Let's break down the problem step by step. ### **Step 1: Data Extraction** - **Purchase price:** $100,000 - **Loan amount (1/4 of purchase price):** $100,000 × 1/4 = **$25,000** - **Own funds:** $75,000 (rest of purchase price) - **Loan interest rate:** 16% compounded annually - **Loan term:** 3 years, equal payments - **System life:** 5 years - **Salvage value at year 5:** $3,000 - **Annual technician cost:** $20,000/year (years 1-5) - **Annual savings:** $50,000/year (years 1-5) - **MARR for reinvestment (external rate):** 14% per year We are to determine the **ERR (External Rate of Return)**. --- ### **Step 2: Loan Payment Calculation** Calculate **annual loan payment** for $25,000 loan at 16% over 3 years. The formula for annual payment \( A \) is: \[ A = P \frac{i(1+i)^n}{(1+i)^n-1} \] where: - \( P = 25,000 \) - \( i = 0.16 \) - \( n = 3 \) \[ A = 25,000 \times \frac{0.16(1+0.16)^3}{(1+0.16)^3-1} \] First, calculate \( (1+0.16)^3 \): \[ (1.16)^3 = 1.560896 \] So, \[ A = 25,000 \times \frac{0.16 \times 1.560896}{1.560896 - 1} \] \[ A = 25,000 \times \frac{0.24974336}{0.560896} \] \[ A = 25,000 \times 0.44533 = \$11,133.25 \] --- ### **Step 3: Cash Flow Diagram** #### **Year 0** - Outflow: -$100,000 (system purchase) - Inflow: +$25,000 (loan) - Net outflow: -$75,000 #### **Years 1-3** - Loan payment: -$11,133.25 (years 1-3) - Technician: -$20,000 (years 1-5) - Savings: +$50,000 (years 1-5) #### **Years 4-5** - Technician: -$20,000 - Savings: +$50,000 #### **Year 5** - Salvage value: +$3,000 --- ### **Step 4: Net Cash Flows** | Year | Net Cash Flow | |------|---------------| | 0 | -75,000 | | 1 | 50,000 - 20,000 - 11,133.25 = **18,866.75** | | 2 | 18,866.75 | | 3 | 18,866.75 | | 4 | 50,000 - 20,000 = **30,000** | | 5 | 50,000 - 20,000 + 3,000 = **33,000** | --- ### **Step 5: Calculate ERR (External Rate of Return)** ERR calculation: - All **outflows** (costs, loan payment, technician) are compounded *forward* at the *MARR* (14%). - All **inflows** (savings, salvage) are discounted *back* to present at the unknown ERR, and set equal. #### **Future value of outflows at year 5 (compounded at 14%)** 1. **Year 0 outflow ($75,000) compounded for 5 years:** \[ 75,000 \times (1.14)^5 = 75,000 \times 1.9254 = 144,405 \] 2. **Yearly outflows (technician + loan payment):** For years 1-3: Outflow = $31,133.25 For years 4-5: Outflow = $20,000 Let's compound each to year 5 at 14%: - Year 1: \( 31,133.25 \times (1.14)^4 = 31,133.25 \times 1.6895 = 52,593 \) - Year 2: \( 31,133.25 \times (1.14)^3 = 31,133.25 \times 1.4826 = 46,140 \) - Year 3: \( 31,133.25 \times (1.14)^2 = 31,133.25 \times 1.2996 = 40,470 \) - Year 4: \( 20,000 \times (1.14)^1 = 20,000 \times 1.14 = 22,800 \) - Year 5: \( 20,000 \times (1.14)^0 = 20,000 \) **Sum of compounded outflows:** \[ 144,405 + 52,593 + 46,140 + 40,470 + 22,800 + 20,000 = 326,408 \] --- #### **Future value of inflows at year 5 (discounted at ERR)** - Inflows: $50,000/year (years 1-5), $3,000 (year 5) Let ERR = \( i \) - Year 1: \( 50,000 \times (1+i)^4 \) - Year 2: \( 50,000 \times (1+i)^3 \) - Year 3: \( 50,000 \times (1+i)^2 \) - Year 4: \( 50,000 \times (1+i)^1 \) - Year 5: \( 50,000 + 3,000 = 53,000 \) Sum all these inflows and set equal to 326,408: \[ 50,000(1+i)^4 + 50,000(1+i)^3 + 50,000(1+i)^2 + 50,000(1+i) + 53,000 = 326,408 \] --- ### **Step 6: Solve for ERR (\( i \))** Let's try the options numerically. #### **Option A: 12.62%** \[ (1.1262): (1.1262)^1 = 1.1262; (1.1262)^2 = 1.2684; (1.1262)^3 = 1.429; (1.1262)^4 = 1.610 \] \[ = 50,000 \times 1.610 + 50,000 \times 1.429 + 50,000 \times 1.2684 + 50,000 \times 1.1262 + 53,000 \] \[ = 80,500 + 71,450 + 63,420 + 56,310 + 53,000 = 324,680 \] This is close to the compounded outflows of **326,408**. #### **Option B: 14.57%** \[ (1.1457): (1.1457)^1 = 1.1457; (1.1457)^2 = 1.3126; (1.1457)^3 = 1.5047; (1.1457)^4 = 1.725 \] \[ = 50,000 \times 1.725 + 50,000 \times 1.5047 + 50,000 \times 1.3126 + 50,000 \times 1.1457 + 53,000 \] \[ = 86,250 + 75,235 + 65,630 + 57,285 + 53,000 = 337,400 \] This is higher than 326,408. #### **Option D: 13.24%** \[ (1.1324): (1.1324)^1 = 1.1324; (1.1324)^2 = 1.2823; (1.1324)^3 = 1.4527; (1.1324)^4 = 1.6460 \] \[ = 50,000 \times 1.6460 + 50,000 \times 1.4527 + 50,000 \times 1.2823 + 50,000 \times 1.1324 + 53,000 \] \[ = 82,300 + 72,635 + 64,115 + 56,620 + 53,000 = 328,670 \] #### **Option C: 16.52%** \[ (1.1652): (1.1652)^1 = 1.1652; (1.1652)^2 = 1.3587; (1.1652)^3 = 1.5840; (1.1652)^4 = 1.8470 \] \[ = 50,000 \times 1.8470 + 50,000 \times 1.5840 + 50,000 \times 1.3587 + 50,000 \times 1.1652 + 53,000 \] \[ = 92,350 + 79,200 + 67,935 + 58,260 + 53,000 = 350,745 \] --- ### **Step 7: Closest Value** The total compounded outflow is **326,408**. - **Option A (12.62%)**: 324,680 (very close) - **Option D (13.24%)**: 328,670 (also close) - **Option B (14.57%)**: 337,400 - **Option C (16.52%)**: 350,745 The closest is **Option A (12.62%)**. --- ## **Final Answer** \[ \boxed{12.62\%} \] **(Option A)** --- **Summary of Steps:** 1. Calculate loan payment. 2. Set up net cash flows for each year. 3. Compound all outflows to year 5 at MARR (14%). 4. Discount all inflows to year 5 at ERR. 5. Set total inflows equal to total outflows. 6. Solve for ERR by checking answer choices. 7. Closest match is 12.62%.