IntroductionInternal rate of return is an interest rate that makes the sum of the cash flows in the present equal to zero.We consider the following problem: given a set of positive values y1, y2, y3,…, yn, on dates 1, 2, 3,…, t, respectively, and an investment at time 0, determine the interest rate i such that the sum of the items at present value is zero. The meaning of making the sum equal to zero shows that the investment amount will generate the given cash flow at the interest rate found.In the case of companies, the internal rate of return is extremely important because it can serve as an evaluator of an investment. An investment is expected to have a rate of return above the savings account yield and preferably above investment in Treasury bonds, since these are extremely safe.Calculation of the internal rate of returnBasically, the problem consists of finding the value of i such that:\[ \text{investment} = y_0 + \frac{y_1}{(1+i)^{1}} + \frac{y_2}{(1+i)^{2}} + ... +\frac{y_t}{(1+i)^{t}} \]Since several investments can be made and at different times, the best way to express the internal rate of return would be:\[ 0 = -\text{investment}_1 -\text{investment}_2 + ... \]\[... + y_0 + \frac{y_1}{(1+i)^{1}} + \frac{y_2}{(1+i)^{2}} + ... +\frac{y_t}{(1+i)^{t}} \]A numerical example:timecash flow0- 5.00012.00022.00032.000We find the IRR from:\[ 0 = -5.000 + \frac{2.000}{(1+i)^{1}} + \frac{2.000}{(1+i)^{2}} + \frac{2.000}{(1+i)^{3}} \]It is necessary to solve a polynomial equation of degree n, which, in general, cannot be done by classical methods (these equations generally do not have a closed-form solution) and is preferably done with the HP12C. Since we will not use a calculator in this course, what we will see is an approximate method of solution.Several IRRs are tested:Interest rateNet Present Value5%44610%-26Since the calculation of the net present value (NPV) was positive for 5% and negative for 10%, it can be seen that the IRR must be between 5% and 10%. It is then solved with linear interpolation, which consists of assuming that the arc generated between the rates is a line segment. Thus, using methods of classical geometry, i can be determined:\[ \frac{446}{i - 5\%}=\frac{26}{10\% - i}\]\[ 44,6 - 446i = 26i - 1,3\]\[ 472i = 43,3 = 9,17\% \]Assuming linearity, the IRR is equivalent to 9.17%. Another more practical formula for calculating it would be simply to use the midpoint, which in this case would be 7.5%.If we were a calculator, we could advance this method again. Interest of 7.5% means an NPV of 201 reais, positive. Since interest of 10% creates an NPV of -26 reais, negative, it is imagined that the IRR is between the two, in this case 8.75%, which generates an NPV of 85 reais.We can apply as many iterations as we want until we get very close to the real value of the IRR. In fact, this is what calculators do, but with several iterations in a few milliseconds due to their nature.ExamplesConsider the following cash flow, calculate its internal rate of returntimecash flow0- 40115220322Several IRRs are tested:Interest rateNet Present Value5%11,4310%6,6915%2,6320%-0,88The IRR is the interest rate that makes the NPV zero. And it is between 15% and 25%, therefore:\[ \frac{2,63}{i - 15\%}=\frac{0,88}{20\% - i}\]\[ i = 18,75\%\]Or we can simply use the midpoint and the IRR will 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. Thus, denoting by i the rate sought, we must have:\[ 0 = 500 - \frac{1.500}{(1+i)^{1}} + \frac{1.000}{(1+i)^{2}} \]\[ 0 = 500·(1+i)^{2} - 1.500·(1+i)^{1} + 1.000 \]\[ 0 = (1+i)^{2} - 3·(1+i)^{1} + 2 \]\[ 0 = i^{2} + 2·i + 1- 3·i -3 + 2 \]\[ 0 = i^{2} - i = i(1 + i) \]\[ i = 0 \text{ or } i = 1\]The IRR is either 0% or 100%. And if we analyze it carefully, the NPV will be positive for rates above 100% and negative for rates between 0% and 100%.Exercises1) A bank grants a company a loan of R\$ 600,000.00 to be paid in three installments due in one, two, and three years, with amounts 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 cash at R\$ 13,000.00 or, alternatively, that amount can be financed with a R\$ 3,000.00 down payment, 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 funds to buy in cash and can earn a return on their investments at a rate of 2% per month, what is their best alternative?4) A machine is sold for R\$ 35 thousand in cash or on credit in six monthly installments, with no down payment, with each of the first three being R\$ 6 thousand and each of the last three being R\$ 8 thousand. What is the interest rate of this financing?5) A plot of land is sold for cash for R\$ 60,000.00 or, alternatively, on credit in six equal monthly installments of R\$ 11,000.00 each, with the first due three months after the purchase. What is the financing interest rate?6) A raw material is sold for R\$ 900.00 in three equal monthly installments, with no surcharge, the first being paid as a down payment. If payment is made in cash, there is a 5% discount on the sale price.a) What is the financing interest rate?b) What is the best payment alternative for the buyer if they can invest their money at a rate of 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 required rate of return is 6% per month, verify whether the investment should be made.9) Assuming that the dividends provided by a stock are R\$ 12.00 and R\$ 14.00, six and 12 months from now, and that the estimated sale value is R\$ 125.00, 12 months from now, calculate:a) The amount to be paid today for this stock so that the investor obtains a return rate of 9% per semester.b) If the investor pays R\$ 120.00 for the stock today, what return rate is obtained?10) A company obtained financing of R\$ 10,000.00 at a rate of 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% p.a.2) 9.70% p.m.3) Buy in cash.4) 5.13% p.m.5) 1.76% p.m.6) a) 5.35% p.m.; b) Cash.7) 21.65% p.m.8) a) 8.85% p.m.; b) It should be made.9) a) R$ 128.00; b) 12.74% per semester.10) R\$ 2,750.00
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