The courseContents: introduction to financial mathematics, percentages, simple and compound interest, simple and compound discounts, equivalence of capitals, annuities (series of payments) and loan amortization.Reading for the studentHAZZAN, Samuel; POMPEO, José Nicolau. Matemática financeira. Saraiva, 2007.Reading for the teacherGIMENES, Cristiano Marchi. Matemática financeira com HP 12C e Excel. 2006.CÉSAR, Benjamin. Matemática Financeira: teoria e 700 questões. Ed. Campus, Rio de Janeiro–RJ, 5th edition, 2005.CARVALHO, Sérgio; CAMPOS, Weber. Matemática Financeira Simplificada para Concursos. 2010.Introduction: the time value of moneyFinancial Mathematics studies the value of MONEY over TIME, in the setting of investments and loan payments. The central idea is simple: R$ 100 (100 Brazilian reais) today is not worth the same as R$ 100 a year from now.Do you remember how much these supermarket products cost a few decades ago? Look at this old Brazilian flyer:The purchasing power of money changes over time, which makes it hard to compare amounts from decades ago with today's amounts. It usually falls: the same banknote buys less and less, because prices go up. This general rise in prices is called inflation. Inflation is not necessarily bad; low and stable inflation is normal in an economy. If your salary rises as much as inflation, or more, your purchasing power is preserved.Low inflation is in fact healthier than deflation (a general fall in prices). With deflation, each unit of money buys more; debtors see the weight of their debts grow, and unchanged salaries become expensive for companies, which tend to cut wages or jobs, causing dissatisfaction and conflict. Moreover, deflation usually comes from weak consumption: people postpone purchases expecting lower prices, which can prolong an economic crisis.On the other hand, very high inflation is also a problem. Hyperinflation brings uncertainty and prevents companies and families from planning. Zimbabwe went through hyperinflation in 2007 and 2008, when it even issued the 100 trillion dollar note below:In a year when inflation was 6%, your salary rose by 4%. What happened to your purchasing power?Answer: It fell, because prices rose more than the salary — what matters is comparing the raise with inflation. Since 4% is less than 6%, the same salary buys fewer things than the year before, even though it grew in money terms.Capital, interest, interest rate and amountWe call CAPITAL any sum of money that a person (an individual or a company) lends to another for a certain time. It can also be seen as the initial value of the transaction.Since the lender gives up using the money during that time, loses purchasing power to inflation and runs the risk of not being paid back (default), the concept of INTEREST arises.INTEREST can be defined as the cost of the loan (for the borrower) or the payment for the use of capital (for the lender).The interest rate is the interest produced in a given unit of time, expressed as a PERCENTAGE of the capital. For example, 2% per month (p.m.) or 22% per year (p.a.).Finally, the sum of the capital and the interest at the end of the transaction is the AMOUNT (or future value), the final value.You lent R$ 2,000 and received R$ 2,300 back one year later. What are the interest, the amount and the annual interest rate?Answer: Interest of R$ 300, amount of R$ 2,300 and rate of 15% p.a. — the amount is the final value (2,300), the interest is the difference (2,300 - 2,000 = 300) and the rate is the interest divided by the capital: 300 / 2,000 = 0.15 = 15%. The 13% comes from dividing 300 by 2,300, which is the amount, not the capital.PercentagePer cent means divided by 100: 25% is the same as 25/100 = 0.25. The % symbol helps you remember this, since it looks like a 100 with its digits jumbled (historically, it came from an abbreviation of the Italian expression per cento).The table above uses the Brazilian notation, with a decimal comma: 0,072 means 0.072.Any number can be written as a percentage; just multiply it by 100: 0.5 is 50%, 3.2 is 320%, and even the number π is about 314.16%. Percentages are used because they make proportions easy to see, since they all share the same base, 100. Which of the sentences below is easier to grasp?\[{1\over2} \text{ is twice } {1\over4}\]\[50\% \text{ is twice } 25\%\]A percentage expresses a relation between two values, which allows us to compare situations of different sizes, something the amount of money alone does not allow. Invest 1,000 reais and earn 100, or invest 500 reais and earn 60? In money terms, the first earns more; in percentage, the first yields 10% and the second, 12%. The percentage makes it clear which investment is better.A practical way to calculate a percentage of a number is the rule of three. For example, 2% of 300:\[{2\over100}={x\over300}\]\[x={2 \cdot 300\over100}=6\]Or, directly, by multiplying by the decimal form: 0.02 · 300 = 6.Investment A earned R$ 150 on R$ 1,200 invested; investment B earned R$ 90 on R$ 600. Which earned more, in percentage terms?Answer: B, with 15%, against 12.5% for A — 150 / 1,200 = 0.125 = 12.5% and 90 / 600 = 0.15 = 15%. A earned more money, but on capital twice as large; this is exactly the comparison percentages are for.ExercisesE1) Convert to a percentage (the images use the Brazilian decimal comma):E2) Convert to a whole number or a decimal, as appropriate:E3) A capital of R$ 8,000 is invested for one year at a rate of 22% p.a. (22% per year).a) What is the interest?b) What is the amount?Answer keyE1) a) 5%; b) 7%; c) 6%; d) 8%; e) 9%; f) 60%; g) 70%; h) 30%; i) 40%; j) 10%; k) 1%; l) 11%; m) 20%; n) 3%; o) 40%; p) 10%; q) 0.1%; r) 2%; s) 30%; t) 80%; u) 4%; v) 63%; w) 84%; x) 840%; y) 52%; z) 71%; aa) 47%; bb) 124.8%; cc) 75.4%; dd) 592.6%; ee) 710.1%; ff) 800%.E2) a) 0.05; b) 0.06; c) 0.07; d) 0.04; e) 0.02; f) 0.02; g) 0.4; h) 0.7; i) 0.8; j) 0.3; k) 0.1; l) 0.4385; m) 0.002; n) 0.0921; o) 0.005; p) 0.001; q) 0.00666; r) 0.00212121; s) 0.2105.E3) a) I = 0.22 · 8,000 = R$ 1,760; b) A = 8,000 + 1,760 = R$ 9,760.
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