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 45-years 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 / (1-1.01^ (-)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 30-year 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 mn-van) 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 30-year perod. She has forecasted that her money wll earn 7.2% compounded monthly. Mare wll spend 25-years 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 10-year 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 10-year perod? 21

7 9. Amercan General offers a 7-year 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 7-year 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 sem-annually, to be repad n 5 years n equal sem-annual 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 mn-van) 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 30-year perod. She has forecasted that her money wll earn 7.2% compounded monthly. Mare wll spend 25-years 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 2-part problem: 1 st calculate the PV for retrement. Then use that value as FV for workng years ( / ) PV / Exercse Amercan General offers a 10-year 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 10-year 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 7-year 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 7-year 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 sem-annually, 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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