Internal rate of return (IRR)

What is the internal rate of return (IRR)?

The internal rate of return, or IRR, is one of the main measures used by investors and finance professionals to assess the profitability of an investment. In mathematical terms, the IRR is the discount rate that makes the net present value (NPV) of all future cash flows equal zero in a discounted cash flow analysis (DCF).

Private Equity (PE) investors find IRR particularly useful because of its focus on cash flows. Since PE managers deploy cash outflows in the initial stages of a project and then harvest cash inflows in later stages, calculating IRR in a PE context is relatively intuitive. One way to conceptualise IRR is as a growth rate that an investment generates annually. In this way, it is similar to a compound annual growth rate (CAGR).

Key takeaways

• IRR is derived directly from the cash flows associated with an investment.
• Initial capital outlays are negative cash flows, while the eventual return cash flow streams are positive cash flows.
• IRR is related to NPV, which in turn is dependent upon the investor’s required rate of return.
• These measures have roots in the concept of time value of money, which has been cited in economics and philosophy for centuries.

IRR formula and calculation

The IRR is a discount rate. The differentiating feature of IRR from other discount rates, such as the required rate of return, is that we do not first calculate the IRR and subsequently apply it to a series of cash flows (as we do when calculating NPV, for example). We do the opposite. Given a series of cash flows, we derive the IRR by working backwards to develop the discount rate. 

IRR Equation:

Where:

• CF₀ — the cash flow at time zero: the initial outlay, usually a negative figure
• CF₁, CF₂ … CFₙ — the cash flows in each subsequent period
• n — the number of periods over which the cash flows extend
• IRR — the internal rate of return: the discount rate we are solving for the exponent on each term (1, 2 … n) is the period in which that cash flow occurs, so cash flows further in the future are discounted more heavily

This is a generalised textbook formula for finding the internal rate of return. We are seeking the discount rate at which the present value of the series of cash flows equals zero, so the left-hand side of the equation is set to 0. On the right-hand side, we have the series of cash flows, beginning with the initial outlay at time zero and extending to n periods.

The cash flows are derived either from actual results — free cash flow per year, over a period of years — or from a set of projected (pro forma) cash flows. We analyse quarterly or annual data, and the number of cash flows might range anywhere from three or four up to ten or fifteen periods. There is no hard and fast rule, and much depends on the investor's time horizon.

The cash flows are the numerators of each entry in the equation, and the discount factor — (1 + IRR) raised to the relevant period — sits in the denominator. We are therefore discounting, as opposed to compounding, the cash flows. With NPV we apply a required rate of return as the discount rate; with IRR we do not know the discount rate at the outset. We are solving for it, and that rate is the IRR.

An example for clarification

Start with a set of four basic cash flows to keep the example simple. The initial outlay at the start of the investment period is $500. For each of the next three years, there is a cash flow at the end of the year: $200 after year 1; $350 after year 2; and $525 as the final cash flow, at the end of year 3.

Over the three years, the cash flows total $1,075 against the $500 outlay. The nominal total is only part of the picture, though — the timing of each cash flow matters as well as its size. Calculating the IRR distils the cash flows down to a single measure, which provides a basis for comparison with other investments.

With the figures filled in, the equation becomes:

0 = -$500 + $200/(1+IRR)¹ + $350/(1+IRR)² + $525/(1+IRR)³

The first term, -$500, is the initial outlay, which is why it carries a negative sign: it is cash paid out rather than received, and it occurs at time zero, so it is not discounted. The subsequent terms place each year's cash flow in the numerator, discounted by the number of periods that have passed.

The IRR takes into account the time value of money, the size of the cash flows, and the number of periods (here, years). The equation now has only one remaining unknown. Because it cannot be rearranged neatly to isolate the IRR, the rate is found by iteration — typically using a financial calculator or a spreadsheet function. In this case, the IRR is 41.70%.

The funds offered on Moonfare's platform have an IRR calculated and provided as part of the due diligence information available when reviewing investments.

Why is the NPV set to zero?

Interpreting IRR on a stand-alone basis can be difficult, so it helps to understand the reasoning behind the calculation. NPV is set to zero because that is how the IRR is defined: it is the discount rate at which the present value of the cash flows nets to zero. Setting NPV to zero is therefore the condition we solve for, not an assumption we add. The advantage is that the resulting rate — as calculated in the earlier example — depends purely on the cash flows from the investment. NPV works the other way around: it may use the same set of cash flows, but the discount rate is supplied as an input to the equation rather than derived from it.

The discount rate used in NPV is often referred to as the required rate of return, and it can be arrived at in a number of ways. It is intended to capture the rate of return an investor requires for a given project or investment. If an investor can earn 10% on an alternative project, they may use 10% as the discount rate in the NPV calculation. On that basis, a positive NPV would indicate that the return exceeds the required rate, while an NPV of zero would mean it exactly meets it.

If NPV is set to zero and the IRR is calculated instead, the IRR can then be compared with the required rate of return. For a conventional series of cash flows like the example above — an initial outlay followed by inflows — an IRR above the required rate corresponds to a positive NPV at that rate. So if the required rate of return is 15% and the IRR is 22%, the cash flow profile exceeds the required rate.

Why is IRR important to private equity?

IRR can be used for a couple of primary purposes. One could compare the IRRs of two different private equity funds. For example, Fund A has consistently posted IRRs around 20%, or a bit higher than that. By comparison, Fund B yields IRRs of 30% and above. Assuming the funds have similar characteristics, then Fund B appears to be providing historically superior performance as measured by IRR.

Private equity and IRR

The Internal Rate of Return (IRR) is a compelling metric in private equity, providing a comprehensive view of the potential profitability and efficiency of investments. It plays an indispensable role in decision-making processes, as it enables investors to compare different investment opportunities, assess their relative risk levels, and plan for optimal capital allocation. When looking at IRR, investors can predict the growth trajectory of their investments over time, providing a foundation for informed strategic decisions. Moreover, it allows for an unbiased comparison of PE funds by standardising returns despite different investment periods. IRR, therefore, offers a valuable lens through which to gauge the financial viability and long-term value creation of private equity investments.