Let's break down the problem step-by-step. --- ## Given Data: | Parameter | Value | |------------------------|-------------------------------------| | MARR (Minimum Attractive Rate of) | 12% per | | Useful life | 14 years | | Initial investment | $5,500 | | Annual receipts | $1,000 per year | --- ## Step 1: Determine the Present Worth (PW) of the receipts The receipts are annual and for 14 years, with a discount rate of 12%. The present value of anuity formula: \[ PW = R \times \frac{1 - (1 + i)^{-n}}{i} \] Where: - R = 1,000 \) - \( i = .12 \) - \( n = 14 \) --- ## Step 2: Calculate the Present Value of Receipts \[ PW = 100 \times \frac{1 - (1 + .12)^{-14}}{.12} \] Calculate \( (1 + .12)^{14} \): \[ (1 + .12)^{14} \approx 1.12^{14} \approx 4.973 \] Now, \( (1 + .12)^{-14} = \frac{1}{4.973} \approx .201 \) Plug into the formula: \[ PW = 100 \times \frac{1 - .201}{.12} = 100 \times \frac{.799}{.12} \] \[ PW \approx 100 \times 6.658 \approx 6,658 \] --- ## **Final Step:** ### **Net Present Value (NPV)** \[ NPV = PW - \text{Initial Investment} = 6,658 - 5,500 = \boxed{\$1,158} \] **Interpretation:** Since the NPV is positive (\$1,158), the project is financially attractive at the 12% MARR. --- ## **Summary:** | Calculation | Result | |--------------|---------| | Present Value of receipts | \$6,658 | | Initial Investment | \$5,500 | | **NPV** | **\$1,158** | The project is likely to be acceptable based on the positive NPV at a 12% discount rate.

# MARR Definition and Explanation ## What is MARR? **MARR** stands for **Minimum Attractive Rate of Return**. It is the lowest return rate that an investor is willing to accept for an investment. In project evaluation, MARR is used as a benchmark to assess whether the returns from a project justify the risks involved. --- # Step-by-Step Explanation of NPV Calculation ## Given Data: - **MARR**: 12% per year - **Useful Life**: 14 years - **Initial Investment**: $5,500 - **Annual Receipts**: $1,000 per year ## Step 1: Present Worth (PW) Calculation ### Formula: To find the present worth of future cash flows, we use the Present Value of Annuity formula: \[ PW = R \times \frac{1 - (1 + i)^{-n}}{i} \] Where: - \( R \) = Annual receipts ($1,000) - \( i \) = Discount rate (12% or 0.12) - \( n \) = Number of years (14) ### Calculation: 1. **Calculate** \( (1 + i)^{-n} \): - \( (1 + 0.12)^{14} \approx 4.973 \) - Thus, \( (1 + 0.12)^{-14} \approx 0.201 \) 2. **Plug values into the formula**: \[ PW = 1,000 \times \frac{1 - 0.201}{0.12} = 1,000 \times \frac{0.799}{0.12} \] \[ PW \approx 1,000 \times 6.658 \approx 6,658 \] ## Step 2: Net Present Value (NPV) Calculation ### Formula: \[ NPV = PW - \text{Initial Investment} \] ### Calculation: 1. **Find NPV**: \[ NPV = 6,658 - 5,500 = 1,158 \] --- # Final Result The **Net Present Value (NPV)** of the project is: \[ \boxed{1,158} \] ## Interpretation Since the NPV is positive (\$1,158), the project is considered financially attractive at the MARR of 12%. This suggests that the project will generate more value than the cost of investment, making it a viable option for investment.