Using Series to Analyze Financial Situations: Present Value


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1 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated values of all the payments. When people thnk about ther retrement, they often hope to have a regular ncome from a penson or from an annuty. A regstered ncome fund (RIF) s one type of annuty. Ths fund s a lump sum of money that s nvested now so that a regular ncome may be drawn from the fund over a long perod at some tme n the future. For example, Blake wants to receve a $1000 payment at the end of the next fve years, so he nvests an amount that s less than $1000 today. The amount that he nvests s less than the future value because the nvestment earns nterest durng the term. The amount that s nvested now to yeld a larger amount n the future s called the present value. In ths secton, you wll calculate the present value of an annuty by addng the present values of all the payments. Part 1: Representng the Present Value of an Ordnary Smple Annuty wth a Seres The St. Charles College Math Club wants to establsh a math scholarshp to reward ts top graduatng math student. The $1000 scholarshp wll be awarded at the end of each year for the next fve years. The club has some money to pay for the future scholarshp payments. Calculate how much money s needed now by fndng the present value of each of the fve $1000 awards and then addng the present values. The trust account pays 9%/a, compounded annually. Here s the tme lne that shows the payments as present values. Year Payment amount $1000 $1000 $1000 $1000 $1000 Present value of each payment P1 P2 P3 P4 P5 Thnk, Do, Dscuss 1. Is $1000 requred now to cover the cost of the $1000 scholarshp that wll be awarded at the end of the frst year? Explan. 156 CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
2 2. The nterest rate s 9%/a, compounded annually. Determne an expresson that would represent the amount that s needed now to provde the frst $1000 scholarshp payment at the end of the frst year. Express the present value of the payment as a product rather than as a quotent by usng negatve exponents. 3. Determne an expresson for the present value of the second $1000 scholarshp. Express ths present value as a product. 4. Repeat step 3 to fnd expressons for the present values of the thrd, fourth, and ffth payments. 5. Copy and complete the tme lne. 6. Use the expressons you found above, begnnng wth the present value of the ffth payment, P5, to wrte the seres that represents the accumulated sum of all the present value payments. Do not fnd the value of each term. 7. What s the frst term? What do you multply the frst term by to get the second term? What s ths type of seres? What s the common rato? How many terms does the seres have? 8. Use the formula you learned earler n ths chapter to fnd the sum of the present values of the payments. How much money would the club need to rase and then depost n the trust account to sustan the scholarshp over fve years? Part 2: Developng a Formula for the Present Value or Amount of an Ordnary Smple Annuty Andrea wants to fnd an algebrac expresson to represent the present, or dscounted, value of an ordnary smple annuty. She lets R be the regular perodc payment of the annuty. She lets n be the total number of nterest converson perods, or the total number of payments. She drew the followng tme lne. Converson perod n 2 n 1 n Payment amount $R $R $R $R $R Present value of each payment $R (1 + ) 1 $R (1 + ) 2 $R (1 + ) n + 2 $R (1 + ) n + 1 $R (1 + ) n Thnk, Do, Dscuss 1. What does represent n the tme lne? Explan why the exponent n the present value of the frst payment s 1. Explan why the exponent n the present value of the last payment s n. 2.8 USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 157
3 2. Gven Andrea s tme lne, wrte the seres of terms, PV, that represents the present or dscounted value of each payment of the annuty, begnnng wth the last payment. What type of seres s PV? 3. What s the frst term of the seres? What s the common rato? Substtute these values n the formula you used n step 8 of Part You can smplfy the equaton n step 3, but before you do, what do you notce about the bases of the powers n ths equaton? What do you do to the exponents when you multply these powers? Use the dstrbutve property to smplfy your equaton. The result s the present value, PV, of an ordnary smple annuty. 5. A math scholarshp awards $500 at the end of each year for ten years. Use the formula you developed n step 4 to fnd the present value that should be deposted n a trust account that pays 11%/a, compounded annually, to sustan the scholarshp. Focus 2.8 Key Ideas The present value, or dscounted value, of an annuty s the value at the begnnng of the term of the annuty. The present value s the sum of all the present values of the payments. Ths tme lne represents the present value of an ordnary smple annuty. Converson perod n 2 n 1 n Payment amount $R $R $R $R $R Present value of each payment $R (1 + ) 1 $R (1 + ) 2 $R (1 + ) n + 2 $R (1 + ) n + 1 $R (1 + ) n The present or dscounted value of an ordnary smple annuty can be wrtten as the geometrc seres R (1 ) n R (1 ) n 1 R (1 ) n 2 R (1 ) 2 R (1 ) 1 where R represents the regular payment n represents the number of nterest converson perods or the total number of payments represents the nterest rate per converson perod 158 CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
4 The present or dscounted value of an ordnary smple annuty s the sum of a geometrc seres, S n a( r n 1), where a R (1 ) r 1 n, r 1, and n s the number of payments. Fnd the present or dscounted value of an ordnary smple annuty, A or PV, usng A PV R 1 (1 ) n Example 1 Clare has just won a lottery. She s offered two prze optons: $ cash today or payments of $1000 each at the end of each month for 20 years. Clare expects a return of 9%/a, compounded monthly, f she nvests the cash prze. (a) Draw a tme lne to represent the present value of the payment opton. (b) Wrte the seres that represents the annuty. (c) Whch opton should Clare choose? Soluton (a) The monthly nterest rate s 0.75% and the term s Converson perod Payment amount $1000 $1000 $1000 $1000 $1000 Present value of each payment $1000 ( ) 1 $1000 ( ) 2 (b) The seres s S (1.0075) (1.0075) (1.0075) (1.0075) (1.0075) 1. (c) $1000 ( ) 238 $1000 ( ) 239 $1000 ( ) 240 r n 1) Method 1: Usng S n a( r 1 It s possble to msnterpret ths problem by comparng the cash value today, $ , to the total money pad under the payment opton, $ $ The seres s geometrc, and a 1000(1.0075) 240, r , and n USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 159
5 r n 1) S n a( Substtute the values for a, r, and n. r 1 S (1.0075) 240 ( ) Smplfy ( ) ( ) The present value of the payment opton s $ Clare should choose the $ f she may nvest t at 9%/a, compounded monthly. Method 2: Usng PV R 1 (1 ) n The seres represents the present or dscounted value of an ordnary smple annuty where R 1000, , and n 240. PV R 1 (1 ) n Substtute the values for R,, and n ( ) 240 Smplfy Usng the formula for the present or dscounted value also yelds $ Example 2 The Sybulsk famly purchase a motor home. They borrow $ at 9.6%/a, compounded monthly, and they wll make payments at the end of each month. They have two choces for the term: seven years or ten years. (a) Draw a tme lne to represent the present value of the sevenyear annuty. (b) Wrte the seres that represents the sevenyear annuty. (c) Fnd the monthly payment under each term. (d) How much would they save n nterest by selectng the shorter term? Soluton (a) The monthly nterest rate s 9.6% % or and the number of perods s for the sevenyear term. Converson perod Payment amount $R $R $R $R $R Present value of each payment $R ( ) 1 $R ( ) 2 $R ( ) 82 $R ( ) 83 $R ( ) CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
6 (b) The seres s S 84 R (1.008) 84 R (1.008) 83 R (1.008) 82 R (1.008) 2 R (1.008) 1, where S (c) Use PV R 1 (1 ). n The present value of the loan s $ Fnd the monthly payment, R, under both terms. For a sevenyear term, PV , 0.8% or 0.08, and n 84. PV R 1 (1 ) n Substtute the values for PV,, and n R 1 ( 1.008) 84 Smplfy and solve for R R For a tenyear term, the values for PV and are the same, but n PV R 1 (1 ) n Substtute the values for PV,, and n R 1 ( ) 120 Smplfy and solve for R R For a sevenyear term, the payment would be $ For a tenyear term, the payment would be $ (d) The total repayment cost of the sevenyear term s $ $ and $ $ for the tenyear term. The famly would save $ n nterest by choosng the shorter term. Example 3 Mr. Fabll deposts $3000 n an RRSP each year. The nterest rate for the RRSP s 10.5%/a, compounded annually. He contrbutes to ths RRSP for 20 years, at the end of whch tme he retres. Upon retrng, Mr. Fabll decdes to transfer ths RRSP to a RIF. How much can he wthdraw from the fund at the end of every three months f the fund earns 9%/a, compounded quarterly, and f he wshes to deplete the fund after 15 years? Use tme lnes to help you solve the problem. Soluton Frst fnd the accumulated or future value of ths RRSP n 20 years. Here s the tme lne, where R 3000, n 20, and 10.5% or Converson perod Depost $3000 $3000 $3000 $3000 $3000 Amount of each depost at end of 20 a $3000 $3000 ( ) $3000 ( ) 2 $3000 ( ) 18 $3000 ( ) USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 161
7 The accumulated or future value of the deposts s the geometrc seres, S (1.105) 3000(1.105) (1.105) (1.105) 19. r n 1) 1 ) n 1 Use ether S n a(, and a 3000, r 1.105, and n 20 or r FV R (1, and R 3000, 0.105, and n 20 to fnd the accumulated value of hs deposts. 5) 20 1 S ( The future value of the RRSP wll be $ Now fnd the quarterly payment that would yeld the present value of $ Here s the tme lne, where n 4 15 or 60, and or Converson perod Payment amount $R $R $R $R $R $R Present value of each payment $R ( ) 1 $R ( ) 2 $R ( ) 58 $R ( ) 59 $R ( ) 60 The present value of the deposts s the geometrc seres S 60 R (1.0225) 60 R (1.0225) 59 R (1.0225) 58 R (1.0225) 2 R (1.0225) 1, where S Solve for R. Use the present value formula PV R 1 (1 ), n where PV , n 4 15 or 60, and or R 1 ( ) 60 Smplfy and solve for R R Therefore, Mr. Fabll can wthdraw $ from the fund at the end of every three months. 162 CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
8 Practse, Apply, Solve 2.8 A 1. Evaluate to sx decmal places. (a) (1.08) 10 (b) (1.12) 6 (c) (1.03) 24 (d) (1.005) 60 (e) (1.075) 20 (f) (1.05) Evaluate to two decmal places. (a) 100(1.1) 24 (1.1) (b) ( (c) 3000(1.05) 10 (1 1.05) (d) ( 0 075) ) Evaluate. (a) 100(1.075) (1.075) (1.075) (1.075) 2 100(1.075) 1 (b) 750(1.001) (1.001) (1.001) (1.001) 2 750(1.001) 1 (c) 1500(1.035) (1.035) (1.035) (1.035) 2 500(1.035) 1 4. For each annuty. fnd the present or dscounted value of each payment at the begnnng of the term. wrte the present or dscounted values of the payments as a seres. fnd the present or dscounted value of the annuty (a) The rate of nterest s 9%/a, compounded annually. Year Depost $8000 $8000 $8000 $8000 $8000 $8000 $8000 (b) The rate of nterest s 8%/a, compounded semannually. Year Depost $300 $300 $300 $300 $300 $300 $300 (c) The rate of nterest s 8%/a, compounded quarterly. Year Depost $750 $750 $750 $750 $750 $750 $750 $ USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 163
9 5. (a) Draw a tme lne to represent the present value of ths annuty: $1200 s deposted at the begnnng of every three months for three years n an account that pays 9%/a, compounded quarterly. (b) Descrbe ths type of annuty. (c) Wrte the present values of all the payments as a seres. (d) Fnd the present value of the annuty. B 6. For each of the followng. draw a tme lne to represent the present value of the annuty. wrte the seres that represents the present value of the annuty. fnd the present value of the annuty (a) $8000 s deposted at the end of every year for 11 years at 7%/a, compounded annually (b) $650 s pad at the end of every 3 months for 6 years at 8%/a, compounded quarterly (c) $60 s deposted at the end of every week for 3 years at 13%/a, compounded weekly (d) $3800 s pad at the end of every 6 months for 8 years at 6.5%/a, compounded semannually 7. The Ontaro Assocaton for Mathematcs Educaton wshes to establsh a math scholarshp to reward the top mathematcs educator n the provnce. The scholarshp wll be worth $2000. The assocaton wll depost a lump sum n a trust account that pays 7.5%/a, compounded annually, for ten years. Determne how much must be deposted now f the frst award wll be gven (a) one year from now (b) mmedately (c) 4 years from now Include a tme lne and a seres to represent each case. 8. Lly wants to buy a snowmoble. She can borrow $7500 at 10%/a, compounded quarterly, f she repays the loan by makng equal quarterly payments for four years. (a) Draw a tme lne to represent the annuty. (b) Wrte the seres that represents the present value of the annuty. (c) Fnd the quarterly payment that Lly must make. 9. Nader plans to buy a used car. He can afford to make payments of $280 each month for a maxmum of three years. The best nterest rate he can fnd s 9.8%/a, compounded monthly. What s the most he can spend on a vehcle? 164 CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
10 10. How much must Mare depost now n a fund payng 6%/a, compounded semannually, f she wthdraws $1000 every sx months, startng sx months from today, for the next ten years? 11. Knowledge and Understandng: Roxanne buys a DVD/CD player for $50 down and ten monthly payments of $40 each. The frst payment s due next month. (a) The nterest rate s 18%/a, compounded monthly. What s the sellng prce of the player? (b) What s the nterest charge? 12. Shmon wants to purchase a speedboat that sells for $22 000, ncludng all taxes. The dealer offers ether a $2000 dscount f Shmon pays the total amount n cash or a fnance rate of 2.4%/a, compounded monthly, f Shmon makes equal monthly payments for fve years. (a) Determne the monthly payment that Shmon must make f he chooses the second offer. (b) What s the total cost of the dealer s fnance plan for the speedboat? (c) To pay for the boat wth cash now, Shmon knows that he can borrow the money from the bank at 6%/a, compounded monthly, over the same fveyear perod. Whch offer should Shmon choose? Justfy your answer. 13. René buys a computer system for $80 down and 18 monthly payments of $55 each. The frst payment s due next month. (a) The nterest rate s 15%/a, compounded monthly. What s the sellng prce of the computer system? (b) What s the fnance charge? 14. The Peca famly buy a cottage on Mantouln Island for $ They plan to depost $5000 and then fnance the remanng amount wth a loan at 9%/a, compounded monthly. The loan payments are monthly. They may choose ether a sevenyear term or a tenyear term. (a) Fnd the monthly payment requred for each term. (b) How much would they save n nterest by selectng the shorter term? (c) What other factors should the Pecas consder when makng ther fnancng decson? 15. Applcaton: A car dealer advertses a new sports car. The sellng prce s $ The dealer offers ths fnance plan: the nterest rate s 2.4%/a, compounded monthly, for fve years wth monthly payments. You can save $3000 by payng cash for the sports car. Suppose you could obtan a loan from a fnancal nsttuton at 5.4%/a, compounded monthly. What s the best way to buy the car? Show your work. 2.8 USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 165
11 16. In Example 3, the nterest rate for the RRSP s 10.5%/a and the nterest rate for the RIF s 9%/a. Determne how much Mr. Fabll would be able to wthdraw from the RIF at the end of every three months f the nterest rate for the RRSP s 9%/a and the nterest rate on hs RIF s 7.5%/a. 17. Communcaton: In Example 3 of secton 2.5, a spreadsheet was used to solve the followng problem: An ad n an electroncs store wndow reads, Bgscreen televson: Was $3270: now $2890 or $500 down and $ for 24 months on approved credt. What s the annual nterest rate, compounded monthly, of the store s payment plan? Dscuss the merts and dffcultes of tryng to solve ths problem usng the present value of an annuty. 18. At the end of each year, Mr. Fox deposts $2600 n an RRSP. The nterest rate s 7.2%/a, compounded annually. He contrbutes to ths RRSP for 20 years, at the end of whch tme he retres. Upon retrng, Mr. Fox decdes to transfer all of ths RRSP to an RIF. The nterest rate for the RIF s 8.4%/a, compounded quarterly. (a) Mr. Fox wshes to deplete the fund after 15 years. Then what wll be the quarterly payment from the RIF? (b) What would be the RIF payment f Mr. Fox had started contrbutng to hs RRSP fve years earler, that s, f he had contrbuted to the RRSP for a total of 25 years? Use tme lnes to help you solve ths problem. 19. A lottery offers wnners two prze choces. Opton A s $1000 each week for lfe and Opton B s $ n one lump sum. The current expected rate of return for a large nvestment s 6.76%/a, compounded weekly. (a) Whch opton would you suggest to a wnner who expects to lve for another 25 years? (b) At what pont n tme s Opton A better than Opton B? 20. Check Your Understandng: The present value of the last payment of an ordnary smple annuty s 2500(1.05) 36. (a) Descrbe two annutes, each wth a dfferent converson perod, that can be represented by the present value of ths last payment. (b) Fnd the present values of all the payments for each annuty n (a). C 21. Thnkng, Inqury, Problem Solvng: Carssa clams that she has found a dfferent method for fndng the present value of a ordnary smple annuty. Instead of fndng the present value of each payment, she fnds the accumulated or future value of each payment. She then fnds the sum of the future values of the payments. Fnally, she fnds the present value of ths total sum. (a) Usng Carssa s method and referrng to Example 1 of secton 2.7, fnd the present value of the annuty where $750 s deposted at the end of every three months n a savngs account that pays 8%/a, compounded quarterly, for ten years. 166 CHAPTER 2 SERIES AND FINANCIAL APPLICATIONS
12 (b) Create another example to show that her clam s true. Include tme lnes. (c) In step 3 of the Thnk, Do, Dscuss n Part 2, you developed the present value formula R(1 ) n [(1 ) n 1. Use ths formula to prove that Carssa s clam works for all smple ordnary annutes. 22. Kyla has just graduated from unversty. She must repay her student loans that total $ She can afford to make monthly payments of $325 each. The bank s nterest rate s 7.2%/a, compounded monthly. Determne how long t wll take Kyla to repay her student loan. 23. Kumar wants to buy a used jeep that costs $7500. He borrows the $7500 at 11.4%/a, compounded monthly. Kumar decdes that he can afford to pay $280 each month toward the loan. How long wll t take to Kumar to repay s loan? Use a spreadsheet to verfy your answer. The Chapter Problem Fnancal Plannng In ths secton, you studed the present value of an annuty. Apply what you learned to answer these questons about the Chapter Problem on page 106. CP13. Can you apply the formula for the present value of an ordnary smple annuty to determne Mr. Sacchetto s new monthly mortgage payment? Explan. CP14. At age 55, Mr. Sacchetto may choose between two RIF optons for nvestng the $ savngs. Opton A provdes monthly payments from an RIF that pays 7.2%/a, compounded monthly. Opton B provdes monthly payments from an RIF that pays 7.25%/a, compounded quarterly. (a) Can you use the formula for the present value of an ordnary smple annuty to solve for the RIF payment under each opton? Explan. (b) For opton A, draw a tme lne to represent the present value of the payments he would receve. (c) Determne the monthly RIF payment he would receve under opton A. CP15. Use the formula for the present value of an ordnary smple annuty to estmate the term of Mr. Sacchetto s car loan. 2.8 USING SERIES TO ANALYZE FINANCIAL SITUATIONS: PRESENT VALUE 167
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