The Internal Rate of Return (IRR) is the discount rate at which the Net Present Value (NPV) of an investment is zero. To approximate the IRR, we use linear interpolation between the two discount rates provided.

Data:

• At 6% discount rate, $\text{NPV} = £12,000$
• At 20% discount rate, $\text{NPV} = -£30,000$

Formula for Linear Interpolation:

$\text{IRR} = r_1 + \frac{\text{NPV}_1}{\text{NPV}_1 - \text{NPV}_2} \times (r_2 - r_1)$ Where:

• $r_1 = 6\% = 0.06$
• $r_2 = 20\% = 0.20$
• $\text{NPV}_1 = £12,000$
• $\text{NPV}_2 = -£30,000$

Substitution:

$\text{IRR} = 0.06 + \frac{12,000}{12,000 - (-30,000)} \times (0.20 - 0.06)$

Simplify:

1. $12,000 - (-30,000) = 12,000 + 30,000 = 42,000$
2. $\frac{12,000}{42,000} = 0.2857$
3. $0.20 - 0.06 = 0.14$
4. $0.2857 \times 0.14 = 0.04$

Final Calculation:

$\text{IRR} = 0.06 + 0.04 = 0.10 \, \text{or} \, 10\%$

Approximate IRR:

The approximate IRR is 10%.

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Would you like additional details or have any follow-up questions? Here are 5 related questions:

1. What is the significance of the IRR in decision-making for investments?
2. How does the IRR compare to other investment appraisal techniques like NPV?
3. What are the limitations of linear interpolation when calculating IRR?
4. Why might a company use multiple discount rates to assess NPV?
5. How could other financial metrics complement the IRR in evaluating this machinery?

Tip: When interpreting IRR, compare it with the company's required rate of return or cost of capital to make informed investment decisions.