Internal Rate of Return (IRR)

Theory

Internal Rate of Return (IRR)

Internal Rate of Return (IRR) is a financial indicator used to evaluate the profitability of an investment. It is the discount rate that makes the Net Present Value (NPV) of a project's cash flows equal to zero.

Formula and Meaning

The IRR is the rate i that satisfies the equation:

$$ \textcolor{var(--primary-color)}{\displaystyle \sum _{t=0}^{n}{\frac {C_{t}}{(1+i)^{t}}}=0} $$

Where:

• C_t is the cash flow at time t.
• i is the IRR.
• t is the number of periods.

In practice, the IRR indicates the percentage return that an investment will generate on an annual basis.

Interpretation

• If the IRR is greater than the cost of capital of the company, the project is acceptable because it generates value.
• If the IRR is less than the cost of capital, the project is not profitable.
• If the IRR is equal to the cost of capital, the project is neutral (neither loss nor net gain).

Applications of IRR

• Evaluation of investment projects (new plants, acquisitions, startups).
• Capital allocation decisions.
• Analysis of returns on bonds and real estate investments.

Limitations of IRR

• IRR multiples:  in the presence of unconventional cash flows (alternating positive and negative), more than one solution may exist.
• Does not consider the scale of the investment:  a project with a high IRR may be less profitable in absolute terms than a project with a higher NPV.
• Does not consider the reinvestment of cash flows:  it is assumed that the flows are reinvested at the same IRR rate, which may not be realistic.

To overcome some limitations, the Modified IRR (MIRR), which assumes a more realistic reinvestment rate.

Practical example to calculate the IRR

Scenario

Suppose a company is evaluating a project that requires an initial investment of €100'000 and generates the following cash flows over the next 4 years:

Calculating IRR

The IRR is the discount rate that makes the Net Present Value (NPV) equal to zero, or it solves this equation:

0 = -100'000 + (30'000 / (1+IRR)^1) + (40'000 / (1+IRR)^2) + (30'000 / (1+IRR)^3) + (50'000 / (1+IRR)^4)

Result

The IRR of the project is about 17.13%.

Interpretation

• If the cost of capital of the company is less than 17.13%, the project is profitable.
• If the cost of capital is greater than 17.13%, the project is not profitable.

Use

To calculate the Internal Rate of Return (IRR), follow these steps:

1. Enter the initial investment

In the Initial Investment field, enter the initial value of the investment:

• Positive= if you received the investment.
• Negative= if you paid the capital.

2. Enter subsequent payments

In the fields Period 1, Period 2, Period 3, ... enter the amounts for subsequent payments:

• Positive= if paid.
• Negative= if received.

3. Add or remove periods

If necessary, you can change the number of periods using the buttons:

• [Add Period] to add a new period.
• [Remove Period] to delete the last period entered.

4. Calculate IRR

Once you have entered all the data, press the [Calculate IRR] button to get the internal rate of return.

5. Read the results

In the Result section you will find:

• The list of payments made each year.
• The value of the Internal Rate of Return (IRR).