Math Problem Statement

How do I calculate the Internal Rate of Return (IRR) for a project with the following cash flows: CF_0 = -9,500, CF_1 = 1,050, CF_2 = 2,250, CF_3 = 1,450, CF_4 = 1,450, CF_5 = 1,250, CF_6 = 1,050?

Solution

To solve this problem, we need to calculate the Internal Rate of Return (IRR) for the project. The IRR is the discount rate that makes the Net Present Value (NPV) of cash flows equal to zero. Here’s how it can be calculated:

Formula for NPV:

NPV=t=0nCFt(1+IRR)tNPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} Where:

  • CFtCF_t = Cash flow at time tt
  • IRRIRR = Internal Rate of Return (what we’re solving for)
  • tt = Time period

Cash Flow Data:

  • CF0=9,500CF_0 = -9,500
  • CF1=1,050CF_1 = 1,050
  • CF2=2,250CF_2 = 2,250
  • CF3=1,450CF_3 = 1,450
  • CF4=1,450CF_4 = 1,450
  • CF5=1,250CF_5 = 1,250
  • CF6=1,050CF_6 = 1,050

Using a financial calculator, Excel, or iterative numerical methods (e.g., trial and error or Newton-Raphson method), we calculate the IRR. I will compute this for you. scipy.optimize import irr

Define the cash flow series

cash_flows = [-9500, 1050, 2250, 1450, 1450, 1250, 1050]

Calculate IRR

internal_rate_of_return = irr(cash_flows) * 100 # Convert to percentage internal_rate_of_returnIt seems I encountered an issue with my calculation environment. However, I can guide you to calculate the IRR using Excel or an online tool:

Using Excel:

  1. Enter the cash flows in a column (e.g., A1:A7):
    • A1: -9500
    • A2: 1050
    • A3: 2250
    • A4: 1450
    • A5: 1450
    • A6: 1250
    • A7: 1050
  2. Use the formula:
    =IRR(A1:A7)
    
    This will give you the IRR.

If you'd like, I can solve this step manually for an approximate answer using interpolation. Would you like me to proceed with that or guide you further on calculating it?

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Math Problem Analysis

Mathematical Concepts

Financial Mathematics
Net Present Value (NPV)
Internal Rate of Return (IRR)

Formulas

NPV = Σ (CF_t / (1 + IRR)^t)
IRR: the rate that makes NPV = 0

Theorems

The IRR is the discount rate that makes the NPV of cash flows equal to zero.

Suitable Grade Level

Grades 10-12, College-level finance