Section 2.3 Present Value of an Annuity; Amortization


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1 Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1, every 6 months for next 3 years. r m 2 A = 1000 = PMT (perodc payment) n 1 A n n A P P A n (1.03) (1.03) (1.03) (1.03) (1.03) (1.03) 6 Years Perod P (1 ) P R n Present Value (PV) of an ordnary annuty: 1 (1 ) PV PMT n : Rate per perod n: Number of perods Notes: Payments are made at the end of each perod. 16
2 Example A car costs $,000. After a down payment of $2,000, the balance wll be pad off n 36 equal monthly payments wth nterest of 6% per year on the unpad balance, Fnd the amount of each payment. Gven: P, 000 2, , 000 n = PV PMT , 000 PMT , PMT n $13, $ / ^ (( )36) Example An annuty that earned 6.5%. A person plans to make equal annual deposts nto ths account for 25 years n order to them make 20 equal annual wthdrawals of $25,000 reducng the balance n the amount to zero. How much must be deposted annually to accumulate suffcent funds to provde for these payments? How much total nterest s earned durng ths entre 45years process? r = annually PMT 25 k 1 25yrs 20yrs Increasng decreasng FV PV 1 (1 ) n PV PMT $275, (1.065) ^ ( ) 20 /.065 FV PV (1 ) n 1 FV PMT PMT FV PMT FV (1 ) n 1 (1.065) 25 1 $4, (1 ) n 1 Wthdraw Depost Total nterest = 20(25000) 25( ) = $
3 Amortzaton Amortzaton debt means the debt retred n gven length (= payment), Borrow money from a bank to buy and agree to payment perod (36 months) Example Borrow $5000 payment n 36 months, compounded r = %. How much payment?..01 n = 36 1 (1 ) n PV PMT PMT (1.01) 36 $ per month 5000*.01 / (11.01^ ()36 )) Example If you sell your car to someone for $2,400 and agree to fnance t at 1% per month on the unpad balance, how much should you receve each month to amortze the loan n 24 months? How much nterest wll you receve? 1 (1 ) n PV PMT PMT (1.01) 24 $1.98 per month Total nterest = amount of all payment ntal loan = 24(1.98) 2400 = $ *.01/(1 (1 +.01) ^ ()24 ) ) 18
4 Amortzaton Schedules Pay off earler last payment (lump sum) = Amortzaton schedules Example If you borrow $500 that you agree to repay n sx equal monthly payments at 1% nterest per month on the unpad balance, how much of each monthly payment s used for nterest and how much s used to reduce the unpad balance PMT $86.27 per month 1 (1.01) The end of the 1 st month nterest due = 500(.01) = $ (.01 / ( )^( ) 6 Pmt # Payment Interest Reducton Unpad Balance 0 $ $ = = $ $ (.01) = = = $ $ (.01) = = = $ $ = $ $ = $ $ = $0.0 Example Construct an amortzaton schedule for a $1,000 debt that s to be amortzed n sx equal monthly payment at 1.25% nterest rate per month on the unpad balance (1.05) 6 PMT 1000 $ per month 1 st month nterest due = 1000(.05) = $ (.05 / ( ) ^ ( ) 6 # Payment Interest Reducton Unpad Balance 0 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $0.0 $ $44.21 Total = $1000
5 Equty Equty = Current net market value Unpad balance Example A famly purchase a home 10 years ago for $80, The home was fnanced by payng 20% down for 30year mortgage at 9%, on the unpad balance. The net market of the house s now $0, and the famly wshes to sell the house. How much equty after makng 0 monthly payments? Equty = Current Net Unpad Balance Unpad balance ( 20 ) 0 10 yrs 30 Down Payment = 20 % Left 80% =.8(80000) = 64, n = (30) = Monthly Payment? PMT PV 1 (1 ) n 64, (1.0075) ,000(.0075 / ( ) ^ ( *30) $ per month Unpad balance 10 years (now) = 20 years 1 (1 ) PV PMT n 1 (1.0075) (( ) ^ 240) / $57, Equty = current unpad balance = 0,000 57,235 = $ 62,
6 Exercses Secton 2.3 Present Value of an Annuty Amortzaton 1. How much should you depost n an account payng 8% compounded quarterly n order to receve quarterly payments of $1,000 for the next 4 years? 2. You have negotated a prce of $25,200 for a new truck. Now you must choose between 0% fnancng for 48 months or a $3,000 rebate. If you choose the rebate, you can obtan a loan for the balance at 4.5% compounded monthly for 48 months. Whch opton should you choose? 3. Suppose you have selected a new car to purchase for $19,500. If the car can be fnanced over a perod of 4 years at an annual rate of 6.9% compounded monthly, how much wll your monthly payments be? Construct an amortzaton table for the frst 3 months. 4. Suppose your parents decde to gve you $10,000 to be put n a college trust fund that wll be pad n equally quarterly nstallments over a 5 year perod. If you depost the money nto an account payng 1.5% per quarter, how much are the quarterly payments (Assume the account wll have a zero balance at the end of perod.) 5. You fnally found your dream home. It sells for $0,000 and can be purchased by payng 10% down and fnancng the balance at an annual rate of 9.6% compounded monthly. a) How much are your payments f you pay monthly for 30 years? b) Determne how much would be pad n nterest. c) Determne the payoff after 100 payments have been made. d) Change the rate to 8.4% and the tme to 15 years and calculate the payment. e) Determne how much would be pad n nterest and compare wth the prevous nterest. 6. Sharon has found the perfect car for her famly (anew mnvan) at a prce of $24,500. She wll receve a $3500 credt toward the purchase by tradng n her old Gremln, and wll fnance the balance at an annual rate of 4.8% compounded monthly. a) How much are her payments f she pays monthly for 5 years? b) How long would t take for her to pay off the car payng an extra $100 per mo., begnnng wth the frst month? 7. Mare has determned that she wll need $5000 per month n retrement over a 30year perod. She has forecasted that her money wll earn 7.2% compounded monthly. Mare wll spend 25years workng toward ths goal nvestng monthly at an annual rate of 7.2%. How much should Mare s monthly payments be durng her workng years n order to satsfy her retrement needs? 8. Amercan General offers a 10year ordnary annuty wth a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annutes f you want to receve payments of $5,000 annually over the 10year perod? 21
7 9. Amercan General offers a 7year ordnary annuty wth a guaranteed rate of 6.35% compounded annually. How much should you pay for one of these annutes f you want to receve payments of $10,000 annually over the 7year perod? 10. You want to purchase an automoble for $27,300. The dealer offers you 0% fnancng for 60 months or a $5,000 rebate. You can obtan 6.3% fnancal for 60 months at the local bank. Whch opton should you choose? Explan. 11. You want to purchase an automoble for $28,500. The dealer offers you 0% fnancng for 60 months or a $6,000 rebate. You can obtan 6.2% fnancal for 60 months at the local bank. Whch opton should you choose? Explan.. Construct the amortzaton schedule for a $5,000 debt that s to be amortzed n eght equal quarterly payments at 2.8% nterest per quarter on the unpad balance. 13. Construct the amortzaton schedule for a $10,000 debt that s to be amortzed n sx equal quarterly payments at 2.6% nterest per quarter on the unpad balance. 14. A loan of $37,948 wth nterest at 6.5% compounded annually, to be pad wth equal annual payments over 10 years 15. A loan of $4,836 wth nterest at 7.25% compounded semannually, to be repad n 5 years n equal semannual payments. 22
8 Secton 2.3 Present Value of an Annuty Amortzaton Exercse How much should you depost n an account payng 8% compounded quarterly n order to receve quarterly payments of $1,000 for the next 4 years? Gven: 1 1 PV PMT PMT 1, 000 r 8%.08, m 4, t 4 r n 4(4) 16 m 4 n $13, ^ 16 /.02 Exercse You have negotated a prce of $25,200 for a new truck. Now you must choose between 0% fnancng for 48 months or a $3,000 rebate. If you choose the rebate, you can obtan a loan for the balance at 4.5% compounded monthly for 48 months. Whch opton should you choose? 0% fnancng: Gven: P 25,200 r 0% 0, t 48 mth PMT 25, $525 Rebate: Gven: P 25,200 Rebate $3,000 r 4.5%.045, m n t 48 mth PV 25, 200 3, 000 $22, PMT 2 PV , 200 $ n ( / ( ^ 48)) Rebate s better and you save = $ per month Or * 48 = $ (over the loan) 22
9 Exercse Suppose you have selected a new car to purchase for $19,500. If the car can be fnanced over a perod of 4 years at an annual rate of 6.9% compounded monthly, how much wll your monthly payments be? Construct an amortzaton table for the frst 3 months / PMT ( / ) 48 I 19500(0.069 / ) (0.069 / ) / (1 ( / ) ^ 48 Pmt # Pmt Amount Interest Reducton Unpad Bal Exercse Suppose your parents decde to gve you $10,000 to be put n a college trust fund that wll be pad n equally quarterly nstallments over a 5 year perod. If you depost the money nto an account payng 1.5% per quarter, how much are the quarterly payments (Assume the account wll have a zero balance at the end of perod.) Gven: PV 10, 000 r 1.5%.015, m 4, t 5 r.015 n 4(5) PMT 10, 000 $ ( ) (0.015) / (1 ( ) ^ 20) 23
10 Exercse You fnally found your dream home. It sells for $0,000 and can be purchased by payng 10% down and fnancng the balance at an annual rate of 9.6% compounded monthly. a) How much are your payments f you pay monthly for 30 years? b) Determne how much would be pad n nterest. c) Determne the payoff after 100 payments have been made. d) Change the rate to 8.4% and the tme to 15 years and calculate the payment. e) Determne how much would be pad n nterest and compare wth the prevous nterest. a) / PMT $ ( / ) b) I 360(916.01) $ c) PV ( / ) $ / d) PMT / $ ( / ) e) I 180( ) $ I1 I Exercse Sharon has found the perfect car for her famly (anew mnvan) at a prce of $24,500. She wll receve a $3500 credt toward the purchase by tradng n her old Gremln, and wll fnance the balance at an annual rate of 4.8% compounded monthly. a) How much are her payments f she pays monthly for 5 years? b) How long would t take for her to pay off the car payng an extra $100 per mo., begnnng wth the frst month? a) b) / PMT $ ( / ) n 1 ( / ) / Dvde both sdes by
11 1 ( ) n n n multply by n ln both sdes n ln1.004 ln n ln / ln ln n 47 mo. n ln1.004 Money s compounded monthly; t can t be compounded at months. Bump to 47mo. Exercse Mare has determned that she wll need $5000 per month n retrement over a 30year perod. She has forecasted that her money wll earn 7.2% compounded monthly. Mare wll spend 25years workng toward ths goal nvestng monthly at an annual rate of 7.2%. How much should Mare s monthly payments be durng her workng years n order to satsfy her retrement needs? Ths s a 2part problem: 1 st calculate the PV for retrement. Then use that value as FV for workng years ( / ) PV / Exercse Amercan General offers a 10year ordnary annuty wth a guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annutes f you want to receve payments of $5,000 annually over the 10year perod? Gven: PMT 5, 000 r 6.65%.0665, m 1, t 10 r.0665 n 1(10) 10 m n 11 PV PMT 1 ( ) 5, $35, (1.0665) ^ 10 /.0665
12 Exercse Amercan General offers a 7year ordnary annuty wth a guaranteed rate of 6.35% compounded annually. How much should you pay for one of these annutes f you want to receve payments of $10,000 annually over the 7year perod? Gven: PMT 10,000 r 6.35%.0635, m 1, t 7 r.0635 n 7 m 1 1 PV PMT n 1 ( ) 7 10, $55, (1.0635) ^ 7 /.0635 Exercse You want to purchase an automoble for $27,300. The dealer offers you 0% fnancng for 60 months or a $5,000 rebate. You can obtan 6.3% fnancal for 60 months at the local bank. Whch opton should you choose? Explan. 0% fnancng: Gven: Rebate: Gven: 27, 300 PMT 1 60 P 27,300 r 0% 0, t 60 mo $ Rebate $5, 000 r 6.3%.063, n 60 PV 27, 300 5, 000 $22, PMT 2 PV , $ n ( / ( / ^ 60)) Rebate s better and you save $455 $ $20.76 per month Or $1, (over the lfe of the loan) 26
13 Exercse You want to purchase an automoble for $28,500. The dealer offers you 0% fnancng for 60 months or a $6,000 rebate. You can obtan 6.2% fnancal for 60 months at the local bank. Whch opton should you choose? Explan. 0% fnancng: Gven: P 28,500 r 0% 0, t 60 mo Rebate: Gven: 28, 500 PMT 1 60 $ Rebate $6, 000 r 6.2%.062, n 60 PV 28, 500 6, 000 $22, PMT 2 PV , $ n Rebate s better and you save $475 $ $37.92 per month Or $2, (over the lfe of the loan) Exercse Construct the amortzaton schedule for a $5,000 debt that s to be amortzed n eght equal quarterly payments at 2.8% nterest per quarter on the unpad balance. Gven: PV 5,000 r 2.8%.028, n 8 PMT PV 1 1 n 5, $ (.028 / ( ^ 8)) 27
14 Pmt Payment Interest Reducton Unpad Balance # 0 $0 $5, $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Total $5, $ $5, $4, $3, $3, $2, $2, $1, $ $0.00 Exercse Construct the amortzaton schedule for a $10,000 debt that s to be amortzed n sx equal quarterly payments at 2.6% nterest per quarter on the unpad balance. Gven: PV 10,000 r 2.6%.026, n 6 PMT PV 1 1 n 10, $1, (.026 / ( ^ 6)) Pmt # Payment Interest Reducton Unpad Balance 0 $0 $10, $1, $ $1, $8, $1, $1, $1, $1, $ $ $ $ $1, $ $1, $6, $1, $5, $1, $3, $1, $1, Total $10, $ $10, $1, $
15 Exercse A loan of $37,948 wth nterest at 6.5% compounded annually, to be pad wth equal annual payments over 10 years Gven: PV 37,948. m 1, r 6.5%.065, n 10 PMT PV 1 1 n 37, $5, (.065 / ( ^ 10)) Pmt # Payment Interest Reducton Unpad Balance 0 $37, $5, $2, $2,8. $35, $5, $2, $2, $32, $5, $2, $3, $28, $5, $1, $3, $25, $5, $1, $3, $21, $5, $1, $3, $18, $5, $1, $4, $13, $5, $ $4, $9, $5, $ $4, $4, $5, $ $4, $0.00 Total $52, $14, $37,
16 Exercse A loan of $4,836 wth nterest at 7.25% compounded semannually, to be repad n 5 years n equal semannual payments. Gven: PV 4,836. m 2, r 7.25%.0725, t 5 r n 2 5 m 2 10 PMT PV 1 1 n 4, $ ( / ( ^ 10)) Pmt # Payment Interest Reducton Unpad Balance 0 $4, $ $ $ $4, $ $ $ $4, $ $ $ $3, $ $9.10 $ $3, $ $1.57 $ $2, $ $95.43 $ $2, $ $77.68 $ $1, $ $59.29 $ $1, $ $40.22 $ $ $ $20.47 $ $0.00 Total $5, $1, $4,
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