Chapter 2 Homework - Purdue Universityjbeckley/q/WD/MA373/S15/S15 373... · The loan has an annual effective interest rate of 6.15%. Determine the amount of Grant’s loan. Solution:

Chapter 2 Homework

February 3, 2015 Copyright Jeffrey Beckley 2014 2015

1. Juan borrows 11,000 at an annual effective rate of 6%. At the end of t years, he repays the loan with a payment of 23,462.21. Determine t . Solution:

11,000(1 0.06) 23,462.21

(1.06) 2.132928182

ln(2.132928182) / ln(1.06) 13

t

t

t

2. Six and a half years ago, John invested 8500 in an account. Today the account is worth 14,358.

Calculate the annual effective interest rate that John has earned over the last six and a half years. Solution:

6.5

6.5

1/6.5

8500(1 ) 14,358

(1 ) 1.689176471

1.689176471 1.083994121 8.4%

i

i

i

3. Matt invests 1000 today at an annual effective rate of 10%.

a. Determine how long it will take for Matt’s money to double.

Solution:

1000(1 .10) 2000

(1.10) 2

ln(2) / ln(1.10) 7.27254

t

t

t

b. If Matt had used the Rule of 72 to estimate how long it would take for his money to

double, what would his estimate have been? Solution:

0.72 0.727.2

0.10i

Chapter 2 Homework


4. Shashank borrows 22,500 to purchase a car. Using the Rule of 72, he estimates that he will need to repay 45,000 at the end five years. Determine the actual amount that Shashank will need to repay at the end of five years. Solution:

5

0.725 0.144

22,500(1 0.144) 44,087.21

ii

5. Simon invests 1000 today. Based on the Rule of 72, Simon expected to have 4000 at the end of

24 years. Determine the interest rate that Simon is earning. Solution:

.7212 0.06

To double twice, it takes 24 years. To double once, it takes 12 years.

ii

6. Grant has agreed to repay a loan with a payment of 2000 at the end of one year and 4000 at the

end of 2 years. The loan has an annual effective interest rate of 6.15%. Determine the amount of Grant’s loan. Solution:

1 22000(1 0.0615) 4000(1 0.0615) 5434.06

7. Yunlu borrows 23,000 and agrees to repay the loan with two payments of P . The first payment will be at the end of one year and the second payment will be at the end of 3 years. Yunlu’s loan has an interest rate of 5% compounded semi-annually.

Determine P . Solution:

2*3 2*2

2 6

0.05 0.0523,000(1 ) (1 )

2 2

0.05 0.0523,000 (1 ) (1 )

2 2

12,678.38

P P

or

P P

P

Chapter 2 Homework


8. Ross borrows 2000 and agrees to make two payments of 1100. The first payment is at the end of one year and the second payment is at the end of two years. Determine the annual effective interest rate that Ross is paying. Solution:

2

2

2

2000(1 ) 1100(1 ) 1100

2000 1100 1100 0

1100 1100 4(2000)( 1100)1.065964601

2(2000)

1

1.065964601 1 6.59646%

i i

x x

x

x i

i

9. Songyan invests 2000 in the STARR Fund. Songyan withdraws 750 from the Fund at the end of

year 1, year 2, and year 3. Songyan has no money left in the fund at the end of three years after making the last withdrawal. Determine Songyan annual effective return on the money in STARR Fund. Solution: Financial calculator: CF0=2000 C01=-750 F01=3 CPT IRR = 6.12856

10. Taylor borrows 10,000 and agrees to repay it with payments of 2700 at the end of each year for the next 3 years. Calculate the annual effective interest rate that Taylor is paying on the loan. Solution: This question is flawed – Do not work

Chapter 2 Homework


11. Rushan borrows 10,000 to be repaid with a payment of 6264.46 at time T and a payment of 6264.46 at time 2T . The annual effective rate on the loan is 7%. Determine T in months. Solution:

0 10,000

01 6264.46

01 2

16.443076%

(1.07) 1.16443076

ln1.164430762.250008 *12 27

ln1.07

T

CF

C

F

IRR CPT

T years months

Chapter 2 Homework


12. Kristen, Sarah, and Grant enter into a financial arrangement. Kristen agrees to pay 2200 to Sarah today. Sarah agrees to pay Kristen 1000 at the end of two years. Sarah also agrees to pay Grant 1400 at the end of two years. Finally, Grant agrees to pay Kristen 1600 at the end of 4 years. Determine the annual effective interest rate for each person in this arrangement. Solution:

Look at Kristen’s cash flows.

-2200 1000 1600

0 2 4

2200 = 1000(1 + i) -2 + 1600(1 + i) -4

Let x = (1 + i) -2 and x2 = (1 + i) -4

1600x2 + 1000x – 2200 = 0

Use quadratic formula.

2

12

2

1000 1000 4 1600 22000.90103049

2 1600

1 0.90103049 0.90103049 1 0.0534896

x

i i

Look at Sarah’s cash flows.

2200 -2400

0 2 4

2 22200 2400(1 ) (1 ) 1.09090909 0.0444659i i i

Look at Grant’s cash flows.

1400 -1600

0 2 4

2 21400 1600(1 i) (1 ) 1.142857143 0.069045i i

Chapter 2 Homework


13. Cong, Erik, and Yifei enter into a financial arrangement. Cong agrees to pay Erik 3000 today. He also agrees to pay Yifei 1000 at the end of one year. At the end of three years, Yifei will pay Cong 4000. At the end of two years, Erik will pay X to Yifei and 1000 to Cong. Using the bottom line approach, the annual yield rate or interest rate is the same for Cong and Yifei. Calculate X . Solution:

Cong: -3000 -1000 +1000 +4000

Erik: +3000 -X-1000

Yifei: +1000 +X -4000

For Cong use the financial calculator:

CF0= -3000

CO1= -1000

CO2=1000

CO3=4000

CPT IRR= 9.14%

Yifei:

1 2 3

3 1

2

0 1000 1.0914 1.0914 4000 1.0914

4000 1.0914 1000 1.0914

1.0914

2573.56

X

X

X

Chapter 2 Homework


14. Haoran, Jiayi, and Yinzhe are business partners. As part of their partnership, Haoran pays Jiayi 100,000 today. Additionally, at the end of 5 years, Jiayi agrees to pay 120,000 to Yinzhe. Finally, at the end of T years, Yinzhe pays Haoran a total of 172,800. Using the bottom line approach, all three partners have the same annual yield. Determine T . Solution: Using Jiayi’s cash flows, we can find the interest rate using our financial calculators.

CF0=100000

C01=0

F01=4

C02= -120000

CPT IRR= 3.7137%

Now using this interest rate we can find T either using Haoran’s or Yinzhe’s cash flows. I will

use Haoran’s:

0 100000 172800 1.037137

0.5787 1.037137

ln(0.5787)15

ln(1.037137)

T

T

T years

Chapter 2 Homework


15. Dylan borrows 50,000 from Bian Bank which will be repaid with two payments of 29,000. The two payments will be made at the end of three years and at the end of six years. When Bian Bank received the payment of at the end of three years, it reinvests that payment at 8% interest. Taking into account reinvestment, determine the annual effective return earned by Bian on this loan. Solution: Dylan will be reinvesting the payments he receives. So at time six his investments will be worth

329000(1.08) 29000 65531.648 .

Setting this value equal to the amount he originally lent to Bian, we can find his annual return.

6

6

65531.648 50000(1 )

65531.6481 4.61168%

50000

i

i

16. Kyle loans Nic an amount of X to be repaid with two payments of 1000. The first payment is two years from the date of the loan and the second payment is 6 years from the date of the loan. Kyle reinvests that first payment at 10%. Taking into account reinvestment, Kyle earned an annual effective rate of 12.23%.

Calculate X . Solution:

6 4(1.1223) 1000(1.10) 1000 1233.12X X

17. Madi borrows 12,000 from Kevin. Madi will repay the loan with a payment of 6500 at the end of

one year and 7000 at the end of two years. Kevin reinvests that first payment at an annual effective interest rate of r . Taking into account reinvestment, Kevin realizes an annual effective return of 7.224%. Determine r . Solution:

212000(1.07224) 6500(1 ) 7000 r 4.56%r

Chapter 2 Homework


18. Elena has an account at Basham Brokerage House. She has made the following deposits and withdrawals over the last two years.

Date Deposits Withdrawals Balance Before Cash Flow

January 1, 2013 5000 0 0

March 1, 2013 4000 0 6000

July 1, 2013 0 6000 9,000

May 1, 2014 2000 0 4,000

December 31, 2014 0 0 7,000

a. EstimateElena’s dollar weighted annual return using simple interest. Solution:

1

1/2

5000

7000

4000 6000 2000 0

5000 0 7000

2000

2000 20000.41379

2 6 16(1 ) 4833.335000 4000(1 ) 6000(1 ) 2000(1 )

24 24 24

1 (1 ) (1.41379) 1 1.189030 18.9030%

t

t

t

T

A C I B

A

B

C

I

I

Ij

A C t

i j i i

Chapter 2 Homework


b. Calculate Elena’s exact annual dollar weighted return using the cash flow functionality in the BA II Plus. Solution: Treat Cash Flows as monthly because cash flows must be evenly spaced:

0 5000

01 0; 01 1

02 5000; 02 1

03 0; 03 3

04 4000; 04 1

05 0; 05 9

06 3000; 06 1

07 0; 07 7

08 15,000; 08 1

1.435966669%

Since the periods are months, the IRR is the monthly

effective intere

CF

C F

C F

C F

C F

C F

C F

C F

C F

IRR CPT

(12)

12(12)

12

st rate which is 1.435966669%12

(1 ) 1 (1 0.01435966669) 18.6598%12

i

ii i

Chapter 2 Homework


19. Jing has a bank account balance of X on January 1, 2013. During the next two years Jing deposits 8000 into his account and withdraws 3000. On January 1, 2015, Jing has a bank account balance of 28,400. Assuming that all cash flows occur on January 1, 2014, Jing estimates his annual dollar weighted return assuming simple interest to be 4.5383664%. Calculate X. Solution:

28,400

0.5

8000 3000 5000

23,400

A X

B

k

C

I X

0.5 2(1 ) (1 ) 1.045383664 1.045383664 1 0.09287005

23,4000.092827005

(1 ) 5000(0.5)

0.092827005( 2500) 23400

21,200

j i j

I Xj

A C k X

X X

X

Chapter 2 Homework


20. Joe invested 50,000 in an account three years ago. One year ago, the account had a balance of 90,000 and Joe withdrew 45,000. Today, Joe has 30,000 in the account. Determine the annual effective time weighted return earned by Joe on this account. Solution:

01

1

22

1 1

1 2

1/ 1/3

90,0001 1.8

50,000

30,0001 0.666666

90,000 45,000

1 (1 )(1 ) (1.8)(.6666666) 1.2

1 (1 ) 1 (1.2) .0626586

TW

TW TW T TW TW

Bj

B

Bj

B C

j j j

i j i i

Chapter 2 Homework


21. Elena has an account at Basham Brokerage House. She has made the following deposits and withdrawals over the last two years.

Date Deposits Withdrawals Balance Before Cash Flow

January 1, 2013 5000 0 0

March 1, 2013 4000 0 6000

July 1, 2013 0 6000 9,000

May 1, 2014 2000 0 4,000

December 31, 2014 0 0 7,000

Calculate Elena’s annual time weighted return. Solution:

01

1

22

1 1

60001 1.2

5000

90001 0.9

6000 4000

Bj

B

Bj

B C

33

2 2

44

3 3

1/2

40001 1.33333

9000 6000

70001 1.16666

4000 2000

1 (1.2)(0.9)(1.33333)(1.16666) 1.68

1 (1.68) 29.6148%

TW

TW TW

Bj

B C

Bj

B C

j

i i

Chapter 2 Homework


22. Evgeny invests in the Marano Mutual Fund. Over the next two years, Evgeny realizes an annual time weighted yield of 12.5%. Evgeny initially invests 100,000 with Marano and has 103,551.14 at the end of two years. During the two year period, Evgeny also withdrew an amount of C to buy a new car. Before the amount was withdrawn, the fund was worth 110,000. Calculate C. Solution: First, let’s find the time weighted yield for the period of two years.

1/

2

(1 ) (1 )

(1.125) 1 0.265625

T

TW TW

TW

j i

j

Now we can solve for C

110000 103551.141 0.265625

100000 110000

110000 103551.14110000 20,000

100000 1.265625

c

c

Chapter 2 Homework


Answers

1. 13 2. 8.4% 3.

a. 7.27254 years b. 7.2 years

4. 44,087.21 5. 6% 6. 5434.06 7. 10,405.74 8. 6.59646% 9. 6.12856% 10. 9.83909% 11. 27 Months 12. Kristen => 5.34896% ; Sarah => 4.44659% ; Grant => 6.90450% 13. 2573.56 14. 15 years 15. 4.61168% 16. 1233.12 17. 4.56% 18.

a. 18.9030% b. 18.6598%

19. 21,200 20. 6.26586% 21. 29.6148% 22. 20,000

Documents

Chapter 2 Homework - Purdue Universityjbeckley/q/WD/MA373/S15/S15 373... · The loan has an annual effective interest rate of 6.15%. Determine the amount of Grant’s loan. Solution: