Think & Done

Internal rate of return IRR Formula is used for IRR calculation with Newton Raphson Method illustrated with example calculation. Newton Raphson method is used to perform IRR calculation with IRR formula that finds the roots of n-degree polynomial.

IRR Calculator

Here you will find an online IRR calculator that calculates internal rate of return given that you provide the series of cash flows

IRR Formula - Newton Raphson Method

The IRR formula listed above is commonly known as Newton Raphson method of finding roots of a polynomial. Finding roots of a n degree polynomial is akin to finding internal rate of return since IRR is the rate at which the IRR Equation equals zero. The initial guess rate or seed value is set to 0.10 or 10% to start out the iterative process. You may opt to use any other value for the seed rate as you wish.

IRR Calculation with Newton Raphson Method

Newton Raphson method is widely used in mathematics to find roots of polynomial, and since a series of discounted net cash flows is a n^th degree polynomial thus finding the IRR with Newton Raphson method is akin to finding the roots of the polynomial. For an n degree polynomial, there may exist as many as n roots or n different IRR values. The Newton Raphson Formula starts off with an initial seed value for the root and the successive iterations depend on the solution from the previous calculation. This is an iterative process that may or may not yield the root of the polynomial or in other words it may not find the IRR value. We terminate our calculations when the solutions from the formula do not converge towards a particular value. One way of knowing whether the successive iterations converge is to use an Error bound usually we will try to see if the Error bound between successive values are within 1/10000th or 0.0001. If we find that solution does not converge to a particular value after running such calculations for a fixed number of times usually up to 20 times, we terminate our calculations and start over by using a different seed value.

Let's get started to use the Newton Raphson method to find IRR with the Formula I listed on top of this page. To keep the calculations to a minimum, we will select a project with just three cash flows -1600, 10000 and -10000. Since IRR is the rate at which sum of discounted net cash flows or the NPV is set to zero, thus we can write a polynomial such as the following:

-1600 + 10000 (1+i)^-1 - 10000 (1+i)^-2

This is a second degree polynomial, as I said earlier IRR Equation can be derived from this polynomial by setting it to zero as follows

-1600 + 10000 (1+i)^-1 - 10000 (1+i)^-2 = 0

Now we can write a function of x such as

f(x) = -1600 + 10000 (1+i)^-1 - 10000 (1+i)^-2

As you may have noticed from the IRR Formula mentioned earlier that we also need the differential or the derivative of this function f(x) which we will denote as f'(x) and is listed below

f'(x) = - 10000 (1+i)^-2 + 20000 (1+i)^-3

Now we are set to use the Newton Raphson method to find internal rate of return with the IRR formula. We will use a seed value of 10% or 0.10 and find successive values until the difference between the successive values is less than 0.0001.

First Iteration

x₀ = 0.10
x₁ = 0.10 - f(x₀)/f'(x₀)

f(x₁) = -1600 + 10000 (1+0.1)^-1 - 10000 (1+0.1)^-2
f(x₁) = -1600 + 10000 (1.1)^-1 - 10000 (1.1)^-2
f(x₁) = -1600 + 9090.909091 - 8264.46281
f(x₁) = -773.553719

f'(x₁) = - 10000 (1+0.1)^-2 + 20000 (1+0.1)^-3
f'(x₁) = - 10000 (1.1)^-2 + 20000 (1.1)^-3
f'(x₁) = - 8264.46281 + 15026.29602
f'(x₁) = 6761.833208

x₁ = 0.10 - [-773.553719 / 6761.833208]
x₁ = 0.10 + 0.1144
x₁ = 0.2144

Error Bound Check

E_b = x₁ - x₀
E_b = 0.2144 - 0.10
E_b = 0.1144

E_b of 0.1144 > 0.0001 thus we continue with the second iteration

Second Iteration

x₁ = 0.2144
x₂ = 0.2144 - f(x₁)/f'(x₁)

f(x₂) = -1600 + 10000 (1+0.2144)^-1 - 10000 (1+0.2144)^-2
f(x₂) = -1600 + 10000 (1.2144)^-1 - 10000 (1.2144)^-2
f(x₂) = -1600 + 8234.519104 - 6780.730488
f(x₂) = -146.2113835

f'(x₂) = - 10000 (1+0.2144)^-2 + 20000 (1+0.2144)^-3
f'(x₂) = - 10000 (1.2144)^-2 + 20000 (1.2144)^-3
f'(x₂) = - 6780.730488 + 11167.21095
f'(x₂) = 4386.48046

x₂ = 0.2144 - [-146.2113835 / 4386.48046]
x₂ = 0.2144 + 0.033332277
x₂ = 0.247732277389817

Error Bound Check

E_b = x₂ - x₁
E_b = 0.247732277389817 - 0.2144
E_b = 0.033332277389817

E_b of 0.033332277389817 > 0.0001 thus we continue with the third iteration

Third Iteration

x₂ = 0.247732277389817
x₃ = 0.247732277389817 - f(x₂)/f'(x₂)

f(x₃) = -1600 + 10000 (1+0.247732277389817)^-1 - 10000 (1+0.247732277389817)^-2
f(x₃) = -1600 + 10000 (1.247732277389817)^-1 - 10000 (1.247732277389817)^-2
f(x₃) = -1600 + 8014.539802 - 6423.284825
f(x₃) = -8.745022083

f'(x₃) = - 10000 (1+0.247732277389817)^-2 + 20000 (1+0.247732277389817)^-3
f'(x₃) = - 10000 (1.247732277389817)^-2 + 20000 (1.247732277389817)^-3
f'(x₃) = - 6423.284825 + 10295.93438
f'(x₃) = 3872.649553

x₃ = 0.247732277389817 - [-8.745022083 / 3872.649553]
x₃ = 0.247732277389817 + 0.00225815
x₃ = 0.249990427001104

Error Bound Check

E_b = x₃ - x₂
E_b = 0.249990427001104 - 0.247732277389817
E_b = 0.002258149611288

E_b of 0.002258149611288 > 0.0001 thus we continue with the third iteration

Fourth Iteration

x₃ = 0.249990427001104
x₄ = 0.249990427001104 - f(x₃)/f'(x₃)

f(x₄) = -1600 + 10000 (1+0.249990427001104)^-1 - 10000 (1+0.249990427001104)^-2
f(x₄) = -1600 + 10000 (1.249990427001104)^-1 - 10000 (1.249990427001104)^-2
f(x₄) = -1600 + 8000.061268 - 6400.098029
f(x₄) = -0.036760973

f'(x₄) = - 10000 (1+0.249990427001104)^-2 + 20000 (1+0.249990427001104)^-3
f'(x₄) = - 10000 (1.249990427001104)^-2 + 20000 (1.249990427001104)^-3
f'(x₄) = - -6400.098029 + 10240.23527
f'(x₄) = 3840.137241

x₄ = 0.249928740361116 - [-0.036760973 /3840.137241]
x₄ = 0.249928740361116 + 0.00000957283
x₄ = 0.249938313

Error Bound Check

E_b = x₄ - x₃
E_b = 0.249938313 - 0.249990427001104
E_b = 0.000009572827832

E_b of 0.000009572827832 < 0.0001 thus we terminate our iterative procedure and can claim that one of the roots of the polynomial or may we say that the IRR is value of x₄ = 0.249938313, in percentage terms it equals 24.99% or 25%.

IRR Calculation Online

This online tool will perform IRR calculation and will display step by step workout similar to one you have seen in this discussion.

All values of IRR

An IRR equation may yield as many IRR values as the degree of the polynomial, thus we can find all values of IRR using the Newton Raphson method just described and illustrated with example. Let us see if we can now find other values for IRR for the same cash flows -1600, 10000, and -10000. To do so with the Newton Raphson method we will need to select a different seed value or guess rate for the IRR. We will select 200% or 2 as a decimal value to start off the calculations.

First Iteration

x₀ = 2
x₁ = 2 - f(x₀)/f'(x₀)

f(x₁) = -1600 + 10000 (1+2)^-1 - 10000 (1+2)^-2
f(x₁) = -1600 + 10000 (3)^-1 - 10000 (3)^-2
f(x₁) = -1600 + 3333.333333 - 1111.111111
f(x₁) = 622.2222222

f'(x₁) = - 10000 (1+2)^-2 + 20000 (1+2)^-3
f'(x₁) = - 10000 (3)^-2 + 20000 (3)^-3
f'(x₁) = - 1111.111111 + 740.7407407
f'(x₁) = -370.3703704

x₁ = 2 - [622.2222222 / -370.3703704]
x₁ = 2 + 1.68
x₁ = 3.68000

Error Bound Check

E_b = x₁ - x₀
E_b = 3.68000 - 2
E_b = 1.68000

E_b of 1.68000 > 0.0001 thus we continue with the second iteration

Second Iteration

x₁ = 3.68000
x₂ = 3.68000 - f(x₁)/f'(x₁)

f(x₂) = -1600 + 10000 (1+3.68000)^-1 - 10000 (1+3.68000)^-2
f(x₂) = -1600 + 10000 (4.68000)^-1 - 10000 (4.68000)^-2
f(x₂) = -1600 + 2136.752137 - 456.5709694
f(x₂) = 80.18116736

f'(x₂) = - 10000 (1+3.68000)^-2 + 20000 (1+3.68000)^-3
f'(x₂) = - 10000 (4.68000)^-2 + 20000 (4.68000)^-3
f'(x₂) = - 456.5709694 + 195.1157989br/> f'(x₂) = - 261.4551705

x₂ = 3.68000 - [80.18116736 / -261.4551705]
x₂ = 3.68000 + 0.306672716
x₂ = 3.98667

Error Bound Check

E_b = x₂ - x₁
E_b = 3.98667 - 3.68000
E_b = 0.30667

E_b of 0.30667 > 0.0001 thus we continue with the third iteration

Third Iteration

x₂ = 3.98667
x₃ = 3.98667 - f(x₂)/f'(x₂)

f(x₃) = -1600 + 10000 (1+3.98667)^-1 - 10000 (1+3.98667)^-2
f(x₃) = -1600 + 10000 (4.98667)^-1 - 10000 (4.98667)^-2
f(x₃) = -1600 + 2005.345161 - 402.1409214
f(x₃) = 3.204239362

f'(x₃) = - 10000 (1+3.98667)^-2 + 20000 (1+3.98667)^-3
f'(x₃) = - 10000 (4.98667)^-2 + 20000 (4.98667)^-3
f'(x₃) = - 402.1409214 + 161.2862701
f'(x₃) = -240.8546512

x₃ = 3.98667 - [3.204239362 / -240.8546512]
x₃ = 3.98667 + 0.00225815
x₃ = 3.99998

Error Bound Check

E_b = x₃ - x₂
E_b = 3.99998 - 3.98667
E_b = 0.01330

E_b of 0.01330 > 0.0001 thus we continue with the third iteration

Fourth Iteration

x₃ = 3.99998
x₄ = 3.99998 - f(x₃)/f'(x₃)

f(x₄) = -1600 + 10000 (1+3.99998)^-1 - 10000 (1+3.99998)^-2
f(x₄) = -1600 + 10000 (4.99998)^-1 - 10000 (4.99998)^-2
f(x₄) = -1600 + 2000.009464 - 400.0037858
f(x₄) = 0.005678673

f'(x₄) = - 10000 (1+3.99998)^-2 + 20000 (1+3.99998)^-3
f'(x₄) = - 10000 (4.99998)^-2 + 20000 (4.99998)^-3
f'(x₄) = - 400.0037858 + 160.0022715
f'(x₄) = -240.0015143

x₄ = 3.99998 - [0.005678673 / -240.0015143]
x₄ = 3.99998 + 0.000023661
x₄ = 4.00000

Error Bound Check

E_b = x₄ - x₃
E_b = 4.00000 - 3.99998
E_b = 0.00002

E_b of 0.00002 < 0.0001 thus we terminate our iterative procedure and can claim that one of the roots of the polynomial or may we say that the IRR is value of x₄ = 4.0, in percentage terms it equals 400%.

Secant Method

To find IRR with Newton Raphson method, one must know about finding derivative or differential of a function. There is another numerical method that helps you calculate internal rate of return without knowing the derivative. This technique is called the secant method that is used in mathematics to find roots of a polynomial. As explained by me previously, the IRR equation forms an n-degree polynomial thus findind it's roots is akin to find internal rate of return which results in a net present value of zero. The formula for secant method is illustrated below with a sample IRR calculation for cash flows of -10000, 5000, 4000, and 3000. The secant method begins with selection of two initial values at which to evaluate the IRR equation. The successive calculations are carried out to check whether if the values converage, if they do we assume to have found our internal rate of return. If the values do not converge we then restart the algorithm by selecting two different initial values.

Secant Method IRR Calculation

f(p) = -10000(1+i)^0 +5000(1+i)^-1 +4000(1+i)^-2 +3000(1+i)^-3 +1000(1+i)^-4

p = p1 - q1 * (p1-p0)/(q1-q0)

p0 = 0.5
p1 = 1
q0 = f(p0) = -3802.469
q1 = f(p1) = -6062.5
p = 1-[-6062.5 x (1-0.5)/(-6062.5--3802.469)]
Error Bound = |-0.341242744964 - 1| = 1.341243 > 0.000001

p1 = 1
p2 = -0.341
q1 = f(p1) = -6062.5
q2 = f(p2) = 22611.607
p = -0.341-[22611.607 x (-0.341-1)/(22611.607--6062.5)]
Error Bound = |0.716424158411 - -0.341| = 1.057667 > 0.000001

p2 = -0.341
p3 = 0.716
q2 = f(p2) = 22611.607
q3 = f(p3) = -5020.769
p = 0.716-[-5020.769 x (0.716--0.341)/(-5020.769-22611.607)]
Error Bound = |0.524247367909 - 0.716| = 0.192177 > 0.000001

p3 = 0.716
p4 = 0.524
q3 = f(p3) = -5020.769
q4 = f(p4) = -3965.628
p = 0.524-[-3965.628 x (0.524-0.716)/(-3965.628--5020.769)]
Error Bound = |-0.198027079339 - 0.524| = 0.722274 > 0.000001

p4 = 0.524
p5 = -0.198
q4 = f(p4) = -3965.628
q5 = f(p5) = 10687.619
p = -0.198-[10687.619 x (-0.198-0.524)/(10687.619--3965.628)]
Error Bound = |0.328777269197 - -0.198| = 0.526804 > 0.000001

p5 = -0.198
p6 = 0.329
q5 = f(p5) = 10687.619
q6 = f(p6) = -2372.23
p = 0.329-[-2372.23 x (0.329--0.198)/(-2372.23-10687.619)]
Error Bound = |0.233086962015 - 0.329| = 0.09569 > 0.000001

p6 = 0.329
p7 = 0.233
q6 = f(p6) = -2372.23
q7 = f(p7) = -1281.815
p = 0.233-[-1281.815 x (0.233-0.329)/(-1281.815--2372.23)]
Error Bound = |0.120600128994 - 0.233| = 0.112487 > 0.000001

p7 = 0.233
p8 = 0.121
q7 = f(p7) = -1281.815
q8 = f(p8) = 413.326
p = 0.121-[413.326 x (0.121-0.233)/(413.326--1281.815)]
Error Bound = |0.148027752281 - 0.121| = 0.027428 > 0.000001

p8 = 0.121
p9 = 0.148
q8 = f(p8) = 413.326
q9 = f(p9) = -51.304
p = 0.148-[-51.304 x (0.148-0.121)/(-51.304-413.326)]
Error Bound = |0.144999207317 - 0.148| = 0.003029 > 0.000001

p9 = 0.148
p10 = 0.145
q9 = f(p9) = -51.304
q10 = f(p10) = -1.818
p = 0.145-[-1.818 x (0.145-0.148)/(-1.818--51.304)]
Error Bound = |0.144887936233 - 0.145| = 0.000111 > 0.000001

p10 = 0.145
p11 = 0.145
q10 = f(p10) = -1.818
q11 = f(p11) = 0.008
p = 0.145-[0.008 x (0.145-0.145)/(0.008--1.818)]
Error Bound = |0.144888442868 - 0.145| = 1.0E-6 < 0.000001

IRR = 14.49%

Alternative Method for Calculating IRR

The procedure in the preceeding paragraph explain the Newton Raphson method for calculating IRR. Although most computer programs such as MS Excel make use of the Newton Raphson method yet when attempting to find IRR with paper and pencil requires that you know how to find derivatives. An alternate approach to finding IRR is with the Linear Interpolation that uses trial and error method yet it is much simpler than the Newton Raphson method. With this method we attempt to find two different interest rates at which the NPV is negative and positive. Once these two rates are ascertained, we use Linear Interpolation formula to approximate the IRR value. Keep in view the IRR value arrived at with Linear Interpolation is an approximation whereas the IRR values calculated with Newton Raphson method is the actual IRR value. In the following paragraphs we will illustrate Linear Interpolation method with example calculation.

IRR Example

Let us illustrate finding Internal Rate of return with an example investment proposal. Let us say you were offered a series of cash inflows at the end of each of the next five years as in amounts of $40,000. Say the initial cash outlay for this proposal is $100,000.

At first we find NPV at two different interest rates, at the lower rate the NPV will be positive and the upper rate the NPV will be negative.

Year	Cash Flow	Present Value @ 26%	Present Value @ 31%
0	-100000	-100000	-100000
1	40000	31746.03	30534.35
2	40000	25195.26	23308.66
3	40000	19996.24	17792.87
4	40000	15870.03	13582.35
5	40000	12595.26	10368.2
	NPV	5402.82	-4413.57

Linear Interpolation

Since at 26% the NPV is 5402.82 and at 31% the NPV is -4413.57, thus the actual IRR lies somewhere between 26% and 31% at which the NPV is zero. We will use linear interpolation as shown below to find the actual IRR value.
iL = 26%
iU = 31%
npvL = 5402.82
npvU = -4413.57

irr = iL + [(iU-iL)(npvL)] / [npvL-npvU]
irr = 0.26 + [(0.31-0.26)(5402.82)] / [5402.82--4413.57]
irr = 0.26 + [(0.05)(5402.82)] / [9816.39]
irr = 0.26 + 270.141 / 9816.39
irr = 0.26 + 0.0275
irr = 0.2875
irr = 28.75%
Thus we have approximated that the actual IRR value is in close proximity of 28.75%, however the actual IRR may just be slightly different from 28.75%. Using an IRR calculator we will find that the actual IRR to be equal to 28.649282902479%.

Frequently Asked Questions

Here I will attempt to answer some of the frequently asked questions pertaining to internal rate of return. This will provide you with an ample material to discern the topic in detail.

Q: What IRR stands for?

A: IRR stands for internal rate of return. An interest rate which sheds light on the health of an investment project.

Q: What is IRR in finance?

A: IRR in finance refers to the investor interest rate of return that makes the investment a worthwhile proposition. As investors incur costs in funding or raising capital for projects, the costs of raising such funds represented as a percentage is compared to the IRR. If the IRR exceeds the cost of capital that would indicate to the investor that investment will be a viable proposition ensuring that there will be a positive return on investment. Yet if IRR lies below the cost of capital that will be indicative of a losing proposition where the investment will yield negative return on investment.

Q: What is IRR calculation?

A: IRR calculation refers to the use of mathematical formula and procedures that helps find internal rate of return. An investor will have to first ascertain the free cash flows. A number of methods are used for IRR calculation some of which calculates an approximate value of IRR as is the case with linear interpolation method. Other methods used in calculating IRR such as the Newton-Raphson method require familiarity with finding differential of a function. This last approach uses iterative technique in solving for the internal rate of return. Many of the popular spreadsheet programs makes use of this iterative technique in finding IRR.

Q: What is IRR formula?

A: In short there is no IRR formula that can find internal rate of return. To understand why there isn't a closed form IRR formula, one has to look at the IRR equation. The IRR equation is a series of discounted cash flows that forms a n-degree polynomial. The discounting factor of each of the cash flow is of the form (1+i)^-t where t is the time period ranging from 0 to n. As the internal rate of return i in the equation is trapped in each part of the series thus it is impossible to solve for i from the IRR equation. This leads us to using a variety of mathematical techniques to find IRR, these techniques include approximation of IRR with linear interpolation, iterative calculation using Newton-Rapshon method or the Secant method.

Q: What IRR means?

A: IRR is an acronym for internal rate of return. IRR helps an investor in judging the financial health of an investment project by comparing it to investor's cost of capital. An IRR higher than the cost of capital will result in a profitable investment.

Q: What does IRR measure?

A: IRR measures the worthiness of an investment. It is a rate of return from an investment that sheds light on whether to take on a particular project or to reject it. It is often used along with other methods such as NPV to make a decision to either accept or reject a project.

Q: What does IRR tell you?

A: As an investor you will have to evaluate capital budgeting projects on behalf of a company. As an analyst you will have to recommend to management which project to undertake. The process of investment analysis in capital budgeting requires analyzing each project to check for expected returns. One of the methods used by analysts is the IRR or internal rate of return. IRR tells you the rate of return that is expected from the project under consideration. This rate of return is compared with opportunity cost or the cost of capital that has to be paid to undertake the project. An IRR higher that this cost of capital would tell you that the investment project will provide net benefits.

Q: What are IRR analysis?

A: IRR analysis includes the investigation in finding financial viability of an investment. The analysis process starts off with finding incremental cash flows to be used in calculating the IRR. The analysis also requires knowing or calculating of cost of capital as this will be compared with IRR to help decide the fate of the project. The critical or crucial part of the IRR analysis is the process of finding the internal rate of return with mathematical techniques such as Newton Raphson method or approximating the IRR with trial and error method such as linear interpolation.

Q: What are IRR projects?

A: IRR projects refer to capital budgeting projects that organizations undertake to purchase, replace or expand equipment, plant, product, services, etc. The company must ensure that the capital projects under consideration are evaluated to see whether it will benefit the organization in terms of bottom line on accounting income statement. The financial analyst has the responsibility to carry out the initial investigation in determining the expected future cash flows from the project. Analyst has to be aware of the cost of capital that is required by the firm to undertake capital projects. A company may use a single cost of capital for all it's investment projects or it may use different cost of capital for each of the project it undertakes. Some of the projects may be approved by department managers yet other projects with large cash outlay may need the approval of the top management and in some cases will require the authorization of board of directors. The idea for new capital projects can come from either managers, employees or directors. The IRR projects would refer to the projects where the decision to accept or reject the capital project is solely based on the internal rate of return when it is compared to the company's cost of capital.

Q: How does IRR work?

A: Working out IRR is a tedious task and in certain cases can consume hours of your time if worked out manually with paper and pencil. Fortunately, there are financial calculators, online calculators and spreadsheet programs that calculate IRR and do all the work for you. These calculators use programming routines that take the cash flows as series of numbers and perform internal calculations and produce IRR.

Q: How does IRR work in Excel?

A: Excel is a popular spreadsheet program used by corporations and individuals. Excel provides accountants and analysts with easy to use layout that makes possible performing complex analysis on data. Excel presents data in a tabular format laid out as rows and columns of cells. The cells can be used to type in numerical amounts along with built-in functions. The functions are available for various categories such as Math and Trigonometry, Statistical, and Financial amongst others. The financial category includes some of the popular and widely used methods such as IRR and NPV. The IRR function in Excel permits you to find internal rate of return. For the Excel IRR function to work, you would have to provide it with a series of cash flows as range of cells. A range in Excel refers to consecutive number of cells. The IRR function in Excel may be provided with an initial guess rate for IRR. Guess is used to fine tune calculation of IRR when Excel fails to calculate IRR using the default guess rate of 10%. The internal working of the IRR function in Excel is beyond the scope of this discussion.

Q: How does IRR affect the cost of capital?

A: IRR does not affect the cost of capital, however, IRR is compared to the cost of capital to check whether it will be worthwhile to invest in a capital project. In brief, an IRR that is higher than the cost of capital would make the investment worthwhile whereas an IRR that is lower than the cost of capital should lead to rejection of the capital investment.

Q: How IRR is calculated?

A: IRR is calculated uses various mathematical techniques including but not limited to trial and error method, Newton-Raphson and Secant method. The trial and error method attempts to find two interest rates that yield a positive and a negative net present values. From there, it uses linear interpolation to approximate the IRR. Newton-Raphson method lays out the IRR equation as a function of x; next step is to determine the derivative f'(x) of function f(x). An iterative calculation is undertaken that uses a guess rate as seed value for the NR algorithm to check whether consecutive calculations converge to a particular value. If so, we have found the IRR if not then the iterative calculations are redone with a different guess rate as a seed value.

Q: How IRR is used?

A: IRR is used to find the worthiness of an investment proposal by comparing it to the company's cost of capital. A cost of capital lower than the IRR is an indication that project will make profits. A cost of capital higher than the IRR will result in a loss. Thus IRR indicates the profitability of a investment project where by we accept a project if it were to have a IRR higher than the company's cost of capital.

Q: How IRR is calculated for a project?

A: To calculate IRR for a project, we start off with determining the incremental cash flow. This process includes finding expected revenue or savings from the new project. The fixed and variable costs are lessened from the expected profits or savings. Any depreciation expense is taken from it as depreciation is a non cash expense. At this stage we have EBIT earning before income and taxation. We lessen tax expense from the EBIT and add back the depreciation. If there were to be any net working capital, it would be included in the initial cost and finally recovered in the last terminal cash flow. Once the free cash flows are ascertained, a variety of methods are used to calculate the internal rate of return. If IRR is found for the given cash flows, it is then compared to the company's cost of capital to make a acceptance or rejection decision.

Q: How IRR is different from NPV?

A: IRR is quite different from NPV as the former is a rate of interest represented as a percentage and the latter is a money amount to be made or lost if the project were to be given a go ahead. In a simple example where the initial cost is $-1000 and incoming cash flow is $1100, the internal rate of return is 10%. If the cost of capital for this investment is 5% then the NPV is $47.62

Q: Why IRR is important to an organization?

A: Most organizations consider IRR as an important measure to value investments. IRR is represented as a percentage rate that shows the return on investment from a capital budgeting project. Organizations use IRR to compare it with organization's cost of capital; the cost of capital is the organization's opportunity cost. It is the cost an organization must pay to undertake the capital budgeting project. If the IRR turns out to be less than the cost of capital then the organization stands to lose out on the money that it must pay to undertake the project. An IRR higher than the cost of capital would lead to a healthy investment. Although an organization may also use other capital budgeting methods such as NPV yet these measures only reflect money amounts and not the interest rates that can be directly compared with organization's cost of capital.

Q: Why is IRR better than NPV?

A: NPV stands for net present value and signifies the net benefits from an investment in terms of money whereas IRR stands for internal rate of return and signifies the rate of return on an investment. An IRR can be directly compared with organization's cost of capital thus providing an organization with a measure to see whether the project covers its cost of capital. What makes IRR better than the NPV is the fact that we can always infer whether the NPV is positive, negative or zero by comparing IRR to organization's cost of capital. At an IRR that is same as the company's cost of capital, the NPV is zero. At an IRR that is higher than the company's cost of capital, the NPV is positive. And at an IRR that is less than the cost of capital, the NPV is negative. The IRR acceptance criteria suggests to accept projects that have an IRR higher than the cost of capital.

Q: Why is IRR a negative?

A: IRR may be negative when the sum of benefits is less than the sum of costs. Take the simple case where cost is $-1000 followed by a benefit of $1000. The IRR in this case is 0%. If the cost was $-1000 followed by benefit of $900, the IRR will be -10%. Only when the sum of benefits exceeds the sum of costs, the IRR will be positive. Say we had a cost of $-1000 followed by a benefit in amount of $1100, the IRR is 10%.

Q: Why is IRR reinvestment assumption IRR?

A: An inherent assumption is that IRR is the reinvestment rate unlike NPV where the reinvestment assumption is the cost of capital. With IRR, it is assumed that the funds can only be reinvested at the internal rate of return. This leads to problems when an IRR is too high or when there are multiple IRR values. Assuming that an IRR is way too high than the company's cost of capital, one can conclude that a company will be unable to reinvest at the IRR.

Q: Why IRR 0?

A: An IRR will be 0% when the costs incurred are the same as sum of benefits. For an example, take the cash flows where we incur a cost of $-1000 followed by two benefits of $500 and $500. Since the costs equal the benefits, the internal rate of return or the IRR is 0%. Only when the benefits surpass the costs that we will have a positive IRR.

Q: Can IRR be negative?

A: Certainly IRR can be negative. Think of it this way when the net benefits are half that of the costs the IRR will be -50%. Here is an example cash flows where IRR is negative. Cash Outflow is $-100 and Cash Inflow is $50. The IRR is -50%

Q: Can IRR be over 100?

A: Definitely IRR can be over 100. Take the example where the costs are $-100 and benefits are $225. The IRR is 125%

Q: Can IRR be greater than 100?

A: As explained earlier in the discussion, IRR can be greater than 100 when the discounted net benefits are twice or more the size of net costs. Let us see this with example calculation or IRR with these cash flows of $-100 $100 $150 $200. The IRR for these cash flows turns out to be over 100 and is equal to 114%

Related DCF analysis methods

Following is a list of related readings that cover other 5 commonly used DCF analysis methods

1. Modified Internal Rate of Return
2. Net Present Value
3. Profitability index
4. Payback period
5. Discounted payback period