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
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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. $178,424 Level of difficulty: Difficult Solution: C.
(1 k )20 1 (1 .15)20 1 FV20 PMT 2 , 000 =$ 2,000(102.4436) $204,887 k .15 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 PV0 PMT $2,000 2,000(6.25933) $12,519 k .15 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.
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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):
FV12 months $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 , 704 , 860 , 602 . 74 380 years B. FV 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
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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 PV PMT $ 4000 $ 57 , 142 . 86 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 PV0 FV $13,548.85 $11,059.90 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 . 38 % For Bank A, k 2
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4
. 0720 0 1 1 7 . 40 % For Bank B, k 4 12
. 0715 0 1 1 7 . 39 % 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
. 5 % QR 9 k 1 1 1 9 . 5 % 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
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B. k 1
12
QR 1 6.1678% FV1 year $50,000 (1.061678) $53,083.90 12
QR C. k 1 365
365
1 6.1831% FV1 year $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
PMT $2 $16.67 k 0.12
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.
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1 1 (1 0.06)4 PV0 $5800 (1 0.06) $21,303.47 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
365 QR f .24 12 k (1 ) 1= (1 ) 1=2.02% m 365 m
B. m=4, f=12.
4 QR f .24 12 k (1 ) 1= (1 ) 1 =1.96% m 4
C. m=3, f=12.
3 QR f .24 12 k (1 ) 1= (1 ) 1 =1.94% m 3
D. m=2, f=12.
k (1
m
m
2 QR f .24 12 ) 1= (1 ) 1 =1.91% m 2
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. FV1 year $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. FV1 year $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. FV10 years $20,000 (1 0.105) $54,281.62 You are ($54,281.62 – $51,874.85) = $2,406.77 10
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:
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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: Ordinary Annuity (Future Value) Level of difficulty: Medium Solution: Taxation of the interest income has the effect of reducing the rate of return. In effect: k 9% (1 0.30) 6.3% . Using this rate of return we find the future value of John’s
(1 0.063)17 1 account to be : FV17 $1000 $28,973.11 0.063 33. John’s parents used a regular savings account to save for his post-secondary education. Based on the amount accumulated (from your answer in Problem 320), how much can John withdraw from the account at the beginning of each year for his four years at university? The account will continue to earn 9 percent per year, but interest income is taxed at a rate of 30 percent. Topic: Annuities Due Level of difficulty: Medium Solution: As in Problem 32, k 9% (1 0.30) 6.3% because of taxation. John’s withdrawals at the beginning of each year are essentially “payments” on an annuity due:
1 1 (1 0.063)4 28,973.11 PMT (1 0.063) PMT $7919.95 0.063 34. Jane’s parents save $1,000 per year for 17 years to pay for her university tuition costs. They deposit the money into a Registered Education Savings Plan (RESP) account so that no tax is Solutions Manual 12 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
payable on the interest income. This RESP account provides a return of 6 percent per year. A. How much will Jane’s account be worth when she begins her university studies? B. As an incentive to save for higher education, the government will add 20 percent to any money contributed to an RESP each year. Including these grants, how much will Jane have in her account? Topic: Ordinary Annuity (Future Value) Level of difficulty: Medium Solution: A. The future value of Jane’s account will be:
(1 0.06)17 1 FV17 $1000 $28,212.88 0.06 B. The grant has the effect of increasing the amount saved from $1,000 to $1,200. The future value of the account will now be:
(1 0.06)17 1 FV17 $1200 $33,855.46 0.06 35. Jane’s parents used an RESP account to save for her post-secondary education. Based on the amount accumulated (from your answer in Problem 340), Jane would like to withdraw the same amount of money at the beginning of each year of her four-year degree program. All funds (interest and principal) withdrawn from this account are taxed at a rate of 15 percent, and the account will earn 6 percent per year on any remaining funds. How much will Jane have available for tuition each year? Topic: Annuities Due Level of difficulty: Medium Solution: Each year, Jane can withdraw:
1 1 (1 0.06)4 33,855.46 PMT (1 0.06) PMT $9217.36 0.06 However, this amount will be subject to income tax at a rate of 15%. The net amount that Jane will have available for tuition is then:
9217.36% (1 0.15) $7834.75 36. Stephen has learned that his great-aunt (see Problem 18Error! Reference source not found.) intends to give him $4,000 each year he is studying at university. Tuition must be paid in advance, so Stephen would like to receive his payments at the beginning of each school year. Solutions Manual 13 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
How much will his great-aunt have to invest today at 7 percent, to make the four annual (start-of-year) payments? Topic: Annuities Due Level of difficulty: Medium Solution: Find the present value of the four-year annuity due:
1 1 1 (1 k )n 1 (1 0.07)4 PV0 PMT (1 k ) $4000 (1 0.07) $14,497.26 k 0.07 Now, discount this amount back three years:
1 1 PV0 FV $14,497.26 $11,834.08 3 3 (1 k ) (1.07) 37. Rather than give her grand-nephew some money each year while he is studying, Stephen’s great-aunt has decided to save the money and pay off Stephen’s student loans when he finishes his degree. The total amount owing at that time will be $16,000. How much will she have to save each year until that time if her investments earn a return of 7 percent per year? Topic: Ordinary Annuities Level of difficulty: Medium Solution: Assuming the savings are invested at the end of each year, find the payment (amount to be saved) for an ordinary annuity with a future value of $16,000.
(1 0.07) 7 1 FV4 $16,000 PMT PMT $1,848.85 0.07 38. Jimmie’s new car (see Problem 20) will cost $29,000. How much will his monthly car payments be if he obtains a loan that is amortized over 60 months, and the nominal interest rate is 8.5 percent per year with monthly compounding? Topic: Effective Interest Rates and Loan Arrangements Level of difficulty: Medium Solution: First, find the effective interest corresponding to the frequency of Jimmie’s car payments (f =12); with monthly compounding, set m=12,
kmonthly
QR 1 m
m
f
12 12
8.5% 1 1 12
1 0.7083%
The 60 car payments form an “annuity” whose present value is the amount of the loan (the price of the car): Solutions Manual 14 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
1 1 (1 0.007083)60 $29,000 PMT PMT $594.98 0.007083 39. Create an amortization schedule for Jimmie’s car loan (see Problem 38Error! Reference source not found.). What portion of the first monthly payment goes towards repaying the principal amount of the loan? What portion of the last monthly payment goes towards the principal? Topic: Loan Arrangements Level of difficulty: Medium Solution: Use the effective monthly interest rate from Problem 38, k=0.7083%
Period
1 2 3 4 5 6 7 8 9 10 11 12 13 ... 35 36 37 ... 59 60
(1) Principal
(2)
Outstanding
Payment
(3) Interest =k*(1)
(4) Principal Repayment = (2)-(3)
Ending Principal = (1)-(4)
29,000.00 28,610.44 28,218.12 27,823.01 27,425.11 27,024.40 26,620.84 26,214.42 25,805.13 25,392.94 24,977.82 24,559.77 24,138.76
594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98 594.98
205.42 202.66 199.88 197.08 194.26 191.42 188.56 185.69 182.79 179.87 176.93 173.97 170.98
389.56 392.32 395.10 397.90 400.72 403.56 406.42 409.29 412.19 415.11 418.05 421.01 424.00
28,610.44 28,218.12 27,823.01 27,425.11 27,024.40 26,620.84 26,214.42 25,805.13 25,392.94 24,977.82 24,559.77 24,138.76 23,714.76
14,083.18 13,587.95 13,089.22
594.98 594.98 594.98
99.76 96.25 92.72
495.22 498.73 502.26
13,587.95 13,089.22 12,586.96
1,177.43 590.79
594.98 594.98
8.34 4.18
586.64 590.79
590.79 0.00
The first monthly payment repays $389.56 of the principal amount of the loan and the last payment repays $590.79. 40. Using the amortization schedule from Problem 39, determine how much Jimmie still owes on the car loan after three years of payments on the five-year loan. What is the present value of this amount? Solutions Manual 15 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
Topic: Loan Arrangements and Discounting Level of difficulty: Medium Solution: After three years, or 36 monthly payments, the principal outstanding is $13,089.22 (from the amortization table). The present value of this amount is:
1 $10,152.30 PV0 $13,089.22 36 ( 1 0 . 007083 ) 41. Jimmie would like to pay off his car loan in three years (see Problem 38), but can only afford monthly payments of $594.98. How big a down-payment must Jimmie make on the $29,000 car if the nominal interest rate is 8.5 percent with monthly compounding? Topic: Loan Arrangements Level of difficulty: Medium Solution: Use the effective monthly interest rate from Problem 38, k=0.7083% Find the present value of Jimmie’s 36 payments:
1 1 (1 0.007083)36 PV0 $594.98 $18,847.95 0.007083 Therefore, Jimmie must make a down payment of
$29,000.00 $18,847.95 $10,152.05 42. Jimmie is offered another loan of $29,000 that requires 60 monthly payments of $588.02 (see Problem 38Error! Reference source not found.). What is the effective annual interest rate on this loan? What would the quoted rate be? Topic: Loan arrangements and Effective Annual Rates Level of difficulty: Medium Solution: The 60 monthly payments form an annuity whose present value is $29,000. Finding the interest rate is most easily done with a financial calculator (TI BAII Plus): N=60, PMT=588.027, PV= -29,000, CPT I/Y = 0.6667% Note that we used N=60 months, so the solution is a monthly interest rate, however, the problem asks for the effective annual rate.
k (1 kmonthly )12 1 (1 0.006667)12 1 8.30% The quoted rate would be:
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QR m [(1 k )
1 12
1] 12 (1 0.0830)
1 12
1 8.00%
Or simply: QR m kmonthly 12 0.006667 8.00% 43. To start a new business, Su Mei intends to borrow $25,000 from a local bank. If the bank asks her to repay the loan in five equal annual instalments of $6,935.24, determine the bank’s effective annual interest rate on the loan transaction. With annual compounding, what nominal rate would the bank quote for this loan? Topic: Determining Rates of Return and Effective Interest Rates Level of difficulty: Medium Solution: Solve the annuity equation to find k, the interest rate:
1 1 (1 k )5 $25,000.00 $6935.24 k ? k The calculations are most easily done with a financial calculator (TI BAII Plus), PV = -25,000, PMT=6935.24, N= 5, FV=0, CPT I/Y = 12% The effective annual interest rate is 12 percent. With annual compounding, the nominal rate (or quoted rate) will also be 12 percent per year. 44. The Business Development Bank is willing to loan Su Mei the $25,000 she needs to start her new company. The loan will require monthly payments of $556.11 over five years. A. What is the effective monthly rate on this loan? B. With monthly compounding, what is the nominal (annual) interest rate on this loan? Topic: Determining Rates of Return and Effective Interest Rates Level of difficulty: Medium Solution: A. There will be 5 x 12 = 60 monthly payments. The calculations are most easily done with a financial calculator (TI BAII Plus), PV = –25,000, PMT=556.11, N= 60, CPT I/Y = 1.0% Because we used monthly payments, and months as the time period, 1.0% is the effective monthly rate. B. The compounding period matches the payment frequency, so the nominal rate, or quoted rate, is:
QR m kmonthly 12 1.0% 12.0% per year. Solutions Manual 17 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
45. Compare the loans in problems 043 and 440. Which is the better deal, and why? Topic: Effective Interest Rates Level of difficulty: Medium Solution: The two loans have the same principal amount; one requires monthly payments, the other annual, so they cannot be compared on that basis. We need an effective rate for the same period of time for comparison. In Problem 43, the effective annual rate was 12%. For Problem 44, the effective annual rate is:
k (1 kmonthly )12 1 (1 0.01)12 1 12.7% . Clearly, the loan in Problem 43 is a better deal because the effective interest rate charged is lower. 46. After losing money playing on-line poker, Scott visits a loan shark for a $750 loan. To avoid a visit from the “collection agency,” he will have to repay $800 in just one week. A. What is the nominal interest rate per week? Per year? B. What is the effective annual interest rate? Topic: Effective Interest Rates Level of difficulty: Medium Solution: A. You will pay interest of (800–750) = $50 after one week. This implies a nominal interest rate of 50/750 = 6.67% per week. With 52 weeks in the year, the nominal rate per year is then 52 x 6.67% = 346.67%. B. The effective annual interest rate is k (1 0.0667) 52 1 27.7210 2,772.10% 2,872.1% – 1=2,772.1% 47. Josephine needs to borrow $180,000 to purchase her new house in Yarmouth, Nova Scotia. She would like to pay off the mortgage in 20 years, making monthly payments. For the initial three-year term, Providence Bank has offered her a quoted annual rate of 6.40 percent. A. What is the effective annual interest rate? B. What is the effective monthly interest rate? C. How much will Josephine’s monthly mortgage payments be? Topic: Mortgage Loans and Effective Interest Rates Level of difficulty: Medium Solution: A. In Canada, fixed-rate mortgages use semi-annual compounding of interest, so m=2. The effective annual rate is therefore:
Solutions Manual 18 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
m
2
QR 0.064 k 1 1 1 1 6.5024% m 2 B. With monthly payments, f=12. We can find the effective monthly interest rate from the effective annual rate, k:
kmonthly 1 k
1
f
1 1 6.5024% 12 1 0.5264% 1
C. The amortization period is 20 years, or 20 x1 2 = 240 months. Josephine’s monthly payments can be computed as:
1 1 (1 0.005264) 240 PMT $1,322.69 $180,000 PMT 0.005264 48. The Yarmouth Credit Union will provide Josephine with a mortgage at a rate of 6.36 percent, but unlike most Canadian mortgages, the compounding will occur monthly. Should Josephine take out the mortgage loan from the Credit Union, or from Providence Bank (see Problem 470)? Can you answer this question without calculating the monthly mortgage payment? Topic: Mortgage Loans and Effective Interest Rates Level of difficulty: Medium Solution: With monthly compounding and payments, the effective monthly interest rate is:
kmonthly
QR 1 m
m
f
12 12
0.0636 1 1 12
1 0.530%
Even though the quoted rate is lower at the Credit Union than at the Bank (see Problem 470), the effective rate is higher. Josephine should take the mortgage loan from the Bank in this case. The monthly payment for the Credit Union mortgage would be $1,327.24, which, as expected, is higher than that at the Bank. 49. Assume Josephine chose the Providence Bank option (see Problem 470). If Josephine can get the same interest rate for a second three-year term as she did originally, how much will her monthly payments be now? Topic: Mortgage Loans Level of difficulty: Medium Solution: If we assume that Josephine does not change the amortization period (now 17 years), then the same interest rate will result in making the same monthly payments. We can confirm this by computing the payment for a mortgage with a principal amount equal to the amount now Solutions Manual 19 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
outstanding ($165,172.38 based on Problem 47), that lasts 204 months (17 years):
1 1 (1 0.005264)204 $165,172.38 PMT PMT $1322.69 0.005264 50. A lakefront house in Kingston, Ontario is for sale with an asking price of $499,000. The real-estate market has been quite active, so the house will almost certainly attract several offers, and may sell for more than the asking price. Charlie is very eager to purchase this house, but is concerned that he may not be able to afford it. He has $130,000 available for a down payment, and can pay up to $1,950.00 per month on a mortgage loan. As a long-time customer, Charlie’s bank has offered him a great mortgage rate of 3.90 percent on a one-year term. If the loan will be amortized over 25 years, what is the most that Charlie can afford to pay for the house? Topic: Mortgage Loans Level of difficulty: Medium Solution: With semi-annual compounding (the norm in Canada) and monthly payments, m=2 and f=12. The effective monthly rate is:
kmonthly
QR 1 m
m
f
0.039 1 1 2
2
12
1 0.3224%
The present value of the mortgage payments over the amortization period (25 years x 12 = 300 months) is:
1 1 (1 0.003224)300 PV0 $1950.00 $374,553.72 0.003224 In addition, Charlie has $130,000 available as a down payment; the most he can pay for the house is, therefore, $374,553.72 + $130,000 = $504,553.72. 51. Timmy sets himself a goal of amassing $1 million in his retirement fund by the time he turns 61. He begins saving $3,000 each year, starting on his 21st birthday (40 years of saving). A. If his savings earn 10 percent per year, will Timmy achieve his goal? B. Will Timmy be able to retire before he turns 60? That is, at what age will the value of his savings plan be worth $1 million? Topic: Investing Early Level of difficulty: Medium Solution: Solutions Manual 20 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
A. Timmy’s savings extend right to age 61 (end of each year), so this is an ordinary annuity.
(1 0.10) 40 1 FV40 $3,000 $1,327,777.67 0.10 Yes, Timmy will achieve his goal by a comfortable margin. B. In the equation for part A set FV=$1,000,000, and solve for the number of years, n. This is easiest done with a financial calculator (TI BAII Plus), FV = –1,000,000, I/Y = 10, PMT = 3000, CPT N = 37.1. Timmy will hit the $1 million dollar mark in just over 37 years, or shortly after his 58 th birthday. 52. Tommy set the same retirement goal as his friend Timmy (see Problem 510). However, there always seemed to be a reason not to save money, so he put it off for many years. Finally, with just 15 years to retirement, he began to save. Fortunately, Tommy’s executive-level job allowed him to save $30,000 per year. If these savings earn 10 percent per year, will Tommy achieve his $1million goal at the desired time? Topic: Investing Early Level of difficulty: Medium Solution: This is an ordinary annuity.
(1 0.10)15 1 FV15 $30,000 $953,174.45 0.10 No, Tommy will not quite achieve his goal by age 61. 53. Jack is 28 years old now and plans to retire in 35 years. He works in a local bank and has an annual after-tax income of $45,000. His expected annual expenditure is $36,000 and the rest of his income will be invested at the beginning of each of the next 35 years at an expected annual rate of return of 12.6 percent. Calculate the amount Jack will receive when he retires. Level of difficulty: Difficult. Solution: Annual investment = Annual income – Annual expenditure = $45,000 – $36,000 = $9,000. This is an annuity due.
(1 k )n 1 FVn PMT (1 k ) k (1 .126)35 1 9,000 (1.126) (9,000)(497.2749)(1.1 2 6) $5,0 3 9,3 8 4 .126 Solutions Manual 21 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
Or using a financial calculator (TI BAII Plus), Hit [2nd] [BGN] [2nd] [Set] N=35, I/Y=12.6, PV=0, PMT= -9,000, CPT FV=5,039,384 54. In Problem 53, if Jack prefers to invest a lump-sum amount today instead of investing annually, but still expects to receive the same amount of money when he retires, how much should he invest today? Level of difficulty: Difficult Solution: There are two ways to get the answer: 1). PV=FV/(1+k)n=5,039,384/(1.126)35=5,039,384/63.6566=$79,165 Or using a financial calculator (TI BAII Plus), N=35, I/Y=12.6, PMT=0, FV=5,039,384, CPT PV= - 79,165 2).
1 1 (1 k )n PV0 PMT (1 k ) k 1 1 (1.126)35 $9,000 (1.126) (9,000)(7.8118311)(1.126) $79,165 .126 Or using a financial calculator (TI BAII Plus), Hit [2nd] [BGN] [2nd] [Set] N=35, I/Y=12.6, PMT= -9,000, FV=0, CPT PV=79,165 55. In Problem 53, if Jack invests the same annual amount at the end of each of the next 35 years instead of at the beginning of the years, how much he will receive when he retires? Explain why this amount is greater or smaller than the result calculated in Problem 53. Level of difficulty: Difficult Solution: Now it is an ordinary annuity.
(1 .126) 35 1 (1 k )35 1 FV35 P M T 9 , 000 = 9,000(497.2749) $4,475,474 k .126 Or using a financial calculator (TI BAII Plus), N=35, I/Y=12.6, PV=0, PMT= –9,000, CPT FV= 4,475,474. So, 4,475,474<5,039,384. As expected, FV of an ordinary annuity is less than an otherwise identical annuity due. Because each annual investment is compounded one year less in an ordinary annuity, where Solutions Manual 22 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
the annual investment is at each year-end, while in the annuity due this amount is invested at the beginning of each year. Alternatively, we can look at Equation 5-4 and Equation 5-6. The difference is due to the compound factor (1+k). 56. A. Determine the month-end payment for a $200,000, 10-year loan with an interest rate of 12 percent, compounded monthly. (Assume there is no down payment) B. Calculate the outstanding loan amount after 18 months. C. Redo part A, assuming it is a mortgage loan with monthly payments. A. Level of difficulty: difficult. Solution: PV=200,000, monthly rate=12%/12=1%, N = (10)(12)=120 months
1 1 (1 .01)120 200,000 PMT .01 1 1 (1 .01)120 PMT 200,000 / .01 So, PMT=$2,869 Or using a financial calculator (TI BAII Plus), N=120, I/Y=1, PV=-200,000, FV=0, CPT PMT=2,869 B. Remaining months to pay=120 – 18=102 months
1 1 (1 .01)102 PV0 2,869 =$182,920 .01 Or using a financial calculator (TI BAII Plus), N=102, I/Y=1, PMT=- 2,869, FV=0, CPT PV=182,920
C. kmonthly= (1
2
.12 12 ) 1 =.9759% 2
Solutions Manual 23 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
1 1 (1 .009759)120 200,000 PMT .009759 1 1 (1 .009759)120 PMT 200,000 / .009759 So, PMT=$2,836 Or using a financial calculator (TI BAII Plus), N=120, I/Y=.9759, PV=-200,000, FV=0, CPT PMT=2,836 57. Investor A just turned 20 years old and currently has no investments. She plans to invest $5,500 at the end of each eight years, beginning in five years. The rate of return on her investment is 15 percent, continuously compounded. Investor B is 40 years old and he just started to invest at the beginning of every year an equal amount of money starting today. He will invest for 10 years. The rate of return on his investment is 16 percent, compounded quarterly. Determine the yearly payment Investor B has to make in order to have the same present value as Investor A. Level of difficulty: Difficult Solution:
Investor A: k=e.15 – 1=16.183424%. 1st, consider an ordinary annuity and the present value of the investment when A turns 25 years old is:
1 1 (1 .16183424)8 PV25 5,500 =$23,749.19 .16183424 Or using a financial calculator (TI BAII Plus), N=8, I/Y=16.183424, PMT=5,500, FV=0, CPT PV=- 23,749.19 2nd, discount this amount for five years back to today when she is 20. PV=FV/(1+k)5=23,749.19/(1.16183424)5=$11,218.3231 Or, N=5, I/Y=16.183424, PMT=0, FV=- 23,749.19, CPT PV=11,218.3231 Investor B: Solutions Manual 24 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
k= (1
.16 4 ) 1=16.985856% 4
1 1 (1 .16985856)10 11,218.3231 PMT (1.16985856) .16985856 PMT=$2,057.38 Or using a financial calculator (TI BAII Plus), Hit [2nd] [BGN] [2nd] [Set] N=10, I/Y=16.985856, PV=11,218.3231, FV=0, CPT PMT= - 2,057.38 Therefore, Investor B has to make an equal amount of $2,057.38 so that the present value of the two investments is the same. 58. Paul and Maria want to have enough money to travel around the world when they retire. They just turned 30 and will retire when they turn 60. They earn a total of $9,000 after taxes each month. Their monthly expenditures include $3,000 in mortgage payments, $850 in car payments, and $1,450 in other expenses. They approached a fund manager and decided to invest the rest of their income at the end of each year. They expect to earn a 10 percent expected annual rate of return for each of the next 30 years. When they retire, they will sell their cottage for an expected price of $50,000. A. Determine how much they will have when they retire. B. How much can Paul and Maria withdraw annually at the beginning of the year for travelling after they retire if they expect to live until they are 90? A. Level of difficulty: Difficult Solution: 1st Calculate their yearly income available for investment Monthly income available=$9,000 – $3,000 – $850 –$1,450=$3,700 Yearly available=$(3,700)(12)=$44,400 2nd Calculate the FV of their investment when they retire:
(1 .1)30 1 FV30 44,400 =$7,303,535 .1 Or using a financial calculator (TI BAII Plus), N=30, I/Y=10, PV=0, PMT=- 44,400, CPT FV=7,303,535 3rd Calculate the amount they will have when they retire: $7,303,535 + $50,000 = $7,353,535 B. Level of difficulty: Difficult Solution: Solutions Manual 25 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
This is an annuity due problem. PV=7,353,535, k=10%, n=30
1 1 (1 .1)30 7,353,535 PMT (1 .1) .1 So, PMT=709,143 Or using a financial calculator (TI BAII Plus), Hit [2nd] [BGN] [2nd] [Set] N=30, I/Y=10, PV=- 7,353,535, FV=0, CPT PMT=709,143 59. Veda has to choose between two investments that have the same cost today. Both investments will ultimately pay $1,300 but at different times, as shown in the table below. If Veda doesn’t choose one of these investments, she could leave the funds in a bank account paying 5 percent per year. Which investment should Veda choose? Year 1 2 3
Investment A $0 $500 $800
Investment B $200 $400 $700
Topic: Discounting Level of difficulty: Difficult Solution: Find the present value of the money paid back to Veda by each investment, using the interest rate on the alternative (the bank account) as the discount rate. For Investment A: PV0
$500 $800 $453.51 $691.07 $1144.58 2 (1 0.05) (1 0.05)3
For Investment B:
PV0
200 400 700 190.48 362.81 604.69 $1157.98 1 2 (1 0.05) (1 0.05) (1 0.05)3
Veda would prefer Investment B, because it has the higher present value. 60. If the cost of each investment in Problem 59 0 is $1,000, should Veda invest in one of them, or simply leave the money in the bank account? Would her decision change if the investments cost $1,200 each? Topic: Discounting and Determining Rates of Return Level of difficulty: Difficult Solutions Manual 26 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
Solution: Observe that the present value of the money returned by both investments, using the 5 percent discount rate, is greater than the $1,000 cost (see Problem 59). Therefore, they both have a positive “net present value.” On this basis, either investment would be preferable to the bank account (returning 5%); Investment B is more preferable. If the cost of the investments was $1,200, neither would be acceptable, as the present value of the money returned is less than the cost (they have a negative net present value). 61. Instead of a $40,000 lump sum, Felix will need $10,000 per year for four years to pay for tuition (see Problem 30). How much will Felix’s parents have to invest at the end of each year for the eight years before he begins his studies if their savings earn compound interest at 7 percent per year? Assume the tuition payments also occur at the end of each year. Topic: Ordinary Annuities Level of difficulty: Difficult Solution: We have two separate annuities to consider: the tuition payments, and the savings amounts. First, find the present value of the four annual tuition payments (at time 8, when Felix is due to begin university studies):
1 1 (1 0.07)4 PV8 10,000 0.07
$33,872.11
This is the amount of savings required at time 8. From the perspective of time 0, this is a future value amount (replaces the $40,000 in Problem 0.) Next, find the annual savings amount:
(1 0.07) 8 1 $33,872.11 PMT PMT $3,301.44 0 . 07 62. Repeat Problem 61 0assuming the tuition payments occur at the beginning of each year (the savings are still invested at the end of each year). Topic: Annuities Due Level of difficulty: Difficult Solution: This problem can be solved in the same two step manner used in Problem 61. However, the first step involves an “annuity due,” as the tuition payments occur at the start of each year:
Solutions Manual 27 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
1 1 (1 0.07)4 PV8 10,000 0.07
(1 0.07) $36,243.16
Or using a financial calculator (TI BAII Plus), Hit [2nd] [BGN] [2nd] [Set] N=4, I/Y=7, PMT=10,000, FV=0, CPT PV= –36,243.16 This answer is used as the future value amount in the second step:
(1 0.07)8 1 $36,243.16 PMT PMT $3532.54 0.07 Or using a financial calculator (TI BAII Plus), (Remember to clear the calculator, or turn off “BGN” mode for this step, as this is an ordinary annuity, not an annuity due) N=8, I/Y=7, PV=0, FV= -36,243.16, CPT PMT= 3532.54
63. Roger has his eye on a new car that will cost $20,000. He has $15,000 in his savings account, earning interest at a rate of 0.5 percent per month. A. How long (to the nearest month) will it be before he can buy the car? B. How long will it be before Roger can buy the car if, in addition to his existing savings, he can save $250 per month? Topic: Determining Holding Periods Level of difficulty: Difficult Solution: A. We know the future value and present value amounts, as well as the monthly interest rate. Finding the number of time periods (months) is most easily done with a financial calculator (TI BAII Plus), PV = 15,000, FV = -20,000, I/Y = 0.5, CPT N = 57.68 It will take nearly 58 months, or close to 5 years before Roger can afford to buy the car! B. Solving the following equation for “n” we get:
20,000
(1.005) n 1 15,000 250 .005 n= 14.86. (1.005) n
Or using a financial calculator (TI BAII Plus), I/Y=0.5, PV=15,000, FV= -20,000, PMT = 250, CPT n = 14.86
64. How many years will it take for an investment to double in value if the rate of return is 9 percent, and compounding occurs: Solutions Manual 28 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
A. annually? B. quarterly? Topic: Effective Interest Rates and Holding Periods Level of difficulty: Difficult Solution: Let’s assume the present value of the investment is $1. The future value, after doubling is then $2. A. Annually: With annual compounding, the effective rate is the same as the quoted rate, 9%. Using a financial calculator (TI BAII Plus), PV = –1, FV = 2, I/Y = 9, CPT N = 8.04 So the investment will double in just over 8 years. B. Quarterly: With quarterly compounding, the effective annual rate is, 0.09 4 k (1 ) 1 9.308% , and a financial calculator allows us to find: 4 PV = -1, FV = 2, I/Y = 9.308, CPT N = 7.79 The higher effective rate means that only 7.79 years are needed to double the value of the investment. 65. Assume Josephine chose the Providence Bank option (see Problem 47). The three-year term on Josephine’s mortgage is now over, and she must renew the loan for another term. How much of the original $180,000 principal amount does she still owe? Topic: Mortgage Loans Level of difficulty: Difficult Solution: The principal outstanding at any time is the present value of the remaining payments. After three years, there are 17 years remaining of the original 20-year amortization period, or 17 x 12 = 204 monthly payments remaining. Based on Problem 47, the payment amount is $1,322.69. Therefore:
PV3 years
1 1 (1 0.005264)204 $1322.69 $165,172.38 0.005264
66. How much interest did Josephine pay over the three-year term of her mortgage loan (see Problem 470)? Topic: Mortgage Loans Level of difficulty: Difficult Solution: Josephine has made 3 x 12 = 36 payments, of $1322.69. The total amount she has paid over Solutions Manual 29 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
the three years is therefore, 36 x $1,322.69 = $47,616.84. Based on the solution to Problem 0, she has paid off ($180,000 – $165,172.38) = $14,827.62 of the principal amount of the mortgage loan. Therefore, of her total payments, ($47,616.84 – $14,827.62) = $32,789.22 was interest. 67. A year has passed since Charlie purchased his house (see Problem 500). Assume the original mortgage amount (from your answer in Problem 500) was to be amortized over 25 years, with an initial one-year term at 3.90 percent. If the interest rate on a one-year term has increased to 5.35 percent, what will be Charlie’s new monthly mortgage payments? Topic: Mortgage Loans Level of difficulty: Difficult Solution: There are 24 years, or 24 x 12 = 288 months remaining on the mortgage. Using the effective monthly interest rate from Problem 50, the outstanding principle amount is:
PV1 year
1 1 (1 0.003224) 288 $365,484.77 $1,950.00 0.003224
The new effective monthly interest rate is:
kmonthly
0.0535 1 2
2
12
1 0.4409% .
Using this rate, and the outstanding principal, the new mortgage payment will be:
1 1 (1 0.004409) 288 PMT $2,243.31 $365,484.77 PMT 0.004409 68. The new and higher mortgage payments calculated in Problem 670 creates a big problem for Charlie, because he can afford to pay only $1,950.00 per month. Rather than sell the house, Charlie has convinced his bank to extend the amortization period on the loan, which will reduce the monthly payments. How long must the amortization period be for this mortgage so that Charlie can afford to make the payments? Topic: Mortgage Loans Level of difficulty: Difficult Solution: Here we must solve the annuity equation to find the amortization period required:
Solutions Manual 30 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
1 1 (1 0.004409)n $365,484.77 $1950.00 n? 0.004409 This can be done algebraically if you are familiar with the logarithm function:
$365,484.77 0.004409 n Ln1 1 Ln(1 0.004409) 397.98 $1950.00 An easier method is to use a financial calculator (TI BAII Plus), I/Y=0.4409, PMT=1950, PV= -365,484.77, FV=0, CPT N= 397.98 We have used a monthly interest rate, so our solution is in terms of months, not years. Our final answer is that Charlie must extend the amortization period to at least 398 months (or 33 years and 2 months) so that he can still afford the monthly payments. 69. Céline has just won a lottery. She will receive a payment of $6,000 each year for nine years. As an alternative, she can choose an immediate payment of $50,000. A. Which alternative should she pick if the interest rate is 5 percent? B. What would the interest rate have to be for Céline to be indifferent about the two alternatives0? C. The lottery offers a third alternative for lottery prize payments: Céline can opt to receive annual payments of $3,000 per year for the rest of her life. Should Céline choose this option to the $50,000 lump sum payment today, if the interest rate is 5 percent per year? Topics: Ordinary Annuities, Determining Rates of Return, Perpetuities Level of difficulty: Difficult Solution: A. The present value of the annual payments can be found with a financial calculator, (TI BAII Plus), N=9, PMT = -6000, I/Y = 5.0, FV=0, CPT PV = 42,646.93 As this is less than $50,000, the immediate payment alternative is better. B. This problem can be solved by trial and error, but the task is much easier with a financial calculator, (TI BAII Plus), N=9, PMT = –6,000, PV = 50,000, FV=0, CPT I/Y = 1.5675%. At an interest rate below 1.5675% per year, the nine-year annuity would be preferable; above the rate the immediate payment is better. C. This alternative is a perpetuity. Its present value is: PMT $3,000 PV0 $60,000 . k 0.05 Solutions Manual 31 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
Clearly this is preferable to a $50,000 lump sum payment. 70. After 20 years, the lottery company in Problem 69 goes out of business, and Céline’s payments of $3,000 annually, which were supposed to continue for the rest of her life, stop. A. What is the present value of the lost payments? B. How much were these lost payments worth 20 years ago when Céline won the prize? C. Had she been able to predict that the lottery company would go out of business, should Céline have chosen the perpetual payments, or the $50,000 lump sum? Topic: Perpetuities and Discounting Level of difficulty: Difficult Solution: A. Even after 20 years, the remaining payments form a perpetuity. PMT $3,000 PV20 years $60,000 k 0.05 B. We must discount the figure from part (a) back 20 years:
PV0
$60,000 $22,613.37 (1 0.05)20
C. The value of 20 years of $3,000 payments (an annuity) can be easily found using a financial calculator (TI BAII Plus), N=20, PMT = -3000, I/Y = 5.0, FV=0, CPT PV = 37,386.63. In hindsight, the $50,000 lump-sum payment would have been preferable. The same result could be found by subtracting the result in part B from that in A; this demonstrates that an annuity can be thought of as the difference between two perpetuities! 71. Alysha has decided to use her $50,000 to make a down payment on a house. She will live in the house for the next two years while still at university, and then sell it when she graduates. The bank has offered her a mortgage rate of 5.1 percent compounded semi-annually on a two-year term, with an amortization period of 25 years. The house she is interested in purchasing costs $280,000. A. If two friends will rent rooms from Alysha for $475 per room, payable at the end of each month, how much additional money must she pay to meet her monthly mortgage payments? B. In two years, Alysha wants to sell the house for a high enough price to cover the remaining principal amount on the mortgage, and return her down payment. What is the minimum sale price she should accept? Topic: Mortgage Loans Level of difficulty: Difficult Solution: A. The effective monthly interest rate is,
Solutions Manual 32 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
kmonthly
0.051 1 2
2
12
1 0.4206%
The amount of the mortgage loan will be ($280,000 – $50,000) = $230,000, and there will be 12 x 25 = 300 monthly payments, the value of which can be found with a financial calculator, (TI BAII Plus), N=300, PV = –230,000, I/Y = 0.4206, CPT PMT = 1350.89. Alysha’s two friends will be paying 2 x $475 = $950 in rent, so she will need an additional $1,350.81 – $950 = $400.81 to make the mortgage payments. B. In two years, Alysha will have made 24 payments, leaving 276. The present value of these payments is the outstanding value of the mortgage loan. Using the calculator again, N=276, I/Y = 0.4206, PMT = 1350.89, CPT PV = 220,336.58. To pay off the loan, and recoup her down payment, Alysha would have to sell the house for at least $220,336.58 + $50,000 = $270,336.58. 72. How much will Tommy have to earn on his savings (see Problem 520) to be able to amass $1 million in 15 years? Topic: Investing Early Level of difficulty: Difficult Solution: In the equation for Problem 52, set FV=$1,000,000, and solve for the interest rate. This is easiest done with a financial calculator (TI BAII Plus), N = 15, PV=0, FV = -1,000,000, PMT = 30,000, CPT I/Y = 10.603 It will take only a little higher return, 10.6%, for Tommy to reach his goal. 73. KashKow Inc. has just declared that its dividend next year will be $3 per share. That rate of payment will continue for an additional four years, after which the dividends will fall back to their usual $2 per share (see Problem 230). What is the present value of all the future dividends? Topic: Multiple Annuities Level of difficulty: Difficult Solution: The dividends for the first five years form an ordinary annuity. Starting in year 6, the reduced dividend stream can be thought of as a perpetuity. However, the value of this perpetuity, as determined by our formula, occurs at year 5 (one year before the first $2 dividend), and must be discounted to the present:
1 1 (1 0.12)5 PV0 $3.00 0.12
1 $2.00 10.81 16.67 0.5674 $20.27 5 0.12 (1 0.12)
Solutions Manual 33 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
74. After one year living in a university residence, Mary-Beth (see Problem 24) decides to rent an apartment for the remaining three years of her degree. She has found a nice location that will cost $450 per month; rent for the first and last month must be paid up front. How much money would Mary-Beth need to have in her bank account right now to be sure she will always have enough for rent? The bank account pays 3.75 percent interest, compounded monthly. Topic: Annuities Due Level of difficulty: Difficult Solution: 0.0375 kmonthly 0.3125% 12 Rent payments are typically made at the start of each month (so this is an annuity due). Over three years, we would expect 36 monthly rent payments. However, the last month’s rent must be paid up front, so the annuity includes only 35 payments; the present value of the last month’s rent is $450 because it will be paid today.
1 1 (1 0.003125)35 PV0 $450 (1 0.003125) $450 $15,393.77 0.003125 75. Suppose that Mary-Beth plans to return home for four months each summer, and will sub-let her apartment for the same amount she pays in rent (see Problem 740). In other words, she will pay $450 rent, but for only eight months of the year, during each of three years. In this situation, how much would she need to have in her bank account, earning 3.75 percent compounded monthly? (Assume that rent payments are made at the start of each month.) Topic: Annuities Due and Multiple Annuities Level of difficulty: Difficult Solution: As in Problem 74, kmonthly 0.3125% . Consider one year, or 8 months of rent payments made at the start of each month:
1 1 (1 0.003125) 8 PV0 $450 0.003125
(1 0.003125) $3,560.99
Mary-Beth will need to have this sum now, plus the same amount one year from now, and again two years from now. In essence, this is a three-year annuity due, whose value must be calculated using the effective annual interest rate.
Solutions Manual 34 Chapter 5 Copyright © 2008 John Wiley & Sons Canada, Ltd. Unauthorized copying, distribution, or transmission is strictly prohibited.
12
0.0375 k 1 1 3.8151% 12 1 1 (1 0.038151) 3 (1 0.038151) $10,295.19 PV0 $3,560.99 0.038151 76. A 65-year-old man intends to use his retirement funds to purchase an annuity from a life insurance company. Given the amount of money the man has available to invest, the insurance company is able to offer two alternatives. The first option is to receive $2,785 each month for as long as he lives; the second option is to receive $3,500 each month, but for only 20 years (payments will be made to his estate if he should die before that time). The relevant interest rate is 6 percent per year. How long must the man live so that the first option is a better deal? Topic: Ordinary Annuities and Multiple Annuities Level of difficulty: Difficult Solution: It is tempting to view the first option as a perpetuity, but this would be incorrect as the man will die at some time, and the payment will then cease. Thus, option one is an ordinary annuity, with an uncertain number of payments. Option two is much easier to value; it includes exactly 240 monthly payments.
kmonthly
0.06 0.5% 12
Using a financial calculator (TI BAII Plus), N = 240, PMT = 3500, I/Y = 0.5, FV=0, CPT PV = –488,532.70 For the first option to be a better deal, it must include enough payments so that its present value is at least as great as for option two. Again using the calculator, PV = –488,532.70, PMT = 2785, I/Y = 0.5, CPT N = 420.3 So option one must continue for 420.3 monthly payments to equal the value of option two, or 421 payments to surpass it. This is just over 35 years. Hence, the man must live to be at least 100 years old for option one to be a better deal!
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