IntroductionOne of the basic premises of financial mathematics is that we cannot compare cash flows in different time periods, because the value of money fluctuates over time.Therefore, two cash flows can only be compared at the same point in time. Since this creates a complication, and it is in our interest to compare flows at different times, we bring all cash flows to the present time and compare them immediately afterward.The value of a cash flow at a time t in the present time is called the present value.Equivalence of two monetary valuesLet us consider two monetary values, x and y, separated by t time periods, for example, the first on date 0 and the second on 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, when we invest x until date t, the amount obtained is equal to y. We also say that x is the present value of y.ExamplesAt a compound interest rate of 2% per month, R\$ 1,500,000.00, three months from now, is equivalent to how much today?Let x be the value today; we must have:\[ x*(1+0,02)^{3} = 1.500.000 \]\[ x = \frac{1.500.000}{(1+0,02)^{3}} \]\[ x = 1.413.483,50 \]In other words, at a rate of 2%, 1,500,000 is the same as 1,413,483.50 in present value.Present value of a set of capitalsLet us consider the values y0, y1, y2,…, yt, on dates 0, 1, 2,…, t, respectively. We call the present value on date 0 (or simply present value) of this set, at an interest rate i, the sum of the equivalent values of these capitals on date 0.The monetary values referred to may be debt payments or income 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 represented by the summation:\[ x = \sum\frac{y_t}{(1+i)^{t}} \]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 should it invest today, at compound interest at a rate of 1.5% per month, to meet these expenses, leaving a zero balance after the last payment?The amount that should be invested today is, by definition, the present value of this set, that is:\[ x = \frac{2000}{(1+0,015)^{1}} + \frac{3000}{(1+0,015)^{2}} + \frac{5000}{(1+0,015)^{3}} = 9.664,01 \]Note that:- By investing R$ 9,664.01 for one month, we will obtain the amount of 9,664.01(1.015) = 9,808.97.- By withdrawing R$ 2,000.00 for the first payment, a balance of R\$ 7,808.97 will remain.- By investing R$ 7,808.97 for one more month, we will obtain the amount of 7,808.97(1.015) = 7,926.11.- By withdrawing R$ 3,000.00 for the second payment, a balance of R\$ 4,926.11 will remain.- By investing R$ 4,926.11 for one more month, we will obtain the amount of 4,926.11(1.015) = 5,000.00.- By withdrawing R$ 5,000.00 for the third payment, the balance will be zero.A store sells a sofa set for a R\$ 500.00 down payment, plus three monthly installments of R\$ 800.00 each. If a buyer can invest their money at a rate of 1.2% per month, how much should they have available today in order to make the purchase?\[ x = 500 + \frac{800}{(1+0,012)^{1}} + \frac{800}{(1+0,012)^{2}} + \frac{800}{(1+0,012)^{3}} = 2.843,53 \]Therefore, if the buyer has R$ 2,843.53 available today (at least), they will be able to buy the sofa set.A television is sold for R\$ 1,500.00 in cash or, alternatively, on credit in three equal monthly installments, with no down payment. What is the value of each installment if the store intends to earn 3% per month on the financing?\[ 1500 = \frac{x}{(1+0,03)^{1}} + \frac{x}{(1+0,03)^{2}} + \frac{x}{(1+0,03)^{3}} \]\[ 1500 = 0,9709x + 0,9426x + 0,9151x = 2,8286x \]\[ x = \frac{1500}{2,8286} = 530,30 \]Sets with equivalent valuesWe say that two sets are equivalent, at a compound interest rate i, if their present values are equal.ExamplesA store sells a refrigerator under the following conditions: a down payment of R\$ 1,000.00 plus one installment of R$ 1,200.00 after one month. A customer proposes paying 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 an interest rate of 3% per month, what is the value of each installment so that the two payment methods are equivalent?\[ x_1 = 1000 + \frac{1200}{(1+0,03)^{1}} = 2.165,05 \]\[ x_2 = 600 + \frac{y}{(1+0,03)^{1}} + \frac{y}{(1+0,03)^{2}} = 2.165,05 \]\[ 0,97087y + 0,942596y = 1.565,05 \]\[ 1,91347y = 1.565,05 \]\[ y = 817,91 \]Analysis of payment alternatives by present valueWe are frequently placed in situations in which several alternatives are offered for the payment of a good or service.If the monetary values are calculated on the same date, we can easily compare them. This can be done by comparing the present values of each alternative.In this case, using the compound interest rate at which we can invest the money, it is possible to calculate the present value of each alternative.ExamplesA house is sold for R\$ 326,000.00 in cash, or on credit for a R\$ 90,000.00 down payment plus three equal monthly installments of R\$ 80,000.00 each, the first due one month after the down payment. What is the best payment alternative for a buyer who can invest their money at a compound interest rate of 1% per month?\[ x_1 = 326.000 \]\[ x_2 = 90.000 + \frac{80.000}{(1+0,01)^{1}}+ \frac{80.000}{(1+0,01)^{2}}+ \frac{80.000}{(1+0,01)^{3}} = 325.278,82 \]Since x2 < x1, it follows that the best alternative for the buyer is payment on credit.Exercises1. A company expects monthly payments of R$ 250,000.00 one, two, and three months from now. How much should it invest today, at minimum, at a compound interest rate of 1.6% per month, to meet these payments?2. A TV set is sold for cash for R$ 1,500.00, or with a 20% down payment plus two equal monthly installments. Knowing that the compound interest rate is 6% per month, what is the value of each installment so that the two forms of payment are equivalent?3. Solve the previous problem assuming there are 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 a compound interest rate of 3.5% per month, what is the value of the down payment?5. A suit is sold in a store for R$ 800.00 down plus one installment of R$ 400.00 after one month. A buyer proposes to pay R$ 200.00 down. Under these conditions, what is the value of the monthly installment so that the two forms of payment are equivalent, knowing that the store operates at a compound interest rate of 4% per month?6. A stereo system is sold for a price P in three equal monthly installments, with no increase, the first being paid as a down payment. If payment is made in cash, there will be a 3% discount on P. What is the best alternative for the buyer, if the investment interest rate is 1.5% per month?7. What is the best alternative for the buyer: to pay R$ 1,200,000.00 in 45 days or three installments of R$ 400,000.00 each, in 30, 45, and 60 days from the purchase, if the compound interest rate for investment is 1.4% per month?8. If a buyer invests their funds at a rate of 1.2% per month, what is their best alternative for a product advertised at the price of R$ 10,000.00: pay in cash with a 3% discount on the advertised price or pay with a 2% discount on the advertised value, in two equal monthly installments, the first due one month after the purchase?9. A plot of land is offered for sale 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 their funds at a rate of 2% per month, what is their best alternative?10. A company has the option of renting a computer for R$ 500.00 per year, for five years (with payment at the end of each year), or buying the same computer for R$ 2,100.00. After five years the computer will have no value. Assuming the data are in real values and that the company invests its funds at a real rate of 10% per year, what is better for the company, to rent or to buy?Answer Key1. R$ 726,624.982. R$ 654.523. R$ 448.934. R$ 758.695. R$ 1,024.006. Cash.7. Pay in 45 days.8. Pay on credit.9. Pay on credit.10. Rent.
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