Question

What is the internal rate-of-return for an investment of $30,000 and uniform annual returns of $5,000 for the next eight years?

Answer 1

The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is:

0 = -$30,000 + $5,000(PVIFA_{IRR, 8})

IRR = 6.88%

Answer 2

Given:

• Initial investment (Outflow): $30,000 (Year 0).

• Uniform annual returns (Inflows): $5,000 for 8 years (Years 1–8).

• 
  

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Step 1: Set Up the IRR Equation

The IRR is the discount rate ($r$) that makes the Net Present Value (NPV) = 0:

$$NPV=-30,000+\sum _{t=1}^8\frac{5,000}{(1+r{)}^t}=0.$$

Step 2: Simplify Using Annuity Formula

The cash inflows form an annuity. The present value of an annuity formula is:

$$P{V}_{\text{annuity}}=\text{PMT}\times \frac{1-(1+r{)}^{-n}}{r}\mathrm{.}$$

Substitute $\text{PMT}=5,000$ and $n=8$:

$$30,000=5,000\times \frac{1-(1+r{)}^{-8}}{r}\mathrm{.}$$

Divide both sides by $5,000$:

$$6=\frac{1-(1+r{)}^{-8}}{r}\mathrm{.}$$

Step 3: Solve for $r$ (IRR)

This equation cannot be solved algebraically; use trial-and-error or a financial calculator.

1. Trial $r=8\mathrm{\%}$:

   $$\frac{1-(1.08{)}^{-8}}{0.08}=5.7466(\text{Too low})\mathrm{.}$$

2. Trial $r=6\mathrm{\%}$:

   $$\frac{1-(1.06{)}^{-8}}{0.06}=6.2098(\text{Too high})\mathrm{.}$$

3. Interpolate between 6% and 8%:

   $$r\approx 6\mathrm{\%}+\left(\frac{6.2098-6}{6.2098-5.7466}\right)\times 2\mathrm{\%}=6.9\mathrm{\%}\mathrm{.}$$

    

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Answer:

The Internal Rate of Return (IRR) is approximately 7.07%.