Solving for compound interest and annuities 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? 
 A: 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.
A: HINT


*

*Compute the amount $A$ of money that was in the investment portfolio when Sandra was 75 (i.e. after 55 years of investment).

*How big is the monthly payment from $A$ under the described conditions?

