7.4. Annuities. Investigate

Transcription

1 7.4 Annutes How would you lke to be a mllonare wthout workng all your lfe to earn t? Perhaps f you were lucky enough to wn a lottery or have an amazng run on a televson game show, t would happen. For most people, however, fantases such as these are not lkely to come true. However, what f you saved money from a very early age? Is t possble to accumulate a mllon dollars n your lfetme? When people nvest, they usually do not smply depost one lump sum and wat several years for t to earn nterest. Most wse nvestors make regular payments, often deducted drectly from ther paycheques. Investments of ths type are called annutes. In ths chapter, you wll encounter only ordnary smple annutes. Investgate regular payments payments of equal value made at equal tme perods annuty a sum of money pad as a seres of regular payments ordnary annuty an annuty for whch the payments are made at the end of each payment perod smple annuty an annuty for whch the compoundng and payment perods are the same How can you determne the amount of an annuty? Coln s awarded $500 per year as long as he mantans a certan average mark for each of hs 4 years of hgh school. Coln plans to depost hs award at the end of each school year nto an account that pays 4% per year, compounded annually. 1. a) Assumng that Coln mantans the necessary average, how many tmes wll he receve $500? b) Wll each of these $500 deposts earn the same nterest, n dollars? Explan. 2. a) Determne a method of calculatng the total amount of Coln s nvestment at graduaton. b) Carry out the method. How much wll be n Coln s account when he graduates? 3. eflect Consder the method that you used to solve ths problem. Would your method be effcent f the number of regular payments was very large? Why or why not? 444 MH Functons 11 Chapter 7

2 Stuatons lke the one descrbed n the Investgate can be represented usng a tme lne. The tme lne shows that a regular payment,, n dollars, s deposted nto an account at the end of each compoundng perod, for n perods. Because these deposts are made at dfferent tmes, they wll each earn dfferent amounts of nterest. For example, the last payment wll earn no nterest, because t wll be receved at the end of the annuty. To determne the amount that each of the other deposts wll earn, apply the compound nterest formula A 5 P(1 ) n to each payment ndvdually. tme lne a dagram used to llustrate the cash flow of an annuty Compoundng Perod Now 1 2 n 2 n 1 n (1 + ) 1 (1 + ) 2 (1 + ) n 2 (1 + ) n 1 The amount, A, of the annuty can be determned by addng the amounts of all the payments. A 5 (1 ) (1 ) 2... (1 ) n 2 (1 ) n 1 Snce ths s a geometrc seres wth frst term a 5 and common rato r 5 1, use the formula for the sum of a geometrc seres. S n 5 a(r n 1) r 1 5 [(1 )n 1] (1 ) 1 5 [(1 )n 1] The total amount, A, at the tme of the last payment of an annuty can be determned usng the formula A 5 [(1 )n 1], where represents the regular payment, n dollars; represents the nterest rate per compoundng perod, as a decmal; and n represents the number of compoundng perods. Ths equaton can also be wrtten n terms of future value as FV 5 [(1 )n 1]. Connectons You studed geometrc seres n Chapter 6 Dscrete Functons. Connectons In ths course, you wll study only ordnary smple annutes. You wll study more complex annutes wth, for example, dfferent compoundng and payment perods, f you choose to study busness at unversty or college. 7.4 Annutes MH 445

3 Example 1 Amount of an Annuty At the end of every month, Hosh deposts $100 n an account that pays 6% per year, compounded monthly. He does ths for 3 years. a) Draw a tme lne to represent ths annuty. b) Determne the amount n the account after 3 years. c) How much nterest wll the annuty have earned? Soluton a) Determne the nterest per compoundng perod and the number of compoundng perods before drawng the tme lne. 5 _ n Tme (months) Now (1.005) (1.005) (1.005) 35 b) Method 1: Use a Scentfc Calculator Substtute the known values nto the formula for the amount of an annuty and evaluate. A 5 [(1 )n 1] 5 100[( )36 1] _ 5 100( ) 100 ( y x 36 1 ) The amount n Hosh s account after 3 years wll be $ MH Functons 11 Chapter 7

4 Method 2: Use a TVM Solver Access the TVM Solver on a graphng calculator and enter the values, as shown. Move the cursor to the FV feld and press ALPHA [SOLVE]. When solvng for an annuty, enter the number of payments for N. The negatve sgn ndcates that payments are beng pad, not receved. Both the number of payments and the compoundng perods per year are 12. The future value of the sum of Hosh s payments s $ Ths s the amount n hs account after 3 years. c) To determne the nterest earned, calculate the dfference between the actual dollar sum of Hosh s payments and the future value of the annuty. Hosh made 36 payments of $100, for a total of $3600. Interest Hosh s annuty earned $ n nterest. Example 2 Determne the egular Payment Sada needs $0 for unversty tuton when she graduates n 2 years. She plans to make deposts nto an account that earns 6.5% per year, compounded b-weekly. a) Draw a tme lne to represent ths annuty. b) How much should she depost b-weekly? Soluton a) B-weekly means every 2 weeks. Snce there are 52 weeks n a year, the number of compoundng perods per year s 52 2, or n Annutes MH 447

5 The regular payment,, s unknown, but the total future value of the annuty, A, s $0. A tme lne representng ths annuty s shown. Tme (2-week perods) Now (1.0025) 1 (1.0025) 2 (1.0025) 51 A = 0 b) Use the formula for the amount of an annuty to solve for the regular payment,. Method 1: Substtute and Then earrange A 5 [(1 )n 1] _ 0 5 [( )52 1] ( ) Sada should depost $72.13 b-weekly to have $0 n 2 years. Method 2: earrange and Then Substtute A 5 [(1 )n 1] A 5 [(1 ) n 1] 5 A (1 ) n 1 _ 0(0.0025) 5 ( ) Substtute the known values. Multply both sdes by Dvde both sdes by Multply both sdes by. Dvde both sdes by (1 + ) n 1. Substtute the known values. Sada should depost $72.13 every 2 weeks to have $0 n 2 years. 448 MH Functons 11 Chapter 7

6 Example 3 Determne the Interest ate Olver plans to nvest $2000 quarterly for 5 years. Hs fnancal advsor nforms hm that hs nvestment wll grow to $ after the 5 years. What annual rate of nterest, compounded quarterly, s Olver s annuty earnng? Soluton Lst the known nformaton for ths ordnary smple annuty. A n Payments are made 4 tmes a year for 5 years. Substtute the known values nto the equaton for the amount of an annuty and solve for. _ [(1 )20 1] Ths equaton s dffcult to solve usng standard algebrac technques. A method of systematc tral could be used, but t would be tme-consumng. Method 1: Apply Graphcal Analyss Graph each sde of the equaton [(1 _ )20 1] as a separate functon usng a graphng calculator, and use the Intersect operaton. The pont of ntersecton s ( , ), whch means that the account wll grow to $ at the end of the annuty when the nterest rate per compoundng perod s , or 1.375%. To determne the annual nterest rate, multply by the number of compoundng perods per year % % Olver s annuty s earnng nterest at a rate of 5.5%, compounded quarterly. 7.4 Annutes MH 449

7 Method 2: Use a TVM Solver Access the TVM Solver on a graphng calculator and enter the values, as shown. Move the cursor to the I% feld and press ALPHA [SOLVE]. ecall that I% represents the annual nterest rate, as a percent. Olver s annuty s earnng nterest at a rate of 5.5%, compounded quarterly. Method 3: Use a TI-Nspre CAS Graphng Calculator Use the solve functon of the TI-Nspre CAS graphng calculator to solve for the nterest rate. From the home screen, select 6:New Document.Then, select 1:Add Calculator. Press b. Select 3:Algebra, and then select 1:Solve. Type the equaton, and then press, I. The nterest rate per compoundng perod s , or 1.375%. To determne the annual nterest rate, multply by the number of compoundng perods per year % % Therefore, Olver s annuty s earnng nterest at a rate of 5.5%, compounded quarterly. Example 4 Vary the Condtons of an Annuty Felca plans to nvest $2600 at 6% per year, compounded annually, for the next 15 years. Compare the effects on the fnal amount f the deposts are made and compoundng perods are annual quarterly monthly weekly 450 MH Functons 11 Chapter 7

8 Soluton Method 1: Use a Graphng Calculator Use a table to organze the values of the varables needed to use the formula for the amount of an annuty A 5 [(1 )n 1]. Note that the regular payment,, must be dvded by the number of payments per year n each scenaro. Compoundng Perod n annual quarterly monthly weekly Calculate the amount for the frst scenaro, annual compoundng, usng a graphng calculator. Felca wll have $ at the end of 15 years f nterest s compounded annually. epeat the calculaton for the other scenaros. Press 2nd [ENTY] to recall the prevous calculaton. Use the cursor keys, the DEL key, and the INS key to modfy the equaton to ft each scenaro. Press ENTE to perform the new calculaton. Technology Tp Press CLEA to start each calculaton wth a new wndow screen. The results of these calculatons are summarzed n the table. Compoundng Perod Amount ($) annual quarterly monthly weekly The amount of the annuty ncreases as the compoundng nterval becomes more frequent. The dfference between weekly and annual compoundng s $ $ , or $ Annutes MH 451

9 Method 2: Use a Spreadsheet Set up a table n a spreadsheet. Te c h n o l o g y Tp To enter a mathematcal expresson n a cell n Mcrosoft Excel, clck on the equal sgn and type the expresson. Connectons Some of the amounts are slghtly dfferent n Method 1 and Method 2. Ths s due to roundng n the calculaton entry steps. Dscrepances such as ths become ncreasngly mportant when large sums of money are nvolved. In general, roundng errors can be avoded or mnmzed by leavng ratonal values n fracton form or carryng addtonal decmal places n the calculaton process. Type the headngs C/Y (compoundng perods per year), n (number of compoundng perods or payments), (nterest rate per compoundng perod), (regular payment), and A (amount). Enter the values 1, 4, 12, and 52 n the C/Y column, to represent the number of compoundng perods n each scenaro. Enter the formulas, startng n cell B2 and workng to the rght: B2 5A2*15 n = 15 number of payments per year = 0.06 number of payments per year C /A2 = $2600 number of payments per year D /A2 E2 5(D2*((1 C2)^B2-1))/C2 Calculate the amount usng the formula A = [(1 + )n 1 _. Use Fll Down to evaluate the remanng calculatons. The results of these calculatons are summarzed n the table. Compoundng Perod Amount ($) annual quarterly monthly weekly The amount of the annuty ncreases as the compoundng nterval becomes more frequent. The dfference between weekly and annual compoundng s $ $ , or $ Key Concepts An annuty s an nvestment n whch regular payments are deposted nto an account. An ordnary smple annuty s one n whch payments are made at the end of every payment perod and nterest s compounded at the end of the same payment perod. [(1 )n 1], where represents the regular payment; represents the nterest rate per compoundng perod, as a decmal; and n represents the number of compoundng perods. The amount, A, of an annuty can be calculated usng the formula A MH Functons 11 Chapter 7 Functons 11 CH07.ndd 452 6/10/09 4:24:21 PM

10 Communcate Your Understandng C1 The tme lne shows an annuty wth an annual nterest rate of 8%. a) How often s nterest compounded? How can you tell? b) What s the duraton of the annuty? How can you tell? c) Explan why ths annuty can be represented as a geometrc seres. d) Identfy the frst term, a, and the common rato, r, of the geometrc seres. Now 1 Compoundng Perods (1.04) 1 (1.04) 2 (1.04) 3 (1.04) 4 (1.04) 5 C2 The graphng calculator screen of a soluton usng the TVM Solver s shown. a) What s the duraton of the annuty? How can you tell? b) Why s the payment value negatve? c) Why s the future value postve? A Practse For help wth questons 1 to 3, refer to Example Calculate the amount of the annuty shown n the tme lne. Now 1 Compoundng Perods To help her granddaughter wth unversty costs, Sasha s grandmother puts $250 nto an account that earns 4.5% per year, compounded annually, at the end of every year for 6 years. a) Draw a tme lne to represent ths annuty. b) Determne the amount of the annuty. c) How much nterest was earned? (1.04) 1 (1.04) 2 (1.04) 3 (1.04) 4 (1.04) 5 3. At the end of every week, for 2 years, Carlo puts $35 nto an account that earns 5.2% per year, compounded weekly. epresentng Connectng easonng and Provng Problem Solvng Communcatng a) Draw a tme lne to represent ths annuty. b) Determne the amount of the annuty. c) How much nterest was earned? For help wth questons 4 and 5, refer to Example 2. Selectng Tools eflectng 4. How much must be nvested at the end of each year, for 4 years, to acheve an amount of $10 000, f nterest s earned at a rate of 6.25% per year, compounded annually? 5. Lucy wants to have $ n her account 3 years from now to buy a car. How much must she nvest per month, f her account earns 7.2% annual nterest, compounded monthly? 7.4 Annutes MH 453

11 B Connect and Apply For help wth questons 6 and 7, refer to Example Donna nvests $75 every 2 weeks n an account that earns compound nterest b-weekly. If she does ths for 7 years, she wll end up wth $ n the account. a) How much total nterest wll have been earned? b) Determne the annual rate of nterest, compounded b-weekly. 7. efer to queston 6. By how much must the nterest rate be ncreased for the amount to grow to $ after 7 years? For help wth questons 8 to 10, refer to Example 4. Use the followng nformaton. Maurce s plannng to depost $160 per month nto an account that earns 4.8% annual nterest, compounded monthly, for 15 years. 8. a) Determne the amount n the account at the end of ths annuty. b) How much nterest wll have been earned? 9. Maurce s fnancal advsor suggests that he would mprove the value of hs annuty f he changed hs payments to $40 per week, at the same nterest rate, compounded weekly. Do you agree or dsagree wth the fnancal advsor? Justfy your response wth mathematcal reasonng. 10. A competng bank offers Maurce epresentng 5% per annum, Problem Solvng compounded Connectng monthly, for hs monthly deposts of Communcatng $160. Whch opton should Maurce choose? Justfy your answer wth mathematcal reasonng. easonng and Provng Selectng Tools eflectng Stck wth hs current arrangement. Follow hs fnancal advsor s suggeston about ncreasng the frequency of hs deposts. Swtch to the competng bank. 11. Lee would lke to retre at age 60 and s consderng two nvestment optons: Opton A: Invest $500 per month begnnng at age 20. Opton B: Invest $1000 per month begnnng at age 40. In both cases, the nterest s 6% per annum, compounded monthly. Whch opton pays more nterest, and by how much? 12. Pnder wants to be a mllonare before he retres. He plans to save a certan amount every week for 40 years. a) If he puts money n an nvestment that earns 7% annual nterest, compounded weekly, what amount must Pnder depost weekly? b) What other strateges could Pnder use to acheve hs goal? Dscuss any assumptons you must make. C Extend 13. Use Technology Use a graphng calculator or graphng software. a) Graph the functon A 5 100(1.05n 1) Descrbe the shape of the graph. b) Interpret ths functon, assumng that t s related to the amount of an annuty. c) Assumng that nterest s compounded annually, dentfy the regular payment and the annual nterest rate. d) Pose and solve two problems related to ths functon. 454 MH Functons 11 Chapter 7

12 14. In queston 13, the functon A5 100(1.05 1) n 0.05 of an annuty. represents the amount a) Wrte a functon to descrbe the total prncpal nvested after n compoundng perods. b) Graph the amount and the prncpal functons on the same set of axes. Descrbe the shape of the prncpal functon. c) Descrbe how the graphs of these two functons represent the nterest earned over tme. Wrte a functon to represent the nterest earned after n compoundng perods. 15. Math Contest Bethany starts nvestng $300 per month at 6% per annum, compounded monthly, for 5 years. After 3 years, the nterest ncreases to 9% per annum, compounded monthly. Determne the amount of her nvestment after 5 years. A $ B $ C $ D $ Math Contest In ABC, a 5 10 mm, b 5 26 mm, and c 5 24 mm. If D s the mdpont of AC and BC s extended to E such that DE 5 24 mm, determne the measure of CED wthout usng a calculator. A 30 B 60 C 45 D Math Contest Gven that a2 (a b) and a and b are whole numbers, determne all possble ordered pars (a, b) that solve ths equaton. 18. Math Contest Gven x2 y , (2)(3)(5)(7)(11) 1, and x y 0, where x and y are ntegers, whch of the followng s true? A only x s dvsble by 11 B only y s dvsble by 11 C both are dvsble by 11 D nether s dvsble by 11 Career Connecton Felcty s an nvestment trader for a large frm n Toronto. She uses her company s money to buy and sell stocks, bonds, and shares to make a proft. Because the stock market s so volatle, Felcty must always have access to the newest nformaton from around the world. Success n her job depends on the use of computers that can perform fast mathematcal calculatons. Watng too long to make a trade could cost her company a lot of money. Felcty traned for her career by takng a 4-year bachelor of admnstratve studes degree at York Unversty. 7.4 Annutes MH 455 Functons 11 CH07.ndd 455 6/10/09 4:24:27 PM