213841945 Chapter 5 Time Value of Money Multiple Choice Questions PDF

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Solutions Manual 1 Chapter 5Chapter 5: Time Value of MoneyMultiple Choice Questions1. What is the total amount accumulated after three years if someone invests $1,000 today with asimple annual interest rate of 5 percent? With a compound annual interest rate of 5 percent?A. $1,150, $1,B. $1,110, $1,C...

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Chapter 5: Time Value of Money Multiple Choice Questions 1. What is the total amount accumulated after three years if someone invests $1,000 today with a simple annual interest rate of 5 percent? With a compound annual interest rate of 5 percent? A. $1,150, $1,103 B. $1,110, $1,158 C. $1,150, $1,158 D. $1,110, $1,103 Level of difficulty: Easy Solution: C. Simple interest rate: $1,000 + ($1,000)(5%)(3) = $1,150 Compound interest rate: $1,000(1.05)3 = $1,158 2. Which of the following has the largest future value if $1,000 is invested today? A. Five years with a simple annual interest rate of 10 percent B. 10 years with a simple annual interest rate of 8 percent C. Eight years with a compound annual interest rate of 8 percent D. Eight years with a compound annual interest rate of 7 percent Level of difficulty: Easy Solution: C. A) $1,000 + ($1,000)(10%)(5) = $1,500 B) $1,000 + ($1,000)(8%)(10) = $1,800 C) $1,000(1.08)8 = $1,851 D) $1,000(1.07)8 = $1,718 Therefore, C is the largest. Interest rates in the following questions are compound rates unless otherwise stated 3. Suppose an investor wants to have $10 million to retire 45 years from now. How much would she have to invest today with an annual rate of return equal to 15 percent? A. $18,561 B. $17,844 C. $20,003 D. $21,345 Level of difficulty: Medium Solution: A. PV=$10,000,000/(1.15)45=10,000,000/538.7693=$18,561 Or using a financial calculator (TI BAII Plus), N=45, I/Y=15, PMT=0, FV=10,000,000, CPT PV= –18,561 Solutions Manual 1 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

4. Which of the following is false? A. The longer the time period, the smaller the present value, given a $100 future value and holding the interest rate constant. B. The greater the interest rate, the greater the present value, given a $100 future value and holding the time period constant. C. A future dollar is always less valuable than a dollar today if interest rates are positive. D. The discount factor is the reciprocal of the compound factor. Level of difficulty: Medium Solution: B. The greater the interest rate, the smaller the present value, given a $100 future value and holding time period constant. 5. Maggie deposits $10,000 today and is promised a return of $17,000 in eight years. What is the implied annual rate of return? A. 6.86 percent B. 7.06 percent C. 5.99 percent D. 6.07 percent Level of difficulty: Medium Solution: A. FV=PV(1+k)n 17,000=10,000(1+ k)8 8ln(1+k)=ln(1.7), therefore k=6.86% Or using a financial calculator (TI BAII Plus), N=8, PV= –10,000, PMT=0, FV=17,000, CPT I/Y=6.86% 6. To triple $1 million, Mika invested today at an annual rate of return of 9 percent. How long will it take Mika to achieve his goal? A. 15.5 years B. 13.9 years C. 12.7 years D. 10 years Level of difficulty: Medium Solution : C. FV=PV(1+k)n (3)(1,000,000)=1,000,000(1.09) n ln(3)=(n)ln(1.09) n=12.7 years Or using a financial calculator (TI BAII Plus), I/Y=9, PV= –1,000,000, PMT=0, FV=3,000,000, CPT N=12.7 Solutions Manual 2 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

7. Which of the following concepts is incorrect? A. An ordinary annuity has payments at the end of each year. B. An annuity due has payments at the beginning of each year. C. A perpetuity is considered a perpetual annuity. D. An ordinary annuity has a greater PV than an annuity due, if they both have the same periodic payments, discount rate and time period. Level of difficulty: Medium Solution: D. The annuity due has a greater PV because it pays one year earlier than ordinary annuity. 8. Jan plans to invest an equal amount of $2,000 in an equity fund every year-end beginning this year. The expected annual return on the fund is 15 percent. She plans to invest for 20 years. How much could she expect to have at the end of 20 years? A. $237,620 B. $176,424 C. $204,887 D. $178,424 Level of difficulty: Difficult Solution: C.

(1  .15) 20 1   (1  k )20  1 2 , 000 =$ FV20  PMT     2, 000(102.4436)  $204,887  .15 k     Or using a financial calculator (TI BAII Plus), N=20, I/Y=15, PV=0, PMT= -2,000, CPT FV=204,887 9. In Problem 8, what is the present value of Jan’s investments? A. $12,625 B. $12,519 C. $14,396 D. $12,396 Level of difficulty: Medium Solution: B.

1  1   1  (1  k )n  1  (1.15)20   $2,000 PV0  PMT  k .15      

    2,000(6.25933) $12,519  

Or using a financial calculator (TI BAII Plus), N=20, I/Y=15, FV=0, PMT= –2,000, CPT PV=12,519 10. What is the present value of a perpetuity with an annual year-end payment of $1,500 and Solutions Manual 3 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

expected annual rate of return equal to 12 percent? A. $14,000 B. $13,500 C. $11,400 D. $12,500 Level of difficulty: Easy Solution: D. PV0=PMT/k=$1,500/.12=$12,500 Practice Problems 11. After a summer of travelling (and not working), a student finds himself $1,500 short for this year’s tuition fees. His parents have agreed to loan him the money for three years at a simple interest rate of 6 percent, with interest due at the end of each year. A. How much interest will he owe his parents after one year? B. How much will he owe, in total, after three years? Topic: Simple Interest Level of difficulty: Easy Solution: A. In one year you will own P x k = $1500 x 6% = $90 of interest. B. After three years, the total (principal and interest) owing will be: P + (n x P x k) = $1500 + (3 x $1500 x 6%) = $1770. 12. Your sister has been forced to borrow money to pay her tuition this year. If she makes annual payments on the loan at year end for the next three years, and the loan is for $2,500 at a simple interest rate of 6 percent, how much will she pay each year? Topic: Simple Interest Level of difficulty: Easy Solution: As the exact amount of interest owing each year will be paid, there is no “compounding.” The amount of each annual payment will be P x k = $2500 x 6% = $150. Unfortunately, these payments never reduce the principal owing, so the loan will never be paid off! 13. Khalil’s summer job has given him $1,200 more than he needs for tuition this year. The local bank pays simple interest at a rate of 0.5 percent per month. How much interest will he earn in one year? Topic: Simple Interest Level of difficulty: Easy Solution: Khalil will be paid interest each month for 12 months, but without compounding. The total interest earned is (n x P x k) = (12 x $1200 x 0.5%) = $72. Solutions Manual 4 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

14. A new Internet bank pays compound interest of 0.5 percent per month on deposits. How much interest will Khalil’s summer savings of $1,200 earn in one year with this online bank account? Topic: Compound Interest Level of difficulty: Easy Solution: The payment of compound interest means that we must compound (or find the future value) of the amount invested (the present value):

FV12months  $1200  (1  0.005) 12  $1274.01 Of this amount, $1,200 was the original amount invested, so $74.01 of interest will be earned. 15. History tells us that a group of Dutch colonists purchased the island of Manhattan from the Native American residents in 1626. Payment was made with wampum (likely glass beads and trinkets), which had an estimated value of $24. Suppose the Dutch had invested this money back home in Europe and earned an average return of 5 percent per year. How much would this investment be worth today, 380 years later, using: A. Simple interest? B. Compound interest? Topic: Simple and Compound Interest Level of difficulty: Easy Solution: A. Value = P + (n x P x k) = $24 + (380 x $24 x 5percent) = $480

    $ 24 ( 1 0 . 05 ) $ 2 , 70 , 86 6 ,. 7 yea rs B. FV 380 380

16. David has been awarded a scholarship that will pay $2,500 one year from now. However, he really needs the money today, and has decided to take out a loan. If the interest rate is 8 percent, how much can he borrow so that the scholarship will just pay off the loan? Topic: Discounting Level of difficulty: Easy Solution: The future value of the loan (the amount to be repaid) is $2,500. The amount that can be borrowed is the present value amount, calculated as:

PV0  FV1 

1 1  $2500   $2314.81 1 (1  k ) (1  .08)1

Or using a financial calculator (TI BAII Plus), N=1, I/Y=8, PMT=0, FV= -2500, CPT PV=2314.81 Solutions Manual 5 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

17. Grace, a retired librarian, would like to donate some money to her alma mater to endow a $4,000 annual scholarship. The university will manage the funds, and expects to earn 7 percent per year. How much will Grace have to donate so that the endowment fund never runs out? Topic: Perpetuities Level of difficulty: Easy Solution: Present value of the perpetual scholarship payment:

1 1         P P V MT $ 4000 $ 57 , 1 . 8 0     k 0 . 07     18. Grace decides that creating a perpetual scholarship is too costly (see Problem 17Error! Reference source not found.). Instead, she would like to support the education of her favourite grand-nephew, Stephen, who plans to begin university in three years. How much will Grace have to invest today, at 7 percent, to be able to give Stephen $4,000 at the end of each year for four years? Topic: Ordinary Annuities Level of difficulty: Easy Solution: Find the present value of the four-year annuity at year 3:

  1 1 (1  k ) n PV3  PMT  k  

1    1 (1  0.07)4    $4000   0.07    

    $13,548.85  

Now, find the present value of this amount today:

 1   1   $11,059.90  $13,548.85   PV0  FV  3  3  (1  k)  (1.07)  19. Bank A pays 7.25 percent interest compounded semi-annually, Bank B pays 7.20 percent compounded quarterly, and Bank C pays 7.15 percent compounded monthly. Which bank pays the highest effective annual rate? Topic: Effective Annual Rates Level of difficulty: Easy Solution: 2

. 0725 0     1 1 7 . 3 %   For Bank A, k  2

Solutions Manual 6 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

4

. 0720 0  1 1 7 . 4 %      For Bank B, k  4 12

. 0715 0   17 . 3 % 1    For Bank C, k   12  Bank B pays the highest effective annual rate. 20. Jimmie is buying a new car. His bank quotes a rate of 9.5 percent per year for a car loan. Calculate the effective annual rate if the compounding occurs: A. Annually B. Quarterly C. Monthly Topic: Determining Effective Annual Rates Level of difficulty: Easy Solution: A. For annual compounding, the effective annual rate will be the same as the quoted rate. To check this: m

1

QR 9 . 5 %          k 9 . 5 % 1 1 1     m   1  B. With quarterly compounding, set m=4, 4

 9.5%  k  1    1  9.84% 4   C. With monthly compounding, set m=12, 12

 9.5%  k  1    1  9.92% 12   21. If Alysha puts $50,000 in a savings account paying 6 percent per year, how much money will she have in total at the end of the first year if interest is compounded: A. Annually? B. Monthly? C. Daily? Topic: Effective vs. Quoted Rates Level of difficulty: Easy Solution: A. k  Quoted Rate  6%  FV1 year  PV0 (1  k )  $50,000  (1.06)  $53,000

Solutions Manual 7 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

12

 QR  B. k   1   1 6.1678% FV1year  $50,000 (1.061678)  $53,083.90 12   QR   C. k   1   365 

365

 1 6.1831% FV1year  $50,000 (1.061831)  $53,091.55

22. Tony started a small business and was too busy to consider saving for retirement. Tony sold the business for $550,000 when he was 55 years old. He thought he could fund his retirement, because this was a lot more than his friend had amassed in his account. Tony can invest this total sum and earn 10 percent per year. How much will his investment be worth in five years? Topic: Investing Early Level of difficulty: Easy Solution:

FV5 years  $550,000 (1  0.10) 5  $885,780.50 Tony will have less than his friend in five years because he is not adding more savings to his account. 23. Public corporations have no fixed life span; as such, they are often viewed as entities that will pay dividends to their shareholders in perpetuity. Suppose KashKow Inc. pays a dividend of $2 per share every year. If the discount rate is 12 percent, what is the present value of all the future dividends? Topic: Perpetuities Level of difficulty: Easy Solution: The value of any perpetual stream of payments can be valued as a perpetuity:

PV0 

$2 PMT   $16.67 0.12 k

Each share is worth $16.67.

24. Mary-Beth is planning to live in a university residence for four years while completing her degree. The annual cost for food and lodging is $5,800 and must be paid at the start of each school year. What is the total present value of Mary-Beth’s residence fees if the discount rate (interest rate) is 6 percent per year? Topic: Annuities Due Level of difficulty: Easy Solution: Because the fees are paid at the start of the year, this is not an ordinary annuity, but rather, an annuity due. Solutions Manual 8 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

1   1  (1 0.06) 4    (1 0.06)  $21,303.47 PV0  $5800   0.06     25. Calculate the effective annual rates for the following: A. 24 percent, compounded daily B. 24 percent, compounded quarterly C. 24 percent, compounded every four months D. 24 percent, compounded semi-annually E. 24 percent, compounded continuously F. Calculate the effective monthly rate for A to D. Level of difficulty: Medium. Solution: A. m = 365:

k  (1 

.24 365 )  1  27.11%. 365

B. m = 4:

k  (1 

.24 4 )  1  26.25%. 4

C. m = 3:

k  (1 

.24 3 )  1  25.97%. 3

D. m = 2:

k  (1 

.24 2 )  1  25.44%. 2

k  e.24 1  27.12%.

E. Continuous compounding: F. The effectively monthly rate is: m

A. m=365, f=12

QR f .24 365 ) 12  1=2.02% )  1= (1  k  (1  m 365 m

4

m

3

m

2

B. m=4, f=12.

QR f .24 k  (1  )  1 = (1  )12 1 =1.96% 4 m

C. m=3, f=12.

k  (1 

D. m=2, f=12.

.24 QR f )  1 = (1  )12 1 =1.91% k  (1  m 2

.24 QR f )  1 = (1  )12 1 =1.94% m 3

26. On the advice of a friend, Gilda invests $20,000 in a mutual fund which has earned 10 percent per year, on average, in recent years. If this rate of return continues, how much will her investment be worth in: Solutions Manual 9 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

A. one year? B. five years? C. ten years? Topic: Compound Interest Level of difficulty: Medium Solution: A. FV1year  $20,000  (1  0.10) 1  $22,000.00 B. FV5 years  $20,000 (1 0.10) 5  $32,210.20 C. FV10 years  $20,000  (1  0.10) 10  $51,874.85 27. Your investment research has turned up an interesting mutual fund. It has had an average annual return 0.5 percent greater than the one Gilda’s friend recommended (see Problem 26). For each time period from Problem 26, calculate how much better off Gilda would be if she invested $20,000 in this mutual fund. Topic: Compound Interest Level of difficulty: Medium Solution: First find the value of the investment after each period of time, and then compare to the values from Problem 26 to determine how much difference a small change in the interest rate can make. A. FV1year  $20,000  (1  0.105)  $22,100.00 . You are ($22,100 – $22,000) = $100 better off 1

after one year. B. FV5 years  $20,000  (1  0.105)  $32,948.94 . You are ($32,948.94 – $32,210.20) = $738.74 5

better off after five years. C. FV10years  $20,000  (1  0.105)

10

 $54,281.62 You are ($54,281.62 – $51,874.85) = $2,406.77

better off after 10 years.

28. When Jon graduates in three years, he wants to throw a big party, which will cost $800. To have this amount available, how much does he have to invest today if he can earn a compound return of 5 percent per year? Topic: Discounting Level of difficulty: Medium Solution: Jon needs $800 in three years; that is the future value amount. The present value equivalent is: Solutions Manual 10 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

PV0  FV3 

1 1  $800   $691.07 3 (1  k ) (1  .05)3

Or using a financial calculator (TI BAII Plus), N=3, I/Y=5, PMT=0, FV= -800, CPT PV=691.07 29. In Problem 28, suppose Jon had only $500 to invest. How much can he plan to spend on the graduation party in three years, if the return on the investment will be: A. simple interest at 5 percent per year? B. compound interest at 5 percent per year? Topic: Simple and Compound Interest Level of difficulty: Medium Solution: A. Jon will earn $500 5%  $25 per year in interest. The value of his investment (or the amount available to spend on the party) will be:

Value  P  ( n  P  k)  $500  (3  500  0.05)  $575 B. The interest earned grows (compounds) each year; the total available in three years is:

FV3  PV0  (1  k)3  $500  (1  .05)3  $578 .81 30. At the age of 10, Felix decided that he wanted to attend a very prestigious (and expensive) university. How much will his parents have to save each year to accumulate $40,000 by the time Felix needs the funds in eight years? Assume Felix’s parents can earn 7 percent (compounded annually) on their savings, and that each year’s savings are deposited at the end of the year. Topic: Ordinary Annuities Level of difficulty: Medium Solution: The future value amount is $40,000. The amount to be saved each year is really the payment on an ordinary annuity:

 (1  0.07)8  1 $40, 000  PMT    PMT  $3898.71 0.07   Or using a financial calculator (TI BAII Plus), N=8, I/Y=7, PV=0, FV= -40,000, CPT PMT= 3898.71 31. Felix’s parents can only afford to save $3,000 per year for his university education, which begins in eight years. What rate of return would they require on these savings if they must accumulate $40,000? Topic: Ordinary Annuities (Solving for IRR) Solutions Manual 11 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.

Level of difficulty: Medium Solution: Solve for the interest rate (or internal rate of return) on an ordinary annuity. This is quite difficult to do algebraically, but is easily handled with a financial calculator (TI BAII Plus). Note that we must use a negative sign for the annual payment (savings) or the future value amount, but not both. N=8, PMT = 3,000, PV=0, FV= –40,000, CPT I/Y=14.2067% 32. Shortly after John was born, his parents began to put money in a savings account to pay for his post-secondary education. They save $1,000 each year, and earn a return of 9 percent per year. However, the interest income is taxed each year at a rate of 30 percent. How much will John’s account be worth after 17 years? Topic: Or...