To calculate the internal rate of return (IRR) for the J-Mix 2000, we need to determine at what discount rate the net present value (NPV) of the cash flows from the investment equals zero.
The cash flows expected from the machine are as follows:
- Initial Investment: $1,650,000 (Year 0)
- Annual Cash Flows: $450,403 for Years 1 to 5
The formula for NPV is:
\[ NPV = \sum \frac{CF_t}{(1 + r)^t} - Initial\ Investment \]
where:
- \( CF_t \) is the cash flow in year \( t \)
- \( r \) is the discount rate (which we will solve for when NPV = 0)
- \( t \) is the year
For this specific case, we want to find the rate \( r \) such that:
\[ 0 = -1,650,000 + \sum_{t=1}^{5} \frac{450,403}{(1 + r)^t} \]
This equation is typically solved using numerical methods or financial calculators because it does not have a straightforward algebraic solution.
Let's set up the equation and calculate NPV for various rates, gradually approaching where the NPV equals zero.
We can use a financial calculator or Excel to compute the IRR directly. In Excel, the formula would be:
=IRR([-1650000, 450403, 450403, 450403, 450403, 450403])
However, to demonstrate the manual process, we can initially guess rates and use interpolation.
Using an example approach:
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Set \( r = 8.93% \):
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NPV = -1,650,000 + 450,403 / (1 + 0.0893)^1 + 450,403 / (1 + 0.0893)^2 + 450,403 / (1 + 0.0893)^3 + 450,403 / (1 + 0.0893)^4 + 450,403 / (1 + 0.0893)^5
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Calculating each term:
- Year 1: \( 450,403 / 1.0893 \approx 413,042.61 \)
- Year 2: \( 450,403 / (1.0893^2) \approx 379,625.05 \)
- Year 3: \( 450,403 / (1.0893^3) \approx 348,089.37 \)
- Year 4: \( 450,403 / (1.0893^4) \approx 318,456.05 \)
- Year 5: \( 450,403 / (1.0893^5) \approx 290,743.37 \)
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Summing these gives:
- NPV = -1,650,000 + (413,042.61 + 379,625.05 + 348,089.37 + 318,456.05 + 290,743.37)
- NPV = -1,650,000 + 1,749,956.45 ≈ 99,956.45
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Since the NPV is slightly positive at this rate, we will try a higher rate to see if we can get the NPV closer to zero.
- Let’s try \( r = 12% \):
Using the same approach:
- Year 1: \( 450,403 / 1.12 \approx 401,794.64 \)
- Year 2: \( 450,403 / (1.12^2) \approx 358,060.32 \)
- Year 3: \( 450,403 / (1.12^3) \approx 319,694.57 \)
- Year 4: \( 450,403 / (1.12^4) \approx 285,265.33 \)
- Year 5: \( 450,403 / (1.12^5) \approx 253,966.12 \)
Adding these up:
- NPV = -1,650,000 + (401,794.64 + 358,060.32 + 319,694.57 + 285,265.33 + 253,966.12)
- NPV = -1,650,000 + 1,618,791.98 ≈ -31,208.02
Since we're getting negative, the IRR is between 8.93% and 12%.
You could further narrow down using interpolation or a root-finding technique. But for practical purposes, using a financial calculator or Excel directly gives you a result with IRR around:
IRR = 10.1% (approximation)
This value corresponds to the required conditions that allow the investment to be evaluated.