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)Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269 or yield rate.
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. Equivalently, it is the interest rate at which the net present value of the future cash flows is equal to the initial investment, 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.
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.
The selection of investments may be subject to budget constraints. There may be mutually exclusive competing projects, or limits on a firm's ability to manage multiple projects. For these reasons, corporations use IRR in capital budgeting to compare the profitability of a set of alternative capital projects. For example, a corporation will compare an investment in a new plant versus an extension of an existing plant based on the IRR of each project. To maximize returns, the higher a project's IRR, the more desirable it is to undertake the project.
There are at least two different ways to measure the IRR for an investment: the project IRR and the equity IRR. The project IRR assumes that the cash flows directly benefit the project, whereas equity IRR considers the returns for the shareholders of the company after the debt has been serviced.
Even though IRR is one of the most popular metrics used to test the viability of an investment and compare returns of alternative projects, looking at the IRR in isolation might not be the best approach for an investment decision. Certain assumptions made during IRR calculations are not always applicable to the investment. In particular, IRR assumes that the project will have either no interim cash flows or the interim cash flows are reinvested into the project which is not always the case. This discrepancy leads to overestimation of the rate of return which might be an incorrect representation of the value of the project.
Fixed income IRR is used to evaluate investments in fixed income securities, using metrics such as the yield to maturity and yield to call.
Liabilities Both IRR and net present value can be applied to liabilities as well as investments. For a liability, a lower IRR is preferable to a higher one.
Capital management Corporations use IRR to evaluate share issues and stock buyback programs. A share repurchase proceeds if returning capital to shareholders has a higher IRR than candidate capital investment projects or acquisition projects at current market prices. Funding new projects by raising new debt may also involve measuring the cost of the new debt in terms of the yield to maturity (internal rate of return).
Private equity IRR is also used for private equity, from the limited partners' perspective, as a measure of the general partner's performance as investment manager. This is because it is the general partner who controls the cash flows, including the limited partners' draw-downs of committed capital.
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 (, ) where is a non-negative integer, the total number of periods , and the , (net present value); the internal rate of return is given by in:
This rational polynomial can be converted to an ordinary polynomial having the same roots by substituting g (gain) for and multiplying by 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, for each root g). Note that in both formulas, is the negation of the initial investment at the start of the project while 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 is the leading coefficient of the ordinary polynomial in g while is the constant term.
The period is usually given in years, but the calculation may be made simpler if 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.
Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); the value obtained is zero if and only if the NPV is zero.
In the case that the cash flows are random variables, such as in the case of a life annuity, the expected values are put into the above formula.
Often, the value of that satisfies the above equation cannot be found analytically. In this case, numerical methods or graphical methods must be used.
Example
If an investment may be given by the sequence of cash flows
Year () Cash flow () 0−123400 1 36200 2 54800 348100
then the IRR 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 . For example, using the secant method, is given by
where is considered the th approximation of the IRR.
This 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: If the function has a single real root , then the sequence converges reproducibly towards . If the function has 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 +∞.
Having when or when may speed up convergence of to .
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 and for IRR, the secant method equation (see above) with always produces an improved estimate . 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 and 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 may occur at any time years after the beginning of the project. 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 can be given by
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).