Internal rate of return

Overview

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) 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.

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 (

n

{\displaystyle n}

,

C

n

{\displaystyle C_{n}}

) where

n

{\displaystyle n}

is a non-negative integer, the total number of periods

N

{\displaystyle N}

, and the

NPV

{\displaystyle \operatorname {NPV} }

, (net present value); the internal rate of return is given by

r

{\displaystyle r}

in:

Example

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

then the IRR

r

{\displaystyle r}

is given by

NPV = − 123400 +

36200

( 1 + r

)

1

+

54800

( 1 + r

)

2

+

48100

( 1 + r

)

3

Numerical solution

Since the above is a manifestation of the general problem of finding the roots of the equation

NPV ⁡ ( r ) = 0

{\displaystyle \operatorname {NPV} (r)=0}

, there are many numerical methods that can be used to estimate

r

{\displaystyle r}

. For example, using the secant method,

r

{\displaystyle r}

is given by

r

n + 1

=

r

n

−

NPV

n

⋅

(

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:

C

0

< 0 ,

C

n

≥ 0

for

n ≥ 1.

{\displaystyle C_{0}<0,\quad C_{n}\geq 0{\text{ for }}n\geq 1.\,}

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

{\displaystyle r_{1}}

and

r

2

{\displaystyle r_{2}}

for IRR, the secant method equation (see above) with

n = 2