Calculating IRR

12 Key excerpts on "Calculating IRR"

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  eBook - ePub

  Strategic Finance for Criminal Justice Organizations

  • Daniel Adrian Doss, William H. Sumrall III, Don W. Jones(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Other considerations of the IRR involve the existence of multiple instances of IRR values spawned from irregularities within cash flows through time. It is beyond the scope and magnitude of this introductory text to address such issues. However, extended discussions may be found among the writings of contemporary financial management texts.

  Some initiatives may require periods that are longer than those that are considered in this text. When these situations occur, it is recommended that IRR calculations be performed through the use of software spreadsheets, proprietary software, or financial calculators. Also, within the context of collegiate finance courses, a tabular solution is also available to solve IRR problems involving a variety of periods. However, for the purposes of this text, the use of the basic formula is appropriate to demonstrate the basic concept of the internal rate of return, and to delineate the calculations through which IRR problems are solved. Future editions of this text, if any, are anticipated to contain the tabular solution methods of IRR problems.

  7.7 Chapter Comments and Summary

  This chapter introduced the internal rate of return (IRR) method of capital budgeting. The methods of capital budgeting encompass perspectives of time, cash value, rate, and profitability potential. The IRR is indicative of a rate-based perspective regarding the rendering of capital budgeting decisions. Further, the IRR method incorporates the time value of money within its primary construct. Derivation of the IRR method may occur through algebraic manipulation of the net present value formula that is given in Chapter 6 .

  Although the outcomes of the IRR method present quantitative findings, the outcomes must be interpreted. The rules governing the interpretation of the IRR outcome are quite basic:

  1. Accept the capital initiative if the value of the calculated IRR outcome is higher than the specified discount rate. 2. Reject the capital initiative if the value of the calculated IRR outcome is lower than the discount rate. 3. When the conditions of mutual exclusion govern the problem domain, select the capital initiative that demonstrates the highest IRR outcome.

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  Finance

  Capital Markets, Financial Management, and Investment Management

  • Frank J. Fabozzi, Pamela Peterson Drake(Authors)
  • 2009(Publication Date)
  • Wiley

    (Publisher)

  The index value is greater than one, which means that the investment produces more in terms of benefits than costs. An advantage of using the profitability index is that it translates the dollar amount of NPV into an indexed value, providing a measure of the benefit per dollar investment. This is helpful in ranking projects in cases in which the capital budget is limited.

  The decision rule for the profitability index depends on the PI relative to 1.0, which means

  INTERNAL RATE OF RETURN

  Suppose an investment opportunity requires an initial investment of $1 million and has expected cash inflows of $0.6 million after one year and another $0.6 million after two years. This opportunity is shown in Figure 14.3 using a time line.

  The return on this investment (denoted by internal rate of return or IRR, in the next equation) is the discount rate that causes the present values of the $0.6 million cash inflows to equal the present value of the $1 million cash outflow, calculated as

  Another way to look at this is to consider the investment’s cash flows discounted at the IRR of 10%. The NPV of this project if the discount rate is 13.0662% (the IRR in this example), is zero:

  An investment’s internal rate of return is the discount rate that makes the present value of all expected future cash flows equal to zero. We can represent the IRR as the rate that solves Going back to Project X, the IRR for this project is the discount rate that solves Using a calculator or a computer, we get the answer of 10.172% per year.

  FIGURE 14.3

  Timeline of Investment Opportunity

  Looking back at the investment profiles of Projects X and Y, each profile crosses the horizontal axis (where NPV = 0) at the discount rate that corresponds to the investment’s IRR. This is no coincidence: by definition, the IRR is the discount rate that causes the project’s NPV to equal zero.

  The IRR is a yield—what is earned, on average, per year. How do you use it to decide which investment, if any, to choose? Let’s revisit Projects X and Y and the IRRs we just calculated for each. If, for similar risk investments, owners earn 10% per year, then both Projects X and Y are attractive. They both yield more than the rate owners require for the level of risk of these two investments:

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  Property Valuation Techniques

  • David Isaac, John O'Leary(Authors)
  • 2013(Publication Date)
  • Bloomsbury Visual Arts

    (Publisher)

  The NPV for the first investment opportunity Discounted Cash Flow 35 suggests that it would earn something above 10 per cent per annum, while the NPV for the second investment suggests that it would earn a rate of return somewhere below 10 per cent. To identify the precise rate of return in this type of situation, the Internal Rate of Return (IRR) needs to be calculated. 3.5 The Internal Rate of Return (IRR) An important piece of information for analysts, valuers and investors is the actual return on capital to be obtained from an investment. This is the Internal Rate of Return (IRR) arising from the interplay of income and expenditure and it is also the discount rate at which the NPV is zero. To illustrate how the IRR is found, the investment opportunity in Example 3.5 has been chosen. That investment was assessed against a target rate of 10 per cent and was found to have an NPV of £3,696. In thin investment markets, such marginal underperformance might be looked at again to establish what the investment would actually earn, i.e. to calculate the IRR. The first step would be to trial the same variables at a lower target rate to try to establish the level above which the investment was performing and which would produce a positive NPV. The calculation of the IRR by formula requires the selection of two discount rates, one generating a positive NPV and the other a negative NPV, and then interpolating between the two. Example 3.6 The NPV for a trial target rate of 8 per cent would be calculated as follows.

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  Fundamentals of Corporate Finance

  • Robert Parrino, David S. Kidwell, Thomas Bates, Stuart L. Gillan(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Thus, using IRR or MIRR, rather than NPV, does not require less effort from financial managers. Summary of Internal Rate of Return (IRR) Method Decision Rule: IRR > Cost of capital ⇨ Accept the project. IRR < Cost of capital ⇨ Reject the project. Key Advantages Key Disadvantage 1. Intuitive and easy to understand. 2. Based on discounted cash flow technique. 1. With nonconventional cash flows, IRR approach can yield no usable answer or multiple answers. 2. A lower IRR can be better if a cash inflow is followed by cash outflows. 3. With mutually exclusive projects, IRR can lead to incorrect investment decisions. 4. IRR calculation assumes cash flows are reinvested at the IRR. Before You Go On 1. What is the internal rate of return (IRR) method? 2. Under what circumstances do the NPV and IRR decision rules always yield the same decision? 3. In capital budgeting, what is a conventional cash flow pattern? 4. Why should the NPV method be the primary decision tool used in making capital investment decisions? LEARNING OBJECTIVE 6. Explain how the profitability index can be used to rank projects when a firm faces capital rationing and describe the limitations that apply to the profitability index. Our discussion of capital budgeting so far has focused on determining whether an individual project creates value for stockholders. Although the analytical methods we have discussed are critical components of the capital budgeting process, they do not tell us what to do when, as is often the case, a firm does not have enough money to invest in all available positive NPV projects. In other words, they do not tell us how to identify the bundle or combination of pos- itive NPV projects that creates the greatest total value for stockholders when there are capital constraints or, as we called it earlier in this chapter, capital rationing. In an ideal world we could accept all positive NPV projects because we would be able to finance them.

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  Fundamentals of Corporate Finance

  • Robert Parrino, David S. Kidwell, Thomas Bates(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  It is generally believed that the cost of capital, which is often lower than 10.5 Internal Rate of Return 351 the IRR, better reflects the rate that firms are likely to earn. Using the IRR may thus involve overly optimistic assumptions regarding reinvestment rates. To eliminate the reinvestment rate assumption of the IRR, some practitioners prefer to calcu-late the modified internal rate of return (MIRR) . In this approach, each operating cash flow is converted to a future value at the end of the project’s life, compounded at the cost of capital. These values are then summed up to get the project’s terminal value (TV). The MIRR is the interest rate that equates the project’s cost (PV ) Cost , or cash outflows, with the future value of the project’s cash inflows at the end of the project (PV ) TV . 2 Because each future value is computed using the cost of capital as the interest rate, the reinvestment rate problem is eliminated. We can set up the equation for the MIRR in the same way we set up Equation 10.4 for the IRR: P V(Cost of the project) PV(Cash inflows) PV PV PV TV (1 MIRR) n Cost TV Cost = = = + (10.5) To compute the MIRR, we have to make two preliminary calculations. First, we need to calculate the value of P V Cost , which is the present value of the cash outflows that make up the investment cost of the project. Since for most capital projects, the investment cost cash flows are incurred at the beginning of the project, t 0 = , there is often no need to calculate the present value. If investment costs are incurred over time t ( 0) > , then the cash flows must be discounted at the cost of capital for the appropriate time period. Second, we need to compute the terminal value (TV). 3 To do this, we find the future value of each operating cash flow at the end of the project’s life, compounded at the cost of capital. We then sum up these future values to get the project’s T V.

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  Economic Feasibility of Projects

  Managerial and Engineering Practice

  • S.L. Tang(Author)
  • 2003(Publication Date)
  • The Chinese University of Hong Kong

    (Publisher)

  3. Discount Cash Flow Method 3.1 The Internal Rate of Return (IRR) In Chapter 2, the methods described for project appraisal were based on a predetermined discount rate i . With a predetermined i for calculating the net present worth or the net annual benefit, the result can either be positive or negative (see Examples 2.1 and 2.2 of Chapter 2). If it is positive, the project is considered viable, and vice versa. However, we might sometimes obtain, by luck, a zero result of net present worth (or net annual benefit) with the use of a predetermined i . In such a case, the project is considered breaking-even since its present worth of total benefits equals its present worth of total costs (or the total annual benefits equal the total annual costs). The discount rate i , in this case, represents exactly the rate of return on the investment over its life, and is called IRR (internal rate of return). Some people like to call it the yield rate or solution rate. Hence, IRR can be defined as the discount rate i that the net present worth of a project is zero if such i is used in the equivalence calculation. In this chapter, the reader will see a full description of the discount cash flow (DCF) method which is a method to determine the IRR of an investment, with costs and benefits known (or estimated) over the time span of the investment. Readers should note that it is not necessary for the DCF method to use a predetermined i . 3.2 Mathematical Presentation of IRR Let C k = total cash outflow (i.e. expenditure or cost) in year k , B k = total cash inflow (i.e. income or benefit) in year k , NCF k = net cash flow in year k . By definition: NCF k = B k – C k (3.1) The cash flows can be tabulated as in Table 3.1:

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  Applied Corporate Finance

  • Aswath Damodaran(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Proponents of the NPV rule argue that it is surplus value, over and above the hurdle rate, no matter what the investment. • The NPV rule does not control for the life of the project. Consequently, when comparing mutually exclusive projects with different lifetimes, the NPV rule is biased toward accepting longer-term projects. Internal Rate of Return The internal rate of return (IRR) is based on discounted cash flows. Unlike the NPV rule, however, it takes into account the project’s scale. It is the discounted cash flow analog to the accounting rates of return. Again, in general terms, the IRR is that discount rate that makes the NPV of a project equal to 0. Investment Decision Rules 205 Figure 5.4 NPV Profile $100.00 $200.00 NPV of Project $300.00 $400.00 $500.00 $600.00 ($100.00) ($200.00) $0.00 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Discount Rate Project 1 Project 2 11% 12% 13% 16% 17% 18% 19% 20% 14% 15% At the internal rate of return, the NPV of this project is 0. The linkage between the NPV and the IRR is most obvious when the NPV is graphed as a function of the discount rate in a net present value profile . In Figure 5.4, we graph the net present values of two projects with five-year lives, with different cash flow profiles, with project 1 generating more cash flows in the later years and project 2 generating higher cash flows in the early years. The NPV profile provides several insights on the project’s viability. First, the internal rate of return is clear from the graph—it is the point at which the profile crosses the x -axis; the IRR for project 2 is 13.57% and the IRR for project 1 is 12.79%. Second, it provides a measure of how sensitive the NPV—and, by extension, the project decision—is to changes in the discount rate. The slope of the NPV profile is a measure of the discount rate sensitivity of the project. In Figure 5.4, the net present value of project 1 is more sensitive to changes in the discount rate than the net present value of project 2.

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  Business Planning and Control

  Integrating Accounting, Strategy, and People

  • Bruce Bowhill(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  The formula to be followed with linear interpolation is shown below: IRR = lowest discount rate + difference in discount rate × NPV at lowest discount rate difference in NPVs For this example: IRR = 10% + (25% − 10%) × £525,000 (£525,000 − ( − £111,000)) = 10% + 12% = 22%. Figure 4.6 Internal rate of return for project A [ 94 ] C H A P T E R 4 Activity 4.5 Calculate the IRR of projects B and C. The advantages and disadvantages of internal rate of return Advantages 1. For most projects IRR actually gives the same project selection as NPV. 2. IRR may be better for communicating than NPV. A comment such as ‘the internal rate of return is 22% on this project’ may seem more meaningful than ‘the NPV of this project is £525,000’. Note that the percentage return is not the same as the ARR. Disadvantages The IRR technique can have more than one IRR, also some projects have no IRR. Activity 4.6 It has been argued that in order to evaluate a project to see if it leads to an increase in shareholder wealth, it is necessary to satisfy the following criteria: 1. The amount of all cash flows should be considered. 2. The timing of the cash flows should be considered. 3. The risk of the cash flows should be considered. Which technique(s) satisfy the three criteria identified above? Decision rule for accepting projects It is possible to accept any combination of projects as long as they comply with the decision rules of the organization (unless the projects are mutually exclusive, i.e. it is only possible to accept one of the options). Payback method : Each organization should decide on how quickly an investment must pay back its cash. If payback must be within 3 years, then 3 years is the cut-off point for the organization. Any investment which took longer than 3 years to pay back would not be accepted. ARR method : Projects with ARRs greater than or equal to the organization’s hurdle rate should be accepted.

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  Net Present Value and Risk Modelling for Projects

  • Martin Hopkinson(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Chapter 1 , there is often a good case for discount factors to be calculated at mid-year points. If this practice is adopted, the effect of inflation should be calculated on the equivalent basis. If Year 1 is the first full year that follows time now, the effect of inflation on cost during the mid-year of Year n should be calculated as:

  Cn:midyear = C0 (1 + inflation rate)n−0.5

  Finally, if inflation has been included directly in the model, but the discount factor has been specified in real terms, the equivalent nominal rate discount factor is calculated as:

  Dnominal = (1 + Inflation rate)(1 + Dreal ) − 1

  Whichever approach is used, project professionals using NPV modelling forecasts should seek assurance that each model is internally consistent. Inconsistent modelling of the effects of inflation and the discount factor is an easy mistake to make.

  Internal Rate of Return

  Internal rate of return IRR is a discounted financial modelling measure that is often used alongside or in place of NPV forecasts. IRR is defined as being the discount rate that would result in a project NPV of zero. It is possible to calculate IRR manually in very simple cases. Excel also has an IRR function. However, the usual procedure for obtaining IRR is to create an NPV model and use trial and error to determine the discount rate that results in an NPV that is sufficiently close to zero for the level of accuracy that is appropriate. As an example, the IRR for Model 2.2 is approximately 17%. A similar procedure can be employed with NPV risk models, using the mean NPV forecast equal to zero as the basis for obtaining the IRR.

  Organisations using IRR usually specify a hurdle rate for projects that must be exceeded in order for them to be approved. If this practice is intended to be consistent with the NPV rule, then the hurdle would be set at the relevant discount rate. However, organisations often choose to set a higher hurdle than this on the grounds of reluctance to accept the risk of proceeding with projects with business cases that are only marginal. Whilst this way of thinking is understandable, it can also be interpreted as being a form of blanket discount rate adjustment which, for reasons described earlier in this chapter, may have counterproductive effects.

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  ACCA Financial Management

  Practice and Revision Kit

  • BPP Learning Media(Author)
  • 2021(Publication Date)
  • BPP Learning Media

    (Publisher)

  166 Financial Management (FM) (b) In most simple accept or reject decisions, IRR and NPV will select the same project. However, NPV has certain advantages over IRR as an investment appraisal technique. NPV and shareholder wealth The NPV of a proposed project, if calculated at an appropriate cost of capital, is equal to the increase in shareholder wealth which the project offers. In this way NPV is directly linked to the assumed financial objective of the company, the maximisation of shareholder wealth. IRR calculates the rate of return on projects, and although this can show the attractiveness of the project to shareholders, it does not measure the absolute increase in wealth which the project offers. Absolute measure NPV looks at absolute increases in wealth and thus can be used to compare projects of different sizes. IRR looks at relative rates of return and in doing so ignores the relative size of the compared investment projects. Non-conventional cash flows In situations involving multiple reversals in project cash flows, it is possible that the IRR method may produce multiple IRRs (that is, there can be more than one interest rate which would produce an NPV of zero). If decision-makers are aware of the existence of multiple IRRs, it is still possible for them to make the correct decision using IRR, but if unaware they could make the wrong decision. Mutually-exclusive projects In situations of mutually-exclusive projects, it is possible that the IRR method will (incorrectly) rank projects in a different order to the NPV method. This is due to the inbuilt reinvestment assumption of the IRR method. The IRR method assumes that any net cash inflows generated during the life of the project will be reinvested at the project's IRR. NPV on the other hand assumes a reinvestment rate equal to the cost of capital. Generally NPV's assumed reinvestment rate is more realistic and hence it ranks projects correctly.

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  Multiple Interest Rate Analysis

  Theory and Applications

  • M. Osborne(Author)
  • 2014(Publication Date)
  • Palgrave Pivot

    (Publisher)

  et al . (2011) contains a classic statement of the arguments and lists four pitfalls.

     1    An IRR, by itself, does not indicate whether a project involves borrowing or lending;

     2    Some cash flows cause the IRR equation to solve for more than one plausible IRR, resulting in ambiguity about which IRR to use as a criterion;

     3    NPV and IRR do not always rank mutually exclusive projects the same; the consensus opinion is that ranking by NPV is reliable, and therefore ranking by IRR is not;

     4    A non-flat yield curve provides more than one cost of capital with which to compare IRR leading to uncertainty about which cost of capital to employ.

  Despite the pitfalls there is considerable empirical evidence that the majority of practitioners continue to use IRR as an investment criterion and performance measure. For example, in the context of capital budgeting there is the study of US data by Graham and Harvey (2001) and the similarly executed study of European data by Brounen et al . (2004). Many studies of capital budgeting practice are published every year containing similar results for various countries. In the context of IRR as a performance measure for private equity firms, hedge funds, and venture capitalists see the works by Phalippou (2008), Phalippou and Gottschalg (2009), Dichev and Yu (2011), and Achleitner et al.

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  Cornerstones of Cost Management

  • Don Hansen, Maryanne Mowen(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  KEY TERMS Accounting rate of return (ARR), 1001 Annuity, 1025 Capital budgeting, 998 Capital investment decisions, 998 Compounding of interest, 1023 Cost of capital, 1003 Discount factor, 1024 Discount rate, 1024 Discounted cash fl ows, 1002 Discounting, 1024 Discounting models, 999 Five-year assets, 1014 Future value, 1023 Half-year convention, 1014 Independent projects, 998 Internal rate of return (IRR), 1005 Modi fi ed accelerated cost recovery system (MACRS), 1014 Mutually exclusive projects, 998 Net present value (NPV), 1003 Nondiscounting models, 999 Payback period, 999 Postaudit, 1018 Present value, 1023 Required rate of return, 1003 Sensitivity analysis, 1020 Seven-year assets, 1014 Three-year assets, 1014 What-if analysis, 1020 APPENDIX A: PRESENT VALUE CONCEPTS An important feature of money is that it can be invested and can earn interest. A dollar today is not the same as a dollar tomorrow. This fundamental principle is the backbone of discounting methods. Discounting methods rely on the relationships between current and future dollars. Thus, to use discounting methods, we must understand these relationships. Future Value Suppose a bank advertises a 4 percent annual interest rate. If a customer invests $100, he or she would receive, after one year, the original $100 plus $4 interest [$100 þ (0.04 × $100) ¼ (1 þ 0.04) × $100 ¼ 1.04 × $100 ¼ $104]. This result can be expressed by the following equation, where F is the future amount, P is the initial or current outlay, and i is the interest rate: F ¼ P (1 þ i ) (19A.1) For the example, F ¼ $100 × (1 þ 0.04) ¼ $100 × 1.04 ¼ $104. Now suppose that the same bank offers a 5 percent rate if the customer leaves the original deposit, plus any interest, on deposit for a total of two years. How much will the customer receive at the end of two years? Again, assume that a customer invests $100. Using Equation 19A.1, the cus-tomer will earn $105 at the end of Year 1 [ F ¼ $100 × (1 þ 0.05) ¼ $100 × 1.05 ¼ $105].

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