The reinvestment rate is the interest rate that the company is assumed to be able to earn by reinvesting the positive cash flows generated by a project until the end of its useful life.The Internal Rate of Return (IRR) is the rate that makes the project's Net Present Value (NPV) equal to zero. In other words, it is the discount rate that satisfies NPV = 0.If the IRR is greater than the Minimum Attractive Rate of Return (MARR), the project tends to be considered feasible.What is the problem with IRR?IRR makes an implicit assumption: all money received during the project can be reinvested at the IRR itself.For example, suppose a project has an IRR = 28% per year. IRR assumes that any cash flow received before the end of the project can also earn 28% per year until closing, which is not always possible in practice.In reality, a company usually reinvests this money at a rate closer to: its weighted average cost of capital (WACC); its Minimum Attractive Rate of Return (MARR); or the return available in other investment opportunities.What is Modified Internal Rate of Return?The Modified Internal Rate of Return (MIRR) corrects this limitation. It uses two different rates: Financing rate: used to bring outflows (negative cash flows) to present value. Reinvestment rate: used to capitalize inflows (positive cash flows) until the end of the project.Thus, MIRR represents the rate that equates the present value of the investments made to the future value of the reinvested inflows.The formula is:\[ \text{MIRR} \left( \frac{FV_{\text{inflows}}}{-PV_{\text{outflows}}}\right)^{1/n}-1 \]ExampleConsider the following project:YearCash Flow0-100,000130,000240,000350,000Assume the reinvestment rate is 10% per year.The receipt of R$ 30,000 in year 1 can be reinvested for two more years:\[ 30,000 \times (1.10)^2 = 36,300 \]The receipt of R$ 40,000 in year 2 can be reinvested for one year:\[40,000 \times (1.10)=44,000 \]The R$ 50,000 received in year 3, however, has no time to earn interest:\[50,000\]Thus, at the end of the project, the accumulated value of the receipts will be:\[36,300+44,000+50,000=130,300\]This is the value that will be used in the MIRR calculation.Example 2Consider an investment of R$ 100,000 that generates:YearFlow0-100,000140,000250,000360,000Assume financing rate = 8%; and reinvestment rate = 10%.The positive cash flows are capitalized until 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:\[ 48,400+55,000+60,000=163,400 \]Since the initial investment is R$ 100,000, the MIRR will be approximately:\[ \left(\frac{163,400}{100,000}\right)^{1/3}-1 \]\[17.8%\ \text{per year} \]Why is this more realistic?Traditional IRR makes an implicit assumption: all money received is reinvested at the IRR itself.For example, if a project's IRR is 35% per year, IRR is assuming that each intermediate receipt can be reinvested at 35% per year, which rarely happens in practice.MIRR replaces this assumption with a more plausible rate, such as:* the company's weighted average cost of capital (WACC);* the return on available financial investments;* the company's minimum attractive rate of return (MARR).An intuitive exampleImagine that you invest R$ 1 million in a factory.* In the first year, it generates R$ 300 thousand in profit.* You do not leave that money sitting idle in a checking account; you invest it in another investment or use it in another company project.The question is: at what rate will that money earn until the end of the project?That is exactly the reinvestment rate. It represents the best estimate of the return the company can obtain by reinvesting the resources generated by the project.That is why MIRR usually provides a more realistic measure of return than traditional IRR.When should each one be used?* IRR: suitable for introductory analyses and when cash flows are conventional (an initial investment followed only by inflows).* MIRR: preferable in professional analyses, because 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.Exercises1) A bakery intends to buy a new industrial oven for R$ 180,000. The equipment is expected to generate the following net cash flows:YearCash Flow (R$)0-180,000155,000265,000370,000480,000The company uses:* financing rate of 9% p.a.* reinvestment rate of 11% p.a.Required:1. Calculate the project's MIRR.2. Should the investment be accepted if the MARR is 10% per year?2) A startup will invest R$ 500,000 in software development.The expected cash flows are:YearFlow (R$)0-500,0001100,0002180,0003250,0004300,000Consider:* financing rate = 12%* reinvestment rate = 8%Required:a) Calculate the MIRR.b) Explain why the reinvestment rate used is different from the financing rate.3) A company only has enough resources to invest in one of the projects below.Project AYearFlow0-250,000190,0002110,0003120,000Project BYearFlow0-250,000140,000290,0003220,000Data:* financing rate = 10%* reinvestment rate = 9%Required:1. Calculate the MIRR of each project.2. Which project has the higher modified return?3. Explain economically why the results are different.4) A company will carry out expansions in two stages.YearFlow0-300,0001-80,0002180,0003210,0004240,000Consider:* financing rate = 10%* reinvestment rate = 12%Required:1. Bring all outflows to present value.2. Take all inflows to future value.3. Calculate the MIRR.5) Which assumption is more realistic?Two analysts evaluate the same project. Analyst A uses IRR. Analyst B uses MIRR. Knowing that:* IRR = 24%* company reinvestment rate = 9%* cost of capital = 11%Answer:a) Which indicator better represents the company's reality?b) Explain why IRR may overestimate profitability.c) In which situations is the MIRR preferable?6) Complete the calculationAfter performing the intermediate calculations, an analyst found:* Present value of cash outflows = R$ 420,000* Future value of cash inflows = R$ 730,000* Project useful life = 5 yearsRequested:1. Calculate the MIRR.2. Economically interpret the result found.7) Write T for True or F for False for the statements below. Justify the false statements. Analyze the statements below.I. ( ) MIRR considers a reinvestment rate defined by the analyst.II. ( ) IRR always produces only one 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 hurdle rate.8) The electric bicycle factory. An industry is evaluating investing in the production of electric bicycles.The initial investment will be R$ 900,000, and the forecast net cash flows are:YearCash Flow0-900,0001180,0002220,0003260,0004310,0005350,000Consider:* financing rate = 11% per year;* reinvestment rate = 9% per year;* hurdle rate = 10% per year.Requested:1. Calculate the future value of the cash inflows.2. Determine the present value of the investment.3. Calculate the MIRR.4. Should the project be accepted? Justify your answer by comparing the MIRR with the hurdle rate.5. Explain why the decision based on MIRR may be different from that based on IRR.Answer Key1) 14.9% p.a.2) 16% p.a.3) Project B has the higher MIRR.4) 17% p.a.8) MIRR = 11.2% p.a.
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