Absolutely! Let’s go step by step as you asked. --- ### **Step 1: Introduction to the Concept** This problem involves finding the present value (PV) of a **growing annuity**, which is a series of payments that grow at a constant rate (here, 3%) over a fixed number of periods (6 years), discounted at a certain interest rate (4%). The company wants to pre-fund the total cost **one year before the first payment**. --- ### **Step 2: Identify the Variables** Let’s list what we know: - First payment at retirement: \( P = \$95,000 \) - Number of payments: \( n = 6 \) - Growth rate per year: \( g = 3\% = 0.03 \) - Discount rate (interest rate): \( r = 4\% = 0.04 \) - Pre-funding occurs **one year before** the first payment. --- ### **Step 3: Present Value of a Growing Annuity Formula** The present value (at the time of the first payment) of a growing annuity is: \[ PV_{0} = P \times \frac{1 - \left(\frac{1+g}{1+r}\right)^n}{r-g} \] Where: - \( P \) = payment at time 1 - \( r \) = discount rate - \( g \) = growth rate - \( n \) = number of payments But since the firm needs to pre-fund **one year before** the first payment, we need to discount this \( PV_0 \) back by one year: \[ PV_{-1} = PV_{0} \times \frac{1}{1+r} \] --- ### **Step 4: Calculate Present Value at Retirement (Time 0)** Plug values into the formula: \[ PV_0 = 95,000 \times \frac{1 - \left(\frac{1.03}{1.04}\right)^6}{0.04 - 0.03} \] First, calculate \( \frac{1.03}{1.04} = 0.9903846 \). Now, \( (0.9903846)^6 \approx 0.9438 \). So, \[ PV_0 = 95,000 \times \frac{1 - 0.9438}{0.01} = 95,000 \times \frac{0.0562}{0.01} = 95,000 \times 5.62 = \$534,050 \] --- ### **Step 5: Discount Back One Year** Now, discount this value one year back: \[ PV_{-1} = PV_0 \times \frac{1}{1.04} = 534,050 \times 0.961538 = \$513,287 \] --- ### **Step 6: Final Answer (Summary)** **The firm would need to invest \$513,287 one year before retirement to fully pre-fund the annuity.** --- ### **Step 7: Brief Explanation of Each Step** - **Step 1:** Introduced the concept of a growing annuity and the need to pre-fund it. - **Step 2:** Defined all variables and clarified the timing of cash flows. - **Step 3:** Presented the growing annuity present value formula and the adjustment for pre-funding. - **Step 4:** Calculated the present value at the first payment date. - **Step 5:** Discounted the sum to the pre-funding date. - **Step 6:** Stated the final answer clearly. - **Step 7:** Summarized what was done in each step. --- **Final concise answer:** The firm must invest \$513,287 one year before retirement to cover the 6-year, 3%-growing, \$95,000 annuity at a 4% interest rate.