IntroductionIn the previous lessons we saw how a set of values can be turned into another, equivalent set for comparison purposes, especially by bringing all the values to the present date.In practice, these sets usually have well-defined characteristics: periodicity, uniformity, growth or decline according to some mathematical law. They are called sequences of cash flows, and the values may be either payments or receipts.Uniform sequence and its present valueConsider the values R1, R2, R3, …, Rn, occurring on dates 1, 2, 3, …, n respectively (the unit of time may be a month, a half-year, a year, etc.). We say that this set is a uniform sequence if all the values are equal to the same R.The present value of this sequence on date 0 (that is, one period before the first payment), at an interest rate i in the same unit of time, is:\[ V = \frac{R}{(1+i)^{1}} + \frac{R}{(1+i)^{2}} + ... + \frac{R}{(1+i)^{n}} \]\[ V = R \left[\frac{1}{(1+i)^{1}} + \frac{1}{(1+i)^{2}} + ... + \frac{1}{(1+i)^{n}}\right] \]The value in brackets could be computed by adding term by term, but for a large n that would take a long time.We can simplify it by noticing that the expression in brackets is the sum of the terms of a geometric progression (GP) whose first term and common ratio are equal:\[ a_1 = q = \frac{1}{1+i} \]Elementary mathematics teaches us that the sum of the first n terms of a GP with a ratio different from 1 is:\[ S = \frac{a_1 \cdot (q^n - 1)}{q - 1} \]Substituting the first term and the ratio above and simplifying, we get:\[ S = \frac{(1 + i)^n - 1}{(1 + i)^n \cdot i} \]Therefore, the present value of the uniform sequence is:\[ V = R \left[\frac{(1 + i)^n - 1}{(1 + i)^n \cdot i}\right] \]The factor in brackets is called the present value factor of a uniform series (or annuity factor). Note: the formula assumes the first payment occurs one period after date 0 (an ordinary annuity, that is, with no down payment).ExamplesA household appliance is sold on credit in four equal monthly payments of R$ 550.00 each, the first due one month after the purchase. If the store finances it at an interest rate of 5% per month, what is its cash price?n = 4i = 5% per monthR = 550\[ V = R \left[\frac{(1 + i)^n - 1}{(1 + i)^n \cdot i}\right] \]\[ V = 550 \left[\frac{(1 + 0.05)^{4} - 1}{(1 + 0.05)^{4} \cdot 0.05}\right] = 550 \cdot 3.545951 = 1{,}950.27 \]The cash price is R$ 1,950.27.A used car is sold for R$ 30,000.00 in cash, but it can also be bought on credit in 12 equal monthly installments, the first due one month after the purchase. Given that the financing interest rate is 2% per month, find the value of each installment.V = 30,000n = 12i = 2% per month\[ 30{,}000 = R \left[\frac{(1 + 0.02)^{12} - 1}{(1 + 0.02)^{12} \cdot 0.02}\right] \]\[ 30{,}000 = R \cdot 10.575341 \]\[ R = \frac{30{,}000}{10.575341} = 2{,}836.79 \]Each installment is R$ 2,836.79.A plot of land is sold in four equal monthly installments of R$ 150,000.00 each, the first one paid as a down payment. If the financing rate is 4% per month, what is the cash price?Here the first installment is paid on date 0, so it is already at present value. The other three form an ordinary uniform sequence:down payment = 150,000R = 150,000n = 3i = 4% per month\[ V = \text{down payment} + R \left[\frac{(1 + i)^n - 1}{(1 + i)^n \cdot i}\right] \]\[ V = 150{,}000 + 150{,}000 \left[\frac{(1 + 0.04)^{3} - 1}{(1 + 0.04)^{3} \cdot 0.04}\right] \]\[ V = 150{,}000 + 150{,}000 \cdot 2.775091 = 566{,}263.65 \]The cash price is R$ 566,263.65.A store sells a phone in 6 monthly installments of R$ 200.00, the first one month after the purchase, with interest of 3% per month. What is the equivalent cash price?Answer: R$ 1,083.44 — the present value factor for n = 6 and i = 3% is 5.417191, and 200 × 5.417191 = 1,083.44. R$ 1,200.00 is the plain sum of the installments, which ignores interest; R$ 1,115.94 would be the price if the first installment were paid at the time of purchase.The same phone is now sold in 6 installments of R$ 200.00 with the first one as a down payment (3% per month). The equivalent cash price becomes:Answer: Higher, because each installment is paid one month earlier and is discounted for one month less — the value becomes 200 + 200 × 4.579707 = R$ 1,115.94, which is the previous value multiplied by 1.03. The down payment does reduce the amount financed, but what is being compared here is the value of the whole sequence on the purchase date.Finding the rate: the IRR of a uniform sequenceWhen we know the present value, the installment and the number of installments, but not the rate, the question becomes: what rate makes what is paid today equal to the present value of what is received? That rate is the internal rate of return (IRR). There is no closed formula to isolate i, so it is found by trial and error (or with a financial calculator or a spreadsheet, using the RATE or IRR function).What is the internal rate of return of an investment of 5,000 reais that then pays 12 monthly installments of 500 reais?investment = 5,000R = 500n = 12We look for the rate i that makes the net present value zero:\[ NPV = -5{,}000 + 500 \left[\frac{(1 + i)^{12} - 1}{(1 + i)^{12} \cdot i}\right] \]Trying a few rates:Rate triedNet present value1%627.542%287.673%-23.00The NPV changes sign between 2% and 3%, so the IRR lies in that interval. It is not at the midpoint (2.5%): since the NPV at 3% is much closer to zero, the IRR is much closer to 3%. A linear interpolation gives a good estimate:\[ IRR \approx 2\% + \frac{287.67}{287.67 + 23.00} \cdot 1\% \approx 2.93\% \]The exact value, obtained with the spreadsheet's IRR function, is 2.92% per month.An investment has an NPV of +287.67 at 2% per month and of −23.00 at 3% per month. What is the best estimate of the IRR?Answer: 2.93% per month — the IRR is where the NPV equals zero; since −23.00 is much closer to zero than +287.67, it is close to 3%: 2% + 287.67 / 310.67 × 1% ≈ 2.93%. The midpoint, 2.50%, ignores the size of each NPV.Exercises1) Find the cash price of a car financed at 3% per month, with 10 monthly installments of R$ 4,500.00 each, the first due one month after the purchase.2) A car is sold for R$ 40,000.00 in cash or in three equal monthly installments with no down payment. What is the value of each installment if the financing interest rate is 1.9% per month?3) A boat is sold for R$ 16,000.00 in cash or with a 20% down payment plus four equal monthly installments. What is the value of each installment if the interest rate is 2% per month?4) A motorcycle is sold in five monthly installments of R$ 2,000.00 each, the first one as a down payment. What is the cash price if the financing interest rate is 2.5% per month?5) A microcomputer is sold for R$ 2,500.00 in cash or in four equal monthly installments, the first one as a down payment. What is the value of each installment if the interest rate is 2.2% per month?6) An apartment with a cash price of R$ 100,000.00 is sold on credit with a 30% down payment and the balance in 100 equal monthly installments, later adjusted for inflation. What is the value of each installment before the adjustment if the financing interest rate is 1% per month?7) The monthly rent of an apartment is R$ 2,000.00, adjusted every six months, and each rent is due at the end of the month it refers to. If the tenant, when signing the lease, wants to pay the first six months in advance, how much must be paid if the interest rate is 2.3% per month?8) A company has to decide whether to buy a machine for R$ 980,000.00 in cash or in 12 monthly installments of R$ 99,630.00 paid at the end of each month. Given that the rate it can earn on investments ranges between 2% and 3% per month, which is the better alternative for the company?9) A bank grants a company a loan of R$ 60,000.00 to be repaid in four equal monthly installments of R$ 16,850.00 each, paid at the end of each month. What is the interest rate of the loan?10) The cash price of a sound system is R$ 820.00. On credit, it can be paid in five equal monthly installments of R$ 198.00 each, paid at the end of each month. What is the monthly interest rate of the financing?11) A large company takes a loan from a bank at 3.5% per month. The amount borrowed is R$ 10,000.00, to be repaid in 24 equal monthly installments, the first due one month after the loan, and the bank charges a credit opening fee equal to 2% of the amount borrowed. Calculate the value of each installment and the rate effectively paid by the company.Answers1) R$ 38,385.912) R$ 13,843.183) R$ 3,361.584) R$ 9,523.955) R$ 645.556) R$ 1,110.607) R$ 11,090.328) Cash: even at 3% per month, the present value of the installments (R$ 991,717.42) is higher than the cash price.9) 4.82% per month10) 6.63% per month11) R$ 622.73; 3.69% per month
ردود الفعل
0
0 تعليقات
كن أول من يعلق
لا شيء هنا بعد.