To evaluate the two projects by calculating the NPV, Payback Period, Discounted Payback Period, Profitability Index, and Net Benefit Cost Ratio (NBCR), we will use a discount rate of 10%.
First, let's compile the cash flows for both projects:
Cash Flows
| Year | Project 1 | Project 2 | |------|-----------|-----------| | 0 | 40000 | 40000 | | 1 | 10000 | 12000 | | 2 | 20000 | 18000 | | 3 | 10000 | 24000 | | 4 | 18000 | 15000 | | 5 | 15000 | 10000 |
A. Net Present Value (NPV)
NPV is calculated using the formula:
\[ NPV = \sum \left( \frac{CF_t}{(1 + r)^t} \right) - Initial , Investment \]
where \( CF_t \) is cash flow at time \( t \), \( r \) is the discount rate, and \( t \) is the year.
For Project 1:
\[ NPV = \frac{40000}{(1 + 0.10)^0} + \frac{10000}{(1 + 0.10)^1} + \frac{20000}{(1 + 0.10)^2} + \frac{10000}{(1 + 0.10)^3} + \frac{18000}{(1 + 0.10)^4} + \frac{15000}{(1 + 0.10)^5} - 40000 \]
\[ = 40000 + \frac{10000}{1.10} + \frac{20000}{1.21} + \frac{10000}{1.331} + \frac{18000}{1.4641} + \frac{15000}{1.61051} - 40000 \]
Calculating each term:
- \( 40000 = 40000 \)
- \( \frac{10000}{1.10} \approx 9090.91 \)
- \( \frac{20000}{1.21} \approx 16528.93 \)
- \( \frac{10000}{1.331} \approx 7513.15 \)
- \( \frac{18000}{1.4641} \approx 12338.68 \)
- \( \frac{15000}{1.61051} \approx 9307.63 \)
Summing these gives us:
\[ NPV \approx 40000 + 9090.91 + 16528.93 + 7513.15 + 12338.68 + 9307.63 - 40000 \] \[ NPV \approx 11370.295 \]
For Project 2:
\[ NPV = \frac{40000}{(1 + 0.10)^0} + \frac{12000}{(1 + 0.10)^1} + \frac{18000}{(1 + 0.10)^2} + \frac{24000}{(1 + 0.10)^3} + \frac{15000}{(1 + 0.10)^4} + \frac{10000}{(1 + 0.10)^5} - 40000 \]
Calculating each term:
- \( 40000 = 40000 \)
- \( \frac{12000}{1.10} \approx 10909.09 \)
- \( \frac{18000}{1.21} \approx 14876.03 \)
- \( \frac{24000}{1.331} \approx 18087.76 \)
- \( \frac{15000}{1.4641} \approx 10243.33 \)
- \( \frac{10000}{1.61051} \approx 6209.31 \)
Summing these gives us:
\[ NPV \approx 40000 + 10909.09 + 14876.03 + 18087.76 + 10243.33 + 6209.31 - 40000 \] \[ NPV \approx 14025.52 \]
Conclusion for A:
- NPV of Project 1: \( 11370.295 \)
- NPV of Project 2: \( 14025.52 \)
Relevant Project: Project 2, because it has a higher NPV.
B. Payback Period
The payback period is the time it takes for a project to recover its initial investment.
For Project 1:
- Year 0: -40000
- Year 1: -40000 + 10000 = -30000
- Year 2: -30000 + 20000 = -10000
- Year 3: -10000 + 10000 = 0
Payback period of Project 1: 3 years.
For Project 2:
- Year 0: -40000
- Year 1: -40000 + 12000 = -28000
- Year 2: -28000 + 18000 = -10000
- Year 3: -10000 + 24000 = 14000
Payback period of Project 2: 3 years (but slightly before end of year 3).
Conclusion for B:
- Payback period of Project 1: 3 years
- Payback period of Project 2: 2.8333 years
Relevant Project: Project 2, as it has a shorter payback period.
C. Discounted Payback Period
The discounted payback period accounts for the time value of money.
For Project 1:
- Year 0 (40000): -40000
- Year 1: 9090.91 => -30909.09
- Year 2: 16528.93 => -14380.16
- Year 3: 7513.15 => -6856.98
- Year 4: 12338.68 => 3481.70 (Completed between year 4 and year 5)
Discounted payback period of Project 1: Between Year 4 and Year 5, approximately 4.5 years.
For Project 2:
- Year 0 (40000): -40000
- Year 1: 10909.09 => -29090.91
- Year 2: 14876.03 => -14214.88
- Year 3: 18087.76 => 3892.88 (Completed between year 3 and year 4)
Discounted payback period of Project 2: Between Year 3 and Year 4, approximately 3.5 years.
Conclusion for C:
- Discounted payback period of Project 1: 4.5 years
- Discounted payback period of Project 2: 3.5 years
Best Project: Project 2, as it has a shorter discounted payback period.
D. Profitability Index (PI)
The Profitability Index is calculated as:
\[ PI = \frac{NPV + Initial , Investment}{Initial , Investment} \]
For Project 1: \[ PI_1 = \frac{11370.295 + 40000}{40000} \approx 1.2843 \]
For Project 2: \[ PI_2 = \frac{14025.52 + 40000}{40000} \approx 1.3506 \]
Conclusion for D:
- Profitability Index of Project 1: \( 1.2843 \)
- Profitability Index of Project 2: \( 1.3506 \)
Relevant Project: Project 2, as it has a higher Profitability Index.
E. Net Benefit Cost Ratio (NBCR)
NBCR is calculated as:
\[ NBCR = \frac{Total , Present , Value , of , Cash , Flows}{Initial , Investment} \]
For Project 1:
\[ \text{Total Present Value of Cash Flows (Project 1)} \approx 51370.295 \] \[ NBCR_1 = \frac{51370.295}{40000} \approx 1.2843 \]
For Project 2:
\[ \text{Total Present Value of Cash Flows (Project 2)} \approx 54025.52 \] \[ NBCR_2 = \frac{54025.52}{40000} \approx 1.3506 \]
Conclusion for E:
- NBCR of Project 1: \( 1.2843 \)
- NBCR of Project 2: \( 1.3506 \)
Relevant Project: Project 2, as it has a higher NBCR.
Summary
- NPV: Project 2 is better.
- Payback Period: Project 2 is better.
- Discounted Payback Period: Project 2 is better.
- Profitability Index: Project 2 is better.
- NBCR: Project 2 is better.
Overall, Project 2 is the superior choice based on all metrics calculated.