## Internal rate of return calculation

Item Quantity Price Action
\$
\$
\$
\$
\$

You can find Internal rate of return by using this calculation software when you provide a series of cash flows, and type of interest compounding. The interest rate is compounded either discretely or continuously.

IRR method is used in DCF analysis to evaluate capital investment projects. There are a number of tools that a financial analyst has at her disposal when it comes to appraising investments. IRR is one such tool, it stands for internal rate of return that refers to the interest rate at which number of conditions are true. The following three statements hold true at internal rate of return:

1. NPV is 0
2. NFV is 0
3. Profitability index is 1
4. Benefit to cost ratio is 1

### IRR formula

If the number of cash flows are limited up to 5, then one can make use of Quadratic formula, Cubic formula and Quartic formula to solve for internal rate of return. Yet for any number of cash flows greater than 5, we are short of any math formula that can find the internal rate of return. We instead have to make use of various other mathematical techniques to find the IRR. Iterative techniques or numerical methods help provide a number of tools that help us in finding roots of polynomials. And since the IRR equation, based on the definition that I listed above, is a n-degree polynomial thus solving for it's root is akin to finding the internal rate of return. And there may be as many solutions for the IRR as the highest degree of the polynomial. The iterative methods only help us find 1 of the many roots and we have to redo the iterative calculations to find more than one IRR.

### Example calculation

I will use Newton-Raphson method to iteratively calculate the internal rate of return. This method requires that we first define a function f(i) and depending on the definition of IRR the function f(i) may either represent NPV, NFV, or profitability index. Once we have defined the function f(i) the next step is to find the derivative or differential of the function such as f'(x). Now we are ready to start the iterative calculations by using a seed value for the IRR. The seed value is referred to as the guess rate in Excel IRR function and 10% or 0.10 is the most commonly value used for the seed. The successive results from iterative calculations are noticed to find convergence, and if the results are within an absolute error bound then we have found one of the IRR values. The following example calculation shows how to find IRR with Newton Raphson method.

f(i) = - 1000 + 400(1+i)^-1 + 300(1+i)^-2 + 200(1+i)^-3 + 150(1+i)^-4 + 50(1+i)^-5 + 25(1+i)^-6
f'(i) = - 400(1+i)^-2 - 600(1+i)^-3 - 600(1+i)^-4 - 600(1+i)^-5 - 250(1+i)^-6 - 150(1+i)^-7

i(N-1) f(x) f'(x) i(N-1) - f(x)/f'(x)
0.1 -90.5568591767 -1781.82046024 0.0491773378983
0.0491773378983 8.82740037481 -2144.6588118 0.0532933307668
0.0532933307668 0.0681459763599 -2111.66335837 0.053325601998
0.053325601998 4.12970978303E-6 -2111.40742795 0.0533256039539
IRR = 5.333%

### Discrete versus continuous compounding

The prior example assumes that interest is compounded discretely or periodically. Discrete compounding of interest refers to APR being divided by number of compounding periods. But in finance and banking, the interest is compounded on contiuous scale meaning interest is paid or earned on ever small fraction of time, thus number of compounding periods are infinite. When interest is compounded continuously the internal rate of return will be lower than the IRR for discrete compounding. Excel IRR function only calculates the internal rate of return when interest is compounded periodically or discretely. Here, I will present to you the IRR calculation for the same cash flows that were used in previous example calculation of internal rate of return and you will find that IRR in this case is lower than the one we found when interest was compounded discretely.

f(i) = - 1000 + 400e^-i + 300e^-2i + 200e^-3i + 150e^-4i + 50e^-5i + 25e^-6i
f'(i) = - 400e^-i - 600e^-2i - 600e^-3i - 600e^-4i - 250e^-5i - 150e^-6i

i(N-1) f(x) f'(x) i(N-1) - f(x)/f'(x)
0.1 -99.6873319325 -1933.81078943 0.0484503176437
0.0484503176437 7.82911699403 -2247.16925808 0.0519343084316
0.0519343084316 0.0402386838009 -2224.11843579 0.0519524004015
0.0519524004015 1.07598381405E-6 -2223.99949108 0.0519524008853
IRR = 5.195%

I am sure you came here to learn to find IRR, it may be that you were given an assignment or a mid-term by your finance professor. I am sure that if you are one of those that are studying financial management at 75% of US colleges and universities then you were taught to use linear interpolation to calculate the IRR. Linear interpolation is easy to explain yet it is a guessing game which makes you seek two interest rates at which NPV is positive and the one at which net present value is negative. This finally leads to using a "IRR formula" to approximate the internal rate of return. If I were a employer seeking fresh business graduates, I would want the potential candidate who can perform financial analysis using tools that produce precise results. I teach at a college where business students are mandated to take math electives such as Linear algebra and numerical methods before they are allowed to take financial management courses. But then colleges like this one charge an annual tuition fee of \$50,000. Now I will give you an online software tool that you will find below this paragraph that will permit you to perform IRR calculation using Newton Raphson method. You will be able to select from type of interest compounding such as dicrete and continuous.

IRR Calculation
Type or copy paste the authorization code in the box located below:
Please type in the following data for IRR calculation:

%

Results from internal rate of return calculations are displayed below:
Results are shown here

## Related Calculations

Other than this calculation, we have a bunch of other calculation tools that you can use to find return on investment. For more details, please visit the investment analysis calculation page.

Feedback

You can let us know what you think of our products and services by filling out the following form. Your comments will help other visitors to our site in judging the quality of our products and services.

Form fields marked with an * are required.