To calculate the Internal Rate of Return (IRR) for Project X, we need to organize the cash flows of the project and solve for the discount rate that makes the Net Present Value (NPV) equal to zero. Here's how we approach this:

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Step 1: Cash Flow Breakdown

1. Initial Investment:

   • Equipment cost: $-428,000$
   • Change in Net Working Capital (NWC) at time 0: $-24,000$
     Net cash flow at time 0 = -428,000 - 24,000 = -452,000.

2. Yearly Operating Cash Flows:

   • Year 1: Operating cash flow $+201,000$, change in NWC $+32,000$ (outflow).
     Net cash flow = 201,000 - 32,000 = +169,000.
   • Year 2: Operating cash flow $-5,000$, change in NWC $-19,000$ (inflow).
     Net cash flow = -5,000 + 19,000 = +14,000.
   • Year 3: Operating cash flow $+234,000$, equipment salvage value $+82,000$, loan repayment $-565,000$, change in NWC $-37,000$ (inflow).
     Net cash flow = 234,000 + 82,000 - 565,000 + 37,000 = -212,000.

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Step 2: Cash Flow Summary

We now have the following cash flows:

Year	Cash Flow ($)
0	$-452,000$
1	$+169,000$
2	$+14,000$
3	$-212,000$

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Step 3: IRR Calculation

The IRR is the discount rate ($r$) that satisfies the equation:

$NPV = \sum_{t=0}^n \frac{CF_t}{(1 + r)^t} = 0$

This is a trial-and-error or computational process. Let’s compute IRR using a numerical approach.

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I will calculate this and provide the IRR.from numpy_financial import irr

Cash flows of the project

cash_flows = [-452000, 169000, 14000, -212000]

Calculate IRR

internal_rate_of_return = irr(cash_flows)

Convert IRR to percentage

internal_rate_of_return_percentage = internal_rate_of_return * 100 internal_rate_of_return_percentagenannansuccessIt seems there was an issue with the calculation, as the IRR function may have difficulty converging due to the unconventional cash flow pattern. Let me try a more nuanced numerical approach to resolve this.import scipy.optimize as opt

Define the NPV function

def npv(rate, cashflows): return sum(cf / (1 + rate)**t for t, cf in enumerate(cashflows))

Define the IRR function (NPV = 0)

def irr_solver(cashflows): return opt.newton(lambda r: npv(r, cashflows), 0