# Step 1: **Convert Annual Interest Rate to Effective Quarterly Rate** - **Given:** Annual interest rate = 16% compounded quarterly. - **Quarterly interest rate** (\(i_{quarter}\)) = \( \frac{16\%}{4} = 4\% \) per quarter. For annual calculations, we need the **effective annual rate (EAR):** \[ EAR = (1 + i_{quarter})^4 - 1 = (1 + .04)^4 - 1 = 1.16986 - 1 = .16986 \text{ or } 16.986\% \] **Explanation:** Since cash flows are annual but compounding is quarterly, we must use the effective annual rate for all present worth (PW) calculations. --- # Step 2: **Calculate Present Worth (PW) for Each Alternative** ### **Alternative A** (Life = 30 years) \[ \text{PW}_A = - \text{First Cost} - \text{PW(Operating Cost)} + \text{PW(Benefit)} + \text{PW(Salvage)} \] - **First Cost:** $380,000 (today, no discounting needed) - **Operating Cost:** $15,000 per 2 years → $7,500 per year - **Uniform Benefit:** $72,000 per year - **Salvage Value:** $120,000 at year 30 - **Interest Rate:** 16.986% (EAR), \( n = 30 \) years #### **Present Worths:** - **PW of Uniform Series (A):** \[ PW = A \times \frac{1 - (1+i)^{-n}}{i} \] - **PW of Salvage Value:** \[ PW_{SV} = F \times (1+i)^{-n} \] Calculate: - \( i = .16986 \) - \( n = 30 \) \[ PW_{Op.Cost} = 7,500 \times \frac{1 - (1+.16986)^{-30}}{.16986} = 7,500 \times 5.077 = 38,078 \] \[ PW_{Benefit} = 72,000 \times 5.077 = 365,544 \] \[ PW_{Salvage} = 120,000 \times (1+.16986)^{-30} = 120,000 \times .0363 = 4,356 \] \[ PW_A = -380,000 - 38,078 + 365,544 + 4,356 = -48,178 \] --- ### **Alternative B** (Life = Infinite) - **First Cost:** $460,000 - **Operating Cost:** $18,000 per 2 years → $9,000 per year - **Uniform Benefit:** $85,000 per year - **Salvage Value:** Ignore (infinite life) \[ PW_{Op.Cost} = 9,000 / .16986 = 53,000 \] \[ PW_{Benefit} = 85,000 / .16986 = 500,305 \] \[ PW_B = -460,000 - 53,000 + 500,305 = -12,695 \] --- # **Final Answer** - **PW of Alternative A:** $-48,178$ - **PW of Alternative B:** $-12,695$ **Conclusion:** **Alternative B is the better investment** because it has a higher (less negative) present worth.