Let's break down the solution step-by-step for each alternative (A, B, C): ### **Given Data:** | | A | B | C | |--------------|---------|---------|---------| | Initial Investment | $9,500 | $18,500 | $22,000 | | Annual Savings | $3,200 | $5,000 | $9,800 | | Annual Costs | $1,000 | $2,750 | $6,400 | | Salvage Value | $6,000 | $4,200 | $14,000 | | Project Life | 8 years | 8 years | 8 years | | Interest Rate | 12% | 12% | 12% | #### **Step 1: Calculate Net Annual Cash Flow** Net Annual Cash Flow = Annual Savings - Annual Costs - **A:** $3,200 - $1,000 = $2,200 - **B:** $5,000 - $2,750 = $2,250 - **C:** $9,800 - $6,400 = $3,400 --- ### **Step 2: Payback Period** Payback Period = Initial Investment / Net Annual Cash Flow - **A:** $9,500 / $2,200 ≈ **4.32 years** - **B:** $18,500 / $2,250 ≈ **8.22 years** - **C:** $22,000 / $3,400 ≈ **6.47 years** --- ### **Step 3: Discounted Payback Period** Discounted Payback Period: Find the time when the sum of discounted net cash flows equals the initial investment. #### Discount Factor, \( DF = \frac{1}{(1+r)^n} \), where \( r = 12\% \) Let's calculate for each year (rounded to 2 decimals): | Year | DF (12%) | |------|----------| | 1 | 0.893 | | 2 | 0.797 | | 3 | 0.712 | | 4 | 0.636 | | 5 | 0.567 | | 6 | 0.507 | | 7 | 0.452 | | 8 | 0.404 | #### **A:** Annual discounted cash flows: - Year 1: $2,200 × 0.893 = $1,965 - Year 2: $2,200 × 0.797 = $1,753 - Year 3: $2,200 × 0.712 = $1,566 - Year 4: $2,200 × 0.636 = $1,399 - Year 5: $2,200 × 0.567 = $1,247 - Year 6: $2,200 × 0.507 = $1,115 - Year 7: $2,200 × 0.452 = $995 - Year 8: $2,200 × 0.404 = $889 Cumulative discounted cash flow each year: 1. $1,965 2. $3,718 3. $5,284 4. $6,683 5. $7,930 6. $9,045 ← **just exceeds $9,500 in year 6 to 7** So, **Discounted Payback Period = 7 years** (since $9,045 after 6 years, need part of year 7): Fraction year = ($9,500 - $9,045)/$995 ≈ 0.46 So, **6.46 years** #### **B:** - Year 1: $2,250 × 0.893 = $2,010 - Year 2: $2,250 × 0.797 = $1,793 - Year 3: $2,250 × 0.712 = $1,602 - Year 4: $2,250 × 0.636 = $1,431 - Year 5: $2,250 × 0.567 = $1,276 - Year 6: $2,250 × 0.507 = $1,141 - Year 7: $2,250 × 0.452 = $1,017 - Year 8: $2,250 × 0.404 = $909 Cumulative discounted cash flow each year: 1. $2,010 2. $3,803 3. $5,405 4. $6,836 5. $8,112 6. $9,253 7. $10,270 8. $11,179 $18,500 is never reached in 8 years. So, **Discounted Payback Period > 8 years** #### **C:** - Year 1: $3,400 × 0.893 = $3,036 - Year 2: $3,400 × 0.797 = $2,711 - Year 3: $3,400 × 0.712 = $2,421 - Year 4: $3,400 × 0.636 = $2,162 - Year 5: $3,400 × 0.567 = $1,927 - Year 6: $3,400 × 0.507 = $1,724 - Year 7: $3,400 × 0.452 = $1,537 - Year 8: $3,400 × 0.404 = $1,374 Cumulative discounted cash flow each year: 1. $3,036 2. $5,747 3. $8,168 4. $10,330 5. $12,257 6. $13,981 7. $15,518 8. $16,892 Still does not reach $22,000. So, **Discounted Payback Period > 8 years** --- ### **Step 4: Net Present Value (NPV)** NPV = \(-\)Initial Investment \(+\) PV of net annual cash flows \(+\) PV of Salvage Value #### **Present Value (PV) of Annuity:** PV = Net Cash Flow × [1 - (1 + r)\^-n]/r For 8 years, r = 0.12 PV Factor = [1 - (1 + 0.12)\^-8]/0.12 = [1 - 0.404]/0.12 = 0.596/0.12 ≈ **4.97** #### **A:** - PV of net cash flows = $2,200 × 4.97 = $10,934 - PV of salvage = $6,000 × 0.404 = $2,424 - NPV = \(-\$9,500\) + $10,934 + $2,424 = **$3,858** #### **B:** - PV of net cash flows = $2,250 × 4.97 = $11,183 - PV of salvage = $4,200 × 0.404 = $1,697 - NPV = \(-\$18,500\) + $11,183 + $1,697 = **\(-\$5,620\)** #### **C:** - PV of net cash flows = $3,400 × 4.97 = $16,898 - PV of salvage = $14,000 × 0.404 = $5,656 - NPV = \(-\$22,000\) + $16,898 + $5,656 = **$554** --- ## **Summary Table** | Alternative | Payback Period | Discounted Payback Period | NPV | |-------------|---------------|--------------------------|---------| | **A** | 4.32 years | 6.46 years | $3,858 | | **B** | 8.22 years | >8 years | -$5,620 | | **C** | 6.47 years | >8 years | $554 | --- ## **Conclusion** - **Alternative A** has the highest NPV and the shortest payback periods. - **Alternative C** has a positive NPV, but longer payback periods. - **Alternative B** is not financially attractive (negative NPV, payback > project life). - **Recommendation:** Select **Alternative A**. --- **If you need detailed calculations or Excel formulas, let me know!**

Certainly! Let's go through each part of the analysis in detailed steps, including all calculations, explanations, and final answers. --- ## **Given Data for Alternatives A, B, and C** | Parameter | A | B | C | |--------------------------|--------------|--------------|---------------| | Initial Investment | $9,500 | $18,500 | $22,000 | | Annual Savings | $3,200 | $5,000 | $9,800 | | Annual Costs | $1,000 | $2,750 | $6,400 | | Salvage Value | $6,000 | $4,200 | $14,000 | | Project Life | 8 years | 8 years | 8 years | | Interest Rate (r) | 12% | 12% | 12% | --- ## **Step 1: Calculate Net Annual Cash Flow** Net Annual Cash Flow = Annual Savings – Annual Costs ### For Alternative A: \[ \text{Net Cash Flow}_A = 3200 - 1000 = \$2,200 \] ### For Alternative B: \[ \text{Net Cash Flow}_B = 5000 - 2750 = \$2,250 \] ### For Alternative C: \[ \text{Net Cash Flow}_C = 9800 - 6400 = \$3,400 \] --- ## **Step 2: Calculate Payback Period** **Payback Period** = Initial Investment / Net Annual Cash Flow ### For Alternative A: \[ \text{Payback}_A = \frac{9500}{2200} \approx 4.32 \text{ years} \] ### For Alternative B: \[ \text{Payback}_B = \frac{18500}{2250} \approx 8.22 \text{ years} \] ### For Alternative C: \[ \text{Payback}_C = \frac{22000}{3400} \approx 6.47 \text{ years} \] --- ## **Step 3: Discounted Payback Period** **Method:** Discount each year's net cash flow with the 12% rate, sum until cumulative discounted cash flows equal or exceed initial investment. ### **Step 3.1: Calculate Discount Factors for each year** Using: \[ DF_n = \frac{1}{(1 + r)^n} \] where \( r = 0.12 \). Compute for each year: | Year | \( DF_n \) | |-------|----------------------------| | 1 | \( 1 / (1.12)^1 = 0.893 \) | | 2 | \( 1 / (1.12)^2 = 0.797 \) | | 3 | \( 1 / (1.12)^3 = 0.712 \) | | 4 | \( 1 / (1.12)^4 = 0.636 \) | | 5 | \( 1 / (1.12)^5 = 0.567 \) | | 6 | \( 1 / (1.12)^6 = 0.507 \) | | 7 | \( 1 / (1.12)^7 = 0.452 \) | | 8 | \( 1 / (1.12)^8 = 0.404 \) | --- ### **Step 3.2: Calculate discounted cash flows for each alternative** **For Alternative A:** | Year | Net Cash Flow | Discount Factor | Discounted Cash Flow | Cumulative Sum | |--------|--------------|----------------|----------------------|----------------| | 1 | \$2,200 | 0.893 | 2200 × 0.893 = \$1,965 | \$1,965 | | 2 | \$2,200 | 0.797 | 2200 × 0.797 = \$1,753 | \$3,718 | | 3 | \$2,200 | 0.712 | 2200 × 0.712 = \$1,566 | \$5,284 | | 4 | \$2,200 | 0.636 | 2200 × 0.636 = \$1,399 | \$6,683 | | 5 | \$2,200 | 0.567 | 2200 × 0.567 = \$1,247 | \$7,930 | | 6 | \$2,200 | 0.507 | 2200 × 0.507 = \$1,115 | \$9,045 | | 7 | \$2,200 | 0.452 | 2200 × 0.452 = \$995 | \$10,040 | | 8 | \$2,200 | 0.404 | 2200 × 0.404 = \$889 | \$10,929 | **Determine when cumulative discounted cash flow equals or exceeds initial investment (\$9,500):** - After 6 years: \$9,045 (less than \$9,500) - After 7 years: \$10,040 (exceeds \$9,500) **Fraction of year needed in year 7:** \[ \text{Fraction} = \frac{9,500 - 9,045}{995} \approx \frac{455}{995} \approx 0.46 \] **Discounted Payback Period for A:** \[ 6 + 0.46 \approx \boxed{6.46\, \text{years}} \] --- **For Alternative B:** | Year | Net Cash Flow | Discount Factor | Discounted Cash Flow | Cumulative Sum | |--------|--------------|----------------|----------------------|----------------| | 1 | \$2,250 | 0.893 | 2,250 × 0.893 = \$2,010 | \$2,010 | | 2 | \$2,250 | 0.797 | 2,250 × 0.797 = \$1,793 | \$3,803 | | 3 | \$2,250 | 0.712 | 2,250 × 0.712 = \$1,602 | \$5,405 | | 4 | \$2,250 | 0.636 | 2,250 × 0.636 = \$1,431 | \$6,836 | | 5 | \$2,250 | 0.567 | 2,250 × 0.567 = \$1,276 | \$8,112 | | 6 | \$2,250 | 0.507 | 2,250 × 0.507 = \$1,141 | \$9,253 | | 7 | \$2,250 | 0.452 | 2,250 × 0.452 = \$1,017 | \$10,270 | - After 6 years: \$9,253 (less than \$18,500) - After 7 years: \$10,270 (less than \$18,500) Since total discounted cash flows after 8 years would be less than initial investment, **the discounted payback period exceeds 8 years**. --- **For Alternative C:** | Year | Net Cash Flow | Discount Factor | Discounted Cash Flow | Cumulative Sum | |--------|--------------|----------------|----------------------|----------------| | 1 | \$3,400 | 0.893 | \$3,036 | \$3,036 | | 2 | \$3,400 | 0.797 | \$2,711 | \$5,747 | | 3 | \$3,400 | 0.712 | \$2,421 | \$8,168 | | 4 | \$3,400 | 0.636 | \$2,162 | \$10,330 | | 5 | \$3,400 | 0.567 | \$1,927 | \$12,257 | | 6 | \$3,400 | 0.507 | \$1,724 | \$13,981 | | 7 | \$3,400 | 0.452 | \$1,537 | \$15,518 | | 8 | \$3,400 | 0.404 | \$1,374 | \$16,892 | Total discounted cash flow after 8 years: \$16,892, which is less than initial investment (\$22,000). Thus, **discounted payback period exceeds 8 years**. --- ## **Step 4: Calculate Net Present Value (NPV)** NPV formula: \[ \text{NPV} = - \text{Initial Investment} + \text{PV of Net Cash Flows} + \text{PV of Salvage Value} \] ### For all alternatives: \[ \text{PV of Annuity} = \text{Net Cash Flow} \times \frac{1 - (1 + r)^{-n}}{r} \] Calculate the annuity factor: \[ \text{Annuity Factor} = \frac{1 - (1.12)^{-8}}{0.12} \approx 4.97 \] --- ### **Alternative A:** - PV of net cash flows: \[ 2200 \times 4.97 \approx \$10,934 \] - PV of salvage value: \[ 6000 \times 0.404 \approx \$2,424 \] - NPV: \[ -9500 + 10,934 + 2,424 = \$3,858 \] --- ### **Alternative B:** - PV of net cash flows: \[ 2250 \times 4.97 \approx \$11,183 \] - PV of salvage value: \[ 4200 \times 0.404 \approx \$1,697 \] - NPV: \[ -18500 + 11,183 + 1,697 = -\$5,620 \] --- ### **Alternative C:** - PV of net cash flows: \[ 3400 \times 4.97 \approx \$16,898 \] - PV of salvage value: \[ 14000 \times 0.404 \approx \$5,656 \] - NPV: \[ -22000 + 16,898 + 5,656 = \$554 \] --- ## **Final Summary:** | Alternative | Payback Period | Discounted Payback Period | NPV | |--------------|----------------|---------------------------|---------| | **A** | **4.32 years** | **6.46 years** | **\$3,858** | | **B** | **8.22 years** | >8 years (not recovered within project life) | -\$5,620 | | **C** | **6.47 years** | >8 years (not recovered within project life) | **\$554** | ## **Conclusion:** - **Alternative A** is the most financially attractive, with the shortest payback, recoverable within project life, and highest positive NPV. - **Alternative C** has a positive NPV but longer payback. - **Alternative B** shows a negative NPV and cannot recover initial investment within 8 years. **Hence, the optimal choice is Alternative A.** --- If you'd like further details or Excel formulas, I can provide them!