# Section 2.2 Future Value of an Annuity

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1 Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment be made at the begnnng of the frst perod. An ordnary annuty requres that the frst payment s made at the end of the frst perod. We wll only deal wth ordnary annutes. \$100 every 6 months, rate r = 0.06 compounded semannually.06 2 t A P 1 100(1.03) 2t 2 Years # of Perod (1.03) 2 100(1.03) (1.03) 100(1.03) 4 100(1.03) 5 100(1.03) S = (1.03) + 100(1.03) (1.03) (1.03) (1.03) 5 = 100( ) \$ n a ar ar ar ar n a r 1 r1 S R R(1 ) R(1 ) 2 R(1 ) 3 R(1 ) n 1 (1 ) n 1 R 1 1 (1 ) n 1 R 10

2 Future Value of an Ordnary Annuty (FV) (1 ) n 1 FV PMT PMT sn PMT: Perodc payment : Rate per perod r m n: Number of payment Example What s the value of an annuty at the end of 10 years f \$1,000 s deposted every 6 months nto an account earnng 8% compounded semannually? How much of ths value s nterest? Gven: 8% compounded semannually a) Annuty (1 ) n 1 FV PMT (1 0.04) \$29, b) How much s the nterest? Deposts = 20(1000) = 20, Interest = Value Depost 29, ,000 = \$ 9, r m 2 11

3 Snkng Funds Snkng Fund s establshed for accumulatng funds to meet future oblgatons or debts. How much I have to pay? PMT FV n 1 1 ; r, n mt m Example A bond ssue s approved for buldng a marna n a cty. The cty s requred to make regular payments every 3 months nto a snkng fund payng 5.4% compounded quarterly. At the end of 10 years, the bond oblgaton wll be retred wth a cost of \$5,000,000. Gven: Cost = \$5,000, n 10 yrs n = mt = 4(10) = 40 r m 4 a) What should each payment be? PMT FV (1 ) n 1 5,000, (1.0135) 40 1 \$95, per quarter (.0135 / (( ) ^ 40-1) b) How much nterest s earned durng the 10 th year? 1 st 9 years = t = 4(9) = 36 (1.0135) 36 1 FV PMT.0135 (1.0135) ,370, ,000,000 4,370, = \$629, after 9 years 3 months (4) PMT (4)( ) = \$380, Interest = 629, , = \$248,628.88

4 Example Experts say the baby boom generaton can t count on a company penson or Socal Securty to provde a comfortable retrement, as ther parents dd. It s recommended that they start to save early and regularly. Sarah, a baby boomer, has decded to depost \$200 each month for 20 years n an account that pays nterest of 7.2% compounded monthly. a) How much wll be n the account at the end of 20 years? b) Sarah beleves she needs to accumulate \$130,000 n the 20-year perod to have enough for retrement. What nterest rate would provde that amount? a) Gven: PMT 200 m r r.072 m n mt (20) 240 (1 ) n 1 FV PMT \$106, b) r r r 1 r r r r r Usng a calculator or program; the annual nterest rate s 8.79%. 13

5 Exercses Secton 2.2 Future Value of an Annuty 1. Recently, Guaranty Income Lfe offered an annuty that pays 6.65% compounded monthly. If \$500 s deposted nto ths annuty every month, how much s n the account after 10 years? How much of ths s nterest? 2. Recently, USG Annuty Lfe offered an annuty that pays 4.25% compounded monthly. If \$1,000 s deposted nto ths annuty every month, how much s n the account after 15 years? How much of ths s nterest? 3. In order to accumulate enough money for a down payment on a house, a couple deposts \$300 per month nto an account payng 6% compounded monthly. If payments are made at the end of each perod, how much money wll be n the account n 5 years? 4. A self-employed person has a Keogh retrement plan. (Ths type of plan s free of taxes untl money s wthdrawn.) If deposts of \$7,500 are made each year nto an account payng 8% compounded annually, how much wll be n the account after 20 years? 5. Sun Amerca recently offered an annuty that pays 6.35% compounded monthly. What equal monthly depost should be made nto ths annuty n order to have \$200,000 n 15 years? 6. Recently, The Hartford offered an annuty that pays 5.5% compounded monthly. What equal monthly depost should be made nto ths annuty n order to have \$100,000 n 10 years? 7. Compu-bank, an onlne bankng servce, offered a money market account wth an APY of 4.86%. a) If nterest s compounded monthly, what s the equvalent annual nomnal rate? b) If you wsh to have \$10,000 n the account after 4 years, what equal depost should you make each month? 8. Amercan Express s onlne bankng dvson offered a money market account wth an APY of 5.65%. a) If nterest s compounded monthly, what s the equvalent annual nomnal rate? b) If you wsh to have \$1,000,000 n the account after 8 years, what equal depost should you make each month? 9. Fnd the future value of an annuty due f payments of \$500 are made at the begnnng of each quarter for 7 years, n an account payng 6% compounded quarterly. 10. A 45 year-old man puts \$2500 n a retrement account at the end of each quarter untl he reaches the age of 60, then makes no further deposts. If the account pays 6% nterest compounded quarterly, how much wll be n the account when the man retres at age 65? 14

6 11. A father opened a savngs account for hs daughter on the day she was born, depostng \$1000. Each year on her brthday he deposts another \$1000, makng the last depost on her 21 st brthday. If the account pays 5.25% nterest compounded annually, how much s n the account at the end of the day on hs daughter s 21 st brthday? How much nterest has been earned?. You deposts \$10,000 at the begnnng of each year for years n an account payng 5% compounded annually. Then you put the total amount on depost n another account payng 6% compounded sem-annually for another 9 years. Fnd the fnal amount on depost after the entre 21- year perod. 13. You need \$10,000 n 8 years. a) What amount should be depost at the end of each quarter at 8% compounded quarterly so that he wll have hs \$10,000? b) Fnd your quarterly depost f the money s deposted at 6% compounded quarterly. 14. You want to have a \$20,000 down payment when you buy a car n 6 years. How much money must you depost at the end of each quarter n an account payng 3.2% compounded quarterly so that you wll have the down payment you desre? 15. You sell a land and then you wll be pad a lump sum of \$60,000 n 7 years. Untl then, the buyer pays 8% smple nterest quarterly. a) Fnd the amount of each quarterly nterest payment on the \$60,000 b) The buyer sets up a snkng fund so that enough money wll be present to pay off the \$60,000. The buyer wll make semannual payments nto the snkng fund; the account pays 6% compounded semannually. Fnd the amount of each payment nto the fund. 15

7 Secton 2.2 Future Value of an Annuty Exercse Recently, Guaranty Income Lfe offered an annuty that pays 6.65% compounded monthly. If \$500 s deposted nto ths annuty every month, how much s n the account after 10 years? How much of ths s nterest? Gven: PMT 500 r 6.65%.0665 m t 10 r.0665 n mt (10) 0 m (1 ) n 1 FV PMT \$84, Total deposts: Interest FV Deposts \$60, , , 000 \$24, Exercse Recently, USG Annuty Lfe offered an annuty that pays 4.25% compounded monthly. If \$1,000 s deposted nto ths annuty every month, how much s n the account after 15 years? How much of ths s nterest? Gven: PMT 1, 000 r 4.25%.0425 m t 15 r.0425 n mt (15) 180 m (1 ) n 1 FV PMT \$251,185.76

8 Total deposts: Interest FV Deposts 1, \$180, , , 000 \$71, Exercse In order to accumulate enough money for a down payment on a house, a couple deposts \$300 per month nto an account payng 6% compounded monthly. If payments are made at the end of each perod, how much money wll be n the account n 5 years? Gven: PMT 300 r 6%.06 m t 5 r n mt (5) 60 m (1 ) n 1 FV PMT \$20, Exercse A self-employed person has a Keogh retrement plan. (Ths type of plan s free of taxes untl money s wthdrawn.) If deposts of \$7,500 are made each year nto an account payng 8% compounded annually, how much wll be n the account after 20 years? Gven: PMT 7,500 r 8%.08 m 1 t 20 r n mt 1(20) 20 m 1 (1 ) n 1 FV PMT , \$343,

9 Exercse Sun Amerca recently offered an annuty that pays 6.35% compounded monthly. What equal monthly depost should be made nto ths annuty n order to have \$200,000 n 15 years? Gven: FV 200, 000 r 6.35%.0635, m, t 15 r.0635 n mt (15) 180 m PMT FV (1 ) n 1 200, \$ per month / / / ^180 1 Exercse Recently, The Hartford offered an annuty that pays 5.5% compounded monthly. What equal monthly depost should be made nto ths annuty n order to have \$100,000 n 10 years? Gven: FV 100,000 r 5.5%.055, m, t 10 r.055 n mt (10) 0 m PMT FV (1 ) n 1 100, \$ per month / / / ^0 1 14

10 Exercse Compu-bank, an onlne bankng servce, offered a money market account wth an APY of 4.86%. a) If nterest s compounded monthly, what s the equvalent annual nomnal rate? b) If you wsh to have \$10,000 n the account after 4 years, what equal depost should you make each month? Gven: APY 4.86%.0486 a) m m APY 1 r 1 m r r / 1 r r 1/ / r Add1 on both sdes ^ 1 / 1 The equvalent annual nomnal rate r 4.75% b) Gven: FV \$10, 000 r.0475, m, t 4 r.0475 n mt (4) 48 m PMT FV (1 ) n 1 10, \$ per month / / / ^

11 Exercse Amercan Express s onlne bankng dvson offered a money market account wth an APY of 5.65%. a) If nterest s compounded monthly, what s the equvalent annual nomnal rate? b) If you wsh to have \$1,000,000 n the account after 8 years, what equal depost should you make each month? Gven: a) m APY 5.65%.0565 m APY 1 r 1 m r r / 1 r r 1/ / r Add1 on both sdes ^ 1 / 1 The equvalent annual nomnal rate r 5.51% b) Gven: FV \$1, 000, 000 r.0551, m, t 8 r.0551 n mt (8) 96 m PMT FV (1 ) n 1 1, 000, 000 \$8, per month / / / ^

12 Exercse Fnd the future value of an annuty due f payments of \$500 are made at the begnnng of each quarter for 7 years, n an account payng 6% compounded quarterly. Gven: PMT 500 r 6%.06 m 4 t 7 r n mt 1 4(7) 1 29 m 4 Snce you put money at the begnnng of each month, we need to add the frst payment. FV (1 ) n 1 PMT \$17, ^ 29 1 /.015 Exercse A 45 year-old man puts \$2500 n a retrement account at the end of each quarter untl he reaches the age of 60, then makes no further deposts. If the account pays 6% nterest compounded quarterly, how much wll be n the account when the man retres at age 65? For the 15 years : PMT 2,500 r 6%.06 m 4 t 15 r n mt 1 4(15) 60 m 4 (1 ) n 1 FV PMT , \$240, For the remanng 5 years, the FV amount s the present amount n A P 1 4(5) 240, \$323, ^ 60 1 /.015 P at 6% compounded quarterly ^ (5* 4) 17

13 Exercse A father opened a savngs account for hs daughter on the day she was born, depostng \$1000. Each year on her brthday he deposts another \$1000, makng the last depost on her 21 st brthday. If the account pays 5.25% nterest compounded annually, how much s n the account at the end of the day on hs daughter s 21 st brthday? How much nterest has been earned? Gven: PMT 1, 000 r 5.25%.0525 m 1 t 21 r n mt 1 1(21) 1 22 m 1 Snce you put money at the begnnng of each year, we need to add the frst payment. FV (1 ) n 1 PMT , \$39, ^ 22 1 /.0525 The Total contrbuton: \$22, The nterest earned: 39, ,00 \$17, Exercse You deposts \$10,000 at the begnnng of each year for years n an account payng 5% compounded annually. Then you put the total amount on depost n another account payng 6% compounded semannually for another 9 years. Fnd the fnal amount on depost after the entre 21-year perod. Gven: FV PMT 10,000 r 5.%.05 m 1 t r n mt m 1 (1 ) n 1 PMT , 000 \$177, Snce the last depost dd mature yet when roll over, then: P 177, , 000 \$167, ^ 13 1 /.05 r n 9(2) 18 m 2 18

14 n A P , \$284, ^ 18 Exercse You need \$10,000 n 8 years. a) What amount should be depost at the end of each quarter at 8% compounded quarterly so that he wll have hs \$10,000? b) Fnd your quarterly depost f the money s deposted at 6% compounded quarterly. a) Gven: FV 10, 000 r 8%.08, m 4, t 8 r n mt 4(8) 32 m 4 PMT FV (1 ) n 1 b) Gven: 10, \$ each quarter FV 10, 000 r 6%.06, m 4, t 8 r n 4(8) 32 m 4 PMT FV (1 ) n 1 10, \$ each quarter / 1.02 ^ / ^

15 Exercse You want to have a \$20,000 down payment when you buy a car n 6 years. How much money must you depost at the end of each quarter n an account payng 3.2% compounded quarterly so that you wll have the down payment you desre? Gven: FV 20,000 r 3.2%.032, m 4, t 6 r n 4(6) 24 m 4 PMT FV (1 ) n 1 20, \$ quarterly / / / ^0 1 Exercse You sell a land and then you wll be pad a lump sum of \$60,000 n 7 years. Untl then, the buyer pays 8% smple nterest quarterly. a) Fnd the amount of each quarterly nterest payment on the \$60,000 b) The buyer sets up a snkng fund so that enough money wll be present to pay off the \$60,000. The buyer wll make semannual payments nto the snkng fund; the account pays 6% compounded semannually. Fnd the amount of each payment nto the fund. Gven: a) I Prt b) Gven: P 60, 000 r 8%.08, m 4, t 7 60, \$1, FV 60, 000 r 6%.06, m 2, t 7 r n 2(7) 14 m 2 PMT FV (1 ) n 1 60, \$ / 1.03 ^

16 Balance.03 Balance Depost I.03* Balance Interest Pmt # Balance Amount Earned 1 \$3, \$0 \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, \$3, *3, \$ * \$ * * * * * \$ \$ \$ \$ \$ * \$ * \$1, * * \$1, \$1, * \$1, * \$1, , , \$7,8.51 \$10, \$14, \$18, \$22, \$26, \$31, \$35, \$40, \$44, \$49, \$54, \$60,

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