Chapter 3 Quiz: Finance Score: $12 / 20 \quad 3 / 5$ answered Question 4 Suppose you want to have $\$ 800,000$ for retirement in 35 years. Your account earns $7 \%$ interest. How much would you need to deposit in the account each month? $\$$ Submit Question
Solution
Step 1 ：Given that the future value (FV) is $800,000, the annual interest rate (r) is 7% or 0.07 in decimal form, the number of compounding periods per year (n) is 12 (monthly deposits), and the number of years (t) is 35, we can use the formula for the future value of an annuity to find the monthly deposit (P).
Step 2 ：The formula for the future value of an annuity is \(FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}\).
Step 3 ：We want to find P, so we rearrange the formula to solve for P: \(P = FV \times \frac{r/n}{(1 + r/n)^{nt} - 1}\).
Step 4 ：Substituting the given values into the formula, we get: \(P = $800,000 \times \frac{0.07/12}{(1 + 0.07/12)^{12 \times 35} - 1}\).
Step 5 ：Simplifying the expression, we get: \(P = $800,000 \times \frac{0.00583333}{(1.00583333)^{420} - 1}\).
Step 6 ：Further simplifying the expression, we get: \(P = $4666.67 / (10.678 - 1)\).
Step 7 ：Finally, simplifying the expression, we get: \(P = $4666.67 / 9.678\).
Step 8 ：Calculating the value of P, we get: \(P = $482.32\).
Step 9 ：So, you would need to deposit approximately \(\boxed{$482.32}\) each month to have $800,000 for retirement in 35 years with a 7% interest rate.
