## Step 1: Convert Interest Rate and Find PW Factors ### **a. Convert nominal annual interest rate to quarterly rate** Given: Annual nominal interest rate, \( i = 16\% \) compounded quarterly Number of quarters per year = 4 Quarterly interest rate (\( i_{quarter} \)) = \( \frac{16\%}{4} = 4\% \) per quarter - For calculations, use \( i_{quarter} = .04 \). ### **b. Find PW factors for each alternative** #### **Alternative A (Life = 30 years = 120 quarters):** - **Uniform Series Present Worth Factor (USPW):** \( P/A, i, n = \frac{1-(1+i)^{-n}}{i} \) For \( n = 120 \), \( i = .04 \) - **Single Payment Present Worth Factor (SPPW) for Salvage:** \( P/F, i, n = (1+i)^{-n} \) #### **Alternative B (Life = Infinite):** - For infinite life, the Present Worth (PW) for a uniform series = \( PW = \frac{A}{i} \) (since \( (1+i)^{-n} \to \) as \( n \to \infty \)). --- **Explanation:** Before comparing alternatives, we must convert all cash flows to present worth using the correct interest rate and period. Since the interest is compounded quarterly, all calculations must be done per quarter. --- ## Step 2: Calculate Present Worth for Each Alternative ### **a. Alternative A:** #### **Convert all cash flows to quarterly:** - **Operating Cost/Benefit:** Given per year, convert to per quarter: - Operating Cost: \( 15,000 / 2 = 7,500 \) per year → \( 7,500 / 4 = 1,875 \) per quarter - Uniform Benefit: \( 72,000 / 4 = 18,000 \) per quarter - **Net Uniform Benefit:** \( = 18,000 - 1,875 = 16,125 \) per quarter - **PW Calculation:** \[ PW_A = -380,000 + (16,125) \times (P/A, 4\%, 120) + 120,000 \times (P/F, 4\%, 120) \] Using tables or calculator: - \( (P/A, 4\%, 120) \approx 25.97 \) - \( (P/F, 4\%, 120) \approx .0291 \) Plug in values: \[ PW_A = -380,000 + 16,125 \times 25.97 + 120,000 \times .0291 \] \[ PW_A = -380,000 + 418,916.25 + 3,492 \] \[ PW_A = \$42,408.25 \] --- ### **b. Alternative B (Infinite Life):** - **Operating Cost:** \( 18,000 / 2 = 9,000 \) per year → \( 9,000 / 4 = 2,250 \) per quarter - **Uniform Benefit:** \( 85,000 / 4 = 21,250 \) per quarter - **Net Uniform Benefit:** \( = 21,250 - 2,250 = 19,000 \) per quarter - **PW Calculation:** \[ PW_B = -460,000 + \frac{19,000}{.04} \] \[ PW_B = -460,000 + 475,000 \] \[ PW_B = \$15,000 \] --- **Explanation:** All cash flows are converted to present worth using the appropriate factors. For finite life, we use the PW factor and for infinite life, the uniform series formula. --- ## **Final Answer** - **PW of Alternative A:** \$42,408.25 - **PW of Alternative B:** \$15,000 **Conclusion:** **Alternative A is the better investment (higher Present Worth).**