Think & Done

IRR formula explained with example calculation of internal rate of return using Newton Raphson, secant and bisection methods by setting NPV to 0, NFV to 0, and profitability index to 1. I will illustrate IRR calculation using a variety of IRR formulas that make use of net present value equation set to zero, net future value equation set to 0 and profitability index equation set to 1. These four equations are used to define internal rate of return IRR as the interest rate of return at which either NPV is zero, NFV is zero or at which profitability index is 1.

IRR formula

If the number of cash flows are limited to 3, 4 or 5 then we can make use of Quadratic formula, Cubic formula or Quartic formula respectively to solve for the internal rate of return. Yet for number of cash flows greater than 5, there exists no closed form formula to calculate IRR, this is so as the variable for IRR is trapped in the underlying IRR equations in a way that one is not able to separate it or to solve for it. For example take the following four forms of IRR equations: first one sets NPV to zero, second one sets NFV to zero, thrid one sets profitability index to one and the last one is the TVM equation for finding IRR for annuity with constant or uniform cash flows. In each instance, the rate is not separable from the other variables.

Thus one is left to utilizing numerical methods to calculate IRR using any of the four IRR equations listed below.

1. IRR with Quadratic Equation using Quadratic Formula Calculator
2. IRR with NPV equation using Newton raphson method and Secant method
   CF₀ + CF₁(1+i)^-1 + CF₂(1+i)^-2 + ... CF_n(1+i)^-n = 0
3. IRR with NFV equation using Newton raphson method and Secant method
   CF₀(1+i)ⁿ⁺¹ + CF₁(1+i)ⁿ + CF₂(1+i)^n-1 + ... CF_n(1+i) = 0
4. IRR with Profitability index equation using Newton raphson method and Secant method
   B₀ + B₁(1+i)^-1 + B₂(1+i)^-2 + ... B_n(1+i)^-n
   ------------------------------------------------------- = 1
   C₀ + C₁(1+i)^-1 + C₂(1+i)^-2 + ... C_n(1+i)^-n
5. IRR with TVM equation using Newton raphson method
   PV (1+i)ⁿ + PMT (1+i*type)[(1+i)ⁿ-1]/i + FV = 0
   or the equivalent
   PV + PMT (1+i*type)[1-(1+i)^-n]/i + FV (1+i)^-n = 0

Newton Raphson IRR formula

Newton Rapshon makes use of the following IRR formula by performing iterative calculations:
i_n = i_n-1 - f(i_n-1)/f'(i_n-1)

Secant IRR formula

And the secant method makes use of the following IRR formula
i_n = i_n-1 - q_n-1 * (i_n-1-i_n-2)/(q_n-1-q_n-2)
where q is either NPV, NFV or profitability index depending on which ever IRR equation is used.

Continuous compounding and IRR

For most investments, it is assumed that interest rate is discounted discretely as in annual discounting, or intra-year discounting as in semi-annual, quarterly, monthly, bi-weekly, fortnightly, weekly or even daily. Yet in a good number of banking and financial applications the interest rate is compounded continuously. Most saving accounts earn interest on basis of continuous compounding; thus if interest was to be compounded or discounted on a continuous scale then we would have to make amendments to the three equations I listed above. The change would reflect the interest factor. In the NPV, and profitability equations, the term (1+i)^-t is the present value interest factor (PVIF) that discounts a $1 at an interest rate of i% for n number of periods. The corresponding present value interest factor for continuous compounding is e^-it where is e is the mathematical constant value of 2.7182818284590452353602874713527 and i is the interest rate and t is the time period at which to discount a particular cash flow. In NFV equation for continuous compounding we replace the discrete compounding interest factor of (1+i)^t by e^it. We would make use of the following equations in our IRR calculations when interest is compounded continuously

1. IRR with NPV equation (continuous compounding of interest)
   CF₀ + CF₁e^-i + CF₂e^-2i + ... CF_ne^-ni = 0
2. IRR with NFV equation (continuous compounding of interest)
   CF₀eⁱ⁽ⁿ⁺¹⁾ + CF₁eⁱⁿ + CF₂e^i(n-1) + ... CF_neⁱ = 0
3. IRR with Profitability index equation (continuous compounding of interest)
   B₀ + B₁e^-i + B_ie^-2i + ... B_ne^-ni
   ------------------------------------------------------- = 1
   C₀ + C₁e^-i + C_ie^-2i + ... C_ne^-ni
4. IRR with TVM equation (continuous compounding of interest)
   PV eⁿⁱ + PMT e^i*type[eⁿⁱ-1]/[eⁱ-1] + FV = 0
   or the equivalent
   PV + PMT e^i*type[1-e^-ni]/[eⁱ-1] + FV e^-ni = 0

Numerical methods to find IRR

I will use a variety of numerical methods to calculate internal rate of return. These methods include Newton Raphson method, secant method and bisection method that will make use of fact that IRR is the rate at which NPV is zero, NFV is zero and profitability index is 1. Follow the links listed below to view detailed exposition of these four methods; on each page you will be presented with detailed workout for numerical method along with a IRR calculator that will let you perform similar IRR calculations for your data.

1. Finding IRR with NPV equation using Newton raphson method
2. Finding IRR with NFV equation using Newton raphson method
3. Finding IRR with Profitability index equation using Newton raphson method
4. Finding IRR with NPV equation using Secant method
5. Finding IRR with NFV equation using Secant method
6. Finding IRR with Profitability index equation using Secant method
7. Finding IRR with TVM equation using Newton raphson method

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 Future Value
3. Net Present Value
4. Profitability index
5. Payback period
6. Discounted payback period