 years old, she redeemed the investment and invested all the money in an account that earned 4.8%, compounded monthly. She received a monthly allowance from the investment. How much will Sandra’s grandmother receive altogether in payments if the investment will have a zero balance at the end of 10 years?

Answer 1

HINT

1. Compute the amount $A$ of money that was in the investment portfolio when Sandra was 75 (i.e. after 55 years of investment).
2. How big is the monthly payment from $A$ under the described conditions?

Answer 2

First the value of the portfolio at age 75. It the sum of geometric series.

$$200 \sum_{a=20}^{74} \cdot 1.088^{75-a} \\ = 200 \frac{1.088 - 1.088^{56}}{1 - 1.088} \\ \approx 200 \cdot 1266.23 \\ \approx 253,247.72 $$ The value of the allowance over the next 10 years should be worth that amount. If the allowance is $a$, each payment is worth $$a \frac{1}{(1 + 0.048/12)^k} = \frac{a}{1.004^k}$$ So the first payment is worth $a$ and last payment is worth $\frac{a}{1.004^{10 \cdot 12}}$. The total allowance is then worth $$a \frac{1 - \frac{1}{1.004^{120}}}{1 - 1.004} \\ \approx 95.53 \cdot a $$.

and $$a \approx \frac{253,247}{95.53} = 2,650.80$$

So at age 75, the 55 payments she made to the fund will be worth about 1266 times a single payment, and the 120 withdraws from the fund will cost about 96 times a single payment.