Let's break down the problem and solve step by step using **only the given formulas**. --- ## **Step 1: Understanding the Problem** - **Goal:** Find the end-of-quarter deposit amount, \( A \). - **Deposits:** Start 3 months after son's 3rd birthday, end on his 18th birthday (inclusive). - **Withdrawals:** $60,000 each on son's 18th, 19th, 20th, and 21st birthdays (4 equal annual withdrawals). - **Interest:** 6% nominal annual, compounded quarterly. --- ## **Step 2: Key Parameters** ### **Interest Rate** - **Nominal annual rate:** \( r = 6\% = 0.06 \) - **Quarterly rate:** \( i = \frac{0.06}{4} = 0.015 \) per quarter ### **Deposits** - **From:** 3 months after 3rd birthday, i.e., at age 3.25 years - **To:** On 18th birthday, i.e., at age 18 years - **Number of years depositing:** \( 18 - 3 = 15 \) years - **Quarters:** \( 15 \times 4 = 60 \) deposits ### **Withdrawals** - **$60,000 on each of the son's 18th, 19th, 20th, and 21st birthdays** - **Time of first withdrawal:** Immediately after the last deposit --- ## **Step 3: Find Present Value Needed at Age 18** At age 18, the account must have enough to pay four $60,000 annual withdrawals: - These are at the beginning of each year (birthday), so treat as ordinary annuity (use \( F = A \left[\frac{(1+i)^N-1}{i}\right] \) if needed, but here, we use present value at the time of first withdrawal). ### **Quarterly Periods Between Withdrawals** - 1 year = 4 quarters, so withdrawals are spaced by 4 quarters. - Effective quarterly interest rate \( i = 0.015 \) #### **Present Value at 18th Birthday** Each $60,000 is withdrawn at 0, 4, 8, and 12 quarters from the 18th birthday. \[ PV = 60,000 + \frac{60,000}{(1+i)^4} + \frac{60,000}{(1+i)^8} + \frac{60,000}{(1+i)^{12}} \] Calculate \( (1+i)^4 = (1.015)^4 \): \[ (1.015)^4 \approx 1.06136 \] \[ (1.06136)^2 = 1.1265 \quad \text{(for 8 quarters)} \] \[ (1.06136)^3 = 1.1956 \quad \text{(for 12 quarters)} \] So, \[ PV = 60,000 + \frac{60,000}{1.06136} + \frac{60,000}{1.1265} + \frac{60,000}{1.1956} \] Calculate each term: - \( 60,000/1.06136 \approx 56,546 \) - \( 60,000/1.1265 \approx 53,282 \) - \( 60,000/1.1956 \approx 50,194 \) Sum: \[ PV = 60,000 + 56,546 + 53,282 + 50,194 = 220,022 \] --- ## **Step 4: Find the Required Quarterly Deposit** Now, we need the quarterly deposit \( A \) such that, after 60 deposits, the amount at the 18th birthday is $220,022. Use the future value of an ordinary annuity formula (from the image): \[ F = A \left[ \frac{(1+i)^N - 1}{i} \right] \] But we want \( A \), so rearrange (from the image): \[ A = F \left[ \frac{i}{(1+i)^N - 1} \right] \] - \( F = 220,022 \) - \( i = 0.015 \) - \( N = 60 \) First, calculate \( (1.015)^{60} \): \[ \ln(1.015) \approx 0.01489 \] \[ 60 \times 0.01489 \approx 0.8934 \] \[ e^{0.8934} \approx 2.443 \] So, \[ (1.015)^{60} \approx 2.443 \] \[ (1.015)^{60} - 1 = 2.443 - 1 = 1.443 \] Plug values into the formula: \[ A = 220,022 \times \frac{0.015}{1.443} \] \[ A = 220,022 \times 0.0104 = 2,288 \] --- ## **Step 5: Final Answer** \[ \boxed{A = \$2,288} \] **Explanation:** To provide $60,000 on each birthday from the 18th to the 21st, the couple must deposit $2,288 at the end of each quarter from the son's 3rd birthday to his 18th birthday, assuming a nominal 6% annual interest rate compounded quarterly. --- ### **Summary Table** | Step | Description | Value | |------|-------------|-------| | 1 | Quarterly interest rate | 0.015 | | 2 | Number of deposits | 60 | | 3 | Present value needed at 18 | $220,022 | | 4 | Quarterly deposit | $\boxed{2,288}$ | --- **Let me know if you need a more detailed explanation for any step!**