LESSON 1 – SUMMARY – TIME VALUE OF MONEY Interest Calculation: Simple Interest Compound Interest Compound and Discount Methods: comparing values ​​ in time Analyzing the interest calculation with different compounding frequency Difference between Stated and Effective annual interest rates 1 FF - LFC/LG/LGM - 2Sem_2014/2015

INTEREST CALCULATION (1/3) There are 2 types of Interest: Simple interest (SI) Compound interest (CI) The difference between the two consists on the fact that with Simple Interest, the interest is not reinvested. 2 FF - LFC/LG/LGM - 2Sem_2014/2015

INTEREST CALCULATION (2/3) Simple Interest the interest is separated from the capital (principal), that is, it is not reinvested. 3 FF - LFC/LG/LGM - 2Sem_2014/2015 Example : Deposit of 10,000 €, 3 years with an annual interest rate of 6% Interest in 1st, 2nd and 3rd years: 10,000 x 0.06 = 600 Total Interest in 3 years: = 1,800 The Deposit is always 10,000 €

INTEREST CALCULATION (3/3) Compound Interest the interest will be added to the capital (principal), that is, each interest payment is reinvested so the interest generates interest. 4 FF - LFC/LG/LGM - 2Sem_2014/2015 Example : Deposit of 10,000 €, 3 years with an annual interest rate of 6% 1st year: Interest: 10,000 x 0.06 = 600 Compound Value at the end of the period: 10, = 10,600 € 2nd year: Interest: 10,600 x 0.06 = 636 Compound Value at the end of the period:10, = 11,236 € 3rd year. Interest: 11,236 x 0.06= Compound Value at the end of the period: 11, = 11, € Total Interest in 3 years: = 1, € The Deposit is not always 10,000 €, on the 2nd year is 10,600 ….

DISCOUNTING AND COMPOUNDING (1/4) The Compound Interest is especially suited to perform comparisons of values in time ​​. Note that the Future Value of the investment in the example can be given by the following formula: FV = 10,000 x ( ) 3 =11, € This operation represents a multi-period compounding in which the future value factor is (1+0.06) 3 5 FF - LFC/LG/LGM - 2Sem_2014/2015 Example : having today 10,000 € is equivalent to have how much within 3 years? The time value is reflected on the interest rate; the question can be posed as follows: If I apply today 10,000 €, what value will I have within 3 years? Using the 6% rate of the previous example: Value at the end of 3 years: 11, €

DISCOUNTING AND COMPOUNDING (2/4) An opposite situation would be to know the present value of a given future cash flow This operation represents a multi-period discounting in which the present value factor is 6 FF - LFC/LG/LGM - 2Sem_2014/2015 Example : In 2 years we are going to receive 20,000 €; that value is equivalent to receive how much today? The same would be to ask: how much do we have to invest today if we want to get 20,000 € in 2 years? We know the future value and we intend to compute the present value (C 0 ). Thus: 20,000 = C 0 x (1+0.06) 2

DISCOUNTING AND COMPOUNDING (3/4) Generalizing, consider : C 0 – Capital at moment 0 C n – Capital at moment n n – Number of periods r - Interest rate of the period 7 FF - LFC/LG/LGM - 2Sem_2014/2015

DISCOUNTING AND COMPOUNDING (4/4) Besides time, the power of discounting and compounding is determined by the level of the interest rate. For example, what is the present value of 10,000€ that will be received within 5, 10 or 20 years, for an interest rate of 5% or 10%? 8 FF - LFC/LG/LGM - 2Sem_2014/2015

STATED AND EFFECTIVE INTEREST RATES (1/4) The number of times the interest is computed in one operation has an important effect on the compound or present value. Example : loan for12 months, amount of 10,000 € and annual interest rate of 10% : A) interest paid quarterly B) interest paid at the end of the year Naturally, these alternatives are not equivalent. As the interest rate is the same for both alternatives, it is better to pay only at the end of the year than to pay quarterly. 9 FF - LFC/LG/LGM - 2Sem_2014/2015

STATED AND EFFECTIVE INTEREST RATES (2/4) We could ask the following: “A stated annual rate of 10%, with interest paid quarterly, is equivalent to what rate with interest paid once at the end of the year?” The answer will be given by applying the well-known concept of compound interest (compounding) quarter by quarter over a year. Quarter Interest = 10,000 x 0.1 x 3/12 = 250 Note: For simplicity of calculations, the computation is made in months, not days. 10 FF - LFC/LG/LGM - 2Sem_2014/2015

STATED AND EFFECTIVE INTEREST RATES (3/4) First, let´s see what is the interest rate for each quarter, recalling that for purpose of interest calculation it is always applied a principle of proportionality: 3 months rate: Then we compound for four periods (quarters); beginning with one money unit of principal, we get the following future value at the end of one year (4 quarters) : As the initial capital was one unit, the interest is By other words, the interest represent an annual rate of 10,38%. Thus, a stated annual rate of 10%, with quarterly interest, becomes an effective annual rate of 10,38%. In other words, it is equivalent to have a rate of 10% with quarterly interest or an annual interest rate of 10,38% with annual interest. 11 FF - LFC/LG/LGM - 2Sem_2014/2015

STATED AND EFFECTIVE INTEREST RATES (4/4) Generalizing and considering: r – Effective annual rate (EAR or AER) r (m) – Stated annual rate for m compounding frequency r m - Effective rate for the sub-period with m frequency: 12 FF - LFC/LG/LGM - 2Sem_2014/2015