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

Size: px

Start display at page:

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

Phoebe Lester
6 years ago
Views:

1 Chapter 3, Sectio 2 1. (S13HW) Calculate the preset value for a auity that pays 500 at the ed of each year for 20 years. You are give that the aual iterest rate is 7% v PV Qa Q i 0.07 Usig your calculator: N 20 I / Y 7 PMT 500 CPT PV *Remember that you should eter the umber, the press the correspodig time value of moey butto. For example, type i 20 the press N. Also, your calculator requires either the paymets or the preset value to be egative. So, eve though the above keystrokes resulted i a egative preset value, your fial aswer is positive Alteratively, you could eter your paymet as egative 500 ad the your resultig preset value will be positive. 2. (S13HW) Calculate the preset value of a auity immediate with mothly paymets of 200 for 10 years usig a iterest rate of 9% compouded mothly. (10) i v PV Qa Q Q i i 0.09 Usig your calculator: N 0 I / Y 0.75 PMT 200 CPT PV *Note that N is always the umber of paymets, ot ecessarily the umber of years. Also, for i mothly paymets, i. Keep this i mid whe isertig umbers ito your calculator!

2 3. (S13HW) Calculate the preset value of a auity immediate with mothly paymets of 200 for 10 years usig a aual effective iterest rate of 9% First, let s fid i sice we have mothly paymets. 1/ i i We ca also use the calculator to fid this value. ICONV EFF 9 I / Y CPT NOM i Now that we have foud our appropriate iterest rate, we may proceed with the problem. PV (10) i i Usig our calculator: N 0 I / Y PMT 200 CPT PV (S13HW) Calculate the accumulated value of a auity immediate with aual paymets of 600 for the ext 25 years usig a aual effective iterest rate of 4% FV Qs Usig your calculator: N 25 I / Y 4 PMT 600 CPT FV *Note: Just like with preset values, the calculator will give you egative aswer for future value if you etered positive paymets. Make sure your fial aswer is positive!

3 5. (S13HW) Calculate the accumulated value of a auity that pays 5000 at the ed of each year for the ext 8 years usig a iterest rate of 10% compouded semi-aually. First, let s fid the aual effective rate. (2) 2 2 i i We ca also do this with our calculator: ICONV NOM 10 I / Y 2 CPT EFF Now we ca calculate the future value: FV Usig your calculator: N 8 I / Y PMT 5000 CPT FV

4 6. (S13HW) Julia is buyig a ew car for 23,000. She is fiacig the etire amout with a loa with mothly paymets over the ext 4 years. The iterest rate o Julia s loa is 8.4% compouded quarterly. Calculate Julia s mothly paymet. First, let s get a appropriate iterest rate for mothly paymets. (4) 4 i i i Or, with your calculator: ICONV NOM 8.4 I / Y 4 CPT EFF / Y CPT NOM i * Notice that we have to use two steps i the calculator: first fid the effect rate, the the mothly rate. Now we ca fid the paymet: (4) 1 ( ) P P With your calculator: N 48 I / Y PV CPT PMT

5 7. (S13HW) Muhamad deposits 150 ito a bak accout at the ed of each quarter for 10 years. Durig this time period, Muhamad ears a aual effective iterest rate of 6.8%. Calculate the amout that Muhamad has at the ed of 10 years. (4) 4 (4) 1/4 i i Fid the appropriate iterest rate, Or, (4) ICONV EFF 6.8 I / Y 4 CPT NOM i *4 The fid the future value: FV 10* Or, with your calculator: N 40 I / Y PMT 150 CPT FV

6 8. (S13HW) Tyler wats to have 1000 i oe year to buy a ew HDTV for the Super Bowl. He decides that he will ivest P at the ed of each moth i a accout earig 8% compouded quarterly i order to accumulate the Calculate P. First, let s get a appropriate iterest rate for mothly paymets. (4) 4 i i i Or, with your calculator: ICONV NOM 8 I / Y 4 CPT EFF / Y CPT NOM i * Notice that we have to use two steps i the calculator: first fid the effect rate, the the mothly rate. Now we ca fid the paymet: ( ) P P With your calculator: N I / Y FV 1000 CPT PMT 80.34

7 9. (S11HW) Michael wo the lottery! He has the followig payout optios: a. Oe millio at the ed of each year for the ext 20 years; or b. A lump sum of 7,469, paid ow. Calculate the aual effective iterest rate at which both optios have the same preset value. This problem eeds to be doe usig your calculator: N 20 PMT PV CPT 1/ Y So your aswer is %. Note that you must eter either the PMT or the PV as a egative umber i order to get the aswer! 10. (SHW) Yi is payig a car loa with paymets of 500 at the ed of each moth. The loa has a mothly effective iterest rate of 1%. If the car loa is for 18,986.98, calculate the umber of paymets that Yi will eed to make. Although you ca calculate this by had, it is much more practical to use the calculator. PMT 500 I / Y 1 PV CPT N (SHW) For a give iterest rate, s = ad a = Calculate. 1 1 a s i 1 1 i % s (0.08) l( ) 13 l(1.08)

8 . (SHW) If d 0.1, calculate a. 14 d i 1d a v i (S11HW) The accumulated value of a year auity is four times the preset value of the same auity. Calculate the accumulated value of 100 i 2 years. (Note: This is NOT askig for the accumulated value of a auity just the accumulated value of a sigle paymet of 100.) Remember the idetity: s (1 i ) a We kow that, s 4a 4 a (1 i) a 4 (1 i). We ca combie these two facts to solve the problem. 2 We wat to fid 100(1 i). This is easy to do usig substitutio: (1 i) 100 (1 i) 100(4) 1600 Chapter 3, Sectio (S13HW) Hua is receivig mothly paymets of 1000 at the start of each moth for 10 years. Calculate the preset value of these paymets usig a aual effective rate of %. First, let s get a appropriate iterest rate for mothly paymets. i i Or, with your calculator:

9 ICONV EFF I / Y CPT NOM i * Now we ca fid the preset value: 0 i 1 1 i PV 1000a i ( ) **Note: We ow have a auity due so we have the extra term to accout for the paymets beig at the begiig of the moth With your calculator: **I BGN Mode!** N 0 I / Y PMT 1000 CPT PV

10 15. (S13HW) Alyssa is the beeficiary of a trust fud which will pay her 1000 at the begiig of each moth for the ext 5 years. Alyssa ivests each paymet ito a accout earig a iterest rate of 7.2% compouded mothly. How much will Alyssa have at the ed of 5 years? FV s Or use your calculator **i BGN mode** N 60 I / Y 0.6 PMT 1000 CPT FV (S13HW) Pigpig wats to have 100,000 whe he retires i 20 years. He deposits level paymets at the begiig of each quarter ito a accout earig a quarterly effective iterest rate of 2%. Determie the amout of the paymet if Pigpig is goig to achieve his goal Ps P P Or use your calculator **i BGN mode** N 80 I / Y 2 FV CPT PMT

11 17. (SHW) You are give a ad a Calculate. We kow that a a (1 i). Therefore, plug the iterest rate ito a to fid a (0.01) 1.01 l (0.01) a (1 i) Now we ca a (S11HW) At the start of each moth for 50 moths, Julia deposits 100 ito her bak accout. At the ed of 50 moths, Julia has Calculate the aual effective iterest rate beig eared by Julia. You eed to use your calculator for this problem. Make sure it is i BGN mode! N 50 PMT 100 FV CPT 1/ Y i (1 i) ( ) i %

12 19. (SHW) Ziche wats to accumulate a sum of moey at age 65 so he ca retire. I order to accomplish this goal, he ca deposit 80 per moth at the begiig of the moth or 81 per moth at the ed of the moth. Calculate the aual effective rate of iterest eared by Ziche. Use the idetity: a a (1 i) but make it applicable to mothly paymets: 80a 81a We get: a 81 i a 80 Now that we have the mothly effective rate, we ca fid the aual effective rate. i 1 i i % i a a 1 Chapter 3 Sectio (SHW) A perpetuity immediate pays 10,000 per moth. The iterest rate eared by the perpetuity is 8% compouded mothly. Calculate the preset value of this perpetuity PV 1000a ( m) i 0.08 m

13 21. (SHW) Mohamad is the beeficiary of a trust fud. Mohamad ad his descedets will receive a paymet of 1000 at the begiig of each moth forever. Usig a aual effective iterest rate of 4%, determie the curret value of these paymets. First, we eed to fid a appropriate iterest rate for mothly paymets i i (1/) Now that we have the mothly iterest rate, we ca calculate the preset value. i PV 1000a * i i NOTE: We had to multiply our preset value by 1 to accout for paymets beig at the begiig of the moth. 22. (S08Q2) Purdue has asked our class to fud a scholarship for actuarial studets. The scholarship will pay 1000 aually with the first paymet beig made immediately. Our cotributios will ear a aual effective rate of 5%. How much must our class doate i order for the scholarship to cotiue forever? Luckily, we were give a aual iterest rate ad we have yearly paymets. However, we do eed to make sure to multiply our preset value by (1+i) to accout for paymets begiig immediately. PV a

14 23. (S11HW) Katie buys a perpetuity due of 1000 per moth for 100,000. Calculate the aual effective rate of iterest used to calculate the price of this perpetuity. i a 1000 i i i i i 99i i 0. Now, we eed to fid the aual effective iterest rate. i 0. (1 i) 1 1 i

15 24. (S09T1) Tyler has iherited $1 millio. He has decided to use his iheritace to purchase oe of the followig: a. A 30 year auity immediate with aual paymets of 106,079.25; or b. A perpetuity due with quarterly paymets of P. Both optios are based o the same iterest rate. Calculate P. First, we ca fid the iterest rate of the 30 year auity. This ca be doe very easily usig our fiacial calculator. PV 1,000,000 PMT 106, N 30 CPT I / Y (Remember to make the preset value or the paymets egative so you do t get a error!) Now, we ca fid the equivalet quarterly iterest rate i order to fid P. (4) i (4) i (4) i ,000,000 Pa P P (4) i P 23, (S11HW) The value of a perpetuity immediate where the paymet is P is 1000 less tha the value of a perpetuity due where the paymet if P. Calculate P. We are give, Pa Pa It will be easier to solve this equatio if we write it i terms of P ad i. P P(1 i) 1000 i i P P Pi 1000i 0 Pi 1000i Pi 1000i P 1000

16 26. (SHW) A perpetuity is fuded by a doatio of 500,000. Paymets of P are to be made at the ed of every secod year. I other words, P will be paid at time 2, 4, 6, etc. If the fud ears a aual effective iterest rate of 8%, calculate P. Note that the paymets are made every secod year. This makes m=1/2 i the i ( m) equatio. A commo mistake would be to make m=2. It is importat to uderstad the differece. (1/2) i /2 (1/2) 2 i , 000 Pa P i (1/2) 0.5 P 500, 000(0.1664) Chapter 3, Sectio (S11HW) A mothly auity immediate pays 100 per moth for moths. Calculate the accumulated value moths after the last paymet usig a omial rate of 4% compouded mothly. We eed to calculate the accumulated value at the ed of the paymets ad the move this value forward moths i s

17 28. (S11HW) A mothly auity due pays 100 per moth for moths. Calculate the accumulated value moths after the last paymet usig a omial rate of 4% compouded mothly. This is essetially the same problem as #30. Because the paymets are ow at the begiig of the i moth, we eed to use s which is equal to s 1. However, i order to move the accumulated value at the ed of moths forward more moths, we really oly eed to multiply 11 i by 1 sice we are already accoutig for oe moth s accumulatio by usig a auity due. Therefore, we get the same aswer. To see this a little more clearly, here is the equatio: i s (S11HW) A mothly auity due pays 100 per moth for moths. Calculate the accumulated value 24 moths after the first paymet usig a omial rate of 4% compouded mothly. We ca us the same formula i #31 except move the accumulated value forward a additioal moth i s (S11HW) Calculate the curret value at the ed of 5 years of a auity due payig aual paymets of 00 for years. The aual effective iterest rate is 6%. To solve this problem, we will fid the accumulated value of the etire auity ad the discout it seve years i order to accout for the fact that ow is at the ed of year s v

18 31. (S11HW) Calculate the preset value of a auity immediate with 20 aual paymets of 500 if auity does ot start util five years have passed. The aual effective iterest rate is 8%. This time, we will fid the preset value of the auity at the start of the paymets ad the discout it 5 years a v 500 (1.08) (S11HW) Joh buys a series of paymets. The first paymet of 50 is i six years. Aual paymets of 50 are made thereafter util 14 total paymets have bee made. Calculate the price Joh should pay to realize a aual effective retur of 7%. We will fid the preset value of a auity due with 14 paymets the discout it six years (1.07) 6 50a v 50 (1.07)(1.07)

19 Chapter 3, Sectio (SHW) Yue bought a house with a 200,000 mortgage for 15 years beig repaid with paymets at the ed of each moth at a iterest rate of 6% compouded mothly. What is the outstadig balace at the ed of 10 years immediately after the 0 th paymet? First, let s fid the amout of the mothly paymet , 000 Pa P P Realize that you ca also do this usig your fiacial calculator: 180 PV 200,000 I / Y 0.5 CPT PMT Now that we have the paymet, we ca fid the OLB. OLB ,000 1 Ps , (SHW) If Yue pays a extra 100 each moth, what is the outstadig balace at the ed of 10 years immediately after the 0 th paymet? Sice Yue decides to pay extra each moth, all we eed to chage from the formula above is the paymet amout OLB 200, ( P 100) s ,

20 35. (SHW) Nacy borrowed moey to buy a ew car. The loa has a iterest rate of 7.8% compouded mothly. Nacy s mothly paymet is 246 ad she has 11 paymets left with the ext paymet due i oe moth. Calculate the outstadig balace o her loa. We kow that if Nacy pays the remaiig 11paymets she will pay off the loa. This meas that her OLB is equal to the preset value of her remaiig paymets OLB 246a (SHW) Phil borrows 100,000 at a aual effective iterest rate of 7%. Phil is repayig the loa with aual paymets of 10,000. Calculate how much Phil still owes immediately after his 10 th paymet. This is a straightforward applicatio of our equatio for OLB. 10 OLB L(1 i) Ps (1.07) (S08T1) Thomas bought a house 5 years ago. I order to buy the house, he borrowed 50,000 to be repaid with 360 mothly paymets of Thomas pays each moth for 60 moths. Calculate outstadig loa balace o Thomas loa immediately after the 60 th paymet. First, let s fid the iterest rate of Thomas loa. 360 PV 50,000 PMT CPT I / Y 0.80 Now that we have all of this iformatio, it is very easy to fid the OLB usig our calculator. Do ot clear aythig out of your calculator, just correct the followig values: 60 CPT FV We could plug the iterest rate i ad calculate the OLB by had, but the calculator saves a lot of time.

21 38. (S09T1) Ashley borrows 30,000 to buy a ew car. Her loa carries a mothly effective iterest rate of 1%. She will repay the loa by makig mothly paymets of Ashley makes k paymets of Immediately after the k th paymet, she pays off the outstadig balace of her loa by makig a paymet of 13, Determie k. Start by fidig. PV I / Y 1 PMT CPT 54. Now determie how may paymets are left. The preset value of future paymet is a 13, PV 13, I / Y 1 PMT CPT k 21 k The 54 k 21 k 33 You ca also trick the calculator ito givig you the aswer directly as follows: PV I / Y 1 PMT FV CPT (S11HW) A loa of 10,000 is beig repaid with 20 o-level aual paymets. The iterest rate o the loa is a aual effective rate of 6%. The loa was origiated 4 years ago. Paymets of 500 at the ed of the first year, 750 at the ed of the secod year, 1000 at the ed of the third year ad 50 at the ed of the fourth year have bee paid. Calculate the outstadig balace immediately after the fourth paymet. We eed to use the retrospective approach , 000(1.06) 500(1.06) 750(1.06) 1000(1.06) (S11HW) Calculate the outstadig balace to the loa i #40 oe year after the fourth paymet immediately before the fifth paymet. Because there are o cash flows i betwee paymets, we just eed to add a year of iterest (1.06)

22 41. (S11T1) Mike bought a ew high defiitio televisio for Mike paid for the televisio usig a 15 moth loa with a iterest rate of 9% compouded mothly. Mike forgot to make the 8 th paymet o the loa. Determie Mike s outstadig loa balace at the ed of the th moth. Oe way to do this problem is to assume that all paymets have bee made ad the add i the missig 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 missig paymet so OLB (1.0075) The differece betwee ad is from roudig. If you do ot roud the paymet to the earest pey, you get Here is aother way proposed by Alliso who is the TA for this class. This questio should be doe usig your calculator because it requires multiple steps. First, fid the level paymets that Mike should make: 15 I / Y 0.75 PV 2000 CPT PMT Secodly, fid the OLB immediately after the 7 th paymet. Do ot clear your calculator betwee steps! 7 CPT FV The, we eed to add iterest for oe moth to the OLB after the 7 th paymet: i Fially, we kow that oce Mike starts makig paymets agai, he will use the same paymet as he did before. To fid the OLB at the ed of the th moth, we eed to add i 4 more paymets. Make

23 sure you have t cleared out your calculator i betwee steps! 4 PV CPT FV

24 Chapter 3, Sectio (S11HW) A auity 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 i = There are two ways to do this problem PV 100a 200a v PV 200a 100a (S11HW) A auity 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 i = There are also two ways to solve this problem i 0.08 FV 100s 1 200s or FV 100s 100s

25 44. (S11HW) A auity 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 right after the 48 th paymet at i = Right after the 48 th paymets, you have received 48 paymets of 100 ad expect to receive 48 more paymets of s 200a Chapter 3, umber (S08T1) Kurt is the beeficiary of a trust. Uder the trust, he will receive paymets at the ed of each year for the ext 20 years. The paymet will be 2000 at the ed of oe year. Each subsequet paymet will icrease by 8%. I other words, the paymet at the ed of the secod year will be 2000(1.08), the paymet at the ed of the third year will be 2000( ), etc. Calculate the preset value of Kurt s paymets uder the trust usig a aual effective iterest rate of 9%. Write out a expressio for the preset value of paymets i order to see that it is the sum of a geometric series: FirstTerm NextAfterLast Recall that the sum of a geometric series= 1 ratio

26 46. (S09T1) Parker has wo the lottery. He will receive 20 aual paymets with the first paymet made ow. The first paymet will be for 25,000. Each subsequet paymet will be 110% of the previous paymet. I other words, the secod paymet will be 25000(1.10) ad the third paymet will be 25000(1.10) 2, etc. Calculate the preset value of Parker s wiigs usig a aual effective iterest rate of 5%. Write out a expressio for the preset value of paymets i order to see that it is the sum of a geometric series: FirstTerm NextAfterLast Recall that the sum of a geometric series= 1 ratio

27 47. (SHW) Today is Zharfa s 25 th birthday. As a birthday gift, his parets have give him a auity immediate with aual paymets. The first paymet is 10,000. Each paymet thereafter is 95% of the prior year s paymet. The last paymet will be made o Zharfa s 65 th birthday. Zharfa ivests each paymet at a aual iterest rate of 8%. How much will Zharfa have o his 65 th birthday. Realize that Zharfa is receivig a total of 40 paymets. A commo mistake is to use 65 paymets because they stop o his 65 th birthday. Also, otice that we are ow fidig the accumulated value! Write out a expressio for the accumulated value of paymets i order to see that it is the sum of a geometric series: FirstTerm NextAfterLast Recall that the sum of a geometric series= 1 ratio , 661,

28 48. (SHW) A auity immediate has geometrically icreasig paymets made aually for 18 years. 1 2 The first paymet is 00. The secod paymet is 00(1.1). The third paymet is 00(1.1) ad paymets cotiue to icrease at a rate of 10% each year. Calculate the preset value of this auity at a aual iterest rate of 10%. Write out a expressio for the preset value of paymets i order to see that it is the sum of a geometric series: Notice that this simplifies to Now we see that this is very simple: (SHW) A auity due has mothly paymets for 8 years. The first paymet is 600 with each successive paymet beig 1% larger tha the previous paymet. Kristi ivests each paymet i a accout that ears % compouded mothly. Calculate the amout that Kristi will have i the accout at the ed of 8 years. i 0. Note that %. This meas that the paymets are growig at the rate of iterest just like i umber 54. Therefore, we wo t eed a sum of a geometric series FV FV *

29 50. (SHW) A perpetuity makes paymets at the ed of each year. The first paymet is Each paymet thereafter is 103% of the previous paymet. Calculate the preset value of this perpetuity at a iterest rate of 8%. We ca also write the value of a perpetuity as a sum of a geometric series: PV We ca say that , so we have

Chapter 4: Time Value of Money

Chapter 4: Time Value of Money FIN 301 Class Notes Chapter 4: Time Value of Moey The cocept of Time Value of Moey: A amout of moey received today is worth more tha the same dollar value received a year from ow. Why? Do you prefer a