7.5. Present Value of an Annuity. Investigate


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1 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on whch to lve. They hre a fnancal advsor, and together they consder whether Owen and Anna have enough money to allow them to lve comfortably for the rest of ther lves by makng regular wthdrawals from an account. To do ths, they calculate the present value of an annuty based on Owen and Anna s projected lvng expenses. regular wthdrawals wthdrawals of equal value drawn at equal perods present value of an annuty the amount of money needed to fnance a seres of regular wthdrawals Investgate How can you determne the present value of an annuty? Owen and Anna have estmated that they wll need to wthdraw $ per month for lvng expenses for the next 20 years, and wonder f they wll have enough to fnance ths. The amount n ther account wll earn 9% annual nterest, compounded monthly.. a) How many wthdrawals have Owen and Anna planned for? How do you know? b) Multply the total number of wthdrawals by $. Do Owen and Anna need to have ths much n ther lfe savngs account on the day they retre? Explan why or why not. 2. Suppose Owen and Anna retre at the end of December. They wll make ther frst wthdrawal from ther lfe savngs account at the end of January. a) Determne the present value of ths frst wthdrawal, usng the formula PV 5 FV ( ). n b) Determne the present value of the second wthdrawal, whch Owen and Anna wll make at the end of February. c) Are the present values of each wthdrawal equal? Explan why or why not. 456 MHR Functons Chapter 7
2 d) Predct whether the present value of the thrd wthdrawal wll be greater than or less than these values. Explan your predcton. Check your predcton by calculatng. 3. Wll the pattern of present values observed n step 2 contnue? Explan your thnkng. 4. Reflect Suggest a method to determne the sum of the present values of all of the wthdrawals that Owen and Anna plan to make after they retre. In Secton 7.4, problems were posed n whch regular payments are made nto an account that grows to a large future amount. Present Value = 0 Regular Payments Future Value In ths secton, problems wll be posed n whch regular wthdrawals wll be made from an account that begns wth a large balance. Present Value Regular Wthdrawals Future Value = 0 To determne the present value requred to fnance a retrement plan such as the one n the Investgate, t s necessary to calculate the present value of each wthdrawal usng the present value formula PV 5 FV ( ). n Tme (months) The present value of the Now annuty can be determned by addng the present values of all the wthdrawals. (.0075) (.0075) 2 (.0075) 3 (.0075) 239 (.0075) Present Value of an Annuty MHR 457
3 PV Snce ths s a geometrc seres wth frst term a 5 and common.0075 rato r 5, the formula for the sum of the frst n terms of a.0075 geometrc seres S n 5 a(r n ) can be used. r Ths process can be generalzed to produce a smplfed result for the present value of an annuty. You can derve ths result n queston 5. The present value, PV, of an annuty can be determned usng the formula PV 5 R[ ( ) n ], where R represents the regular wthdrawal; represents the nterest rate per compoundng perod, as a decmal; and n represents the number of compoundng perods. Example Present Value of an Annuty Josh s puttng hs summer earnngs nto an annuty from whch he can draw lvng expenses whle he s at unversty. He wll need to wthdraw $ per month for 8 months. Interest s earned at a rate of 6%, compounded monthly. a) Draw a tme lne to represent ths annuty. b) How much does Josh need to nvest at the begnnng of the school year to fnance the annuty? Soluton a) Represent the known nformaton on a tme lne. R 5 n _ Now (.005) (.005) 2 Tme (months) (.005) 3 (.005) 7 (.005) MHR Functons Chapter 7
4 b) Method : Use a Scentfc Calculator Substtute the known values nto the formula for the present value of an annuty and evaluate. PV 5 R[ ( ) n ] _ [ ( 0.005) _ 5 ( ) (.005 y x 8 ) Josh needs to nvest $ at the begnnng of the school year to fnance the annuty. Method 2: Use a TVM Solver Calculator key strokes may vary. Access the TVM Solver on a graphng calculator and enter the values, as shown. Move the cursor to the PV feld and press ALPHA [SOLVE]. The present value of ths annuty s $ , whch s the amount that Josh must nvest at the begnnng of the school year. Example 2 Determne the Regular Wthdrawal Fona s lfe savngs total $ when she decdes to retre. She plans an annuty that wll pay her quarterly for the next 30 years. If her account earns 5.2% annual nterest, compounded quarterly, how much can Fona wthdraw each quarter? Soluton Determne the number of compoundng perods and the nterest rate per compoundng perod. n PV Use the formula for the present value of an annuty to solve for the regular wthdrawal, R. 7.5 Present Value of an Annuty MHR 459
5 Method : Substtute and Then Rearrange PV R[ ( ) n] R[ ( 0.03) 20] _ R(.03 20) 3 R5 R (.03 20) Substtute the known values. Multply both sdes by Dvde both sdes by Fona can wthdraw $ every quarter for 30 years. Method 2: Rearrange and Then Substtute PV 5 R[ ( ) n] (PV) 5 R[ ( ) n] R5 (PV) R5 0.03( ) _ R ( ) n ( 0.03) 20 Multply both sdes by. Dvde both sdes by ( + ) n. Substtute the known values. Fona can wthdraw $ every quarter for 30 years. Te c h n o l o g y Tp When workng wth annutes usng a TVM Solver, you must specfy the actual number of payments or wthdrawals, N. The graphng calculator software does not automatcally assume that ths concdes wth the number of compoundng ntervals. Method 3: Use a TVM Solver Access the TVM Solver on a graphng calculator and enter the values, as shown. Note that the present value s negatve, ndcatng that ths amount s pad nto the account. Move the cursor to the PMT feld and press ALPHA [SOLVE]. The amount of the regular wthdrawals s $ Note that ths value s postve, ndcatng that Fona wll receve these payments. Key Concepts The present value of an annuty s the total amount that can fnance a seres of regular wthdrawals over a specfc perod of tme. R[ ( ) n] The present value, PV, of an annuty can be calculated usng the formula PV 5, where R represents the regular wthdrawal; represents the nterest rate per compoundng perod, as a decmal; and n represents the number of compoundng perods. 460 MHR Functons Chapter 7 Functons CH07.ndd 460 6/0/09 4:24:36 PM
6 Communcate Your Understandng C The tme lne shows an annuty from whch semannual wthdrawals are made for 0 years. Interest s compounded semannually. a) What s the annual rate of nterest? How can you tell? b) How many wthdrawals wll be made, n total? How can you tell? Now (.035) Tme (6month perods) c) Explan why ths annuty can be represented as a geometrc seres. (.035) 2 d) Identfy the frst term, a, and the common rato, r, of the geometrc seres. (.035) 3 (.035) 9 (.035) 20 C2 The graphng calculator screen of a soluton usng the TVM Solver s shown. Descrbe ths annuty fully. A Practse For help wth questons to 3, refer to Example.. Calculate the present value of the annuty shown. Now (.035) (.035) 2 (.035) 3 (.035) 9 (.035) 20 Tme (6month perods) Brandon plans to wthdraw $ at the end of every year, for 4 years, from an account that earns 8% nterest, compounded annually. a) Draw a tme lne to represent ths annuty. b) Determne the present value of the annuty. 3. Lauren plans to wthdraw $650 at the end of every 3 months, for 5 years, from an account that earns 6.4% nterest, compounded quarterly. a) Draw a tme lne to represent ths annuty. b) Determne the present value of the annuty. c) How much nterest s earned? 7.5 Present Value of an Annuty MHR 46
7 For help wth questons 4 and 5, refer to Example An annuty has an ntal balance of $8000 n an account that earns 5.75% nterest, compounded annually. What amount can be wthdrawn at the end of each of the 6 years of ths annuty? 5. After graduatng from hgh school, Karen works for a few years to save $ for unversty. She deposts her savngs nto an account that wll earn 6% nterest, compounded quarterly. What quarterly wthdrawals can Karen make for the 4 years that she wll be at unversty? B Connect and Apply 6. How much should be n an account today so that wthdrawals n the amount of $5 000 can be made at the end of each year for 20 years, f nterest n the account s earned at a rate of 7.5% per year, compounded annually? 7. Jule just won Reasonng and Provng $ n Representng a lottery! She Problem Solvng estmates that to Connectng lve comfortably she wll need to Communcatng wthdraw $5000 per month for the next 50 years. Her savngs account earns 4.25% annual nterest, compounded monthly. Selectng Tools Reflectng a) Can Jule afford to retre and lve off her lottery wnnngs? b) What s the mnmum amount that Jule must wn to retre n comfort mmedately? Dscuss any assumptons you must make. 8. An annuty has an ntal balance of $5000. Annual wthdrawals are made n the amount of $800 for 9 years, at whch pont the account balance s zero. What annual rate of nterest, compounded annually, was earned over the duraton of ths annuty? 9. Shen has nvested $5 000 nto an annuty from whch she plans to wthdraw $500 per month for the next 3.5 years. If at the end of ths tme perod the balance of the annuty s zero, what annual rate of nterest, compounded monthly, dd ths account earn? 0. Jordan has $6000 to nvest n an Representng annuty from whch he plans Connectng to make regular wthdrawals over the next 3 years. He s consderng two optons: Reasonng and Provng Problem Solvng Communcatng Opton A: Wthdrawals are made every quarter and nterest s earned at a rate of 8%, compounded quarterly. Opton B: Wthdrawals are made every month and nterest s earned at a rate of 7.75%, compounded monthly. a) Determne the regular wthdrawal for each opton. b) Determne the total nterest earned for each opton. c) Dscuss the advantages and dsadvantages of each opton from Jordan s perspectve. Selectng Tools Reflectng. Ronald and Benjamn need to take out a small loan to help expand ther petgroomng busness, Fluff n Shne. They estmate that they can afford to pay back $250 monthly for 3 years. If nterest s 6%, compounded monthly, how much of a loan can Ronald and Benjamn afford? 2. Chapter Problem Chloe has acheved her ntal fnancal goal of growng her nvestments to $2 000 after 4 years. She deposts ths amount nto a new account that earns 5% per year, compounded quarterly. Now her ntenton s to make regular wthdrawals from ths account every 3 months for the next 2 years. How much wll Chloe s regular wthdrawals be? 462 MHR Functons Chapter 7
8 3. Jose plans to nvest $0 000 at the end of each year for the 25 years leadng up to her retrement. After she retres, she plans to make regular wthdrawals for 25 years. Assume that the nterest rate over the next 50 years remans constant at 7% per year, compounded annually. a) Once she retres, whch amount do you predct that Jose wll be able to wthdraw per year? less than $0 000 $0 000 more than $0 000 Explan your answer. b) Estmate how much she wll be able to wthdraw. Provde reasonng for your estmate. c) Determne the amount of Jose s nvestment annuty on the day she retres. d) Use ths amount to determne the regular wthdrawal she can make at the end of each year for 25 years after retrement. e) Compare your answer n part d) to your estmate n part b). How close was your estmate? Achevement Check 4. Abraham s grandparents plan to set up an annuty to help hm when he moves nto an apartment to attend college. Abraham wll be able to wthdraw $3000 at the end of each year for 4 years. The frst wthdrawal wll be made year from now, when Abraham begns college. If the annuty earns 7% nterest, compounded annually, how much should Abraham s grandparents nvest now to fnance the annuty? Use two dfferent methods to solve ths problem. C Extend 5. For a smple annuty wth regular payment R and nterest rate,, as a decmal, per compoundng perod, use the formula for the sum of a geometrc seres, S n 5 a(r n ), to derve the formula r for the present value of the annuty, PV 5 R[ ( ) n ]. 6. By law, mortgage nterest rates must be stated as an annual rate compounded semannually. However, payments are usually made monthly, so the nterest rate must be converted to a monthly rate. A $ mortgage has an amortzaton perod of 25 years, at an nterest rate of 5%, compounded semannually. a) Use the compound nterest formula to determne the equvalent nterest rate compounded monthly. b) Determne the monthly payment requred to pay off ths mortgage over 25 years. Connectons A mortgage s a type of loan by whch a house or other real estate property s used as a guarantee of repayment of the debt. The total tme over whch the loan s repad s called the amortzaton perod. 7. A mortgage of $ s amortzed over 25 years wth an nterest rate of 6.7%, compounded semannually. a) What s the monthly payment? b) Suppose you choose to make weekly payments nstead of monthly payments. What s the weekly payment? c) Calculate the total nterest pad wth the weekly payments. 7.5 Present Value of an Annuty MHR 463
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