To appraise the profitability of the proposed investment using both the Net Present Value (NPV) and the Profitability Index (PI) methods, let's break it down step by step.

1. Net Present Value (NPV) Calculation

The formula for NPV is:

$NPV = \sum \frac{C_t}{(1 + r)^t} - C_0$

Where:

• $C_t$ = Cash inflow at time $t$
• $r$ = Discount rate (10% in this case)
• $t$ = Year (from 1 to 4)
• $C_0$ = Initial investment ($50,000)

Cash Flows:

• Year 1: $20,000
• Year 2: $15,000
• Year 3: $25,000
• Year 4: $10,000

The NPV formula applied here:

$NPV = \frac{20000}{(1+0.1)^1} + \frac{15000}{(1+0.1)^2} + \frac{25000}{(1+0.1)^3} + \frac{10000}{(1+0.1)^4} - 50000$

Breaking this down:

$NPV = \frac{20000}{1.1} + \frac{15000}{1.21} + \frac{25000}{1.331} + \frac{10000}{1.4641} - 50000$

$NPV = 18181.82 + 12396.69 + 18785.57 + 6831.68 - 50000$

$NPV = 56195.76 - 50000$

$NPV = 6195.76$

Thus, NPV = $6,195.76.

Since the NPV is positive, this indicates that the project adds value and should be considered profitable.

2. Profitability Index (PI) Calculation

The formula for the Profitability Index is:

$PI = \frac{\sum \frac{C_t}{(1 + r)^t}}{C_0}$

We already calculated the present value of future cash flows (in the NPV calculation):

$\text{Present Value of Cash Inflows} = 56195.76$

Thus:

$PI = \frac{56195.76}{50000} = 1.1239$

The Profitability Index (PI) = 1.1239.

A PI greater than 1 suggests that the investment is profitable, confirming that the project is worth considering.

Conclusion

• NPV = $6,195.76, which means the project is expected to generate more than the initial investment, making it profitable.
• PI = 1.1239, indicating that the project returns 12.39% more than the initial cost, thus a good investment.

Would you like further details on any of the steps?

Related Questions

1. How would a change in the discount rate affect the NPV and PI?
2. What are the limitations of using NPV and PI in project appraisal?
3. Can you compare NPV with Internal Rate of Return (IRR)?
4. What happens if the initial investment was financed through a loan at 10% interest?
5. How would delayed cash inflows (e.g., starting from year 2) affect profitability?

Tip: A positive NPV indicates that the investment should add value, but you should also compare it with other projects or alternative investments.