To evaluate and compare the two mutually exclusive projects, let's break down each method of assessment: Payback Rule, Average Rate of Return (ARR), and Net Present Value (NPV).
### 2.1 Payback Rule
The payback period is the time it takes for the cumulative cash flows to equal the initial investment.
**Project A:**
- Initial Investment = -200,000
- Year 1: Cash Flow = 18,000
- Year 2: Cash Flow = 28,000
- Year 3: Cash Flow = 28,000
- Year 4: Cash Flow = 300,000
Cumulative cash flows by year:
- End of Year 1: 18,000
- End of Year 2: 18,000 + 28,000 = 46,000
- End of Year 3: 46,000 + 28,000 = 74,000
- End of Year 4: 74,000 + 300,000 = 374,000
To recover the initial investment of 200,000:
\[ 74,000 \text{ (by the end of Year 3)} + x = 200,000 \]
where \( x \) is the needed portion of Year 4's cash flow to reach the investment recovery:
\[ x = 200,000 - 74,000 = 126,000 \]
Since we get 300,000 in Year 4:
\[ \text{Fraction of Year 4} = \frac{126,000}{300,000} = 0.42 \]
So, payback period for Project A is:
\[ 3 + 0.42 = 3.42 \, \text{years} \]
**Project B:**
- Initial Investment = -20,000
- Year 1: Cash Flow = 10,000
- Year 2: Cash Flow = 9,000
- Year 3: Cash Flow = 10,000
- Year 4: Cash Flow = 8,000
Cumulative cash flows by year:
- End of Year 1: 10,000
- End of Year 2: 10,000 + 9,000 = 19,000
- End of Year 3: 19,000 + 10,000 = 29,000
To recover the initial investment of 20,000:
\[ 19,000 \text{ (by the end of Year 2)} + x = 20,000 \]
where \( x \) is the needed portion of Year 3's cash flow to reach the investment recovery:
\[ x = 20,000 - 19,000 = 1,000 \]
Since we get 10,000 in Year 3:
\[ \text{Fraction of Year 3} = \frac{1,000}{10,000} = 0.1 \]
So, payback period for Project B is:
\[ 2 + 0.1 = 2.1 \, \text{years} \]
**Conclusion**: According to the payback rule, Project B is more lucrative as its payback period (2.1 years) is shorter than that of Project A (3.42 years).
### 2.2 Average Rate of Return (ARR)
ARR is calculated using the formula:
\[ \text{ARR} = \frac{\text{Average Annual Profit}}{\text{Initial Investment}} \]
**Project A:**
- Total Profit = \(18,000 + 28,000 + 28,000 + 300,000 - 200,000 = 174,000\)
- Average Annual Profit = \( \frac{174,000}{4} = 43,500 \)
- Initial Investment = 200,000
\[ \text{ARR} = \frac{43,500}{200,000} \times 100 = 21.75\% \]
**Project B:**
- Total Profit = \(10,000 + 9,000 + 10,000 + 8,000 - 20,000 = 17,000\)
- Average Annual Profit = \( \frac{17,000}{4} = 4,250 \)
- Initial Investment = 20,000
\[ \text{ARR} = \frac{4,250}{20,000} \times 100 = 21.25\% \]
**Conclusion**: According to the ARR, Project A is more viable as its ARR (21.75%) is slightly higher than that of Project B (21.25%).
### 2.3 Net Present Value (NPV)
NPV is the present value of cash inflows minus the initial investment.
\[ \text{NPV} = \sum \frac{C_t}{(1 + r)^t} - C_0 \]
where \( C_t \) is the cash flow at time \( t \), \( r \) is the discount rate, and \( C_0 \) is the initial investment.
**Project A:**
\[ \text{NPV} = \frac{18,000}{(1 + 0.14)^1} + \frac{28,000}{(1 + 0.14)^2} + \frac{28,000}{(1 + 0.14)^3} + \frac{300,000}{(1 + 0.14)^4} - 200,000 \]
\[ \text{NPV} = \frac{18,000}{1.14} + \frac{28,000}{1.14^2} + \frac{28,000}{1.14^3} + \frac{300,000}{1.14^4} - 200,000 \]
\[ \text{NPV} = 15,789.47 + 21,543.30 + 19,346.76 + 176,321.05 - 200,000 \]
\[ \text{NPV} = 233,000.58 - 200,000 \]
\[ \text{NPV} = 33,000.58 \]
**Project B:**
\[ \text{NPV} = \frac{10,000}{(1 + 0.14)^1} + \frac{9,000}{(1 + 0.14)^2} + \frac{10,000}{(1 + 0.14)^3} + \frac{8,000}{(1 + 0.14)^4} - 20,000 \]
\[ \text{NPV} = \frac{10,000}{1.14} + \frac{9,000}{1.14^2} + \frac{10,000}{1.14^3} + \frac{8,000}{1.14^4} - 20,000 \]
\[ \text{NPV} = 8,771.93 + 6,922.47 + 6,742.09 + 4,908.20 - 20,000 \]
\[ \text{NPV} = 27,344.69 - 20,000 \]
\[ \text{NPV} = 7,344.69 \]
**Conclusion**: According to the NPV rule, Project A is more lucrative as its NPV (33,000.58) is higher than that of Project B (7,344.69).
### Overall Conclusion:
- Using the payback rule, Project B is more lucrative.
- Using the ARR, Project A is more viable.
- Using the NPV rule, Project A is more lucrative.
Given that the NPV method is generally regarded as the most accurate indicator of a project's profitability, Project A should be preferred based on its higher NPV.
The following 2 mutually exclusive projects (Project A and Project B) are available :
Year/s Cash Flows (A) Cash Flows (B)
0 -200 000 -20 000
1 18 000 10 000
2 28 000 9 000
3 28 000 10 000
4 300 000 8 000
NB: 1. The company requires a rate of return of 14% on its investment.
2. Assume profits equal cash flows
2.1. Applying the payback rule, which project is more lucrative? (8)
2.2. Using the average rate of return (ARR), determine which project is more viable (6)
2.3. Determine which project is more lucrative if the NPV rule is applied. (11)
1 answer
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