The Internal Rate of Return (IRR) is the discount rate that makes a project's Net Present Value (NPV) equal to zero. If the IRR is higher than the Minimum Attractive Rate of Return (MARR, the lowest return a company accepts on a new investment; in Brazil it is called TMA), the project tends to be considered viable.IRR is one of the most widely used indicators in investment analysis, but it carries a hidden assumption about what happens to the money the project returns before it ends. The Modified Internal Rate of Return (MIRR) exists to make that assumption explicit and more realistic. To understand it, we need one concept: the reinvestment rate, the interest rate the company is assumed to earn by reinvesting the positive cash flows generated by the project until the end of its useful life.What is the problem with IRR?IRR makes an implicit assumption: all the money received during the project can be reinvested at the IRR itself.For example, suppose a project has an IRR of 28% per year. IRR assumes that any cash flow received before the end of the project can also earn 28% per year until the project ends, which is rarely possible in practice. The higher the IRR, the less realistic this assumption becomes.In reality, a company usually reinvests that money at a rate closer to:its weighted average cost of capital (WACC);its Minimum Attractive Rate of Return (MARR);the return available from other investment opportunities or financial investments.There is also a second problem: when the cash flows change sign more than once (for example, an investment, then inflows, then another large outflow), the IRR equation may have more than one solution, or none. In that case, there is no way to know which IRR to use.A project has an IRR of 35% per year, and the company can invest its cash at 10% per year. What does the IRR assume about the project's intermediate inflows?Answer: That they are reinvested at 35% per year, the IRR itself — this is the implicit assumption in the IRR calculation. Since the company can only earn 10% per year, the IRR tends to overstate the real return; MIRR corrects this by using the 10% as the reinvestment rate.What is the Modified Internal Rate of Return?The Modified Internal Rate of Return (MIRR) corrects these limitations. It uses two different rates:Financing rate: used to bring the outflows (negative cash flows) to present value, at date 0.Reinvestment rate: used to compound the inflows (positive cash flows) to the end of the project, at date n.After that, the project is reduced to two values: a total outflow at date 0 and a total inflow at date n. MIRR is the rate that turns one into the other, that is, the rate that equates the present value of the investments with the future value of the reinvested inflows. Since there is only one inflow and one outflow, the answer is always unique.The formula is:\[ \text{MIRR} = \left( \frac{FV_{\text{inflows}}}{-PV_{\text{outflows}}}\right)^{1/n}-1 \]where \(FV_{\text{inflows}}\) is the future value of the inflows at date n (at the reinvestment rate), \(PV_{\text{outflows}}\) is the present value of the outflows at date 0 (at the financing rate, with a negative sign, hence the minus sign) and \(n\) is the number of periods of the project.Example 1: reinvestment at 10% per yearConsider the following project:YearCash Flow (R$)0-100,000130,000240,000350,000Suppose the reinvestment rate is 10% per year. Since the only outflow occurs at date 0, it is already at present value: R$ 100,000.The R$ 30,000 received in year 1 can be reinvested for two more years:\[ 30,000 \times (1.10)^2 = 36,300 \]The R$ 40,000 received in year 2 can be reinvested for one year:\[40,000 \times 1.10 = 44,000 \]The R$ 50,000 received in year 3 has no time left to earn interest:\[50,000\]So, at the end of the project, the accumulated value of the inflows will be:\[36,300+44,000+50,000=130,300\]Applying the formula:\[ \text{MIRR} = \left(\frac{130,300}{100,000}\right)^{1/3}-1 \approx 9.2\% \text{ per year} \]For comparison, this project's IRR is about 8.9% per year. Here MIRR came out above IRR because the reinvestment rate (10%) is higher than the IRR itself: in conventional projects with the whole investment at date 0, MIRR always lies between the IRR and the reinvestment rate.Example 2: different financing and reinvestment ratesConsider an investment of R$ 100,000 that generates:YearCash Flow (R$)0-100,000140,000250,000360,000Assume a financing rate of 8% per year and a reinvestment rate of 10% per year.The only outflow is at date 0, so the financing rate changes nothing here: the present value of the outflows is R$ 100,000. (It only matters when there are outflows at later dates, as in exercise 4.)The positive cash flows are compounded to year 3:Year 1: \(40,000 \times (1.10)^2 = 48,400\)Year 2: \(50,000 \times 1.10 = 55,000\)Year 3: \(60,000\)Total future value:\[ 48,400+55,000+60,000=163,400 \]Therefore, the MIRR will be:\[ \text{MIRR} = \left(\frac{163,400}{100,000}\right)^{1/3}-1 \approx 17.8\% \text{ per year} \]The IRR of this same project is about 21.6% per year. The gap of almost 4 percentage points comes precisely from the reinvestment assumption: IRR assumes the R$ 40,000 and the R$ 50,000 earn 21.6% per year; MIRR, only the 10% the company can actually earn.A project requires R$ 200,000 today and, after compounding all inflows at the reinvestment rate, accumulates R$ 320,000 at the end of 4 years. What is the approximate MIRR?Answer: 12.5% per year — MIRR = (320,000 / 200,000)^(1/4) − 1 = 1.6^(0.25) − 1 ≈ 12.5%. The 15% comes from dividing the 60% gain by 4 years, which ignores compound interest.Why is MIRR more realistic?MIRR replaces the assumption of reinvestment at the IRR itself with a more plausible rate, such as:the company's weighted average cost of capital (WACC);the return on available financial investments;the minimum attractive rate of return (MARR) adopted by the company.An intuitive exampleImagine you invest R$ 1 million in a factory.In the first year, it generates R$ 300 thousand in cash.You do not leave that money sitting in a checking account: you put it into another investment or use it in another company project.The question is: at what rate will that money grow until the end of the project?That is exactly the reinvestment rate. It represents the best estimate of the return the company can earn by reinvesting the resources generated by the project. That is why MIRR usually provides a more realistic measure of return than the traditional IRR, especially when the IRR is very high.A project's IRR is 18% per year and the company's reinvestment rate is 9% per year. What can be expected of the MIRR?Answer: It will lie between 9% and 18% per year — MIRR always lies between the IRR and the reinvestment rate, because the intermediate inflows now earn 9% instead of 18%, but the project still has its own return. It would only equal 9% for a project that earned nothing above that.When to use each one?IRR: suitable for introductory analyses and when cash flows are conventional (an initial investment followed only by inflows) and the IRR is not far from the rates the company actually earns.MIRR: preferable in professional analyses, since it uses more realistic reinvestment assumptions and eliminates the problem of multiple IRRs.In general, MIRR is considered a more reliable measure of a project's profitability, especially when the company knows its financing rate and its reinvestment rate. Even so, because it is a rate, it does not show the size of the gain: to choose between projects of different sizes, NPV remains the deciding criterion. In Excel and Google Sheets, the function is MIRR(values, finance_rate, reinvest_rate).In which of these situations is MIRR especially preferable to IRR?Answer: When the cash flows change sign more than once and the IRR has two solutions — MIRR reduces the project to one outflow at date 0 and one inflow at date n, so it always has a single answer. To know how many years the investment takes to pay back, the indicator is the payback period.Exercises1) A bakery plans to buy a new industrial oven for R$ 180,000. The equipment is expected to generate the following net cash flows:YearCash Flow (R$)0-180,000155,000265,000370,000480,000The company uses:a financing rate of 9% p.a.;a reinvestment rate of 11% p.a.Required:a) Calculate the project's MIRR.b) Should the investment be accepted if the MARR is 10% per year?2) A startup will invest R$ 500,000 in developing software.The expected cash flows are:YearCash Flow (R$)0-500,0001100,0002180,0003250,0004300,000Consider:financing rate = 12% p.a.;reinvestment rate = 8% p.a.Required:a) Calculate the MIRR.b) Explain why the reinvestment rate used is different from the financing rate.3) A company has funds to invest in only one of the projects below.Project AYearCash Flow (R$)0-250,000190,0002110,0003120,000Project BYearCash Flow (R$)0-250,000140,000290,0003220,000Data:financing rate = 10% p.a.;reinvestment rate = 9% p.a.Required:a) Calculate the MIRR of each project.b) Which project has the higher modified return?c) Explain in economic terms why the results differ.4) A company will carry out an expansion in two stages.YearCash Flow (R$)0-300,0001-80,0002180,0003210,0004240,000Consider:financing rate = 10% p.a.;reinvestment rate = 12% p.a.Required:a) Bring all outflows to present value.b) Take all inflows to future value.c) Calculate the MIRR.5) Which assumption is more realistic?Two analysts evaluate the same project. Analyst A uses IRR; Analyst B uses MIRR. It is known that:IRR = 24% p.a.;the company's reinvestment rate = 9% p.a.;cost of capital = 11% p.a.Answer:a) Which indicator better represents the company's reality?b) Explain why IRR may overstate profitability.c) In which situations is MIRR preferable?6) Complete the calculation.After the intermediate calculations, an analyst found:present value of the outflows = R$ 420,000;future value of the inflows = R$ 730,000;useful life of the project = 5 years.Required:a) Calculate the MIRR.b) Interpret the result in economic terms.7) Mark T (true) or F (false) for the statements below and justify the false ones.I. ( ) MIRR uses a reinvestment rate defined by the analyst.II. ( ) IRR always produces a single solution.III. ( ) MIRR eliminates the problem of multiple IRRs.IV. ( ) IRR assumes that all positive cash flows are reinvested at the IRR itself.V. ( ) The reinvestment rate is usually the company's cost of capital or MARR.8) The electric bicycle factory. A manufacturer is considering investing in the production of electric bicycles.The initial investment will be R$ 900,000, and the forecast net cash flows are:YearCash Flow (R$)0-900,0001180,0002220,0003260,0004310,0005350,000Consider:financing rate = 11% per year;reinvestment rate = 9% per year;MARR = 10% per year.Required:a) Calculate the future value of the inflows.b) Determine the present value of the investment.c) Calculate the MIRR.d) Should the project be accepted? Justify by comparing the MIRR with the MARR.e) Explain why a decision based on MIRR may differ from one based on IRR.Answer key1) FV of the inflows ≈ R$ 313,006; MIRR ≈ 14.8% p.a. Since 14.8% > 10%, the project should be accepted.2) a) FV of the inflows ≈ R$ 905,923; MIRR ≈ 16.0% p.a. b) The financing rate is what it costs to obtain the money; the reinvestment rate is what the company can earn by investing the cash the project returns. They are different markets and, in general, borrowing costs more than investing yields.3) a) MIRR of A ≈ 11.5% p.a. (FV ≈ R$ 346,829); MIRR of B ≈ 13.5% p.a. (FV ≈ R$ 365,624). b) Project B. c) B has larger total inflows (R$ 350,000 versus R$ 320,000) but receives them later; A brings the money in earlier. Since early inflows only earn 9% per year until the end of the project, A's earlier timing does not make up for B's extra R$ 30,000. If the reinvestment rate were much higher, receiving early would weigh more.4) a) PV of the outflows = 300,000 + 80,000/1.10 ≈ R$ 372,727. b) FV of the inflows = 180,000 × 1.12² + 210,000 × 1.12 + 240,000 ≈ R$ 700,992. c) MIRR ≈ 17.1% p.a.5) a) MIRR, because it uses the rate at which the company can actually reinvest (9%). b) Because IRR assumes the intermediate inflows earn the full 24% per year, which the company cannot achieve. c) When the IRR is very different from the rates the company earns, when the cash flows change sign more than once (multiple IRRs) and when the company knows its financing and reinvestment rates.6) a) MIRR = (730,000 / 420,000)^(1/5) − 1 ≈ 11.7% p.a. b) Investing R$ 420,000 today is equivalent to obtaining R$ 730,000 in 5 years, a return of about 11.7% per year already accounting for reinvestment of the inflows at the company's rate; the project is attractive if this rate exceeds the MARR.7) I. T. II. F: with cash flows that change sign more than once, IRR can have several solutions (or none). III. T. IV. T. V. T.8) a) FV of the inflows ≈ R$ 1,535,797. b) R$ 900,000 (the investment is already at date 0). c) MIRR ≈ 11.3% p.a. d) Yes: 11.3% > 10% (the NPV at 10% is also positive, ≈ R$ 69,853). e) IRR (≈ 12.7% p.a.) assumes reinvestment at 12.7%; MIRR uses the company's 9%. In a project whose IRR barely exceeds the MARR, the MIRR may fall below the MARR and reverse the decision.
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