Internal rate of return

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Internal Rate of Return (IRR) is a foundational financial metric used in capital budgeting and investment analysis to measure the expected annualized rate of return generated by an investment project. It is formally defined as the discount rate that results in a net present value (NPV) of zero for all projected cash flows of the project.

Key moments

19th centuryIRR-adjacent calculation methods begin appearing in early engineering economy literature for infrastructure project evaluation
Mid-20th centuryIRR becomes a standard, widely taught tool for corporate capital budgeting and investment decision making
Late 20th centuryModified Internal Rate of Return (MIRR) is developed to address key flaws of the traditional IRR method

Core Application and Decision Rule

The standard IRR decision rule is straightforward: if a project's calculated IRR is higher than the investor's minimum required rate of return (typically the cost of capital or a hurdle rate), the project is considered acceptable. It is widely used because it provides a simple, intuitive percentage figure that makes it easy to compare the relative profitability of different investment opportunities, and it is especially common in private equity and real estate to communicate expected returns to stakeholders.

Key Inherent Limitations

Traditional IRR has well-documented flaws. It assumes all interim positive cash flows can be reinvested at the same IRR, which is often unrealistic in practice, especially for high-return projects. For projects with non-conventional cash flows (where negative cash flows occur multiple times after initial investment), there can be multiple valid IRR values, leading to ambiguous conclusions. As a relative ratio metric, it also ignores project scale: a small high-IRR project may deliver less total profit than a larger lower-IRR project.

Modern Perspective on IRR

While net present value (NPV) is widely considered theoretically superior for capital budgeting decisions, IRR remains a popular complementary metric. Most finance practitioners now use IRR alongside NPV rather than as the primary decision tool, and modified versions like MIRR have been adopted in some contexts to correct for the reinvestment assumption bias. Despite its flaws, IRR's simplicity makes it a persistent standard in investment reporting and analysis.

Internal rate of return (IRR) is a method of calculating an investment's rate of return. The term internal refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or financial risk.

The method may be applied either ex-post or ex-ante. Applied ex-ante, the IRR is an estimate of a future annual rate of return. Applied ex-post, it measures the actual achieved investment return of a historical investment.

It is also called the discounted cash flow rate of return (DCFROR)[1] or yield rate.[2]

Definition

The IRR of an investment or project is the "annualized effective compounded return rate" or rate of return that sets the net present value (NPV) of all cash flows (both positive and negative) from the investment equal to zero.[2][3] Equivalently, it is the interest rate at which the net present value of the future cash flows is equal to the initial investment,[2][3] and it is also the interest rate at which the total present value of costs (negative cash flows) equals the total present value of the benefits (positive cash flows).

IRR represents the return on investment achieved when a project reaches its breakeven point, meaning that the project is only marginally justified as valuable. When NPV demonstrates a positive value, it indicates that the project is expected to generate value. Conversely, if NPV shows a negative value, the project is expected to lose value. In essence, IRR signifies the rate of return attained when the NPV of the project reaches a neutral state, precisely at the point where NPV breaks even.[4]

IRR accounts for the time preference of money and investments. A given return on investment received at a given time is worth more than the same return received at a later time, so the latter would yield a lower IRR than the former, if all other factors are equal. A fixed income investment in which money is deposited once, interest on this deposit is paid to the investor at a specified interest rate every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. An investment which has the same total returns as the preceding investment, but delays returns for one or more time periods, would have a lower IRR.

Uses

Savings and loans

In the context of savings and loans, the IRR is also called the effective interest rate.

Profitability of an investment

The IRR is an indicator of the profitability, efficiency, quality, or yield of an investment. This is in contrast with the NPV, which is an indicator of the net value or magnitude added by making an investment.

To maximize the value of a business, an investment should be made only if its profitability, as measured by the internal rate of return, is greater than a minimum acceptable rate of return. If the estimated IRR of a project or investment - for example, the construction of a new factory - exceeds the firm's cost of capital invested in that project, the investment is profitable. If the estimated IRR is less than the cost of capital, the proposed project should not be undertaken.[5]

The selection of investments may be subject to budget constraints.

Calculation

Given a collection of pairs (time, cash flow) representing a project, the NPV is a function of the rate of return. The internal rate of return is a rate for which this function is zero, i.e. the internal rate of return is a solution to the equation NPV = 0 (assuming no arbitrage conditions exist).

Given the (period, cash flow) pairs (n, C_n) where n is a non-negative integer, the total number of periods N, and the, (net present value); the internal rate of return is given by r in:

[2][3]

This rational polynomial can be converted to an ordinary polynomial having the same roots by substituting g (gain) for 1+r and multiplying by g^N to yield the equivalent but simpler condition

The possible IRR's are the real values of r satisfying the first condition, and 1 less than the real roots of the second condition (that is, r = g-1 for each root g). Note that in both formulas, C_0 is the negation of the initial investment at the start of the project while C_N is the cash value of the project at the end, equivalently the cash withdrawn if the project were to be liquidated and paid out so as to reduce the value of the project to zero. In the second condition C_0 is the leading coefficient of the ordinary polynomial in g while C_N is the constant term.

The period n is usually given in years, but the calculation may be made simpler if r is calculated using the period in which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter.

Problems with use

Comparison with NPV investment selection criterion

As a tool applied to making an investment decision on whether a project adds value or not, comparing the IRR of a single project with the required rate of return, in isolation from any other projects, is equivalent to the NPV method. If the appropriate IRR (if such can be found correctly) is greater than the required rate of return, using the required rate of return to discount cash flows to their present value, the NPV of that project will be positive, and vice versa. However, using IRR to sort projects in order of preference does not result in the same order as using NPV.

Maximizing NPV

One possible investment objective is to maximize the total NPV of projects.

When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects. In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints).

When the objective is to maximize total value, IRR should not be used to compare projects of different duration. For example, the NPV added by a project with longer duration but lower IRR could be greater than that of a project of similar size, in terms of total net cash flows, but with shorter duration and higher IRR.

Practitioner preference for IRR over NPV

Mathematics

Mathematically, the value of the investment is assumed to undergo exponential growth or decay according to some rate of return (any value greater than −100%), with discontinuities for cash flows, and the IRR of a series of cash flows is defined as any rate of return that results in a NPV of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow).

Thus, internal rate(s) of return follow from the NPV as a function of the rate of return. This function is continuous. Towards a rate of return of −100% the NPV approaches infinity with the sign of the last cash flow, and towards a rate of return of positive infinity the NPV approaches the first cash flow (the one at the present). Therefore, if the first and last cash flow have a different sign there exists an IRR. Examples of time series without an IRR:

In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, the resulting function of the rate of return is continuous and monotonically decreasing from positive infinity (when the rate of return approaches −100%) to the value of the first cash flow (when the rate of return approaches infinity), so there is a unique rate of return for which it is zero. Hence, the IRR is also unique (and equal). Although the NPV-function itself is not necessarily monotonically decreasing on its whole domain, it is at the IRR.

Similarly, in the case of a series of exclusively positive cash flows followed by a series of exclusively negative ones the IRR is also unique.

Finally, by Descartes' rule of signs, the number of internal rates of return can never be more than the number of changes in sign of cash flow.

The reinvestment debate

It is often stated that IRR assumes reinvestment of all cash flows until the very end of the project. This assertion has been a matter of debate in the literature.

Sources stating that there is such a hidden assumption have been cited below.[14][17] Other sources have argued that there is no IRR reinvestment assumption.[18][19][20][21][22][23]

In personal finance

The IRR can be used to measure the money-weighted performance of financial investments such as an individual investor's brokerage account. For this scenario, an equivalent,[24] more intuitive definition of the IRR is, "The IRR is the annual interest rate of the fixed rate account (like a somewhat idealized savings account) which, when subjected to the same deposits and withdrawals as the actual investment, has the same ending balance as the actual investment." This fixed rate account is also called the replicating fixed rate account for the investment. There are examples where the replicating fixed rate account encounters negative balances despite the fact that the actual investment did not.[24] In those cases, the IRR calculation assumes that the same interest rate that is paid on positive balances is charged on negative balances. It has been shown that this way of charging interest is the root cause of the IRR's multiple solutions problem.[25][26] If the model is modified so that, as is the case in real life, an externally supplied cost of borrowing (possibly varying over time) is charged on negative balances, the multiple solutions issue disappears.[25]

Unannualized internal rate of return

In the context of investment performance measurement, there is sometimes ambiguity in terminology between the periodic rate of return, such as the IRR as defined above, and a holding period return. The term internal rate of return (IRR) or Since Inception Internal Rate of Return (SI-IRR) is in some contexts used to refer to the unannualized return over the period, particularly for periods of less than a year.[27]

External links

Economics Interactive Lecture from University of South Carolina

References

Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269^
Stephen G. Kellison. The theory of interest McGraw-Hill Irwin, 2009^
Samuel A. Broverman. Mathematics of investment and credit ACTEX Publications, Inc, 2010^

Example

If an investment may be given by the sequence of cash flows

then the IRR r is given by

In this case, the answer is 5.96% (in the calculation, that is, r = 0.0596).

Numerical solution

Since the above is a manifestation of the general problem of finding the roots of the equation, there are many numerical methods that can be used to estimate r. For example, using the secant method, r is given by

where r_n is considered the nth approximation of the IRR.

This r can be found to an arbitrary degree of accuracy. Different accounting packages may provide functions for different accuracy levels. For example, Microsoft Excel and Google Sheets have built-in functions to calculate IRR for both fixed and variable time-intervals; "=IRR(...)" and "=XIRR(...)".

The convergence behavior is as follows:

Having when or when may speed up convergence of r_n to r.

If the function has a single real root r, then the sequence converges reproducibly towards r.
If the function has n real roots, then the sequence converges to one of the roots, and changing the values of the initial pairs may change the root to which it converges.
If function has no real roots, then the sequence tends towards +∞.

Numerical solution for single outflow and multiple inflows

Of particular interest is the case where the stream of payments consists of a single outflow, followed by multiple inflows occurring at equal periods. In the above notation, this corresponds to:

In this case the NPV of the payment stream is a convex, strictly decreasing function of interest rate. There is always a single unique solution for IRR.

Given two estimates r_1 and r_2 for IRR, the secant method equation (see above) with n=2 always produces an improved estimate r_3. This is sometimes referred to as the Hit and Trial (or Trial and Error) method. More accurate interpolation formulas can also be obtained: for instance the secant formula with correction

(which is most accurate when ) has been shown to be almost 10 times more accurate than the secant formula for a wide range of interest rates and initial guesses. For example, using the stream of payments {−4000, 1200, 1410, 1875, 1050} and initial guesses r_1 = 0.25 and r_2 = 0.2 the secant formula with correction gives an IRR estimate of 14.2% (0.7% error) as compared to IRR = 13.2% (7% error) from the secant method.

If applied iteratively, either the secant method or the improved formula always converges to the correct solution.

Both the secant method and the improved formula rely on initial guesses for IRR. The following initial guesses may be used:

where

Here, refers to the NPV of the inflows only (that is, set and compute NPV).

Exact dates of cash flows

A cash flow C_n may occur at any time t_n years after the beginning of the project. t_n may not be a whole number. The cash flow should still be discounted by a factor. And the formula is

For numerical solution we can use Newton's method

where is the derivative of and given by An initial value r_1 can be given by

Maximizing long-term return

Maximizing total value is not the only conceivable possible investment objective. An alternative objective would for example be to maximize long-term return. Such an objective would rationally lead to accepting first those new projects within the capital budget which have the highest IRR, because adding such projects would tend to maximize overall long-term return.

Example

To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise up to 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year.

Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time.

Solution

The cost of capital for both investors is 10 percent.

Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: and the IRR of each is (of course) greater than the cost of capital: so the IRR of Big-Is-Best is 32 percent, and so the IRR of Small-Is-Beautiful is 37.5 percent.

Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two?

As an example, Max Value selects Big-Is-Best, which has the higher net present value of US$20,000, while Max Return selects Small-Is-Beautiful for its higher return of 37.5 percent, compared to the 32 percent return offered by Big-Is-Best. The result is that each decision-maker chooses a different project without conflict.

How can this be rational for both investors? The answer lies in the fact that the investors do not have to invest the full 100,000 US dollars. Max Return is content to invest only 10,000 US dollars for now. After all, Max Return may rationalize the outcome by thinking that maybe tomorrow there will be new opportunities available to invest the remaining 90,000 US dollars the bank is willing to lend Max Return, at even higher IRRs. Even if only seven more projects come along which are identical to Small-Is-Beautiful, Max Return would be able to match the NPV of Big-Is-Best, on a total investment of only 80,000 US dollars, with 20,000 US dollars left in the budget to spare for truly unmissable opportunities. Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing.

Solution

The cost of capital for both investors is 10 percent.

WorldBrand briefing

Core Application and Decision Rule

Key Inherent Limitations

Modern Perspective on IRR

Definition

Uses

Savings and loans

Profitability of an investment

Calculation

Problems with use

Comparison with NPV investment selection criterion

Maximizing NPV

Practitioner preference for IRR over NPV

Mathematics

The reinvestment debate

In personal finance

Unannualized internal rate of return

See also

Further reading

External links

References

Fixed income

Liabilities

Capital management

Private equity

Example

Numerical solution

Numerical solution for single outflow and multiple inflows

Numerical solution

Numerical solution for single outflow and multiple inflows

Exact dates of cash flows

Maximizing long-term return

Example

Solution

Solution

Example

Solution

Solution

Multiple IRRs

Limitations in the context of private equity

Modified internal rate of return

Average internal rate of return (AIRR)