0.07. i PV Qa Q Q i n. Chapter 3, Section 2

Transcription

1 Chapter 3, Secto 2 1. (S13HW) Calculate the preset value for a auty that pays 500 at the ed of each year for 20 years. You are gve that the aual terest rate s 7% v PV Qa Q Usg your calculator: N 20 I / Y 7 PMT 500 CPT PV *Remember that you should eter the umber, the press the correspodg tme value of moey butto. For example, type 20 the press N. Also, your calculator requres ether the paymets or the preset value to be egatve. So, eve though the above keystrokes resulted a egatve preset value, your fal aswer s postve Alteratvely, you could eter your paymet as egatve 500 ad the your resultg preset value wll be postve. 2. (S13HW) Calculate the preset value of a auty mmedate wth mothly paymets of 200 for 10 years usg a terest rate of 9% compouded mothly. (10) v PV Qa Q Q Usg your calculator: N 0 I / Y 0.75 PMT 200 CPT PV *Note that N s always the umber of paymets, ot ecessarly the umber of years. Also, for mothly paymets,. Keep ths md whe sertg umbers to your calculator!

2 3. (S13HW) Calculate the preset value of a auty mmedate wth mothly paymets of 200 for 10 years usg a aual effectve terest rate of 9% Frst, let s fd sce we have mothly paymets. 1/ We ca also use the calculator to fd ths value. ICONV EFF 9 I / Y CPT NOM Now that we have foud our approprate terest rate, we may proceed wth the problem. PV (10) Usg our calculator: N 0 I / Y PMT 200 CPT PV (S13HW) Calculate the accumulated value of a auty mmedate wth aual paymets of 600 for the ext 25 years usg a aual effectve terest rate of 4% FV Qs Usg your calculator: N 25 I / Y 4 PMT 600 CPT FV *Note: Just lke wth preset values, the calculator wll gve you egatve aswer for future value f you etered postve paymets. Make sure your fal aswer s postve!

3 5. (S13HW) Calculate the accumulated value of a auty that pays 5000 at the ed of each year for the ext 8 years usg a terest rate of 10% compouded sem-aually. Frst, let s fd the aual effectve rate. (2) We ca also do ths wth our calculator: ICONV NOM 10 I / Y 2 CPT EFF Now we ca calculate the future value: FV Usg your calculator: N 8 I / Y PMT 5000 CPT FV

4 6. (S13HW) Jula s buyg a ew car for 23,000. She s facg the etre amout wth a loa wth mothly paymets over the ext 4 years. The terest rate o Jula s loa s 8.4% compouded quarterly. Calculate Jula s mothly paymet. Frst, let s get a approprate terest rate for mothly paymets. (4) Or, wth your calculator: ICONV NOM 8.4 I / Y 4 CPT EFF / Y CPT NOM * Notce that we have to use two steps the calculator: frst fd the effect rate, the the mothly rate. Now we ca fd the paymet: (4) 1 ( ) P P Wth your calculator: N 48 I / Y PV CPT PMT

5 7. (S13HW) Muhamad deposts 150 to a bak accout at the ed of each quarter for 10 years. Durg ths tme perod, Muhamad ears a aual effectve terest rate of 6.8%. Calculate the amout that Muhamad has at the ed of 10 years. (4) 4 (4) 1/ Fd the approprate terest rate, Or, (4) ICONV EFF 6.8 I / Y 4 CPT NOM * 4 The fd the future value: FV 10* Or, wth your calculator: N 40 I / Y PMT 150 CPT FV

6 8. (S13HW) Tyler wats to have 1000 oe year to buy a ew HDTV for the Super Bowl. He decdes that he wll vest P at the ed of each moth a accout earg 8% compouded quarterly order to accumulate the Calculate P. Frst, let s get a approprate terest rate for mothly paymets. (4) Or, wth your calculator: ICONV NOM 8 I / Y 4 CPT EFF / Y CPT NOM * Notce that we have to use two steps the calculator: frst fd the effect rate, the the mothly rate. Now we ca fd the paymet: ( ) P P Wth your calculator: N I / Y FV 1000 CPT PMT 80.34

7 9. (S11HW) Mchael wo the lottery! He has the followg payout optos: a. Oe mllo at the ed of each year for the ext 20 years; or b. A lump sum of 7,469, pad ow. Calculate the aual effectve terest rate at whch both optos have the same preset value. Ths problem eeds to be doe usg your calculator: N 20 PMT PV CPT 1/ Y So your aswer s %. Note that you must eter ether the PMT or the PV as a egatve umber order to get the aswer! 10. (SHW) Y s payg a car loa wth paymets of 500 at the ed of each moth. The loa has a mothly effectve terest rate of 1%. If the car loa s for 18,986.98, calculate the umber of paymets that Y wll eed to make. Although you ca calculate ths by had, t s much more practcal to use the calculator. PMT 500 I / Y 1 PV CPT N (SHW) For a gve terest rate, s = ad a = Calculate. 1 1 a s % s (0.08) l( ) 13 l(1.08)

8 . (SHW) If d 0.1, calculate a. 14 d d a v (S11HW) The accumulated value of a year auty s four tmes the preset value of the same auty. Calculate the accumulated value of years. (Note: Ths s NOT askg for the accumulated value of a auty just the accumulated value of a sgle paymet of 100.) Remember the detty: s (1 ) a We kow that, s 4a. We ca combe these two facts to solve the problem. 4 a (1 ) a 4 (1 ) 2 We wat to fd 100(1 ). Ths s easy to do usg substtuto: (1 ) 100 (1 ) 100(4) Book Problem 3.2, Number 2 Gary 3xa 3a Tracy xa a v 3 v v v 3(1 v ) v (1 v ) (1 v ) (1 ) 3(1 )(1 ) v

9 Chapter 3, Secto (S13HW) Hua s recevg mothly paymets of 1000 at the start of each moth for 10 years. Calculate the preset value of these paymets usg a aual effectve rate of %. Frst, let s get a approprate terest rate for mothly paymets Or, wth your calculator: ICONV EFF I / Y CPT NOM * Now we ca fd the preset value: PV 1000a ( ) **Note: We ow have a auty due so we have the extra term to accout for the paymets beg at the begg of the moth Wth your calculator: **I BGN Mode!** N 0 I / Y PMT 1000 CPT PV

10 16. (S13HW) Alyssa s the beefcary of a trust fud whch wll pay her 1000 at the begg of each moth for the ext 5 years. Alyssa vests each paymet to a accout earg a terest rate of 7.2% compouded mothly. How much wll Alyssa have at the ed of 5 years? FV s Or use your calculator ** BGN mode** N 60 I / Y 0.6 PMT 1000 CPT FV (S13HW) Pgpg wats to have 100,000 whe he retres 20 years. He deposts level paymets at the begg of each quarter to a accout earg a quarterly effectve terest rate of 2%. Determe the amout of the paymet f Pgpg s gog to acheve hs goal Ps P P Or use your calculator ** BGN mode** N 80 I / Y 2 FV CPT PMT

11 18. (SHW) You are gve a ad a Calculate. We kow that a a (1 ). Therefore, plug the terest rate to a to fd a (0.01) 1.01 l (0.01) a (1 ) Now we ca a (S11HW) At the start of each moth for 50 moths, Jula deposts 100 to her bak accout. At the ed of 50 moths, Jula has Calculate the aual effectve terest rate beg eared by Jula. You eed to use your calculator for ths problem. Make sure t s BGN mode! N 50 PMT 100 FV CPT 1/ Y (1 ) ( ) %

12 20. (SHW) Zche wats to accumulate a sum of moey at age 65 so he ca retre. I order to accomplsh ths goal, he ca depost 80 per moth at the begg of the moth or 81 per moth at the ed of the moth. Calculate the aual effectve rate of terest eared by Zche. Use the detty: a a (1 ) but make t applcable to mothly paymets: 80a 81a We get: a a 80 Now that we have the mothly effectve rate, we ca fd the aual effectve rate % 21. Book Problem 3.3, Number 6 a 2 21 a 2 1 v a2 1 1 d v v a 1 v 1v 1v d 21 9 v v 21 1 v a a 1 a v 1 d d 48 the a sce v d 1 1 d

13 22. Book Problem 3.3, Number 7 a S but S 1 S 1 S d d a S a S d sce d a 1 S ( ) l l Chapter 3 Secto (SHW) A perpetuty mmedate pays 10,000 per moth. The terest rate eared by the perpetuty s 8% compouded mothly. Calculate the preset value of ths perpetuty PV 1000a ( m) 0.08 m

14 24. (SHW) Mohamad s the beefcary of a trust fud. Mohamad ad hs descedets wll receve a paymet of 1000 at the begg of each moth forever. Usg a aual effectve terest rate of 4%, determe the curret value of these paymets. Frst, we eed to fd a approprate terest rate for mothly paymets (1/) Now that we have the mothly terest rate, we ca calculate the preset value PV 1000a * NOTE: We had to multply our preset value by 1 to accout for paymets beg at the begg of the moth. 25. (S08Q2) Purdue has asked our class to fud a scholarshp for actuaral studets. The scholarshp wll pay 1000 aually wth the frst paymet beg made mmedately. Our cotrbutos wll ear a aual effectve rate of 5%. How much must our class doate order for the scholarshp to cotue forever? Luckly, we were gve a aual terest rate ad we have yearly paymets. However, we do eed to make sure to multply our preset value by (1+) to accout for paymets begg mmedately. PV a

15 26. (S11HW) Kate buys a perpetuty due of 1000 per moth for 100,000. Calculate the aual effectve rate of terest used to calculate the prce of ths perpetuty a Now, we eed to fd the aual effectve terest rate. 0. (1 )

16 27. (S09T1) Tyler has herted $1 mllo. He has decded to use hs hertace to purchase oe of the followg: a. A 30 year auty mmedate wth aual paymets of 106,079.25; or b. A perpetuty due wth quarterly paymets of P. Both optos are based o the same terest rate. Calculate P. Frst, we ca fd the terest rate of the 30 year auty. Ths ca be doe very easly usg our facal calculator. PV 1,000,000 PMT 106, N 30 CPT I / Y (Remember to make the preset value or the paymets egatve so you do t get a error!) Now, we ca fd the equvalet quarterly terest rate order to fd P. (4) (4) (4) ,000,000 Pa P P (4) P 23, (S11HW) The value of a perpetuty mmedate where the paymet s P s 1000 less tha the value of a perpetuty due where the paymet f P. Calculate P. We are gve, Pa Pa It wll be easer to solve ths equato f we wrte t terms of P ad. P P(1 ) 1000 P P P P 1000 P 1000 P 1000

17 29. (SHW) A perpetuty s fuded by a doato of 500,000. Paymets of P are to be made at the ed of every secod year. I other words, P wll be pad at tme 2, 4, 6, etc. If the fud ears a aual effectve terest rate of 8%, calculate P. Note that the paymets are made every secod year. Ths makes m=1/2 the ( m) equato. A commo mstake would be to make m=2. It s mportat to uderstad the dfferece. (1/2) /2 (1/2) , 000 Pa P (1/2) 0.5 P 500, 000(0.1664) Chapter 3, Secto (S11HW) A mothly auty mmedate pays 100 per moth for moths. Calculate the accumulated value moths after the last paymet usg a omal rate of 4% compouded mothly. We eed to calculate the accumulated value at the ed of the paymets ad the move ths value forward moths s

18 31. (S11HW) A mothly auty due pays 100 per moth for moths. Calculate the accumulated value moths after the last paymet usg a omal rate of 4% compouded mothly. Ths s essetally the same problem as #30. Because the paymets are ow at the begg of the moth, we eed to use s whch s equal to s 1. However, order to move the accumulated value at the ed of moths forward more moths, we really oly eed to multply 11 by 1 sce we are already accoutg for oe moth s accumulato by usg a auty due. Therefore, we get the same aswer. To see ths a lttle more clearly, here s the equato: s (S11HW) A mothly auty due pays 100 per moth for moths. Calculate the accumulated value 24 moths after the frst paymet usg a omal rate of 4% compouded mothly. We ca us the same formula #31 except move the accumulated value forward a addtoal moth s (S11HW) Calculate the curret value at the ed of 5 years of a auty due payg aual paymets of 00 for years. The aual effectve terest rate s 6%. To solve ths problem, we wll fd the accumulated value of the etre auty ad the dscout t seve years order to accout for the fact that ow s at the ed of year s v

19 34. (S11HW) Calculate the preset value of a auty mmedate wth 20 aual paymets of 500 f auty does ot start utl fve years have passed. The aual effectve terest rate s 8%. Ths tme, we wll fd the preset value of the auty at the start of the paymets ad the dscout t 5 years a v 500 (1.08) (S11HW) Joh buys a seres of paymets. The frst paymet of 50 s sx years. Aual paymets of 50 are made thereafter utl 14 total paymets have bee made. Calculate the prce Joh should pay to realze a aual effectve retur of 7%. We wll fd the preset value of a auty due wth 14 paymets the dscout t sx years (1.07) 6 50a v 50 (1.07)(1.07)

20 Chapter 3, Secto (SHW) Yue bought a house wth a 200,000 mortgage for 15 years beg repad wth paymets at the ed of each moth at a terest rate of 6% compouded mothly. What s the outstadg balace at the ed of 10 years mmedately after the 0 th paymet? Frst, let s fd the amout of the mothly paymet , 000 Pa P P Realze that you ca also do ths usg your facal calculator: 180 PV 200,000 I / Y 0.5 CPT PMT Now that we have the paymet, we ca fd the OLB. OLB ,0001 Ps , (SHW) If Yue pays a extra 100 each moth, what s the outstadg balace at the ed of 10 years mmedately after the 0 th paymet? Sce Yue decdes to pay extra each moth, all we eed to chage from the formula above s the paymet amout OLB 200, ( P 100) s ,

21 38. (SHW) Nacy borrowed moey to buy a ew car. The loa has a terest rate of 7.8% compouded mothly. Nacy s mothly paymet s 246 ad she has 11 paymets left wth the ext paymet due oe moth. Calculate the outstadg balace o her loa. We kow that f Nacy pays the remag 11paymets she wll pay off the loa. Ths meas that her OLB s equal to the preset value of her remag paymets OLB 246a (SHW) Phl borrows 100,000 at a aual effectve terest rate of 7%. Phl s repayg the loa wth aual paymets of 10,000. Calculate how much Phl stll owes mmedately after hs 10 th paymet. Ths s a straghtforward applcato of our equato for OLB. 10 OLB L(1 ) Ps (1.07) (S08T1) Thomas bought a house 5 years ago. I order to buy the house, he borrowed 50,000 to be repad wth 360 mothly paymets of Thomas pays each moth for 60 moths. Calculate outstadg loa balace o Thomas loa mmedately after the 60 th paymet. Frst, let s fd the terest rate of Thomas loa. 360 PV 50,000 PMT CPT I / Y 0.80 Now that we have all of ths formato, t s very easy to fd the OLB usg our calculator. Do ot clear aythg out of your calculator, just correct the followg values: 60 CPT FV We could plug the terest rate ad calculate the OLB by had, but the calculator saves a lot of tme.

22 41. (S09T1) Ashley borrows 30,000 to buy a ew car. Her loa carres a mothly effectve terest rate of 1%. She wll repay the loa by makg mothly paymets of Ashley makes k paymets of Immedately after the k th paymet, she pays off the outstadg balace of her loa by makg a paymet of 13, Determe k. Start by fdg. PV I / Y 1 PMT CPT 54. Now determe how may paymets are left. The preset value of future paymet s a 13, PV 13, I / Y 1 PMT CPT k 21 k The 54 k 21 k 33 You ca also trck the calculator to gvg you the aswer drectly as follows: PV I / Y 1 PMT FV CPT (S11HW) A loa of 10,000 s beg repad wth 20 o-level aual paymets. The terest rate o the loa s a aual effectve rate of 6%. The loa was orgated 4 years ago. Paymets of 500 at the ed of the frst year, 750 at the ed of the secod year, 1000 at the ed of the thrd year ad 50 at the ed of the fourth year have bee pad. Calculate the outstadg balace mmedately after the fourth paymet. We eed to use the retrospectve approach ,000(1.06) 500(1.06) 750(1.06) 1000(1.06) (S11HW) Calculate the outstadg balace to the loa #40 oe year after the fourth paymet mmedately before the ffth paymet. Because there are o cash flows betwee paymets, we just eed to add a year of terest (1.06)

23 44. Book Problem 3.6, Number 1 A. We just eed to use the preset value of the last four paymets. (Preset value take rght after the sxth paymet). By Had: OLB Pa Usg Calculator: 4 I / Y 6 PMT 1516 CPT PV B. Ths tme we wll eed to take to cosderato both paymet levels. We wat the preset value of future paymets take mmedately after the 3 rd paymet (1.06) 1 (1.06) OLB a a v Note: Make sure to brg the 1516 paymets back three extra years!

24 45. (S11T1) Mke bought a ew hgh defto televso for Mke pad for the televso usg a 15 moth loa wth a terest rate of 9% compouded mothly. Mke forgot to make the 8 th paymet o the loa. Determe Mke s outstadg loa balace at the ed of the th moth. Oe way to do ths problem s to assume that all paymets have bee made ad the add the mssg paymet. If we assume that all paymets have bee made: 15 I / Y 0.75 PV 2000 CPT PMT d Amort P1 1 P2 BAL Now add the mssg paymet so OLB (1.0075) The dfferece betwee ad s from roudg. If you do ot roud the paymet to the earest pey, you get Here s aother way proposed by Allso who s the TA for ths class. Ths questo should be doe usg your calculator because t requres multple steps. Frst, fd the level paymets that Mke should make: 15 I / Y 0.75 PV 2000 CPT PMT Secodly, fd the OLB mmedately after the 7 th paymet. Do ot clear your calculator betwee steps! 7 CPT FV The, we eed to add terest for oe moth to the OLB after the 7 th paymet: Fally, we kow that oce Mke starts makg paymets aga, he wll use the same paymet as he dd before. To fd the OLB at the ed of the th moth, we eed to add 4 more paymets. Make sure you have t cleared out your calculator betwee steps! 4 PV CPT FV

25 Chapter 3, Secto (S11HW) A auty pays 100 at the ed of each moth for 4 years ad the 200 a moth at the ed of each moth for the ext four years. Calculate the preset value at = There are two ways to do ths problem PV 100a 200a v PV 200a 100a (S11HW) A auty pays 100 at the ed of each moth for 4 years ad the 200 a moth at the ed of each moth for the ext four years. Calculate the accumulated value at = There are also two ways to solve ths problem FV 100s 1 200s or FV 100s 100s

26 48. (S11HW) A auty pays 100 at the ed of each moth for 4 years ad the 200 a moth at the ed of each moth for the ext four years. Calculate the curret value rght after the 48 th paymet at = Rght after the 48 th paymets, you have receved 48 paymets of 100 ad expect to receve 48 more paymets of s 200a Book Problem 3.7, Number s 1 Qs Qs Q

27 50. Book Problem 3.7, Number 2 Let s treat ths problem as the sum of two separate autes. The frst wll be the 100 paymets made yearly startg ow, ad the other wll be the 300 paymets made yearly startg sx moths. We wll eed to make sure to brg the secod auty back sx moths order to get the preset value today a 300a v Chapter 3, umber (S08T1) Kurt s the beefcary of a trust. Uder the trust, he wll receve paymets at the ed of each year for the ext 20 years. The paymet wll be 2000 at the ed of oe year. Each subsequet paymet wll crease by 8%. I other words, the paymet at the ed of the secod year wll be 2000(1.08), the paymet at the ed of the thrd year wll be 2000( ), etc. Calculate the preset value of Kurt s paymets uder the trust usg a aual effectve terest rate of 9%. Wrte out a expresso for the preset value of paymets order to see that t s the sum of a geometrc seres: FrstTerm NextAfterLast Recall that the sum of a geometrc seres= 1 rato

28 52. (S09T1) Parker has wo the lottery. He wll receve 20 aual paymets wth the frst paymet made ow. The frst paymet wll be for 25,000. Each subsequet paymet wll be 110% of the prevous paymet. I other words, the secod paymet wll be 25000(1.10) ad the thrd paymet wll be 25000(1.10) 2, etc. Calculate the preset value of Parker s wgs usg a aual effectve terest rate of 5%. Wrte out a expresso for the preset value of paymets order to see that t s the sum of a geometrc seres: FrstTerm NextAfterLast Recall that the sum of a geometrc seres= 1 rato

29 53. (SHW) Today s Zharfa s 25 th brthday. As a brthday gft, hs parets have gve hm a auty mmedate wth aual paymets. The frst paymet s 10,000. Each paymet thereafter s 95% of the pror year s paymet. The last paymet wll be made o Zharfa s 65 th brthday. Zharfa vests each paymet at a aual terest rate of 8%. How much wll Zharfa have o hs 65 th brthday. Realze that Zharfa s recevg a total of 40 paymets. A commo mstake s to use 65 paymets because they stop o hs 65 th brthday. Also, otce that we are ow fdg the accumulated value! Wrte out a expresso for the accumulated value of paymets order to see that t s the sum of a geometrc seres: FrstTerm NextAfterLast Recall that the sum of a geometrc seres= 1 rato , 661,

30 54. (SHW) A auty mmedate has geometrcally creasg paymets made aually for 18 years. 1 2 The frst paymet s 00. The secod paymet s 00(1.1). The thrd paymet s 00(1.1) ad paymets cotue to crease at a rate of 10% each year. Calculate the preset value of ths auty at a aual terest rate of 10%. Wrte out a expresso for the preset value of paymets order to see that t s the sum of a geometrc seres: Notce that ths smplfes to Now we see that ths s very smple: (SHW) A auty due has mothly paymets for 8 years. The frst paymet s 600 wth each successve paymet beg 1% larger tha the prevous paymet. Krst vests each paymet a accout that ears % compouded mothly. Calculate the amout that Krst wll have the accout at the ed of 8 years. 0. Note that %. Ths meas that the paymets are growg at the rate of terest just lke umber 54. Therefore, we wo t eed a sum of a geometrc seres FV FV *

31 56. (SHW) A perpetuty makes paymets at the ed of each year. The frst paymet s Each paymet thereafter s 103% of the prevous paymet. Calculate the preset value of ths perpetuty at a terest rate of 8%. We ca also wrte the value of a perpetuty as a sum of a geometrc seres: PV We ca say that , so we have Chapter 3, Secto (SHW) A 30 year auty makes aual paymets at the ed of each year. The frst paymet s The secod paymet s The thrd paymet s 10,000 ad each subsequet paymet s 1000 greater tha the pror paymet. Calculate the preset value of ths auty at 7% terest. ( I a) : pq, P 8000; Q 1000; 0.07; 30 Q ( I pq, a) Pa a v

32 58. (S09Q3)A auty due makes mothly paymets for 15 years. The frst paymet s $10. Each subsequet paymet s $10 larger tha the prevous paymet. Calculate the preset value of ths auty usg a omal terest rate of % compouded mothly. Q We kow that ( I pq, a) Pa a v. We do have to adjust ths formula for the fact that paymets are at the begg of the moth. So: 1%; P Q 10; 180 Q PV Pa a v (1 ) 10a 180 a 180 v 180 (1.01) (10)( ) (1.01) 54, a v Alteratvely, we kow that whe P=Q, ( Ia). We wll use ths formula but we wll also accout for the paymets beg at the begg of the moth. 1%; P Q 10; a v 0.01 ( Ia) Now we eed to multply ths value by $10 ad accout for paymets at the begg of the moth: (10)(1.01)

33 59. (SHW) Chrsta has a scholarshp that wll pay her mothly for the ext 4 years. The frst paymet s 100. Each paymet after the frst paymet s 25 greater tha the prevous paymet. I other words, the secod paymet s 5, the thrd paymet s 150, etc. Chrsta takes each paymet ad vests the paymet a fud earg 6% compouded mothly. Calculate the amout that Chrsta wll have the fud oe moth after the last paymet. Ths questo s askg for the accumulated value of ths auty oe moth after the last paymet. Ths s the same as askg for the accumulated value of a auty due. We kow that the preset Q value of a auty mmedate s: ( I pq, a) Pa a v. If we multply ths by (1 ) we get the accumulated value o the date of the last paymet. We the multply by (1 ) to get the value oe moth after the last paymet. Therefore: Q AV Pa a v (1 ) (1 ) 0.06 P 100; Q 25; 0.005; AV 100a 48 a 48 v 48 (1.005) (1.005) (100)( ) (1.005) (1.005) 36, Alteratvely, Q ( Is) Ps s ( Is) But, we wat the amout Chrsta s accout oe moth after the last paymet: ( 1.005)

34 60. (SHW) Ally s recevg auty paymets at the ed of each quarter for 20 years. The frst paymet s P. The secod paymet s 2P. The thrd paymet s 3P ad paymets cotue to crease the same patter. The accumulated value of Ally s auty s 100,000 usg a quarterly effectve terest rate of 4%. Determe P. We kow that the accumulated value of ths auty s 100,000 ad that P Q, Further, sce the (4) quarterly effectve terest rate s 4%, ths meas that The 4 Q P AV 100, 000 Pa a v (1 ) Pa a 80 v (1.04) , 000 P (1.04) P(, ) 100, 000 P 8.11, Alteratvely, we kow that whe P=Q we ca use P( Is) s P 0.04 (4) 0.04; P Q; 4 ( Is) ; P (1.04) 80 s P P

35 61. (SHW) Brooke s the beefcary of a trust fud that wll make a paymet o each of her brthdays wth the fal paymet o her 60 th brthday. Today s Brooke s 20 th brthday ad she wll receve the frst paymet of 50,000. Each subsequet paymet wll be 1000 less tha the pror paymet. I other words, she wll receve 49,000 o her 21 st brthday, 48,000 o her 22 d brthday, etc. Calculate the preset value of Brooke s paymets at a aual effectve terest rate of 5%. Frst we eed to recogze the umber of paymets. She wll receve a paymet today o her 20 th brthday ad paymets wll cotue utl the fal paymet s o her 60 th brthday. That s 41 paymets. It s temptg to thk that there are 40 paymets (60 mus 20), but that would ot clude the paymet at age 20. Ths s a auty due sce the frst paymet s ow. So: ( I a ) ( I a ) (1 ) P, Q P, Q For ths auty P 50,000 ad Q 1000 Q PV Pa ( a v ) (1 ) 1000 PV a a v (50, 000) ( 41 ) (1.05) ; I / Y 5; PMT 1; CPT PV a v PV 1000 (50, 000)( ) (( ) ) ( ) 661,250.05

36 62. (S08T1) Sarah s recevg a perpetuty of 1000 payable at the begg of each year. Joh s recevg a perpetuty mmedate that pays 200 at the ed of year oe, 400 at the ed of year two, 600 at the ed of year three, etc. The preset value of Sarah s perpetuty s equal to the preset value of Joh s perpetuty f the preset values are calculated at. Calculate. Sarah's Perpetuty = Joh's Perpetuty (1 ) 2 Multply both sdes by b b 4ac 8 8 4(10)( 2) a 2(10) 63. (S11HW) A auty pays 10 at the ed of year 2, ad 9 at the ed of year 4. The paymets cotue decreasg by 1 each two year perod utl 1 s pad at the ed of year 20. Calculate the preset value of the auty at a aual effectve terest rate of 5%. For ths auty, the paymets are every two years so we eed the terest rate for a two year perod ad the we just calculate everythg else the same as ay other auty. We eed (1/2) (1/2) 1/2 (1/2) (1.05) / 2 1/ 2 1/ 2 Q 1 PV Pa a v a a v ( ) 10 ( 10 )

37 64. Book Problem 3.9, Number 2 We are gve that the fal accumulated amout s 100,000. We also kow that 100,000 10(5,000) X Is where X=5000(). Therefore we ca solve for X 10(1.1) (1.1) X X Book Problem 3.9, Number (1.1) X 1 (1.1) We are gog to splt ths to a perpetuty ad a auty. 1. Perpetuty: Startg at tme 23, the paymets wll be $980 forever. 22 (980) PV (1.04) Auty: Ths wll be a 22 year creasg auty PV 320a 22 a 22v Now we ca add the two values together. We wll also wat to brg the total value forward by 1 year to be the value at the frst paymet (1.04) 19,591.87

38 66. (S11HW) Jula vests 1000 at the ed of each year for 20 years to Fud A. Fud A ears a aual effectve terest rate of 5%. At the ed of ed of each year, the terest s removed from Fud A ad vested Fud B. Fud B ears a aual effectve terest rate of 8%. What s the total amout that Jula wll have at the ed of 20 years. For Fud A, the amout at the ed of 20 years wll be (1000)(20)=20,000 sce the terest s removed each year. The terest amout that s deposted to Fud B s 1000(0.05) 50 at the ed of year 2. There s ot terest the frst year sce the frst depost s at the ed of the frst year. The terest the thrd year s 2000(0.05) 100. The terest cotues to crease by 50 each year. At the ed of twety years, we wll have twety paymets of 1000 plus the amout that terest paymets of 50 each year wll have accumulated to. 1000(20) 50( Is) (1.08) 50 1 (1.08) , (1.08) (1.08) 36, (S11HW) Brado vests 100,000 Fud A today whch ears a aual effectve terest rate of 8% terest. The terest o Fud A s pad at the ed of each year to Fud B whch ears a aual effectve terest rate of 9%. The terest o Fud B s also pad out at the ed of each year to Fud C whch ears a aual effectve terest rate of 10%. At the ed of years, Brado lqudates all three Fuds. How much does Brado have? To uderstad the problem, summarze the three fuds: A. 100,000 at tme 0, wth terest take out. B. Yearly paymets of 100,000*0.08=8000 wth terest take out. C. Yearly paymets of 8000*0.09=720 creasg by 720 each tme. We wll ed up wth: (1.1) (1.1) , 000 8, (1.1) (1.1) 263, Note that there are oly 11 $720 paymets because they do t beg utl Fud B accumulates terest.

39 Chapter 3 Secto Book Problem 3.11, Number 1 Ths problem ca be doe completely usg your calculator. Frst, fd the umber of paymets before the drop paymet: PV 200,000 PMT 25,000 I / Y 1 CPT N Ths meas that the 9 th paymet wll be the drop paymet. Leave everythg your calculator. Go to 2ND AMORT P1 1 P2 8 BAL (NOTE: you always roud N from above dow to put to P2.). Ths s the OLB 8. We ca ow fd the drop paymet; 9, (1.01) 9, Book Problem 3.11, Number 2 Set up the tal equato the same way wth your calculator: PV 200,000 PMT 25,000 I / Y 1 CPT N Go to 2ND AMORT P1 1 P2 8 BAL We ca ow fd the balloo paymet: 25,000 9, ,

40 Chapter 3, Secto 13 You are gve the followg table for use wth Problems 60 to 72: Year 1 Year2 Year 3 Year 4 Year 5 Portfolo Year (S08T2) Elmer vests 3000 o Jauary 1, 2001 a fud whch credts terest usg the vestmet year method. Herma vests 3000 o Jauary 1, 2000 a fud whch credts terest usg the portfolo method. O December 31, 2007, how much wll Herma s fud exceed Elmer s fud? Elmer s Balace /31/07: 3000(1.06)(1.0575)(1.056)(1.054)(1.053)(1.052)(1.0525)=4, Herma s Balace /31/07: 3000(1.073)(1.0705)(1.068)(1.065)(1.061)(1.057)(1.052)(1.0525)=4, Herma s Balace exceeds Elmer s Balace by $ (SHW) A fud credts terest usg the vestmet year method. Kayla vests 100 o Jauary 1, She also vests 300 o Jauary 1, Calculate the amout that she wll have o December 31, (1.055)(1.053)(1.051)(1.05)(1.051)= (1.045)(1.047)(1.0495)= Total= =$473.33

41 72. (SHW) Ratk s recevg a four year auty wth aual paymets 2000 o the Jauary 1 of each year. The frst paymet s made Jauary 1, Ratk vests each paymet a fud usg the vestmet year method. How much does Ratk have o December 31, 2007? 2000(1.045)(1.047)(1.0495)(1.052) +2000(1.05)(1.0535)(1.055) +2000(1.055)(1.0575) +2000(1.06) =