IntroductionThe internal rate of return (IRR) is the interest rate at which the sum of all cash flows, brought to present value, equals zero.Consider the following problem: we have an investment made at date 0 and a set of positive values y₁, y₂, y₃, …, yₜ received at dates 1, 2, 3, …, t. We want to find the interest rate i that makes the sum of all these values, in present value, equal to zero (the investment enters with a negative sign). Making this sum zero means that investing the initial amount at rate i would generate exactly those receipts: the IRR is therefore the return of the investment itself.For companies, the IRR is very important because it serves to evaluate an investment. A project is expected to have a rate of return above the yield of a savings account and, preferably, above government Treasury bonds, which are very safe investments: if it does not earn more than they do, the project's risk is not worth taking.What does it mean to say that a project's IRR is 12% per year?Answer: That the present value of the project's cash flows, discounted at 12% per year, equals zero — that is the definition of the IRR: the rate that makes the NPV zero, that is, the project's compound return. It is neither a total profit on the investment nor a simple-interest return.Calculating the internal rate of returnIn the simplest case, with a single investment at date 0, the problem is to find the value of i such that:\[ \text{investment} = \frac{y_1}{(1+i)^{1}} + \frac{y_2}{(1+i)^{2}} + ... +\frac{y_t}{(1+i)^{t}} \]Since there may be several investments on different dates, the most general form is to treat each cash flow with its own sign (outflows negative, inflows positive) and require the net present value (NPV) to be zero:\[ 0 = y_0 + \frac{y_1}{(1+i)^{1}} + \frac{y_2}{(1+i)^{2}} + ... +\frac{y_t}{(1+i)^{t}} \]A numerical example:timecash flow0- 5,00012,00022,00032,000The IRR (in Portuguese, TIR) is the rate i that satisfies:\[ 0 = -5000 + \frac{2000}{(1+i)^{1}} + \frac{2000}{(1+i)^{2}} + \frac{2000}{(1+i)^{3}} \]This requires solving a polynomial equation of degree t, which in general cannot be done by formula (equations of degree above 4 have no general formula, and even those of degree 3 and 4 have impractical ones). In practice, people use the HP 12C financial calculator or a spreadsheet's IRR function. Since we will not use a calculator in this course, we will look at an approximate method: testing rates.Testing a few rates:Interest rateNet present value (NPV)5%44610%-26Since the NPV is positive at 5% and negative at 10%, the IRR lies between 5% and 10%. To estimate it, we use linear interpolation: we assume that, between these two rates, the NPV curve is a straight line segment. By similar triangles:\[ \frac{446}{i - 5\%}=\frac{26}{10\% - i}\]\[ 446 · (0.10 - i) = 26 · (i - 0.05)\]\[ 44.6 - 446i = 26i - 1.3\]\[ 472i = 45.9 \Rightarrow i = 0.0972 = 9.72\% \]By linear interpolation, the IRR is approximately 9.72% (the exact value is 9.70%: the interpolation errs little because the NPV curve is almost straight in this stretch).Another way, simpler and less precise, is to take the midpoint between the two rates, in this case 7.5%, and repeat the process (this is called the bisection method). At 7.5%, the NPV is 201 reais, positive. Since at 10% the NPV is -26 reais, negative, the IRR lies between the two; the new midpoint is 8.75%, which gives an NPV of 85 reais, still positive, so the IRR lies between 8.75% and 10%, and so on.We can repeat these iterations as many times as we like until we get very close to the true value of the IRR. That is exactly what calculators do, except that they perform dozens of iterations in a few milliseconds.ExamplesConsider the following cash flow and calculate its internal rate of returntimecash flow0- 40115220322Testing a few rates:Interest rateNet present value (NPV)5%11.4310%6.6915%2.6320%-0.88The IRR is the interest rate that makes the NPV zero. Since the NPV changes sign between 15% and 20%, the IRR lies in that interval. Interpolating:\[ \frac{2.63}{i - 15\%}=\frac{0.88}{20\% - i}\]\[ i = 15\% + 5\% · \frac{2.63}{2.63 + 0.88} = 18.75\%\]By interpolation, the IRR is approximately 18.75% (the exact value is 18.68%). Using only the midpoint, the first estimate would be 17.5%.Calculate the internal rate of return of the investment given by the following cash flowstimecash flow05001-1,50021,000This is a non-conventional investment, because the cash flow changes sign more than once (inflow, outflow, inflow). Denoting the rate we seek by i, we must have:\[ 0 = 500 - \frac{1500}{(1+i)^{1}} + \frac{1000}{(1+i)^{2}} \]Multiplying everything by (1 + i)² and then dividing by 500:\[ 0 = 500·(1+i)^{2} - 1500·(1+i)^{1} + 1000 \]\[ 0 = (1+i)^{2} - 3·(1+i)^{1} + 2 \]\[ 0 = i^{2} + 2·i + 1- 3 - 3·i + 2 \]\[ 0 = i^{2} - i = i(i - 1) \]\[ i = 0 \text{ or } i = 1\]The IRR is either 0% or 100%: two rates make the NPV zero. Looking more closely, the NPV is negative for rates between 0% and 100% and positive for rates above 100%. This is the weak point of the IRR: when the cash flow changes sign more than once, there may be more than one IRR (or none), and in that case it is better to decide by the NPV.An investment of R$ 1,000.00 (R$ is the Brazilian real) returns a single payment of R$ 1,210.00 at the end of two years. What is its IRR?Answer: 10% per year — the IRR is the rate i such that 1,000 · (1 + i)² = 1,210, so (1 + i)² = 1.21 and 1 + i = 1.1. The option 21% is the total return over the two years; 10.5% is that return divided by two, as if it were simple interest.A project's NPV is R$ 30.00 at a 10% rate and -R$ 20.00 at a 15% rate. By linear interpolation, what is the approximate IRR?Answer: 13% — the total distance between the NPVs is 30 + 20 = 50; zero lies 30/50 = 60% of the way from 10% to 15%, so i = 10% + 0.6 · 5% = 13%. The option 12.5% is the midpoint, which ignores that the NPV at 10% is farther from zero.How to use the IRR to make decisionsTo decide whether an investment is worthwhile, we compare the IRR with the minimum attractive rate of return (MARR, in Portuguese TMA), which is the lowest return the investor accepts — usually what could be earned in a safe investment, such as Treasury bonds, plus a risk premium.If the IRR is greater than the MARR, the investment earns more than the alternative and should be accepted (the NPV calculated at the MARR is positive).If the IRR is less than the MARR, the investment should be rejected.If they are equal, it makes no difference.The same reasoning applies to borrowers: the IRR of a financing deal is its effective cost, and financing is only worthwhile if that cost is lower than what the buyer could earn by investing the money (see exercises 3 and 6 below).A project has an IRR of 12% per year, and the company only accepts projects that earn at least 15% per year (its MARR). What should it do?Answer: Reject, because the IRR is lower than the MARR — the project earns 12% and the company requires 15%. Discounted at 15%, the NPV of this conventional project would be negative, not positive; a positive IRR on its own does not say whether the project is worthwhile.Exercises1) A bank grants a company a loan of R$ 600,000.00 to be repaid in three installments, due in one, two and three years, of R$ 200,000.00, R$ 300,000.00 and R$ 400,000.00, respectively. What is the interest rate of this loan?2) A piece of equipment is sold for R$ 13,000.00 in cash or, alternatively, for R$ 3,000.00 down plus three monthly installments of R$ 4,000.00 each. What is the interest rate of this financing?3) If, in the previous problem, the buyer has the money to pay in cash and can earn 2% per month on investments, what is the best option?4) A machine is sold for R$ 35 thousand in cash or in six monthly installments with no down payment, the first three of R$ 6 thousand each and the last three of R$ 8 thousand each. What is the interest rate of this financing?5) A plot of land is sold for R$ 60,000.00 in cash or in six equal monthly installments of R$ 11,000.00 each, the first due three months after the purchase. What is the interest rate of the financing?6) A raw material is sold for R$ 900.00 in three equal monthly installments with no surcharge, the first paid as a down payment. If payment is made in cash, there is a 5% discount on the sale price.a) What is the interest rate of the financing?b) What is the best payment option for the buyer if the money can be invested at 1.7% per month?7) A person invested R$ 500,000.00 and received R$ 200,000.00 after one month, R$ 250,000.00 after two months and R$ 300,000.00 after three months. What is the internal rate of return of this investment?8) By investing R$ 120,000.00, a person expects to receive R$ 40,000.00, R$ 60,000.00 and R$ 90,000.00 after three, five and seven months, respectively.a) What is the internal rate of return of this investment?b) Assuming the investor's minimum attractive rate of return is 6% per month, determine whether the investment should be made.9) Assuming that the dividends paid by a share are R$ 12.00 and R$ 14.00 six and 12 months from now, and that its estimated sale price is R$ 125.00 twelve months from now, calculate:a) The price to be paid today for this share so that the investor earns a rate of return of 9% per half-year.b) If the investor pays R$ 120.00 for the share today, what rate of return is obtained?10) A company obtained financing of R$ 10,000.00 at 120% per year compounded monthly (that is, an effective rate of 10% per month). The company paid R$ 6,000.00 at the end of the first month and R$ 3,000.00 at the end of the second month. Calculate the amount to be paid at the end of the third month to settle the debt.Answer key1) 20.61% per year.2) 9.70% per month.3) Buy in cash.4) 5.13% per month.5) 1.76% per month.6) a) 5.36% per month; b) Pay in cash.7) 21.65% per month.8) a) 8.85% per month; b) It should be made.9) a) R$ 128.00; b) 12.74% per half-year.10) R$ 2,750.00
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