Let's break down the problem and solve it step-by-step: --- ### **1. Understand the Problem** - **Goal:** Make equal end-of-quarter deposits from son's 3rd birthday (first deposit 3 months after 3rd birthday) to his 18th birthday (last deposit at 18th birthday), such that there will be **$60,000 available on each birthday from the 18th to the 21st** for college. - **Interest:** 6% compounded quarterly. - **Withdrawal:** 4 withdrawals of $60,000 each (ages 18, 19, 20, 21). --- ### **2. Find the Quarterly Interest Rate** - **Nominal Rate (annual, compounded quarterly):** \( r = 6\% \) - **Quarterly Interest Rate:** \( i = \frac{0.06}{4} = 0.015 \) --- ### **3. Timeline and Number of Periods** - **Deposits:** From age 3 (after 3rd birthday) to age 18 (at 18th birthday). - Number of years = \( 18 - 3 = 15 \) years - Number of quarters = \( 15 \times 4 = 60 \) - **Number of deposits:** 60 (one each quarter, last one at 18th birthday) - **Withdrawals:** 1 each at 18th, 19th, 20th, and 21st birthday. - 4 annual withdrawals --- ### **4. Calculate the Present Value Required at 18th Birthday** We need enough money at the 18th birthday to fund four annual withdrawals of $60,000 each (at ages 18, 19, 20, 21). Let’s treat this as the **present value** of an ordinary annuity (annual withdrawals, but interest is quarterly). - **Each withdrawal:** $60,000 - **Number of withdrawals:** 4 - **Effective interest per year (compounded quarterly):** \[ (1 + i)^4 - 1 = (1.015)^4 - 1 \approx 0.06136 \] So, the annual effective interest rate is about 6.136%. #### **Present Value at 18th birthday:** Using the formula for the present value of an ordinary annuity (from the given formulas, in terms of annual payments and effective annual rate): \[ P = A \left[ \frac{1 - (1 + i)^{-N}}{i} \right] \] But since withdrawals are annual and interest is compounded quarterly, use the annual effective rate \( i_{eff} = 0.06136 \). \[ P = 60,000 \left[ \frac{1 - (1 + 0.06136)^{-4}}{0.06136} \right] \] Calculate: - \( (1 + 0.06136)^{-4} = (1.06136)^{-4} \approx 0.7823 \) - \( 1 - 0.7823 = 0.2177 \) - \( \frac{0.2177}{0.06136} \approx 3.548 \) - \( 60,000 \times 3.548 \approx \$212,880 \) **So, at the 18th birthday, we need $212,880 available.** --- ### **5. Find the End-of-Quarter Deposit (A) Needed** Now, we have to find the quarterly deposit \( A \) that will accumulate to $212,880 at the end of 60 quarters (from just after age 3 to just at 18th birthday). Use the future value formula for an ordinary annuity: \[ F = A \left[ \frac{(1 + i)^N - 1}{i} \right] \] Where: - \( F = 212,880 \) - \( i = 0.015 \) - \( N = 60 \) Solve for \( A \): \[ A = F \left[ \frac{i}{(1 + i)^N - 1} \right] \] Calculate \( (1.015)^{60} \): - \( \ln(1.015) \approx 0.01489 \) - \( 0.01489 \times 60 = 0.8934 \) - \( e^{0.8934} \approx 2.443 \) - \( (1.015)^{60} - 1 = 2.443 - 1 = 1.443 \) Now substitute: \[ A = 212,880 \times \frac{0.015}{1.443} \] \[ A = 212,880 \times 0.0104 \approx \$2,216.20 \] --- ## **Final Step-by-Step Answer** ### **Summary Table** | Step | Calculation | Result | |------|-------------|--------| | 1 | Quarterly rate \( i \) | 0.015 | | 2 | Effective annual rate \( i_{eff} \) | 0.06136 | | 3 | PV at 18th birthday | \$212,880 | | 4 | Number of deposits | 60 | | 5 | Quarterly deposit \( A \) | \$2,216.20 | ### **Final Answer** \[ \boxed{ \text{The end-of-quarter deposit that must be made is approximately } \$2,216.20. } \] --- **Explanation:** From the 3rd to the 18th birthday (60 quarters), depositing \$2,216.20 at the end of each quarter at 6% compounded quarterly will provide \$60,000 on each birthday from the 18th to 21st.