04 Annuities and Perpetuities - Matem\u00e1ticas Financieras I ACT-22305-001 M.A Fernando Jes\u00fas Mart\u00ednez Eissa Investment Projects Firms\u2019 Valuation

04 Annuities and Perpetuities - Matemu00e1ticas...

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Investment Projects & Firms’ Valuation Fernando Martínez, MBA, CRM MMXX-II
IV. Annuities & Perpetuities Fernando Jesús Martínez Eissa, MBA MMXX-II School of Economic and Business Sciences
An arithmetic sequence , is a series of terms which difference between two consecutive terms is a constant, for example: 1. 6,11,16,21,26,31,36,41. The constant difference ( d ) is: 5 2. 54,50,46,42,38,24,30,26,22,18. The constant difference d =-4 Arithmetic Sequences To build a 5-term sequence considering a as its first term and d , its difference, then the sequence shall be: d a d a d a d a a 4 , 3 , 2 , , + + + + We can observe that for an “ n ” terms sequence, the last term could be written as: ࠵? = ࠵? + ࠵? − 1 ࠵?
The sum of an arithmetic sequence is: ) * +,*- ./0 1 ࠵? = ࠵? 2࠵? + ࠵? − 1 ࠵? 2 = ࠵? 2 ࠵? + ࠵? Arithmetic Sequences
Example 1 Find 15 th term and the addition of the first 10 terms of the arithmetic sequence : 6, 11, 16, 21,… Solution We have: a =6, d =5 and n =15. Also we know: ( ) ( ) 285 51 6 5 2 = + = + = u a n s To add the first 10 terms we need to find the 10 th term, applying the as we do before: a =6, d =5 and n =10, therefore n =51. Therefore the 15 th term is: Then, the sum of the first 10 terms is: ࠵? = ࠵? + ࠵? − 1 ࠵? ࠵? = 6 + 15 − 1 5 ࠵? = 76 Arithmetic Sequences
Example 2 Juan borrows $4,000 and agrees to pay $400 each month at 2.5% simple interest rate. Find the total amount of interest to be repaid. Solution Mes Pagos Saldo Insoluto Intereses 4,000 100 1 400 3,600 90 2 400 3,200 80 3 400 2,800 70 4 400 2,400 60 5 400 2,000 50 6 400 1,600 40 7 400 1,200 30 8 400 800 20 9 400 400 10 10 400 0 0 550 The amount of interest paid is: (4,000)*2.5%=100; (3,600)*2.5%=90; (3,200)*2.5%=80. It is also known that Juan will pay 10 installments (4,000/400=10). Therefore, the sum of interest paid is: ( ) [ ] [ ] 550 90 200 5 1 2 2 = - = - + = d n a n s Arithmetic Sequences
A geometric sequence , is a series of terms that can be obtained multiplying the previous term by one fix number called a rate, for example: 1. 4,-8,16,-32,64,-128,256,-512. The rate ( r ) is: -2 2. 729,486,324,216,144,96,64. The rate ( r ) is: r =2/3 8 7 6 5 4 3 2 , , , , , , , , ar ar ar ar ar ar ar ar a To build a 9-term geometric sequence, considering “ a as the first term and “ r as the increasing rate, we have: We can observe that for an “ n-terms ” the last term could be expressed as: ( ) ( ) i ar u n 1 - = Geometric Sequences
Let s be the sum of the geometric sequence: Then, 1 3 2 , , , , , - n ar ar ar ar a !

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