Your brother wants to borrow $ 10 comma 000 from you. He has offered to pay you back $ 12 comma 750 in a year. If the cost of capital of this investment opportunity is 9 %​, what is its​ NPV? Should you undertake the investment​ opportunity? Calculate the IRR and use it to determine the maximum deviation allowable in the cost of capital estimate to leave the decision unchanged.



To determine whether you should undertake the investment opportunity, we need to calculate the Net Present Value (NPV) and the Internal Rate of Return (IRR). Here are the steps:

1. **Calculate the NPV:**

   The formula for NPV is:
   \[
   \text{NPV} = \sum \frac{C_t}{(1 + r)^t} - C_0
   \]
   where:
   - $C_t$ is the cash inflow at time $t$
   - $r$ is the discount rate (cost of capital)
   - $C_0$ is the initial investment

   Given:
   - $C_0 = -10,000$
   - $C_1 = 12,750$
   - $r = 0.09$

   Thus,
   \[
   \text{NPV} = \frac{12,750}{(1 + 0.09)} - 10,000
   \]

   Let's calculate it:

   \[
   \text{NPV} = \frac{12,750}{1.09} - 10,000
   \]
   \[
   \text{NPV} = 11,697.25 - 10,000
   \]
   \[
   \text{NPV} = 1,697.25
   \]

2. **Calculate the IRR:**

   The IRR is the rate that makes the NPV of the investment zero. For this investment, we solve for $IRR$ in the following equation:
   \[
   0 = \frac{12,750}{(1 + IRR)} - 10,000
   \]

   Rearranging and solving for $IRR$:
   \[
   10,000 = \frac{12,750}{(1 + IRR)}
   \]
   \[
   1 + IRR = \frac{12,750}{10,000}
   \]
   \[
   1 + IRR = 1.275
   \]
   \[
   IRR = 0.275 \text{ or } 27.5\%
   \]

3. **Determine the maximum deviation allowable in the cost of capital:**

   The IRR is 27.5%. The cost of capital is 9%. The investment decision remains unchanged as long as the cost of capital is less than the IRR.

   Therefore, the maximum allowable deviation is:
   \[
   27.5\% - 9\% = 18.5\%
   \]

### Conclusion:

- **NPV**: $ \$1,697.25 $
- **IRR**: $ 27.5\% $
- **Maximum Deviation Allowable in Cost of Capital**: $ 18.5\% $

Since the NPV is positive, and the IRR is significantly higher than the cost of capital, you should undertake the investment opportunity.

Would you like further details or have any questions?

Here are some related questions you might consider:
1. How do you calculate the future value of an investment?
2. What is the difference between IRR and NPV?
3. How do you interpret the payback period of an investment?
4. What are the risks associated with high IRR investments?
5. How can sensitivity analysis be applied to investment decisions?

**Tip:** When evaluating investment opportunities, always consider both the NPV and IRR to get a comprehensive view of the potential returns and risks.

Learn how to calculate the Net Present Value (NPV) and Internal Rate of Return (IRR) for a $10,000 investment opportunity with a cost of capital of 9%. Determine whether to undertake the investment based on NPV and IRR calculations. Explore the maximum allowable deviation in cost of capital for decision stability.

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