APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES Example: Brado s Problem Brado, who is ow sixtee, would like to be a poker champio some day. At the age of twety-oe, he would like to make a ame for himself i the world of poker by eterig ad wiig a big-time poker touramet i Las Vegas. The oly problem is that he eeds a lot of cash to realize this dream. Besides payig for the $5000 touramet etry fee, Brado also eeds to save a additioal $10000 to cover travellig costs ad livig expeses. After learig about compoud iterest ad auities i Mr. Nolfi s class, Brado decided it might be a good idea to ivest some moey. He cosiders several possibilities as outlied below. (a) $10000 Caada Savigs Bod that pays iterest at a rate of 4% per aum, ot compouded (Simple ) (b) $10000 Caada Savigs Bod that pays iterest at a rate of 4% per aum, compouded aually (Compoud ) (c) Brado the cosiders makig mothly deposits to a auity. To achieve his goal by the age of twety-oe, how much would he eed to deposit at the ed of each moth if the auity pays iterest at the rate of 6% aum compouded mothly? A ivestmet i which deposits/paymets are made at regular itervals is called a Auity. If the deposit/paymet is made at the ed of the paymet period, the auity is called a Ordiary Auity. If istead the deposits/paymets are made at the begiig of the paymet period, the auity is called a Auity Due. This course will deal oly with ordiary auities. Which of these ivestmets will allow Brado to realize his dream? Solutio Because the give iformatio is icomplete, we eed to make a reasoable assumptio before we attempt to solve this problem. Although we are told that Brado is ow sixtee, we do ot kow his exact age. He could have just tured sixtee, he may have tured sixtee a few moths ago or he may be about to tur sevetee. To avoid this cofusio, let s assume that he has just tured sixtee ad that he would like to have the moey by his twety-first birthday. This gives him exactly five years to save the moey. (a) Simple : is paid directly to the ivestor, that is, it is ot added to the value of the ivestmet. This meas that iterest paymets remai costat over time. P = pricipal = origial amout ivested = $10000.00 r = aual rate of iterest = 4% = 0.04 t = time i years = 5 By doig a few simple calculatios, we ca easily see that Brado will fall far short of his goal! Paymet # Paymet Accout Balace Total Paid How Accout Balace is Calculated $10,000.00 1 $400.00 $10,000.00 $400.00 The accout balace does ot chage because iterest paymets are 2 $400.00 $10,000.00 $800.00 made to the ivestor, ot to the accout! 3 $400.00 $10,000.00 $1,200.00 4 $400.00 $10,000.00 $1,600.00 Total value of the ivestmet = $10000.00 + $2000.00 < $15000.00 5 $400.00 $10,000.00 $2,000.00 Easier way to perform the calculatios: I = total iterest paid after t years I A = total value of ivestmet after t years = Prt = $10000.00( 0.04)( 5 ) = $2000.00, ( ) ( ) A = P 1 + rt = $10000.00 1 + 0.04 5 = $12000.00
(b) Compoud : paid to the accout, which meas that iterest paymets icrease with time. P = pricipal = origial amout ivested = $10000.00 r = aual rate of iterest = 4% = 0.04 = total umber of compoudig periods = total umber of times iterest is calculated ad paid to the accout 51 = 5 = ( ) i = periodic rate of iterest = iterest rate per compoudig period = r (# of compoudig periods per year) = 0.04 1 Paymet # = 0.04 Paymet Accout Balace $10,000.00 Total Paid 1 $400.00 $10,400.00 $400.00 $10,000 1.04 How Accout Balace is Calculated 2 $416.00 $10,816.00 $816.00 $10,400 1.04 = ( $10,000 1.04) 1.04 = $10,000 1.04 2 3 $432.64 $11,248.64 $1,248.64 $10,816 1.04 = ($10,000 1.04 2 ) 1.04 = $10,000 1.04 3 4 $449.95 $11,698.59 $1,698.59 $11,248.64 1.04 = ($10,000 1.04 3 ) 1.04 = $10,000 1.04 4 5 $467.94 $12,166.53 $2,166.53 $11,698.59 1.04 = ($10,000 1.04 4 ) 1.04 = $10,000 1.04 5 We ca see that the accout balace is calculated by multiplyig the previous balace by 1.04. The fial balace of $12,166.53 (i.e. the future value) is calculated by multiplyig the origial amout of $10,000.00 (i.e. the preset value) by 1.04, five times. I other words, Substitutig ito this formula we obtai $12,166.53 = $10,000.00 (1.04)(1.04)(1.04)(1.04)(1.04) = $10,000.00 (1.04) 5 A= P(1 + i) = 10000(1 +.04) 5 = 10000(1.04) = 12166.53 Very Importat Observatio Notice that the compoud iterest formula is a example of the geeral term of a geometric sequece: t Usig the above example as a guide, we ca make the followig observatio: P represets the pricipal (origial amout ivested) A represets the value of the ivestmet after compoudig periods i represets the periodic iterest rate, the 1 = ar t 1 + = ar Sice A P ( ) A= P(1 + i) This equatio is kow as the compoud iterest formula. = 1 + i, A= t + 1, P= a, 1+ i= r 5
(c) To uderstad this part of the example, it s importat to comprehed the termiology i the followig table: Termiology Value i this Example Explaatio PV = Preset Value $0.00 Amout of moey with which Brado started. FV = Future Value $15,000.00 = Total Number of Compoudig Periods 12 5 = 60 The amout of moey Brado wats to have at the ed of the five year period. This is the total umber of compoudig periods over the course of the ivestmet (or loa). r = Aual Rate 6% = 0.06 Rate of iterest payable per year i = Periodic Rate r 0.06 0.005 c = 12 = Rate of iterest payable per compoudig period. d = Deposit/Paymet made at Regular Itervals Ukow The amout deposited/paid at regular itervals. p = Paymets per Year (Paymet Frequecy) 12 The umber of paymets/deposits made per year. c = Compoudig Periods per Year (Compoudig Frequecy) 12 The umber of times per year that the iterest compouds. Note: For the sake of simplicity, i this course it will be assumed that p = c i all cases. However, this eed ot be the case. For example, for all mortgages i Caada, c = 2 (i.e. iterest must be compouded twice per year or semi-aually) but p ca have may differet values such as 12 (mothly), 24 (semi-mothly) ad 52 (weekly). The followig diagram, which is called a time-lie, illustrates this situatio. Although this seems somewhat more complicated tha the sceario i part (b), i reality, it is merely a simple applicatio of the ideas i (b). The oly differece is that i this case, the method of part (b) eeds to be applied multiple times. The amout deposited each moth is d dollars. Each deposit of d dollars ears iterest, which causes its value to grow over time. After m compoudig periods (which happes to be m moths), the value of a deposit of d dollars grows to d (1.005) m dollars (by applyig the compoud iterest formula). Sum of future values of all the deposits Moth 0 1 2 3 57 58 59 d d d d d d 60 d + d(1.005) 2 + d (1.005) d + d(1.005) + d(1.005) + + d(1.005) 59 + d (1.005) 2 59 Future Value of 60 th Deposit Future Value of 59 th Deposit Future Value of 58 th Deposit Future Value of 1 st Deposit So it appears that the crux of this problem is to figure out how to add up the future values of each deposit of d dollars. Sice we kow that Brado eeds to have $15,000.00 by the time he is 21, the sum of the future values of each deposit of d dollars will have to be $15,000.00. Kowig this will allow us to solve for d ad fially determie exactly how much moey Brado eeds to pay to the auity each moth.
Gladly, we already kow how to fid such a sum. Careful examiatio of the expressio reveals that it is othig more tha a geometric series. I this geometric series, a = d, r = 1.005 ad = 60. Therefore, 60 60 2 59 d(1.005 1) d(1.005 1) d + d(1.005) + d(1.005) + + d(1.005) = =. 1.005 1 0.005 Also recall that Brado eeds a future value of $15,000 to realize his dream of becomig a poker champio. Therefore, d 60 (1.005 1) 0.005 60 1.005 1 = 15000 15000(0.005) d = 214.99 Therefore, Brado must deposit $214.99 at the ed of each moth. Summary of Brado s Ivestmet Optios The followig is a graphical summary of the three ivestmets cosidered by Brado. Clearly, the auity is the oly ivestmet that ca satisfy his eeds. I additio to the faster growth rate, the auity has the advatage of ot requirig a large amout of cash to be ivested up frot. All Brado eeds to do is to be able to put away $214.99 each moth. Compoud Simple Ordiary Auity I this graph, the simple iterest ad compoud iterest curves appear to merge ito oe. This is due etirely to the coarse scale used. Zoomig i reveals that the graphs are ot idetical (see graph at the left). Summary of Ivestmet Equatios Type of Ivestmet Equatios Meaig of Variables Simple I = Prt A = P( 1+ rt) Compoud A= P( 1+ i) Ordiary Auity ( i) 1+ 1 FV = R i A P= = A + i ( 1+ i) ( i) ( 1 ) 1 1+ PV = R i r = aual rate of iterest i = periodic rate of iterest = r (# compoudig periods per year) = total umber of compoudig periods t = time i years (simple iterest oly) R = amout deposited/paid at regular itervals PV = preset value FV = future value I = amout of iterest eared/paid after t years (simple iterest) A = value of ivestmet after t years (simple iterest) or value of ivestmet after compoudig periods (compoud iterest)
More Examples 1. Simple 2. Compoud
3. Ordiary Auity (Future Value)
4. Ordiary Auity (Preset Value)
Terms you Need to Kow Term Deposit/Paymet Frequecy Daily Every day: 365 times per year Weekly Every week: 52 times per year Semi-Mothly Twice per moth: 24 times per year Mothly Every moth: 12 times per year Bi-Mothly Every two moths: 6 times per year Quarterly Every three moths: 4 times per year Semi-Aually Every six moths: 2 times per year Aually Every year: 1 time per year