Solutions to text book questions ch1to10 PDF

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Solutions to text book questions ch1to10, needed for the practice, good indication of the exam.
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Brealey 6CE Solutions to Chapter 5

Note: Unless otherwise stated, assume that cash flows occur at the end of each year. 1.

a. b. c. d.

100/(1.08)10 100/(1.08)20 100/(1.04)10 100/(1.04)20

2.

a. b. c. d.

100  (1.08)10 100  (1.08)20 100  (1.04)10 100  (1.04)20

3.

With simple interest, you earn 4% of $1000, or $40 each year. There is no interest on interest. After 10 years, you earn total interest of $400, and your account accumulates to $1400. With compound interest, your account grows to 1000  (1.04)10 = $1480.24 Therefore $80.24 is interest on interest.

4.

FV = 700

= = = =

$46.32 $21.45 $67.56 $45.64 = = = =

$215.89 $466.10 $148.02 $219.11

PV = 700/(1.05)5 = $548.47

5. Present Value

Years

Future Value

a.

$400

11

$684

b.

$183

4

$249

c.

$300

7

$300

Interest Rate* 1/11 5% = () –1 1/4 8% = () –1 1/7 0% = () – 1

To find the interest rate, we rearrange the equation 1/n FV = PV  (1 + r)n to conclude that r = () -1 To use a financial calculator for (a) enter PV= (-)400, FV = 684, PMT = 0, n = 11 and compute the interest rate.

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6.

You should compare the present values of the two annuities. Discount Rate a. b.

Present Value of 10-year, $1000 annuity

5% 20%

7721.73 4192.47

Present Value of 15-year, $800 annuity 8303.73 3740.38

When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less severe. When the interest rate is high, as in part (b), the shorter but higher annuity is more valuable. In this case, with the 20 percent interest rate, the present value of more distant payments is substantially reduced, making it better to take the shorter but higher annuity.

7.

PV = 200/1.05 + 400/1.052 + 300/1.053 = 190.48 + 362.81

8.

+ 259.15 = $812.44

In these problems, you can either solve the equation provided directly, or you can use your financial calculator setting PV = ()400, FV = 1000, PMT = 0, i as specified by the problem. Then compute n on the calculator. a.

400  (1 + .04)t = 1,000

t = 23.36 periods

b.

400  (1 + .08)t = 1,000

t = 11.91 periods

c.

400  (1 + .16)t = 1,000

t = 6.17 periods

Note: To solve directly, use the natural log function, ln. For example, for (a), ln[ (1.04)t ] = ln[1000/400] t × ln[1.04] = 0.91629 t = 0.91629/.03922 = 23.36 period. Using the calculator: PV = (-)400, FV = 1000, i = 4, compute n to get n = 23.36. 9.

a. b. c. d.

PV = 100 × PVIFA(.08,10) = 100 × 6.7101 = 671.01 PV = 100 × PVIFA(.08,20) = 100 × 9.8181 = 981.81 PV = 100 × PVIFA(.04,10) = 100 × 8.1109 = 811.09 PV = 100 × PVIFA(.04,20) = 100 × 13.5903 = 1,359.03

10.

a. b. c. d.

FV = 100 × FVIFA(.08,10) = 100 × 14.4866 = 1,448.66 FV = 100 × FVIFA(.08,20) = 100 × 45.7620 = 4,576.20 FV = 100 × FVIFA(.04,10) = 100 × 12.0061 = 1,200.61 FV = 100 × FVIFA(.04,20) = 100 × 29.7781 = 2,977.81

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11. APR

Compounding Period

Per Period Rate, APR/m

Effective annual rate

a.

12%

1 month (m = 12/yr) .12/12 =.01

1.0112 1 = .1268 = 12.68%

b.

8%

3 months (m = 4/yr) .08/4 = .02

1.024  1 = .0824 = 8.24%

c.

10%

6 months (m = 2/yr) .10/2 = .05

1.052  1 = .1025 = 10.25%

12. Effective Annual Rate, EAR

Compounding Period

Number of Periods per year, m

a.

10.0%

1 month

b.

6.09%

c.

8.24%

Per period rate, (1+EAR)1/m -1

APR, m × per period rate

12

1.11/12 – 1 = .008

12×.008 = .096 = 9.6%

6 months

2

1.06091/2  1 = .03

2×.03 = .06 = 6%

3 months

4

1.08241/4  1 = .02

4×.02 = .08 = 8%

13.

We need to find the value of n for which 1.08n = 2. You can solve to find that n = 9.01 years. On a financial calculator you would enter PV = ()1, FV = 2, PMT = 0, i = 8 and then compute n.

14.

Semiannual compounding means that the 8.5 percent loan really carries interest of 4.25 percent per half year. Similarly, the 8.4 percent loan has a monthly rate of .7 percent. APR

Period

m

Effective annual rate = (1 + per period rate)m – 1

8.5% 8.4%

6 months 1 month

2 12

(1.0425)2  1 = .0868 = 8.68% (1.007)12  1 = .0873 = 8.73%

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Choose the 8.5 percent loan for its slightly lower effective rate.

15.

APR = 1%  52 = 52% EAR = (1 + .01)52  1 = .6777 = 67.77%

16.

Our answer assumes that the investment was made at the beginning of 1900 and now it is the end of 2011. Thus the investment was for 112 years (2011 – 1900 + 1). a. b.

17.

1000  (1.05)112 = $236,157.37 PV  (1.05)112 = 1,000,000 implies that PV = $4,234.46

$1000  1.05 = $1050.00 $1050  1.05 = $1102.50

First-year interest = $50 Second-year interest = $1102.50  $1050 = $52.50

After 9 years, your account has grown to 1000  (1.05)9 = $1551.33 After 10 years, your account has grown to 1000  (1.05)10 = $1628.89 Interest earned in tenth year = $1628.89  $1551.33 = $77.56 18.

Method 1: If you earned simple interest (without compounding) then the total growth in your account after 25 years would be 4% per year  25 years = 100%, and your money would double. With compound interest, your money would grow faster, and therefore would require less than 25 years to double. Method 2: Another quick way to answer the question is with the Rule of 72. Dividing 72 by 4 gives 18 years, which is less than 25. The exact answer is 17.673 years, found by solving 2000 = 1000  (1.04)n. [On your calculator, input PV = (-) 1000, FV = 2000, i = 4, PMT = 0, and compute the number of periods.]

19.

20.

We solve 422.21  (1 + r)10 = 1000. This implies that r = 9%. [On your calculator, input PV = (-)422.21, FV = 1000, n = 10, PMT = 0, and compute the interest rate.] The number of payment periods: n = 12 × 4 = 48. If the payment is denoted PMT, then 5-4

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PMT  annuity factor( %, 48 periods) = 8,000 PMT = $202.90. The monthly interest rate is 10/12 = .8333 percent. Therefore, the effective annual interest rate on the loan is (1.008333)12  1 = .1047 = 10.47 percent. 21.

a.

PV = 100  annuity factor(6%, 3 periods) = 100  = $267.30

b.

22.

a.

If the payment stream is deferred by an extra year, each payment will be discounted by an additional factor of 1.06. Therefore, the present value is reduced by a factor of 1.06 to 267.30/1.06 = $252.17. This is an annuity problem with PV = (-)80,000, PMT = 600, FV = 0, n = 20  12 = 240 months. Use a financial calculator to solve for i, the monthly rate on this annuity: i = .5479%. EAR = (1 +.005479)12  1 = .06776 = 6.777% APR = 12 × monthly interest rate = 12 × .5479% = 6.5748%, compounded monthly

23.

b.

Again use a financial calculator and enter n = 240, i = .5%, FV = 0, PV = ()80,000 and compute PMT = $573.14

a.

With PV = 9,000 and FV = 10,000, the annual interest rate is defined by 9,000  (1 + r) = 10,000, which implies that r = 11.11%.

b.

Your present value is 10,000 (1  d), and the future value you pay back is 10,000. Therefore, the annual interest rate is determined by: PV  (1 + r) = FV [10,000 (1 – d)]  (1 + r) = 10,000

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1+r=  r= 1=>d Since 0 < d < 1, then 1 – d < 1 and d/(1 – d) > d. So r must be greater than d. c.

With a discount interest loan, the discount is calculated as a fraction of the future value of the loan. In fact, the proper way to compute the interest rate is as a fraction of the funds borrowed. Since PV is less than FV, the interest payment is a smaller fraction of the future value of the loan than it is of the present value. Thus, the true interest rate exceeds the stated discount factor of the loan.

24.

If we assume cash flows come at the end of each period (ordinary annuity) when in fact they actually come at the beginning (annuity due), we discount each cash flow by one period too many. Therefore we can obtain the PV of an annuity due by multiplying the PV of an ordinary annuity by (1 + r). Similarly, the FV of an annuity due also equals the FV of an ordinary annuity times (1 + r). Because each cash flow comes at the beginning of the period, it has an extra period to earn interest compared to an ordinary annuity.

25.

a.

Solve for i in the following equation: 10,000 = 275 × PVIFA(i, 48) Using the calculator, set PV = -10,000, PMT = 275, FV = 0, n = 48 and solve for i i= 1.19544% per month APR = 12 × 1.19544% = 14.3453% EAR = (1 + .0119544)12 – 1 = .153271, or 15.3271%

b.

Annual payment = 12 × 275 = 3,300 Repeat the steps in (a) to find the EAR of this car loan to see which loan is charging the lower interest rate: Solve for i in the following equation: 10,000 = 3,330 × PVIFA(i, 4) Using the calculator, set PV = -10,000, PMT = 3,300, FV = 0, n = 4 and solve for i i= 12.11% per year Little Bank's loan interest rate of 12.11% is less than the EAR of 15.53% on Big Bank's loan. With a lower interest rate, Little Bank's loan is better.

c.

Find the annual loan payment, P, such that 10,000 = X × PVIFA(15.3271%, 4) Using the calculator, set PV = -10,000, FV = 0, n = 4, i = 15.3271 and solve for PMT = $3,525.86. By comparison, 12 times $275 per month is $3,300. The annual payment on a 4-year loan equivalent to $275 per month for 48 months is greater than 12 times the monthly payment of $275 because of the benefit of delaying payment to the end of each year. The borrower gets to delay payment and therefore is better off. If Little Bank doesn't charge at least $3,525.86 annually, it earns less on its loan than Big Bank earns on its loan.

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26.

27.

a.

Compare the present value of the lease to cost of buying the truck. PV lease = 8,000 × PVIFA(7%, 6) = -$38,132.32 It is cheaper to lease than buy because by leasing the truck will cost only $38,132.32, rather than $40,000. Of course, the crucial assumption here is that the truck is worthless after 6 years. If you buy the truck, you can still operate it after 6 years. If you lease it, you must return the truck and replace it.

b.

If the lease payments are payable at the start of each year, then the present value of the lease payments are: PV annuity due lease = 8,000 + 8,000 × PVIFA(7%, 5) = 8,000 + 32,801.58 = $40,801.58. Note too that PV of an annuity due = PV of ordinary annuity  (1 + r). Therefore, with immediate payment, the value of the lease payments increases from its value in the previous problem to $38,132  1.07 = $40,801 which is greater than $40,000 (the cost of buying a truck). Therefore, if the first payment on the lease is due immediately, it is cheaper to buy the truck than to lease it.

a.

Compare the PV of the payments. Assume the product sells for $100. Installment plan: Down payment = .25 × 100 = 25 Three installments of .25 × 100 = 25 PV = 25 + 25  annuity factor(6%, 3 years) = $91.83. Pay in full: Payment net of discount = $90. Choose this payment plan for its lower present value of payments. Note: The pay-in-full payment plan will have the lowest present value of payments, regardless of the chosen product price.

b.

28.

Installment plan: PV = 25  annuity factor(6%, 4 years) = $86.63. Now the installment plan offers the lower present value of payments.

a.

PMT  annuity factor(12%, 5 years) = 1000 PMT  3.6048 = 1000 PMT = $277.41

b.

If the first payment is made immediately instead of in a year, the annuity factor will be greater by a factor of 1.12. Therefore PMT  (3.6048  1.12) = 1000. PMT = $247.69.

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29.

This problem can be approached in two steps. First, find the PV of the $10,000, 10-year annuity as of year 3, when the first payment is exactly one year away (and is therefore an ordinary annuity). Then discount the value back to today. Using a financial calculator, 1) PMT = 10,000; FV = 0; n = 10; i = 6%. Compute PV3 = $73,600.87 2) PV0 = = = $61,796.71 A second way to solve the problem is the take the difference between a 13-year annuity and a 3-year annuity, valued as of the end of year 0: PV of delayed annuity = 10,000 × PVIFA(6%,13) – 10,000 × PVIFA(6%,3) = 10,000 × (8.852683 – 2.673012) = 10,000 × 6.179671 = $61,796.71

30.

Note: Assume that this is a Canadian mortgage. The monthly payment is based on a $175,000 loan with a 300-month (12 × 25 years) amortization. The posted interest rate of 6 percent has a 6-month compounding period. Its EAR is (1 + .06/2)2 – 1 = .0609, or 6.09%. The monthly interest rate equivalent to 6.09% annual is (1.0609)1/12 – 1 = .004939, or 0.4939%. PMT  annuity factor(.4939%, 300) = 175,000 PMT = $1,119.71. When the mortgage expires in 5 years, there will be 20 years remaining in the amortization period, or 240 months. The loan balance in five years will be the present value of the 240 payments: Loan balance in 5 years = $1,119.71  Annuity factor (.4939%, 240 periods) = $157,215.

31.

The EAR of the posted 7% rate is (1 + .07/2)2 – 1 = .071225. The monthly interest rate equivalent is (1.071225)1/12 – 1 = .00575, or 0.575%. The payment on the mortgage is computed as follows: PMT  annuity factor (.575%, 300 periods) = 350,000 PMT = $2,451.44 per month. If you pay the monthly mortgage payment in two equal installments, you will pay $2,451.44/2, or $1,225.72 every two weeks. Thus each year you make 26 payments. The bi-weekly equivalent of the 7% posted interest rate is (1.071225)1/26 – 1 = .002649,

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or .2649% every two weeks. Now calculate the number of periods it will take to pay off the mortgage: $1,225.72  Annuity factor (.2649%, n periods) = $350,000 Using the calculator: PMT = 1,225.49, PV = (-)350,000, i = .2649 and compute n = 533.84. This is the number of bi-weekly periods. Divide by 26 to get the number of years: 533.84/26 = 20.5 years. If you pay bi-weekly, the mortgage is paid off 5.5 years sooner than if you pay monthly.

32.

a.

Input PV = (-)1,000, FV = 0, i = 8%, n = 4, compute PMT which equals $301.92

b. Time 0 1 2 3 4 c.

33.

Loan Balance $1,000.00 $778.08 $538.41 $279.56 0

Year End Interest Due on Balance $ 80.00 $62.25 $43.07 $22.37 0

Year End Payment $301.92 $301.92 $301.92 $301.92 —

Amortization of Loan $221.92 $239.67 $258.85 $279.56 —

301.92  annuity factor (8%, 3 years) = 301.92 × 2.5771 = $778.08, which equals the loan balance after one year.

The loan repayment is an annuity with present value $4248.68. Payments are made monthly, and the monthly interest rate is 1%. We need to equate this expression to the amount borrowed, $4248.68, and solve for the number of months, n. [On your calculator, input PV = () 4248.68, FV = 0, i = 1%, PMT = 200, and compute n.] The solution is n = 24 months, or 2 years. The effective annual rate on the loan is (1.01)12  1 = .1268 = 12.68%

34.

The present value of the $2 million, 20-year annuity, discounted at 8%, is $19,636,295 If the payment comes one year earlier, the PV increases by a factor of 1.08 to $21,207,198.

35.

The real rate is zero. With a zero real rate, we simply divide her savings by the years of retirement: $450,000/30 = $15,000 per year.

36.

Per month interest = 6%/12 = .5% per month 5-9

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FV in 1 year (12 months) = 1000  (1.005)12 = $1,061.68 FV in 1.5 years (18 months) = 1000  (1.005)18 = $1,093.93 37.

You are repaying the loan with an annuity of payments. The PV of those payments must equal $100,000. Therefore, 804.62  annuity factor(r, 360 months) = 100,000 which implies that the interest rate is .750% per month. [On your calculator, input PV = ()100,000, FV = 0, n = 360, PMT = 804.62, and compute the interest rate.] The effective annual rate is (1.00750)12  1 = .0938 = 9.38%. If the lender is a Canadian financial institution, the quoted rate will be the APR for a 6-month compounding period: (1 + )2 – 1 = .0938 = (1.0938)1/2 -1 = .04585 APR = 2 × [(1.0938)1/2 -1] = .0917 or 9.17%, which is lower than the effective annual rate. Note: A simpler APR calculation is .750%  12 = 9%. However, this is not how Canadian mortgage lenders calculate their APRs.

38.

EAR = e.04 -1 = 1.0408 -1 = .0408 = 4.08%

39. The PV of the payments under option (a) is 11,000, assuming the $1,000 rebate is paid immediately. The PV of the payments under option (b) is $250  annuity factor(1%, 48 months) = $9,493.49 Option (b) is the better deal. 40.

100  e.10×6 = $182.21 100  e.06×10 = $182.21

41. Your savings goal is 30,000 = FV. You currently have in the bank 20,000 = PV. The PMT = (-) 100 and r = .5%. Solve for n to find n = 44.74 months. 5-10 Copyright © 2016 McGraw-Hill Ryerson Limited

Note: You may have to solve this by trial-and-error if your calculator cannot handle these numbers. 42.

The present value of your payments to the bank equals: $100  annuity factor(8%, 10 years) = $671.01 The present value of your receipts is the value of a $100 perpetuity deferred for 10 years:  = $578.99 This is a bad deal if you can earn 8% on your other investments.

43.

If you live forever, you will receive a $100 perpetuity which has present value 100/r. Therefore, 100/r = 2500, which implies that r = 4 percent

44.

r = 10,000/125,000 = .08 = 8 percent.

45.

Suppose the purchase price is $1. If you pay today, you get the discount and pay only $.97. If you wait a month, you must pay $1. Thus, you can view the deferred payment as saving a cash flow of $.97 today, but paying $1 in a month. The monthly rate is therefore .03/.97 = .0309, or 3.09%. The effective annual rate is (1.0309)12  1 = .4408 = 44.08%.

46.

You borrow $1000 and repay the loan by making 12 monthly payments of $100. We find that r = 2.923% by solving: 100  annuity factor(r, 12 months) = 1000 [On your calculator, input PV = ()1,000, FV = 0, n = 12, PMT = 100, and compute the interest rate.] The APR is therefore 2.923%  12 = 35.08% and the effective annual rate is (1.02923)12  1 = .4130 = 41.30% How do we know that the true rates...