IntroductionOne of the basic premises of financial mathematics is that we cannot compare cash flows that occur on different dates, because the value of money changes over time.So two cash flows can only be compared when they are on the same date. Since we often need to compare amounts due on different dates, we bring all of them to today's date and only then compare them.Today's value equivalent to a cash flow occurring at date t is called its present value (in Portuguese-language exams you will also see "valor atual"; both names mean the same thing).Equivalence of two amounts of moneyConsider two amounts of money, x and y, separated by t periods of time: the first at date 0 and the second at date t. We say that x and y are equivalent, at a compound interest rate i, if:\[ x · (1+i)^{t} = y \]In other words, x is equivalent to y if, by investing x until date t, the resulting amount equals y. We also say that x is the present value of y. To find x, simply divide: x = y / (1 + i)^t. This operation of bringing a value from the future back to today is called discounting it.ExampleAt a compound interest rate of 2% per month, R$ 1,500,000.00 (R$ is the Brazilian real) three months from now is equivalent to how much today?If x is today's value, we must have:\[ x · (1+0.02)^{3} = 1500000 \]\[ x = \frac{1500000}{(1+0.02)^{3}} \]\[ x = 1413483.50 \]That is, at 2% per month, receiving R$ 1,500,000.00 three months from now is the same as receiving R$ 1,413,483.50 today.At a compound interest rate of 10% per month, what is the value today of a payment of R$ 1,210.00 due two months from now?Answer: R$ 1,000.00 — x = 1,210 / 1.1² = 1,210 / 1.21 = 1,000. The option R$ 1,008.33 comes from discounting with simple interest (1,210 / 1.2), and R$ 1,100.00 discounts only one month.Present value of a set of amountsConsider the values y₀, y₁, y₂, …, yₜ at dates 0, 1, 2, …, t, respectively. The present value at date 0 (or simply the present value) of this set, at an interest rate i, is the sum of the values equivalent to these amounts at date 0.These values may be payments on a debt or income to be received.\[ x = y_0 + \frac{y_1}{(1+i)^{1}} + \frac{y_2}{(1+i)^{2}} + ... +\frac{y_t}{(1+i)^{t}} \]Which can be written as a summation:\[ x = \sum_{k=0}^{t}\frac{y_k}{(1+i)^{k}} \]ExamplesA company expects to pay R$ 2,000.00 one month from now, R$ 3,000.00 two months from now and R$ 5,000.00 three months from now. How much must it invest today, at compound interest of 1.5% per month, to cover these expenses, ending with a zero balance after the last payment?The amount to be invested today is, by definition, the present value of this set, that is:\[ x = \frac{2000}{(1+0.015)^{1}} + \frac{3000}{(1+0.015)^{2}} + \frac{5000}{(1+0.015)^{3}} = 9664.01 \]Let us check it month by month:Investing R$ 9,664.01 for one month gives an amount of 9,664.01 × 1.015 = 9,808.97.Withdrawing R$ 2,000.00 for the first payment leaves a balance of R$ 7,808.97.Investing R$ 7,808.97 for another month gives an amount of 7,808.97 × 1.015 = 7,926.10.Withdrawing R$ 3,000.00 for the second payment leaves a balance of R$ 4,926.10.Investing R$ 4,926.10 for another month gives an amount of 4,926.10 × 1.015 = 4,999.99, that is, R$ 5,000.00 (the one-cent difference comes from rounding).Withdrawing R$ 5,000.00 for the third payment, the balance will be zero.A store sells a sofa set for a down payment of R$ 500.00 plus three monthly installments of R$ 800.00 each. If a buyer can invest money at 1.2% per month, how much must the buyer have today to make the purchase?\[ x = 500 + \frac{800}{(1+0.012)^{1}} + \frac{800}{(1+0.012)^{2}} + \frac{800}{(1+0.012)^{3}} = 2843.53 \]Therefore, if the buyer has at least R$ 2,843.53 today, the sofa set can be bought: the buyer pays the down payment and invests the rest, which will earn enough for the three installments.A television is sold for R$ 1,500.00 in cash or, alternatively, in three equal monthly installments with no down payment. What is the value of each installment if the store wants to earn 3% per month on the financing?\[ 1500 = \frac{x}{(1+0.03)^{1}} + \frac{x}{(1+0.03)^{2}} + \frac{x}{(1+0.03)^{3}} \]\[ 1500 = 0.9709x + 0.9426x + 0.9151x = 2.8286x \]\[ x = \frac{1500}{2.8286} = 530.30 \]Each installment will be R$ 530.30.A debt will be paid with R$ 1,100.00 one month from now and R$ 1,210.00 two months from now. At compound interest of 10% per month, what is the present value of this debt?Answer: R$ 2,000.00 — 1,100 / 1.1 = 1,000 and 1,210 / 1.1² = 1,000, for a total of 2,000. The option R$ 2,310.00 is the plain sum of the installments, which ignores the time value of money; R$ 2,100.00 discounts both installments by only one month.Equivalent sets of amountsTwo sets of amounts are said to be equivalent, at a compound interest rate i, if their present values are equal.ExampleA store sells a refrigerator on the following terms: a down payment of R$ 1,000.00 plus one installment of R$ 1,200.00 after one month. A customer proposes a down payment of R$ 600.00 plus two equal monthly installments, the first due one month after the purchase. If the store finances at 3% per month, what must each installment be for the two payment plans to be equivalent?Present value of the store's plan:\[ x_1 = 1000 + \frac{1200}{(1+0.03)^{1}} = 2165.05 \]Present value of the customer's plan, which must be equal:\[ x_2 = 600 + \frac{y}{(1+0.03)^{1}} + \frac{y}{(1+0.03)^{2}} = 2165.05 \]\[ 0.970874y + 0.942596y = 1565.05 \]\[ 1.913470y = 1565.05 \]\[ y = 817.91 \]Each installment must be R$ 817.91.A product costs R$ 1,000.00 in cash. The store accepts a down payment plus one installment of R$ 550.00 a month later, charging compound interest of 10% per month. What must the down payment be for the two options to be equivalent?Answer: R$ 500.00 — the present value of the installment is 550 / 1.1 = 500, so the down payment is 1,000 - 500 = 500. The option R$ 450.00 comes from subtracting the installment without discounting it (1,000 - 550).Comparing payment options by present valueWe are often offered several ways to pay for a good or service: cash, a down payment, installments.If all amounts are brought to the same date, the comparison becomes easy. The most practical way is to calculate the present value of each option, using as the interest rate the one we can earn by investing our money. The option with the lowest present value is the best for the payer.ExampleA house is sold for R$ 326,000.00 in cash, or on credit for a down payment of R$ 90,000.00 plus three equal monthly installments of R$ 80,000.00 each, the first due one month after the down payment. Which is the best payment option for a buyer who can invest money at compound interest of 1% per month?\[ x_1 = 326000 \]\[ x_2 = 90000 + \frac{80000}{(1+0.01)^{1}}+ \frac{80000}{(1+0.01)^{2}}+ \frac{80000}{(1+0.01)^{3}} = 325278.82 \]Since x₂ is less than x₁, the best option for the buyer is paying in installments, which costs about R$ 721.18 less in today's money.A refrigerator costs R$ 1,000.00 in cash or R$ 1,100.00 in a single payment one month from now. For someone who can invest money at 5% per month, which is the better option?Answer: Pay in cash, because R$ 1,100.00 one month from now is worth about R$ 1,047.62 today — 1,100 / 1.05 = 1,047.62, more than the R$ 1,000.00 cash price. The store charges 10% for the month, and the buyer can only earn 5% by investing: investing R$ 1,000.00 would yield R$ 1,050.00, not enough to pay R$ 1,100.00.Exercises1. A company expects monthly payments of R$ 250,000.00 one, two and three months from now. What is the minimum it must invest today, at compound interest of 1.6% per month, to cover these payments?2. A TV set is sold for R$ 1,500.00 in cash, or for a 20% down payment plus two equal monthly installments. Given a compound interest rate of 6% per month, what must each installment be for the two payment plans to be equivalent?3. Solve the previous problem assuming three monthly payments in addition to the down payment.4. A set of cabinets is sold for R$ 3,000.00 in cash or with a down payment plus three monthly installments of R$ 800.00 each. If the store finances its sales at compound interest of 3.5% per month, what is the down payment?5. A suit is sold in a store for a down payment of R$ 800.00 plus one installment of R$ 400.00 after one month. A buyer proposes a down payment of R$ 200.00. Under these conditions, what must the installment, paid one month later, be for the two payment plans to be equivalent, given that the store operates at compound interest of 4% per month?6. A sound system is sold for a price P 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 3% discount on P. What is the best option for the buyer if the investment rate is 1.5% per month?7. What is the best option for the buyer: paying R$ 1,200,000.00 in 45 days or three installments of R$ 400,000.00 each at 30, 45 and 60 days after the purchase, if the compound interest rate for investments is 1.4% per month?8. If a buyer invests at 1.2% per month, what is the best option for a product advertised at R$ 10,000.00: paying in cash with a 3% discount on the advertised price, or paying with a 2% discount on the advertised price in two equal monthly installments, the first due one month after the purchase?9. A plot of land is offered for R$ 400,000.00 in cash or on credit with a 20% down payment plus two quarterly installments of R$ 164,000.00 each. If the buyer invests at 2% per month, what is the best option?10. A company can either rent a computer for R$ 500.00 per year for five years (paying at the end of each year) or buy the same computer for R$ 2,100.00. After five years the computer will have no value. Assuming the figures are in real terms and that the company invests at a real rate of 10% per year, which is better for the company, renting or buying?Answer key1. R$ 726,624.982. R$ 654.523. R$ 448.934. R$ 758.695. R$ 1,024.006. Pay in cash.7. Pay in 45 days.8. Pay in installments.9. Pay in installments.10. Rent.
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