Understanding Annuities. Some Algebraic Terminology.

Transcription

1 Understandng Annutes Ma 162 Sprng 2010 Ma 162 Sprng 2010 March 22, 2010 Some Algebrac Termnology We recall some terms and calculatons from elementary algebra A fnte sequence of numbers s a functon of natural numbers 1, 2,, n Thus, the formula a k = 2k + 1 for k = 1, 2,, 10 descrbes a sequence 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 We may also let a sequence run out to nfnty as n 1, 1 2, 1 3,, 1 n, Here the sequence can also be descrbed where n = 1, 2, as 1 n A sequence may also be called a progresson Two progressons are mportant, the Arthmetc Progresson and the Geometrc Progresson

2 AP and GP Arthmetc progresson: Ths s a sequence whch has a startng number a and successve numbers are obtaned by addng a number d (called the common dfference) Thus, ts n-th term s a + (n 1)d Example: Take a = 3, d = 4 The sequence s 3, 7, 11, 15, 19,, 3 + 4(n 1), The n-th term can be better wrtten as 4n 1 Geometrc progresson: The geometrc progresson has a startng number a and successve terms are obtaned by multplyng by a common rato r Thus, ts n-th term s ar (n 1) Example: Take a = 2 and r = 1 2 The sequence s: 2, 1, 1 2, 1 4,, 2, Note that the n-th term s better 2 (n 1) 1 wrtten as 2 (n 2) Arthmetc Seres We need the formula for the sum of terms n AP The sum of the AP a, a + d, a + 2d,, a + (n 1)d s called an Arthmetc Seres and s wrtten as (a + (k 1)d) Its sum s gven by the formula: a + a + (n 1)d a + (k 1)d = n 2 ( = n a + n 1 ) d 2 An alternate way to remember t s ( number of terms ) ( average of the frst and the last term )

3 Geometrc Seres We need the formulas for the sum of terms n GP The sum of the GP a, ar, ar 2,, ar (n 1) s called a Geometrc Seres and s wrtten as Its sum s gven by the formula: (ar (k 1) ) = a ( r n ) 1 = a r 1 (ar (k 1) ) ( 1 r n 1 r If r < 1, then we can make sense of the formula even for an nfnte GP and wrte; ) (ar(k 1) ) = a ) ( 1 1 r Basc Annuty What s an annuty? An annuty s a combnaton of nvestments (or payments) For convenence,we assume the followng condtons whch are vald n most practcal stuatons A fxed amount R s nvested exactly m tmes a year Ths gves exactly m perods n a year and each s 1 m-th part of the year Each payment s made at the end of ts perod The payments are made for a perod of t years and thus the number of payments s exactly mt = n For each perod, the nterest rate s the same r% annual and thus n each perod, the nterest earned by 1 dollar s exactly r m = Ths s called the perodc rate

4 Basc Annuty Formula Wth the notaton as explaned above, how much money wll be accumulated by makng a perodc nvestment of R dollars at the end of each of the n perods when the perodc rate s and the nterest s compounded n each perod? The answer comes out as a geometrc seres Here s how we reason t out The payment at the end of the frst perod s compounded for (n 1) perods and hence becomes worth R(1 + ) (n 1) The payment at the end of the second perod s compounded only for (n 2) perods and becomes worth R(1 + ) (n 2) Contnung, the very last payment s worth R(1 + ) (n n) = R In other words, t acqures no nterest! Addng up the terms n reverse,s = R + R(1 + ) + + R(1 + ) (n 1), or S = R (1+)n 1 (1+) 1 = R (1+)n 1 A homework problem If you nvest $300 per month at 59% compounded monthly, how much wll your nvestment be worth n 25 years? You notce that you are gven R = 300, r = 59% and t = 25 You deduce that = r m = 59 You want to fnd S 1200 We use( the formula: (1 + ) n ) 1 S = R to get $ and n = (25) (12) = 300

5 Another homework problem How much dd you nvest each month at 620% compounded monthly, f 25 years later the nvestment s worth $178, 67936? You notce that you are not gven R, but you know r = 629% and t = 25 You also know S = 178, You deduce that = r m = and n = (25) (12) = 300 We use the formula ( to wrte: (1 + ) n ) = R The quantty (1 + )n 1 evaluates to Usng ths value, we get: R = = So, $250 s a reasonably accurate answer Present Value of an Annuty Often, the perodc nvestments are just payments - lke mortgage - aganst borrowed funds What s the relaton between the perodc payment R and the borrowed amount P, when the nterest rate s r% and the payment s m tmes a year? As usual, we let be the perodc rate and n the number of perods or the total number of payments Thnk lke the lender and fnd out what sngle nvestment of P dollars would yeld the same accumulaton n same number of years and same rate Ths gves us the equaton: P (1 + ) n = S = R (1+)n 1 thus the formula: P = R (1+)n 1 (1+) = R 1 (1+)( n) n Ths gves the needed formula R = P 1 (1+) ( n) and

6 Usng the Annuty Formulas We now have the basc formulas needed to answer all questons about perodc nvestments or payments Example of a Trust Fund If a trust s set up so that you take 6 years to travel and pursue other nterests Suppose that you wll make b-weekly wthdrawals of $2, 000 from a money market account that pays 400% compounded b-weekly How much should the fund be? Answer: Imagne the trust fund to be a lender and your wthdrawls as mortgage payments to you Thus, we use the formula: P = R 1 (1+)( n) Here R = 2000, = and n = 26 6 = 156 The formula yelds or $277, More Examples of Annutes Snkng Fund Ths means a fund set up wth perodc nvestments to be sunk or used up at the end of the n perods Example You plan on buyng equpment worth 30, 000 dollars n 3 years Snce you frmly beleve n not borrowng, you plan on makng monthly payments nto an account that pays 400% compounded monthly How much must your payment be? You have to fnd out the value of R, but know that S, the expected accumulaton s 30, 000 wth t = 3 and r = 004 Moreover m = 12 (from the word monthly!!) and hence = = and n = 12 3 = 36 Usng S = R (1+)n 1 we get ( R = (1+) n 1 ) = Thus, the reported answer s whch actually yelds $

7 Contnued Examples About Accuracy In the above calculaton, the evaluaton of (1 + ) n 1 = ( ) s nvolved If you calculate ths and dvde nto 30000, you need to keep many dgts of accuracy Try varous approxmatons to see how to get the most accurate answer (to the penny) You wll fnd that you need to keep at least four accurate decmal places the the frst answer Thus, as a general prncple, n these problems, you should not copy down ntermedate answers, but store and recall them, so that maxmum accuracy s mantaned Further Examples of Annuty As another example, consder ths problem If you can afford a monthly payment of $1010 for 33 years and f the avalable nterest rate s 410%, what s the maxmum amount that you can afford to borrow? You note that R = 1010, = r m = and m = 12 wth t = 33, so that n = = 396 But you don t want S, the future accumulaton! You want the money now, to be pad back over the years So, you use the formula for P, the present value Thus, you evaluate: P = R 1 (1+)( n) = = Note that due to the large numbers nvolved, your fracton needs 10 dgt accuracy! Thus, the hardest part s always to fgure out whch formula s approprate!