8.4. Annuities: Future Value. INVESTIGATE the Math Annuities: Future Value

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1 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at the end of each year n a Canada Savngs Bond earnng 6.5%/a compounded annually. Her frst depost s on Decemer 3, 007.? How much wll her nvestments e worth 0 years later, on January, 07? A. Copy the tmelne shown. How would you calculate each of the future values FV to FV0? = 6.5%/a compounded annually Payment $0 $000 $000 $000 $000 $000 $000 FV0 FV9 FV8 FV3 FV FV B. Set up a spreadsheet wth columns as shown. Copy the data already entered. Complete the entres under Date Invested up to Dec. 3, A Date Invested Dec. 3, 06 Dec. 3, 05 Dec. 3, 0 B C D Numer of Years Amount Invested Invested Value on Jan., 07 $ $ $ $ Annutes: Future Value C. Fll n cells D3 and D to show what the nvestments made on Dec. 3, 05, and Dec. 3, 0, respectvely wll e worth on Jan., 07. D. How s the value n cell D3 (FV9) related to the value n cell D (FV0)? How s the value n cell D (FV8) related to the value n cell D3 (FV9)? E. Use the pattern from part D to complete the rest of the entres under Value on Jan., 07. F. What type of sequence do the values on Jan., 07 form?

2 G. Calculate the total amount of all the nvestments at the end of 0 years for ths annuty. Reflectng H. The values of all of the nvestments at the end of each year for 0 years formed a specfc type of sequence. How s the total value of the annuty at the end of 0 years related to a seres? I. How could you use the related seres to solve prolems nvolvng annutes? APPLY the Math annuty a seres of payments or nvestments made at regular ntervals. A smple annuty s an annuty n whch the payments concde wth the compoundng perod, or converson perod. An ordnary annuty s an annuty n whch the payments are made at the end of each nterval. Unless otherwse stated, each annuty n ths chapter s a smple, ordnary annuty. EXAMPLE Representng the future value of an annuty earnng compound nterest as a seres a) Hans plans to nvest $000 at the end of each 6-month perod n an annuty that earns.8%/a compounded sem-annually for the next 0 years. What wll e the future value of hs annuty? ) You plan to nvest $R at regular ntervals n an annuty that earns % compounded at the end of each nterval. What wll e the future value, FV, of your annuty after n ntervals? Barara s Soluton a) n Snce the nterest s pad sem-annually, I calculated the nterest rate per compoundng perod and the numer of compoundng perods =.8%/a compounded sem-annually Payment $0 $000 $000 $000 $000 $000 $000 $ ( + 0.0) 000( + 0.0) 000( + 0.0) ( + 0.0) ( + 0.0) ( + 0.0) 39 I drew a tmelne of the nvestments for each compoundng perod, and I represented the amount of each nvestment. The last $000 nvestment earned no nterest ecause t was deposted at the end of the term. The frst $000 nvestment earned nterest over 39 perods. It ddn t earn nterest durng the frst compoundng perod ecause t was deposted at the end of that perod. Chapter 8 Dscrete Functons: Fnancal Applcatons 505

3 ) 000, 000(.0), 000(.0),..., 000(.0) 38, 000(.0) 39 S (.0) 000(.0) (.0) (.0) 39 S n 5 a(r n ) r S (.00 ).0 8 $ The future value of Hans s annuty at the end of 0 years s $ Payment S n 5 a(r n ) r 5 R 3( )n ( ) 0 3 R, R( ), R( ),..., R( ) n, R( ) n S n 5 R R( ) R( )... R( ) n R( ) n 5 R 3 a ( )n % per compoundng perod n n n $0 $R $R $R $R $R $R R R( + ) R( + ) R( + ) n R( + ) n 3 R( + ) n The future value of an annuty n whch $R s nvested at the end of each of n regular ntervals earnng % of compound nterest per nterval s FV 5 R 3 a ( )n, where s expressed as a decmal. The future values of all of the nvestments form a geometrc sequence wth frst term $000 and common rato.0. The total amount of all these nvestments s the frst 0 terms of a geometrc seres. I used the formula for the sum of a geometrc seres to calculate the total amount of all of Hans s nvestments. I rounded to the nearest cent. I drew a tmelne of the nvestments for each compoundng perod to show the amount of each nvestment. The last $R nvestment earned no nterest. The frst $R nvestment earned nterest n tmes. The values of all of the nvestments form a geometrc sequence wth frst term R and common rato. The total amount of all these nvestments s the frst n terms of a geometrc seres. I used the formula for the sum of a geometrc seres to calculate the total amount of all the nvestments Annutes: Future Value

4 EXAMPLE Selectng a strategy to determne the future value of an annuty Che puts away $500 every 3 months at 5.%/a compounded quarterly. How much wll her annuty e worth n 5 years? Kew s Soluton: Usng a Geometrc Seres n Payment S (.0300 ) , 500(.03), 500(.03),..., 500(.03) 98, 500(.03) 99 S (.03) 500(.03) (.03) (.03) 99 S n 5 a(r n ) r 8 $ $0 $500 $500 $500 $500 $500 $500 $500 $500 $500 The total amount of all of Che s nvestments at the end of 5 years wll e $ = 5.%/a compounded quarterly ( ) 500( ) 500( ) 3 500( ) 500( ) ( ) ( ) ( ) 99 Frst I calculated the nterest rate per compoundng perod and the numer of compoundng perods. I drew a tmelne of the nvestments for each quarter to show the amounts of each nvestment. I calculated the value of each nvestment at the end of 5 years. The values form a geometrc sequence wth frst term $500 and common rato.03. The total amount of all these nvestments s the frst 00 terms of a geometrc seres. I used the formula for the sum of a geometrc seres to calculate the total amount of all of Che s nvestments. I rounded to the nearest cent. Chapter 8 Dscrete Functons: Fnancal Applcatons 507

5 Tna s Soluton: Usng the Formula for the Future Value of an Annuty R 5 $ n FV 5 R 3 a ( )n a ( 0.03) $ The future value of Che s annuty wll e $ I calculated the nterest rate per compoundng perod and the numer of compoundng perods. I susttuted the values of R,, and n nto the formula for the future value of a smple, ordnary annuty. I rounded to the nearest cent. EXAMPLE 3 Selectng a strategy to determne the regular payment of an annuty Sam wants to make monthly deposts nto an account that guarantees 9.6%/a compounded monthly. He would lke to have $ n the account at the end of 30 years. How much should he depost each month? Chantal s Soluton I calculated the nterest rate per compoundng perod and the numer of compoundng n perods. FV 5 $ FV 5 R 3 a ( )n R 3 a ( 0.008) The future value of the annuty s $ I susttuted the values of FV,, and n nto the formula for the future value of an annuty Annutes: Future Value

6 R R R 5 $0.80 Sam would have to depost $0.80 nto the account each month n order to have $ at the end of 30 years. To solve for R, I dvded oth sdes of the equaton y I rounded to the nearest cent. Support Tech For help usng a spreadsheet to enter values and formulas, and to fll down, see Techncal Appendx, B-. EXAMPLE Selectng a strategy to determne the term of an annuty Nahd orrows $ to uy a cottage. She agrees to repay the loan y makng equal monthly payments of $750 untl the alance s pad off. If Nahd s eng charged 5.%/a compounded monthly, how long wll t take her to pay off the loan? Zak s Soluton 3 A B C D E Payment Numer Payment Interest Pad Prncpal Pad Balance $ $7.50 $3.50 $ "=E3*0.05/" "=B-C" "=E3-D" I set up a spreadsheet to calculate the alance after every payment. The nterest s always charged on the alance and s of 5.% snce t s compounded monthly. The part of the prncpal that s pad off wth each payment s $750, less the nterest. The new alance s the old alance, less the part of the prncpal that s pad A B C D E Payment Numer Payment Interest Pad Prncpal Pad Balance $7.50 $6.05 $.59 $3.3 $.66 $6.55 $3.5 $9.9 $6.6 $3.6 $3.50 $33.95 $35. $36.87 $38.3 $733.5 $ $70.06 $73.39 $76.7 $ $ $ $9 08. $ $ $ 9.9 $ 08.7 $ 68. $7.7 $.0 I used the FILL DOWN command to complete the spreadsheet untl the alance was close to zero. t months 8 8 months Nahd can pay off the loan after 88 payments, whch would take aout 5 years and 8 months. After 88 payments, the alance s close to zero. I calculated the numer of years needed to make 88 payments y dvdng y, snce there are payments each year. I got a value greater than 5. The 5 meant 5 years, so I had to fgure out what of a year was. Chapter 8 Dscrete Functons: Fnancal Applcatons 509

7 In Summary Key Ideas The future value of an annuty s the sum of all regular payments and nterest earned. % per compoundng perod 0 3 n n n Payment $0 $R $R $R $R $R $R R R( + ) R( + ) R( + ) n 3 R( + ) n R( + ) n The future value can e wrtten as the geometrc seres FV 5 R R( ) R( )... R( ) n R( ) n where FV s the future value; R s the regular payment; s the nterest rate per compoundng perod, expressed as a decmal; and n s the numer of compoundng perods. The formula for the sum of a geometrc seres can e used to determne the future value of an annuty. Need to Know A varety of technologcal tools (spreadsheets, graphng calculators) can e used to solve prolems nvolvng annutes. The payment nterval of an annuty s the tme etween successve payments. The term of an annuty s the tme from the frst payment to the last payment. The formula for the future value of an annuty s FV 5 R 3 a ( )n where FV s the future value; R s the regular payment each compoundng perod; s the nterest rate per compoundng perod, expressed as a decmal; and n s the numer of compoundng perods Annutes: Future Value

8 CHECK Your Understandng. Each year, Erc nvests $500 at 8.%/a compounded annually for 5 years. a) Calculate the value of each of the frst four nvestments at the end of 5 years. ) What type of sequence do the values form? c) Determne the total amount of all of Erc s nvestments.. Calculate the future value of each annuty. a) ) c) d) Rate of Compound Interest Regular Payment per Year Tme $00 per month 3.6% monthly 50 years $500 per quarter 6.% quarterly 5 years $500 every 6 months 5.6% sem-annually 8 years $000 per year.5% annually 0 years 3. Los nvests $650 every 6 months at.6%/a compounded sem-annually for 5 years. How much nterest wll she earn after the 5th year?. Josh orrows some money on whch he makes monthly payments of $5.3 for 3 years. If the nterest rate s 5.%/a compounded monthly, what wll e the total amount of all of the payments at the end of the 3 years? PRACTISING 5. Calculate the future value of each annuty. K a) ) c) d) Rate of Compound Interest Regular Payment per Year Tme $500 per year 6.3% annually 0 years $50 every 6 months 3.6% sem-annually 3 years $00 per quarter.8% quarterly 7 years $5 per month 8% monthly 35 years 6. Mke wants to nvest money every month for 0 years. He would lke to have A $ at the end of the 0 years. For each nvestment opton, how much does he need to nvest each month? a) 0.%/a compounded monthly ) 5.%/a compounded monthly Chapter 8 Dscrete Functons: Fnancal Applcatons 5

9 7. Kk has several optons for nvestng $00 per year: a) ) c) d) Rate of Compound Interest Regular Payment per Year $00 per month 7.% monthly $300 per quarter 7.% quarterly $600 every 6 months 7.% sem-annually $00 per year 7.% annually Wthout dong any calculatons, whch nvestment would e est? Justfy your reasonng. 8. Kenny wants to nvest $50 every three months at 5.%/a compounded quarterly. He would lke to have at least $6500 at the end of hs nvestment. How long wll he need to make regular payments? 9. Sonja and Anta want to make equal monthly payments for the next 35 years. At the end of that tme, each person would lke to have $ Sonja s ank wll gve her 6.6%/a compounded monthly. Anta can nvest through her work and earn 0.8%/a compounded monthly. a) How much more per month does Sonja have to nvest? ) If Anta decdes to nvest the same monthly amount as Sonja, how much more money wll she have at the end of 35 years? 0. Jamal wants to nvest $50 every month for 0 years. At the end of that tme, T he would lke to have $ At what annual nterest rate, compounded monthly, does Jamal need to nvest to reach hs goal? Round your answer to two decmal places.. Draw a mnd map for the concept of future value of annutes. Show how t s C related to nterest, sequences, and seres. Extendng. Carmen orrows $0 000 at.8%/a compounded monthly. She decdes to make monthly payments of $50. a) How long wll t take her to pay off the loan? ) How much nterest wll she pay over the term of the loan? 3. Greg orrows $3 000 for the purchase of a house. He plans to make regular monthly payments over the next 0 years to pay off the loan. The ank s chargng Greg 6.6%/a compounded monthly. What monthly payments wll Greg have to make?. How many equal monthly payments would you have to make to get 00 tmes the amount you are nvestng each month f you are earnng 8.%/a compounded monthly? 5 8. Annutes: Future Value