10.5 Future Value and Present Value of a General Annuity Due

Transcription

1 Chapter 10 Autes Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the car at the ed of the lease? 6. Valeta leases a car worth $8,000 at 4.75% compouded mothly. She agrees to make 48 lease paymets of $450 each at the begg of every moth. What s the buyout prce (resdual value of the car at the ed of the lease? 7. Etha coverted hs RRSP to a RRIF that pays hm $1500 at the begg of every moth for three years followed by $000 at the begg of every moth for the ext fve years. If the rate of retur of the auty s 3.85% compouded mothly durg the etre perod, what was the value of the RRIF? 8. Samuel s cosderg purchasg a auty wth a rate of retur of 6% compouded quarterly whch pays 0 equal paymets of $1000 at the begg of every 3 moths, followed by aother 0 equal paymets of $1500 at the begg of every 3 moths. How much should he pay for the auty? 10.5 Future Value ad Preset Value of a Geeral Auty Due I a geeral auty due, paymets are made at the begg of each paymet perod, ad the compoudg perod s ot equal to the paymet perod. I ths secto, you wll lear how to calculate the future value ad preset value of a geeral auty due. Smlar to the ordary geeral autes, we frst eed to calculate the equvalet perodc terest rate per paymet perod ( usg Formulas 10.3(a ad 10.3(b: Number of compoudg perods per year Number of paymets per year = (1 + c - 1 The, substtutg the value of ' ' for '' the smple auty due Formula 10.4(a ad Formula 10.4(b, solve for 'FV Due ' ad 'PV Due ', as follows: ^1+ h ^1+ h FV Due = PMT ; E (1+ ad PV Due = PMT ; - E (1+ Note: I usg the Texas Istrumets BA II Plus calculator for auty due calculatos, set the calculator to begg of perod calculatos ('BGN' mode as descrbed Secto 10.. Example 10.5(a Calculatg the Future Value, Amout Ivested, ad Total Iterest of a Geeral Auty Due Tao vested $5000 a fud at the begg of every three moths for fve years. The fud was earg a terest rate of 5% compouded mothly. ( What was the total amout vested? ( What was the accumulated value of the vestmet? ( What was the terest eared over the perod? Ths s a geeral auty due as: Paymets are made at the begg of each paymet perod (quarter Compoudg perod (mothly paymet perod (quarter

2 37 Chapter 10 Autes cotued ( The total amout vested the fud = (PMT = 0 # = $100, ( The accumulated value of the vestmet Number of compoudg perods per year = 1 Number of paymets per year 4 = (1 + c - 1 = ( (1/4-1 = per quarter Usg Formula 10.4(a ad substtutg for, ^1+ h - 1 FV Due = PMT ; E (1+ ^ h - 1 = 5000 ; E ( = 5000 [ ] ( = 114, Therefore, the accumulated value of hs vestmet would be $114, ( The terest eared over the perod = FV Due - (PMT = 114, ,000 = $14,89.31 Therefore, the terest eared over the perod s $14, Example 10.5(b Calculatg the Preset Value of a Geeral Auty Due Alexadra herted moey that was vested a accout whch provded her $4500 at the begg of every moth for 30 years. If the terest rate o the savgs accout was 4% compouded sem-aually, what was the amout of the hertace? Roud your aswer to the earest hudred dollars. Ths s a geeral auty due as: Paymets are made at the begg of each paymet perod (mothly Compoudg perod (sem-aually paymet perod (mothly Number of compoudg perods per year = Number of paymets per year 1

3 Chapter 10 Autes 373 cotued = (1 + c - 1 = ( (/1-1 = per moth Usg Formula 10.4(b ad substtutg for, 1- ^1+ h - PV Due = PMT ; E ( ^ fh = 4500 ; E f ( = 4500 [ ]( = $949, Therefore, she herted $949, (rouded to the earest $100. Example 10.5(c Book value of a asset (that s leased s ts value o a specfc date ad ca be calculated by fdg the preset value of the remag lease paymets at that tme. Choosg Betwee Lease ad Buy Optos ad Calculatg the Book Value A suppler of machery provded a compay wth the followg two optos to lease or buy equpmet: Buy Opto: Pay $17,500 mmedately to ow the equpmet. Lease Opto: Make lease paymets of $180 at the begg of every moth for four years. At the ed of four years, pay $10,000 to ow the equpmet. The cost of borrowg s 3% compouded aually. ( Whch opto s ecoomcally better for the compay? ( I the lease opto, what wll be the book value of the equpmet at the ed of three years? The lease paymets form a geeral auty due as: Paymets are made at the begg of each paymet perod (mothly Compoudg perod (aually paymet perod (mothly ( PV Lease opto = PV PMT + PV Resdual Value Evaluatg Lease or Buy Optos Number of compoudg perods per year 1 = Number of paymets per year 1 = ( (1/1-1 = per moth Substtutg ' ' Formula 10.(b 1- ^1+ h - PV Due = PMT ; E (1+ 48 PV PMT = 180 < 1 - ] g - F( = 180[ ]( =

4 374 Chapter 10 Autes Usg the facal calculator, you ca ether compute PV separately ad add the aswers together, or you ca do the etre calculato oe step as show here. Usg Formula 9.1(b, PV = FV(1 + - PV $10,000 = 10,000( (1#4 = 10,000( = PV Lease Opto = PV PMT + PV $10,000 = , = $17, The preset value of the lease opto s less tha the purchase prce of $17,500. Therefore, the lease opto s ecoomcally better. ( Calculatg the book value at the ed of three years = Remag lease paymets = 1 mothly paymets Book Value Three years = PV Remag paymets + PV $10,000 1 = 180 < 1 - ] g - F( ( ,000( = = 11, = $11, Therefore, the lease opto, the book value of the equpmet at the ed of three years wll be $11, Exercses Aswers to the odd-umbered problems are avalable at the ed of the textbook 1. Calculate the future value of the followg geeral autes due: Perodc Paymet Paymet Perod Term of Auty Iterest Rate Compoudg Frequecy a. $3750 Every year 16 years 4.80% Mothly b. $1950 Every 6 moths 1.5 years 4.00% Quarterly c. $1500 Every 3 moths 9 years ad 9 moths 3.75% Daly d. $100 Every moth 6 years 7 moths 3.90% Sem-aually. Calculate the future value of the followg geeral autes due: Perodc Paymet Paymet Perod Term of Auty Iterest Rate Compoudg Frequecy a. $3600 Every year 6 years 4.50% Daly b. $150 Every 6 moths 9.5 years 5.10% Quarterly c. $750 Every 3 moths 5 years ad 9 moths 4.60% Mothly d. $1950 Every moth 1 years 8 moths 6.00% Sem-aually 3. Calculate the preset value of each of the geeral autes due Problem Calculate the preset value of each of the geeral autes due Problem.

5 Chapter 10 Autes What would the accumulated value of a fud be f $500 was deposted at the begg of every three moths for years ad 9 moths ad the fud was earg.85% compouded mothly? 6. As part of hs New Year's resoluto, Clve commtted to save $500 per moth a mutual fud accout. He deposted the moey at the begg of every moth ad projected that he would ear 8.5% compouded aually o the fuds. At the ed of 1 moths, how much dd Clve accumulate? 7. At the begg of every 3 moths, $450 s deposted to a savgs accout for 15 years. The moey remaed the accout for aother 5 years ad eared a average of 3.45% compouded mothly durg the 0 year perod. Calculate the accumulated amout ad the terest eared the accout. 8. A gradmother opeed a savgs accout for her graddaughter o the day she was bor ad deposted $1000. Each year o her brthday, she deposted $1000 for 16 years. If the accout eared 4.38% compouded daly, how much was the accout o her graddaughter's 1 st brthday ad what s the terest eared? 9. What wll be the value of a lease cotract f t requres you to make lease paymets of $450 at the begg of each moth for fve years? Assume a terest rate of 6% compouded sem-aually. 10. What should be the amout a RRSP that s earg 5.5% compouded quarterly f t ca be coverted to a RRIF that wll provde $3000 at the begg of every moth for te years? 11. Marl secured a lease o a mache by payg $4500 as a dow paymet ad lease paymets of $750 at the begg of every moth for a perod of fve years. At the ed of fve years, he would have to pay $5000 to ow the mache. If moey s worth 5.55% compouded quarterly, calculate the prce of the mache. 1. Na leased a car for four years by makg a dow paymet of $1500 ad lease paymets of $180 at the begg of every moth for four years. She could ow the car by payg $4000 at the ed of the lease. If moey s worth 4% compouded quarterly, calculate the prce of the car. 13. Lyda deposted $10,000 to a savgs accout at the begg of every year for 10 years. The accout was growg at 3% compouded mothly. After the 10 year perod, she left the accumulated moey the accout to grow for aother year. a. What was the balace the accout at the ed of 11 years? b. What was the total terest eared? 14. Chase has bee cotrbutg $1500 to a retremet fud at the begg of each quarter for the past 15 years. He decded to stop makg paymets ad to allow hs vestmet to grow for aother 5 years. If moey ca ear 8% compouded mothly, how much terest would he have eared over the 0-year perod? 15. Your retremet fud curretly has a balace of $40,000 ad you depost to ths fud $400 at the begg of every moth startg from today for the ext 15 years. Calculate the accumulated value of the fud ad the amout of terest eared f the fud ears 4.5% compouded quarterly. 16. You depost $5,000 to a savgs accout yeldg 3.5% compouded sem-aually. I addto, you pla to depost $1000 at the begg of every three moths for te years to ths accout. Determe the total amout you wll have at the ed of te years ad the terest eared. 17. A cotract requres lease paymets of $400 at the begg of every moth for te years. a. What s the value of the cotract f the cost of borrowg s 4.5% compouded aually? b. What s the value of the cotract f the cost of borrowg s 4.5% compouded daly? 18. Soog decded to retre Caada ad wats to receve a retremet come of $4000 at the begg of every moth for 15 years from her savgs. If a bak Vacouver s offerg her a terest rate of 5% compouded aually, how much would she have to vest to get her plaed retremet come? What s the facal beeft for her to vest a bak Nagara Falls f they are offerg her a terest rate of 4.99% compouded quarterly?

6 376 Chapter 10 Autes 19. How much lesser wll the value of a RRSP accout be at the ed of te years f you cotrbute $00 at the begg of every moth stead of $400 at the begg of every year? Assume the accout ears a terest rate of 5. compouded sem-aually both cases. 0. Natha deposted $500 at the begg of each moth to a fud. Hs wfe deposted $1500 at the begg of every three moths to a smlar fud. If they cotue these deposts, whch fud wll have a larger balace at the ed of 15 years? The fuds ear terest at 6.% compouded sem-aually. 1. How much should Ma Corp. pay for a auty that would gve t $0,000 at the begg of every year for the frst fve years ad $30,000 at the begg of every year for the ext three years, f the terest rate s 7% compouded mothly throughout the term?. What s the purchase prce of a auty that provdes begg-of-moth paymets of $400 for the frst three years ad $600 for the ext two years? Assume that the terest rate s 4% compouded quarterly throughout the tme perod. 3. The mothly ret o a apartmet s $1750 per moth, payable at the begg of each moth. What sgle paymet advace o the frst day of retal would be equal to three years' ret? Assume a terest rate of 4.38% compouded daly. 4. Machery ca be leased for your maufacturg plat wth a paymet of $4500 at the begg of every three moths. What sgle paymet advace o the frst day of the lease would be equal to four years of lease paymets? Assume a terest rate of 4.4% compouded mothly. 5. Jasme, a offce admstrator, was evaluatg the followg quotato that she receved for the purchase of a prtg mache for her compay: Lease Opto: Make paymets of $750 at the begg of every moth for two years. At the ed of two years pay $3000 to ow the prter. Purchase Opto: Make a paymet of $19,500 mmedately. The cost of borrowg s 6% compouded sem-aually. a. Whch opto s ecoomcally better for the compay? b. I the lease opto, what wll be the book value of the prter at the ed of oe year? 6. Madso had the opto of purchasg a mache for $59,000 or makg lease paymets of $100 at the begg of every moth for four years ad the payg $15,000 at the ed of the fourth year to ow t. The cost of borrowg moey s 7% compouded quarterly. a. Whch opto s ecoomcally better for her? b. I the lease opto, what wll be the book value of the mache at the ed of three years? 7. Ryder vested $1000 at the begg of every three moths to hs RRSP accout from age 35 to 45 ad left the moey to grow utl hs retremet at the age of 65. Bll vested $1000 at the begg of every three moths to a smlar RRSP accout from age 45 utl hs retremet at the age of 65. Assumg moey ears 5.8% compouded aually both RRSP accouts, calculate who had the greater accumulated value ad by how much whe they retred? 8. Cha made deposts of $50 to a fud at the begg of every moth from age 40 to 48. He left the moey to accumulate the fud utl he reached the age of 60. Davd made deposts of $50 at the begg of every moth to a RRSP from age 48 utl he reached the age of 60. Assumg moey ears 6.5% compouded semaually both cases, calculate who had the greater accumulated value ad by how much whe they reached 60 years of age?

7 Chapter 10 Autes A lottery wer s offered a choce of ether recevg $0,500 at the begg of every year for te years or recevg $18,000 at the begg of every sx moths for fve years. If the terest rate s 4.4% compouded quarterly, whch choce s ecoomcally better ( curret value ad by how much more? 30. You have a choce of ether recevg $15,000 at the begg of every year for te years or recevg $50 at the begg of every moth for fve years. If the terest rate s 4.8% compouded sem-aually, whch choce s ecoomcally better ( curret value ad by how much more? Calculatg Perodc Paymet (PMT 3. If the future value of a auty, the perodc terest rate, '', ad the umber of paymet perods, '', are kow, we ca use the future value formula to solve for the ukow 'PMT'. Smlarly, f the preset value of a auty, the perodc terest rate, '', ad the umber of paymet perods, '', are kow, we ca use the preset value formula to solve for the ukow 'PMT'. Example 10.6(a Calculatg the Perodc Paymet of a Ordary Smple Auty, Gve Preset Value Rodey receved a $35,000 home mprovemet loa from hs bak at a terest rate of 6% compouded mothly. ( How much would he eed to pay every moth to settle the loa te years? ( What was the amout of terest charged? Ths s a ordary smple auty as: Paymets are made at the ed of each paymet perod (mothly Compoudg perod (mothly = paymet perod (mothly ( Usg Formula 10.(b, - PV = PMT 1- ^1+ h ; E - 1- ^ h 10 35,000 = PMT ; E Solvg for 'PMT', we obta, , , 000 PMT = -10 = 1- ^ h 690. = $ ; E Therefore, he would eed to pay $ every moth to settle the loa te years. ( Iterest Charged over the tme perod = (PMT - PV = 10( ,000 = $11,68.40 Therefore, the amout of terest charged durg the perod of the loa would be $11,68.40.