Math 373 Chapter 3 Homework Spring 2015jbeckley/q/WD/MA373/S15/S15... · 2015-02-24 · Chapter 3 Homework Spring 2015 1. Trey is receiving an annuity immediate which pays 150 each

February 24, 2015 Copyright Jeffrey Beckley 2014 2015

Math 373 Chapter 3 Homework

Spring 2015

1. Trey is receiving an annuity immediate which pays 150 each year for 20 years. Calculate the present value of this annuity at an annual effective interest rate of 5%.

Solution:

1 n

n

va

i

201

11.05

150 1869.3315510.05

Or with the calculator: N=20, I/Y=5, PMT=-150, CPT PV=1869.31551

2. Davis is the beneficiary of an annuity immediate pays 3000 each year for 15 years. Calculate the

accumulated value of this annuity at an annual effective interest rate of 8.2%.

Solution:

(1 ) 1n

n

is

i

15(1.082) 13000 82735.4774

.082

With the calculator: N=15, I/Y=8.2, PMT=-3000, CPT FV=82735.4774


3. An annuity due pays 250 each year for 12 years. Calculate the present value of this annuity at an annual effective interest rate of 6%.

Solution:

121

11.06

(1.06)(250) 2221.7186440.06

With the calculator set to BGN: N=12, I/Y=6, PMT=-250, CPT PV=2221.718644

4. An annuity due pays 325 each year for 18 years. Calculate the accumulated value of this annuity

at an annual effective interest rate of 4%.

Solution:

18

1.04 1(325) 8668.149555

0.04

1.04

With the calculator set to BGN: N=18, I/Y=4, PMT=-325, CPT FV=8668.149555

5. Madi is the beneficiary of an annuity that will pay her 60 at the end of each month for 8 years.

Calculate the present value of this annuity at 6% compounded monthly.

Solution:

12 8

11

0.061

1260 4565.713095.06

12

N=96, I/Y=.5, PMT=-60, CPT PV=4565.713095


6. Madi is the beneficiary of an annuity that will pay her 60 at the end of each month for 8 years.

Calculate the present value of this annuity at an annual effective interest rate of 6%.

Solution: 12

(12)

(12)

(1.06) 112

0.00486755112

i

i

81

11.06

60 4592.7115540.004867551

N=96, I/Y=0.4867551, PMT=-60, CPT PV=4592.711

7. Colin is buying a house for 113,450. He is going to finance the entire amount with a loan which

he will repay with monthly payments for 15 years. The interest rate on Colin’s loan is 9.6%

compounded monthly.

Calculate the amount of Colin’s monthly payment. Solution:

12 15

11

.0961

12 113,450.096

12

95.1385953 113,450

1191.528214

P

P

P

N=180, I/Y=.8, PV=-113450, CPT PMT=1191.528214


8. Cameron is taking a loan of L . He will repay the loan with monthly payments of 97 for 10 years.

Calculate L , the amount of the loan, assuming that the loan has an annual effective interest rate of 6.9%.

Solution:

12(12)

(12)

10

1.069 112

.0055757912

11

1.06997 8469.992766

.00557579

i

i

N=120, I/Y=.557579, PMT=-97, CPT PV=8469.99

9. Mitchell has inherited 250,000. He decides to take his inheritance and buy an annuity with quarterly payments for the next 20 years. The first payment will be made immediately.

Calculate the amount of the quarterly payment if the interest rate is 6% compounded quarterly. Solution:

4 20

11

0.061

.064250,000 10.06 4

4

250,000 47.10343335

5307.468739

P

P

P

Calculator set to BGN

N=80, I/Y=1.5, PV=-250,000, CPT PMT=5307.468739


10. Pri wants to accumulate 30,000 to buy a new car at the end of five years. She deposits D into a

bank account at the start of each month over the next five years. The bank account earns an annual effective interest rate of 9%.

Calculate D in order that she will have the 30,000 at the end of 5 years.

Solution: 12

(12)

(12)

5 12

1.09 112

.00720732312

(1.007207323) 1(1.007207323) 30,000

.007207323

398.5572601

i

i

D

D


N=60, I/Y=.7207323, FV=-30,000, CPT PMT=398.5572601

11. Laura has a loan which requires payments of 1000 semi-annually for n years. The amount of the loan is 13,003.17 and the loan has an interest rate of 12% compounded semi-annually.

Compute n .

Solution: 2

2

2

11

.121

121000 13,003.170.12

2

11 .7801902

1.06

1ln ln .2198098

1.06

12 ln ln .2198098

1.06

2 26.000017

13

n

n

n

n

n

n


12. Antony is receiving an annuity with annual payments of 250 at the end of each year for 20 years. The present value of this annuity is 3323.59.

Calculate the interest rate for this annuity.

Solution: N=20, PMT=-250, PV=3323.59, CPT I/Y=4.25%

13. Sam deposits 1000 into a bank account at the end of each year for 17 years. During this 17 year

period, the bank pays an annual effective interest rate of i . At the end of 17 years, James has

accumulated 30,000.

Determine i . Solution:

N=17, PMT=-1000, FV=30,000, CPT I/Y=6.69116992

14. Jie has 100,000 and uses it to purchase an annuity due which pays 18,000 annually for 7 years. Calculate the annual effective interest rate for this annuity.

Solution:

Calculator set to BGN N=7, PMT=-18,000, PV=100,000, CPT I/Y=8.485631304

15. If 0.1d , calculate 12

a .

Solution:

12

14

0.10.11111

1 0.9

11

1.111116.458134178

0.11111

di

d

a

16. The accumulated value of an n year annuity is four times the present value of the same annuity.

Calculate 2100(1 ) ni .

Solution:

an(1+ i)n = 4a

n® (1+ i)n = 4

(1+ i)2n = 42 = 16

100(1+ i)2n = 100(16) = 1600


17. Sandra wants to accumulate a sum of money at age 65 so she can retire. In order to accomplish this goal, she can deposit 80 per month at the beginning of the month or 81 per month at the

end of the month. Calculate the annual effective rate of interest earned by Sandra.

Solution: The number of payments n is not given, but this is true for any n so we just set it up and n will cancel out.

(12)

80 81 but (1 ) 80(1 ) 81

Now divide both sides by 80(1 ) 81 1 81/ 80

Since payments are monthly, the i above is really but we need the annual effective interest r12

n n n n n n

n

a a a i a i a a

a i i

i

12 12(12)

ate

81(1+i)= 1+ 1.160754518 16.075%

12 80

ii

18. Tyler is the beneficiary of a trust fund which pays him 1000 at the end of each month forever.

At an interest rate of 9.6% compounded monthly, calculate the present value of Tyler’s

perpetuity.

Solution:

1000125,000

0.096( )

12

19. Kevin wants to fund a scholarship at Purdue. He wants the scholarship to pay 3000 at the start

of each year with the first payment being made now. These payments are to continue into perpetuity. If the annual effective interest rate is 4%, determine the amount that Kevin must donate to Purdue to fund this scholarship.

Solution:

30003000 78,000

.04


20. Drew is the beneficiary of a trust fund that has 100,000 in the fund. At the end of each month he or his decendents will receive 1000 forever.

Calculate the annual effective interest rate earned by the trust fund.

Solution:

(12)

(12)

1000100,000

( )12

1000 100,00012

i

i

(12)

12

1000.01

12 100,000

(1.01) 1 1.126825 1

12.6825%

i

i

i

21. Tyler has inherited $1 million. He has decided to use his inheritance to purchase one of the following:

a. A 30 year annuity immediate with annual payments of 106,079.25; or b. A perpetuity due with quarterly payments of P.

Both options are based on the same interest rate.

Calculate P.

Solution: N=30, PMT=-106,079.25, PV=1,000,000 CPT I/Y=10%

4(4)

(4)

1.1 14

0.02411374

11 1,000,000

0.0241137

23,545,92

i

i

P

P


22. The value of a perpetuity immediate where the payment is P is 1000 less than the value of a perpetuity due where the payment if P. Calculate P.

Solution:

1000 (1 )

1000

1000

P P Pi P

i i i

P P P PP

i i i i

P

23. A perpetuity is funded by a donation of 500,000. Payments of P are to be made at the end of

every second year. In other words, P will be paid at time 2, 4, 6, etc. If the fund earns an annual

effective interest rate of 8%, calculate P. Solution:

1.08 = 1+i(.5)

.5

æ

èçö

ø÷

i(.5)

.5= .1664

P

i(.5)

.5

= 500,000

P

.1664= 500,000

P = 83,200

24. A deferred annuity has 18 annual payments of 100 where the first payment is made in 7 years.

Calculate the present value of this deferred annuity using an annual effective interest rate of 5.75%.

Solution:

6

18100a v N=18, I/Y=5.75, PMT=-100, CPT PV = 1103.381873

61

1103.381873 788.93929661.0575


25. An annuity immediate has 12 annual payments of 600. The annual effective interest rate is 10%.

Calculate the accumulated value of this annuity 5 years after the last payment.

Solution:

600a12

(1.10)5 ®N=12, I/Y=10, PMT=-600, CPT FV=12830.57

12830.57(1.10)5 = 20,663.76

26. An annuity due has 12 annual payments of 600. The annual effective interest rate is 10%.

Calculate the accumulated value of this annuity 5 years after the last payment.

Solution:

27. A deferred annuity has 18 annual payments of 100 where the first payment is made in 7 years. Calculate the current value of this deferred annuity at the end of ten years using an annual

effective interest rate of 5.75%. Solution:

4 14100( )s a

4100s = N=4, I/Y=5.75, PMT=-100, CPT FV=435.8415109

14100a = N=14, I/Y=5.75, PMT=-100, CPT PV=944.0576449

435.8415109+944.0576449=1379.899156

28. John buys a series of payments. The first payment of 50 is in six years. Annual payments of 50 are made thereafter until 14 total payments have been made. Calculate the price John should pay to realize an annual effective return of 7%.

Solution:

50a14v5 ®N=14, I/Y=7, PMT=-50, CPT PV=437.2734

437.27341

1.07

æ

èçö

ø÷

5

= 311.77


29. Anji is the beneficiary of a perpetuity which pays 1000 per year with the first payment at the end of ten years.

Calculate the present value of Anji’s perpetuity at an annual effective interest rate of 8%.

Solution:

9 91000 112500( ) 6253.112089

0.08 1.08v

30. For a given interest rate, n

s = 21.4953 and n

a = 7.90378. Calculate n.

Solution:

1 1

1 18%

7.90378 21.4953

1 11 ( ) 1 ( )

1 1.087.90378.08

10.3676976 ( )

1.08

1ln(0.3676976) *ln( ) 13

1.08

n n

n n

n

n

ia S

i i

iai

n n

31. You are given that 10.17847n

a and 19.01987n

s . Calculate 2n

a .

Solution:

28

14*2

1 1

1 14.567%

10.17847 19.01987

1 11 ( ) 1 ( )

1 1.0456710.17847.04567

10.535149275 ( )

1.04567

1ln(0.535149275) *ln( ) 14

1.04567

11 ( )

1.04567 15.625471390.04567

n n

n n

n

n

ia S

i i

iai

n n

a


32. You are given 37.3537n

a and 37.7272n

a . Calculate n .

Solution:

(1 )

37.3537(1 ) 37.7272 0.01

11 ( )

1.0137.3537.01

10.626463 ( )

1.01

1ln(0.626463) *ln( ) 47

1.01

n n

n

n

n

a i a

i i

a

n n

33. Yuanzheng has a mortgage loan on his house. The loan was for 100,000 is being repaid with

level monthly payments for 15 years. The interest rate on the loan is 9% compounded monthly.

Calculate the outstanding loan balance right after the 120th payment.

Solution: First find amount of payments.

15 12100,000Pa

N=180, I/Y=9/12=.75, PV=-100,000, CPT PMT=1014.266584

120 180 120 601014.266584OLB Qa a

N=60, I/Y=.75, PMT=-1014.266584, CPT

PV=48860.64301

34. If Yuanzheng (from Problem immediately above) pays an extra 200 per month, what is his

outstanding balance right after the 120th payment? Solution:

120

120 1201014.266584 200 1214.266584 100,000(1.0075) 1214.266584OLB s

1201214.266584s N=120, I/Y=.75, PMT=-1214.266584, CPT FV=234977.9202

120100,000(1.0075) 234977.9202 10,157.78761


35. Robert borrowed 30,000 to be repaid with annual payments of 2000 for n years followed by a smaller payment at the end of 1n years.

The annual effective interest rate on this loan is 3.75%.

Calculate the outstanding loan balance right after the 10th payment.

Solution: If we use the retrospective approach, we do not need to know n or the amount of the smaller

payment.

10

10 10(30,000)(1.0375) 2000(s )

(30,000)(1.445043943) 2000(11.86783847) 19615.64135

OLB

36. Aaron is repaying a loan with monthly payments of 200. The interest rate on the loan is 6.9% compounded monthly.

Xiao has 14 payments remaining with the next payment due in one month.

Calculate the outstanding balance on this loan. Solution:

14200OLB a N=14, I/Y=6.9/12=.575, PMT=-200, CPT PV=2682.864348

37. Tracy borrows 30,000 to buy a new car. Her loan carries a monthly effective interest rate of 1%.

She will repay the loan by making monthly payments of 721.70.

Tracy makes k payments of 721.70. Immediately after the kth payment, she pays off the

outstanding balance of her loan by making a payment of 13,608.94. Determine k.

Solution:

OLBk =13,608.94

First find N

I/Y=1, PMT=-721.70, PV=30,000, CPT N=54

54721.70 13,608.94k N k k

OLB Qa a

I/Y=1, PMT=-721.70, PV=13,608.94, CPT N=21

54-k = 21

k=33


38. A loan of 10,000 is being repaid with 20 non-level annual payments. The interest rate on the loan is an annual effective rate of 6%. The loan was originated 4 years ago. Payments of 500 at

the end of the first year, 750 at the end of the second year, 1000 at the end of the third year and 1250 at the end of the fourth year have been paid. Calculate the outstanding balance

immediately after the fourth payment. Solution:

410000(1.06) accumulated value of payments made

3 2

4

500(1.06) 750(1.06) 1000(1.06) 1250 3748.208

10000(1.06) 3748.208 8876.56

39. Calculate the outstanding balance to the loan in Number 38 one year after the fourth payment

immediately before the fifth payment.

Solution:

5 4 3 210,000(1.06) 500(1.06) 750(1.06) 1000(1.06) 1250(1.06) 9409.16

40. Linhan bought a new high definition television for 2000. Linhan paid for the television using a

15 month loan with an interest rate of 9% compounded monthly.

LInhan forgot to make the 8th payment on the loan.

Determine Linhan’s outstanding loan balance at the end of the 12th month.

Solution:

Pa15

= 2000®N=15, I/Y=9/12, PV=-2000, CPT PMT=141.47

OLB12 =141.47a3®N=3, I/Y=.75, PMT=-141.47, CPT PV=418.123

4418.123 141.47(1.0075) 563.90

41. An annuity immediate has annual payments for 30 years. The payments for the first five years are 5000 each. The payments during the second five years are 4000 each. The payments during the last 20 years are 1000 each.

Calculate the present value at an annual effective interest rate of 6.5%.

Solution:

30 10 51000 3000 1000a a a

N=30, I/Y= 6.5, PMT = -1000, CPT PV = 13,058.67591

N=10, I/Y= 6.5, PMT = -3000, CPT PV = 21,566.49067

N=5, I/Y= 6.5, PMT = -1000, CPT PV = 4,155.679438

13,058.67591 + 21,566.49067 + 4,155.679438 = 38,780.84602


42. An annuity due has annual payments for 30 years. The payments for the first five years are 1000 each. The payments during the second five years are 3000 each. The payments during the last

20 years are 4000 each.

Calculate the present value at an annual effective interest rate of 4%. Solution:

30 10 54000 1000 2000a a a


N=30, I/Y= 4, PMT = -4000, CPT PV = 71,934.85853

N=10, I/Y= 4, PMT = -1000, CPT PV = 8,435.331611

N=5, I/Y= 4, PMT = -2000, CPT PV = 9,259.790449

71,934.85853 - 8,435.311611 – 9,259.790449 = 54,239.75647

43. Joseph is receiving an annuity with monthly payments for 10 years. The payments at the end of the first 60 months are 200 and the payments at the end of the second 60 months are 100. Joseph takes each payment and invests them in a fund earning 6% compounded monthly.

How much will Joseph have in the fund at the end of 10 years?

Solution:

120 60200 100s s

N=120, I/Y=6/12=.5, PMT=-200, CPT FV=32,775.87

N=60, I/Y=.5, PMT=-100, CPT FV=6977.00

32,775.87 – 6977.00 = 25,798.87

44. A 30 year annuity immediate with annual payments pays 2000 in years 1, 3, 5, …, 29 and pays

6000 in years 2, 4, 6, …, 30.


Solution: Convert i to biannual rate.

1+i(.5)

.5

æ

èçö

ø÷

.5

= 1.08 ®i(.5)

.5= .1664

2000a30 .08

+ 4000a15 .1664

N=30, I/Y=8, PMT=-2000, CPT PV=22, 515.57

N=15, I/Y=16.64, PMT=-4000, CPT PV=21,649.53

22,515.57 + 21, 649.53 = 44,165.15


45. Reid wants to buy an annuity immediate for his parents. The annuity will have annual payments

for 30 years. The payments will increase each year. The payment in the first year will be 8,000.

The payment in the second year will be 105% of the payments in the first year. The payment in

the third year will be 105% of the payments in the second year. This same pattern will continue

for all 30 years.

At an interest rate of 7% calculate the price that Reid should pay for this annuity. (The price is

the same as the present value.)

Solution:

1 30 318000(1.07) 8000(1.05) (1.07)172,895.56

1 1.05 /1.07

46. Zhe has won the lottery and will receive annual annuity payments at the beginning of each year

for 20 years. The first payment is 50,000. Each subsequent payment is 10% larger than the prior

payment. In other words, the first payment is 50,000. The second payment is 50,000(1.1). The

third payment is 50,000(1.1)2, etc.

Calculate the present value of Zhe’s winnings at an annual effective interest rate of 6%.

Solution:

201.10

11.06

50,000 1,454,407.831.10

11.06

47. If Zhe took each payment in the problem immediately above and invested it in a fund earning

5% interest, how much would he have at the end of 20 years.

Solution:

20

20

1.101

1.0550,000 (1.05) 4,277,912.36

1.101

1.05


48. Joanna is receiving an annuity due with 28 payments. The first payment is 13,000. Each

subsequent payment is 95% of the previous payment.

Calculate the present value of Joanna’s annuity at an annual effective rate of 4.5%.

Solution: 28

0.951

1.04513,000 133,083.90

0.951

1.045

49. An annuity immediate has geometrically increasing payments made annually for 24 years. The

first payment is 1000. The second payment is 11000(1.08) . The third payment is 21000(1.08)

and payments continue to increase at a rate of 8% each year.

Calculate the present value of this annuity at an annual interest rate of 8%.

Solution:

1 2 23

2 3 24

1 1 1 1

1000 1000(1.08) 1000(1.08) 1000(1.08)...

1.08 (1.08) (1.08) (1.08)

1000 1000 1000 1000...

(1.08) (1.08) (1.08) (1.08)

1000(24)22,222.22

1.08

PV


50. An annuity due has monthly payments for 7 years. The first payment is 800 with each successive payment being 1% larger than the previous payment. Tony invests each payment in

an account that earns 12% compounded monthly.

Calculate the amount that Tony will have in the account at the end of 7 years. Solution:

(12)(12)

1 2 8384 84

1 2 83

84 84

0.12 0.0112

800(1.01) 800(1.01) 800(1.01)(1.01) 800 ... (1.01)

(1.01) (1.01) (1.01)

800 800 800 ... 800 (1.01) 800(84)(1.01) 155,011.77

ii

AV PV

51. A perpetuity makes payments at the end of each year. The first payment is 2000. Each payment

thereafter is 103% of the previous payment. Calculate the present value of this perpetuity at an

interest rate of 8%.

Solution:

12000 0

1.0840,000

1.031

1.08


Chapter 3, Section 3.9

52. Brian earned a scholarship which pays him a monthly payment at the end of each month for 48

months. The first payment is 50. The second payment is 60. The third payment is 70. The

same pattern continues with each payment being 10 greater than the previous payment.

Calculate the present value of the scholarship using an interest rate of 9% compounded

monthly.

Solution:

48

48 48

10 150 48

0.09 /12 1 0.09 /12

n

n n

QPa a nv

i

a a

N=48, I/Y=.75, PMT=-1, CPT PV=40.18478

50(40.18478) + 1333.33(40.18478 - 33.533478)= 10,877.64

53. As a retirement benefit, Jeff receives an annuity due with makes annual payments for 20 years.

The first payment is 10,000. Each payment thereafter, the payment is 1500 greater than the

prior payment. In other words, the first payment is 10,000. The second payment is 11,500. The

third payment is 13,000, etc.

Calculate the present value of this retirement benefit at an annual effective rate of 5%.

Solution:

20

20 20

150010000 20 (1.05)

.05a a v

N=20, I/Y=5, PMT=-1, CPT PV=12.4622

10000(12.4622)+1500

.0512.4622 - 20v20( )é

ëêù

ûú(1.05) = 285,972.46


54. An annuity pays 100 at the end of the first quarter, 200 at the end of the second quarter, 300 at

the end of the third quarter, etc. The payments continue for 5 years.

Calculate the accumulated value of this annuity using an interest rate of 8% compounded

quarterly.

Solution:

20

20

20 20

100 1 .08100 20 1

.08.08 / 4 41

4

a a

N=20, I/Y=2, PMT=-1 CPT PV=16.3514

20

20100 1100(16.3514) 16.3514 20 1.02

.02 1.02

16095.18(1.02)20 = 23,916.59

55. An annuity due makes annual payments for 15 years. The first payment is 15,000. The second

payment is 14,000. The payments continue in the same pattern until the last payment of 1000

is made.


Solution:

15

15 15

100015000 15 (1.07)

0.07a a v

N=15, I/Y=7, PMT=-1, CPT PV=9.107914

15100015000(9.107914) 9.107914 15 (1.07) 90,064.74

.07v


56. Spenser is receiving annuity payments at the end of each quarter for 20 years. The first payment is P . The second payment is 2P . The third payment is 3P and payments continue to

increase in the same pattern.

The accumulated value of Spenser’s annuity is 75,000 using a quarterly effective interest rate of 2%.

Determine P . Solution:

Pa80

+P

.02a

80- 80v80( )é

ëêù

ûú(1.08)80 = 75,000

N=80, I/Y=2, PMT=-1, CPT PV=39.7445

(39.7445P +1166.78677P)(1.02)80 = 75,000

5882.37P = 75,000

57. Queenie is the beneficiary of a trust fund that will make a payment on each of her birthdays with the final payment on her 60th birthday. Today is Queenie’s 20th birthday and she will

receive the first payment of 50,000. Each subsequent payment will be 1000 less than the prior payment. In other words, she will receive 49,000 on her 21s t birthday, 48,000 on her 22nd birthday, etc.

Calculate the present value of Queenie’s payments at an annual effective interest rate of 5%.

Solution:

41

41 41

100050,000 41 (1.05)

.05a a v

N=41, I/Y=5, PMT=-1, CPT PV=17.29437

41100050,000(17.29437) 17.29437 41 (1.05) 661,250.05

0.05v


58. Sarah is receiving a perpetuity of 1000 payable at the beginning of each year. John is receiving a perpetuity immediate that pays 200 at the end of year one, 400 at the end of year two, 600 at

the end of year three, etc. The present value of Sarah’s perpetuity is equal to the present value of John’s perpetuity if the present values are calculated at i. Calculate i.

Solution:

2

2

2

2

2

2

1000 200 200(1 )

1000 1000 200 200

1000 1000 200 200

1000 800 200 0

.8 .2 0

( 1)( .2) .2

ii i i

ii

i i i

i i i

i i

i i

i i i

59. An annuity pays 10 at the end of year 2, and 9 at the end of year 4. The payments continue decreasing by 1 each two year period until 1 is paid at the end of year 20. Calculate the present value of the annuity at an annual effective interest rate of 5%.

Solution:

.5(.5) (.5)

10

10 10

(1.05) 1 0.10250.5 0.5

110 10

0.1025

i i

a a v

N=10, I/Y=10.25, PMT=-1 CPT PV=6.07913

10110(6.07913) 6.07913 10 38.2524

0.1025v


60. A perpetuity immediate makes annual payments. The first payment is 500. Each subsequent payment is 25 greater than the prior payment until the payments reach 750. Thereafter, all

payments will be 750.

Calculate the present value of this perpetuity at an annual interest rate of 6%. Solution:

Break this into two parts: 11 year P & Q annuity and a perpetuity discounted by 11 years.

11

11 11

25500 11

0.06a a v

N=11, I/Y=6, PMT=-1, CPT PV=7.88687

11

11

25500(7.88687) 7.88687 11 4815.1922

0.06

7506584.84

0.06

4815.1922 6584.84 11400.04

v

v

You could also break it up into a 10 year P&Q formula and perpetuity discounted 10 years.


61. Suyi invests 1000 at the end of each year for 20 years into Fund A. Fund A earns an annual effective interest rate of 5%. At the end of end of each year, the interest is removed from Fund

A and invested in Fund B. Fund B earns an annual effective interest rate of 8%. What is the total amount that Suyi will have at the end of 20 years.

Solution:

19 19

19 19

1000(0.05) 50

5050 19 (1.08)

0.08a a v

N=19, I/Y=8, PMT=-1, CPT PV=9.6036

19 1950 A= 50(9.6036) (9.6036 19 ) (1.08) 16,101.23

.08

Fund B=1000(20) 20,000 20,000 16,101.23 36,101.23

Fund v

Total

62. Ryan invests 100,000 in Fund A today which earns an annual effective interest rate of 8%

interest. The interest on Fund A is paid at the end of each year into Fund B which earns an

annual effective interest rate of 9%. The interest on Fund B is also paid out at the end of each year into Fund C which earns an annual effective interest rate of 10%. At the end of 12 years,

Ryan liquidates all three Funds. How much does Ryan have? Solution:

11 11

11 11

100,000(0.08) 8000

8000(0.09) 720

720720 11 (1.10)

0.10a a v

N=11, I/Y=10, PMT=-1, CPT PV=6.4951

11 11720720(6.4951) 6.4951 11 (1.10) 67,566.84

.1

Fund A + Fund B+Fund C=100,000 12(8000) 67,566.84 263,566.84

v


63. Colleen invests P into Fund A at the start of each year for 25 years. Fund A earns an annual

effective interest rate of 10%.

At the end of each year, Colleen withdraws the interest from Fund A and deposits it into Fund B

which earns an annual effective interest rate of 6%.

At the end of 25 years, Colleen has a total of 135,000 in both accounts.

Determine P .

Solution:

25 25

25 0.06 25 0.06

25 25

25 0.06 25 0.06

After 25 years, Fund A will have 25P.

After 25 years, Fund B will have:

0.10.1 25(1.06) (1.06)

0.06

0.125 0.1 25(1.06) (1.06) 135,000

0.06

PPa a

PP Pa a

P

25 0.06

25 25

1; / 6; 25; 12.78335616

135,0001682.02

0.125 0.1(12.78335616) (12.78335616 25(1.06) (1.06)

0.06

MT I Y N CPT PV a

P


64. Will has a loan of 50,000 which is being repaid with annual payments of 6000 followed by a smaller drop payment. The loan has an interest rate of 7%. Calculate the amount of the drop

payment.

Solution: PV=50,000, PMT=-6000, I/Y=7, CPT N=12.9395

So use N=12

12

1250,000(1.07) 6000 5,278.87

5278.87(1.07) 5648.39

s

65. Daniel borrowed 30,000 to be repaid with monthly payments of 1000 with the final payment being a balloon payment. The interest rate on the loan is 14.4% compounded monthly.

Calculate the amount of the balloon payment.

(12) 0.144.012

12 12

i

PV=30,000, PMT=-1000, I/Y=1.2, CPT N=37.413 So use N=36

36

3630,000(1.012) 1000 1393.10

1393.10(1.012) 1409.82

s

66. Mingjing has a loan of 50,000 which is being repaid with monthly payments of 1500. The final

payment will be a Balloon payment of B. The interest rate on the loan is 18% compounded monthly.

Aiman has an identical loan except the last payment is a Drop payment of D.

Calculate B-D. Solution:

PV=50,000, PMT=-1500, I/Y=18/12, CPT N=46.555

For B use N=45, for D use N=46

45

45

46

46

: 50,000(1.015) 1500 2289.35(1.015) 2323.69

: 50,000(1.015) 1500 823.69(1.015) 836.04

2323.69 836.04 1487.65

B s

D s

B D

Documents

Math 373 Chapter 3 Homework Spring 2015jbeckley/q/WD/MA373/S15/S15... · 2015-02-24 · Chapter 3 Homework Spring 2015 1. Trey is receiving an annuity immediate which pays 150 each