Internal Rate of Return

The internal rate of return (IRR) is a key concept in financial analysis, representing the interest rate at which an investment’s net present value (NPV) equals zero.

• interest rate at which the cash flows of a project are equivalent
• rate of return earned on the unrecovered balance of an investment over its lifetime. Specifically, it is the rate paid on the unpaid balance of borrowed money (for borrowing), or the rate earned on the unrecovered balance of an investment (for investment), so that the final payment or receipt brings the balance to exactly zero with interest considered.
• also the discount rate that makes the present worth of cash inflows equal to the present worth of cash outflows.
• also known as Rate of Return (ROR), Return on Investment (ROI), Yield, and Marginal Efficiency of Capital.  The term ROI is common in large capital projects.
• The IRR is often viewed as a break-even interest rate, where the profitability of an investment is neither positive nor negative.
• It is an internal rate because it specifically refers to the return on the portion of the investment that remains within the project.
• Note that the IRR is not the return on the initial amount of the investment; rather it is on the unrecovered balance, which changes each time period.

The decision rule for evaluating a single project based on the IRR is:

• If the IRR > Minimum Attractive Rate of Return (MARR), accept the project.
• If the IRR = MARR, remain indifferent.
• If the IRR < MARR, reject the project. It is important to note that this rule applies specifically to simple investments. When comparing mutually exclusive projects, one cannot simply select the project with the highest IRR. Instead, an incremental analysis approach is necessary.

Note: Mutually Exclusive Project is a set of investment opportunities where choosing one project means rejecting the others.

Calculating IRR

The calculation of IRR involves finding the interest rate that makes the net present worth of all cash flows equal to zero. It is calculated as:

where:

C_t = Net cash inflow during the period t
C_o​=Total initial investment costs
IRR=The internal rate of return
n = The number of time periods​

This can be achieved through several methods:

• Trial and error by testing different interest rates in the PW equation until the net present worth is close to zero.
• Direct solution for particular cash flows. Might lead to multiple roots while solving for non-simple cash flows.
• Utilizing software like Microsoft Excel, which has a built-in IRR function.

The IRR is a valuable tool when used correctly for evaluating an investment’s profitability and comparing different investment options. It gives a sense of the return rate that is internal to the project rather than a total dollar amount.

Limitations of IRR:

i) Multiple IRR for non-conventional/non-simple cash flows:

Depending on the cash flow sequence, there may be more than one real-number root to the IRR equation, resulting in multiple IRR values. IRR calculations and interpretations depend on whether the investment is simple or non-simple.

• Multiple IRRs and Non-Conventional Cash Flows: IRR may yield multiple solutions or no solution when cash flows have multiple sign changes, leading to ambiguity in interpreting results.

• A simple investment has an initial cash outflow followed by only inflows, or a single change of sign in the net cash flow series (from negative to positive). In a simple investment, the IRR is the unique interest rate that results in a zero net present worth.
• A non-simple investment, on the other hand, has multiple sign changes in the net cash flow series, which can lead to multiple IRRs. Because of the possibility of having multiple rates of return, it is recommended that the IRR analysis be abandoned and either the present worth (PW) or annual equivalent (AE) analysis be used to make an accept-or-reject decision.
• Alternatives to using IRR for non-simple investments are EROR (External Rate of Return) methods which include the modified internal rate of return (MIRR) or the return on invested capital (RIC or ROIC), .

ii) Assumptions about Reinvestment of Cash Flows: IRR assumes that cash flows are reinvested at the same rate as the IRR, which may not be realistic and can lead to overestimation or underestimation of project profitability.

iii) Inability to Compare Projects of Different Scales: Since IRR is a percentage measure, it cannot compare investments with different scales; other metrics like Net Present Value (NPV) are needed for such comparisons.

iv) Ignores Timing and Absolute Size of Returns: IRR does not account for the timing or absolute size of returns, potentially favoring projects with high rates but low dollar returns over those with lower rates but higher absolute returns.

Incremental IRR Analysis:

It is a crucial step in rate of return analysis when choosing between competing projects, especially when the project with the highest overall IRR may not be the most economically beneficial.

• When selecting from two or more mutually exclusive alternatives on the basis of IRR, an incremental IRR analysis must be used.
• The incremental IRR value between two alternatives is identified as i*,
• The selection guideline is that if i* is greater >= MARR, then the larger investment alternative is selected; otherwise the smaller investment alternative is selected.

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