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Brealey 5CE

Solutions to Chapter 6

Note: Unless otherwise stated, assume all bonds have $1,000 face (par) value.

1. a. The coupon payments are fixed at $60 per year.

Coupon rate = coupon payment/par value = 60/1000 = 6%, which remains unchanged.

b. When the market yield increases, the bond price will fall. The cash flows are discounted at a higher rate.

c. At a lower price, the bonds yield to maturity will be higher. The higher

yield to maturity on the bond is commensurate with the higher yields

available in the rest of the bond market.

d. Current yield = coupon payment/bond price. As coupon payment remains the same and the bond price decreases, the current yield increases.

2. When the bond is selling at a discount, $970 in this case, the yield to maturity is greater than 8%. We know that if the discount rate were 8%, the bond would sell at par. At a price below par, the YTM must exceed the coupon rate.

Current yield equals coupon payment/bond price, in this case, 80/970. So, current yield is also greater than 8%.

3. Coupon payment = .08 x 1000 = $80

Current yield = 80/bond price = .07

Therefore, bond price = 80/.07 = $1,142.86

4. Par value is $1000 by assumption.

Coupon rate = $80/$1000 = .08 = 8%

Current yield = $80/$950 = .0842 = 8.42%

Yield to maturity = 9.1185%

[Enter in the calculator: N = 6; PV= -950; FV = 1000; PMT = 80]

5. To sell at par, the coupon rate must equal yield to maturity. Since Circular bonds

yield 9.1185%, this must be the coupon rate.

6. a. Current yield = annual coupon/price = $80/1,100 = .0727 = 7.27%.

b. On the calculator, enter PV = -1100, FV = 1000, n = 10, PMT = 80

Then compute I/Y (or i) and will get YTM = 6.6023%.

7. When the bond is selling at par, its yield to maturity equals its coupon rate. This firms bonds are selling at a yield to maturity of 9.25%. So the coupon rate on the new bonds must be 9.25% if they are to sell at par.

8. The current bid yield on the bond was 3.09%. To buy the bond, investors pay the ask price. The investor would pay 108.21 percent of par value. With $1,000 par value, this means paying $1,082.1 to buy a bond.

9. Coupon payment = interest = .05 1000 = 50

Capital gain = 1100 1000 = 100

Rate of return = interest + capital gainpurchase price = 50 + 1001000 = .15 = 15%

10. Tax on interest received = tax rate interest = .3 50 = 15

After-tax interest received = interest tax = 50 15 = 35

Fast way to calculate:

After-tax interest received = (1 tax rate) interest = (1 .3) 50 = 35

Tax on capital gain = .5 .3 100 = 15

After-tax capital gain = 100 15 = 85

Fast way to calculate:

After-tax capital gain = (1 tax rate) capital gain = (1 .5.3)100 = 85

After-tax rate of return = after-tax interest + after-tax capital gainpurchase price

= 35 + 851000 = .12 = 12%

11. Bond 1

year 1: PMT = 80, FV = 1000, i = 10%, n = 10; Compute PV0 = $877.11

year 2: PMT = 80, FV = l000, i = 10%, n = 9; Compute PV1 = $884.82

Rate of return = 80 + (884.82 877.11)877.11 = .10 = 10%

Bond 2

year 1: PMT = 120, FV = 1000, i = 10%, n = 10; Compute PV0 = $1122.89

year 2: PMT = 120, FV = l000, i = 10%, n = 9; Compute PV1 =$1115.18

Rate of return = 120 + (1115.18 1122.89)1122.89 = .10 = 10%

Both bonds provide the same rate of return.

12. Accrued interest =

Coupon payment number of days from last coupon to purchase date number of days in coupon period

= 22.5 136184 = $16.63

Dirty bond price= clean bond price + accrued interest = $990+ $16.63= $1006.63

The quoted clean price is $990. The bond pays semi-annual interest. The last $22.5 coupon was paid on March 1, 2011, and the next coupon will be paid on September 1, 2011. The number of days from the last coupon payment to the purchase date is 136 (from March 1 to July 15) and the total number of days in the coupon period is 184 (from March 1 to September 1). The accrued interest is $16.63, and the total cost of buying one bond is $1006.63.

13. a. If YTM = 8%, price will be $1000.

b. Rate of return = interest + capital gainoriginal price

= = -0.0182 = -1.82%

c. Real return = 1 + nominal interest rate 1 + inflation rate 1

= 1 = .0468 = 4.68%

14. a. With a par value of $1000 and a coupon rate of 8%, the bondholder receives 2 payments of $40 per year, for a total of $80 per year.

b. Assume it is 9%, compounded semi-annually. Per period rate is 9%/2, or 4.5%

Price = 40 annuity factor (4.5%, 18 periods) + 1000/1.04518 = $939.20

c. If the yield to maturity is 7%, compounded semi-annually, the bond will sell above par, specifically for $1,065.95:

Per period rate is 7%/2 = 3.5%

Price = 40 annuity factor(3.5%, 18 periods) + 1000/1.03518 = $1,065.95

15. On your calculator, set N = 30, FV =1000, PMT = 80.

a. Set PV = -900 and compute the interest rate to find that YTM = 8.9708%

b. Set PV = -1000 and compute the interest rate to find that YTM = 8%.

c. Set PV = -1100 and compute the interest rate to find that YTM = 7.1796%

16. On your calculator, set N=60, FV=1000, PMT=40.

a. Set PV = -900 and compute the interest rate to find that the (semiannual) YTM =4.4831%. The bond equivalent yield to maturity is therefore 4.4831 2 = 8.9662%.

b. Set PV = -1000 and compute the interest rate to find that YTM = 4%. The annualized bond equivalent yield to maturity is therefore 4 2= 8%.

c. Set PV = -1100 and compute the interest rate to find that YTM = 3.5917%. The annualized bond equivalent yield to maturity is therefore 3.5917 2 = 7.1834%.

17. In each case we solve this equation for the missing variable:

Price= 1000/(1 + YTM)maturity

Price Maturity (years) YTM

300 30.0 4.095%

300 15.64 8.0%

385.54 10.0 10.0%

Alternatively the problem can be solved using a financial calculator:

Solving the first question: PV = (-)300, PMT = 0, n = 30, FV = 1000, and compute i.

18. PV of perpetuity = coupon payment/rate of return.

PV = C/r = 60/.06 = $1000

If the required rate of return is 10%, the bond sells for:

PV = C/r = 60/.1 = $600

19. Because current yield = .098375, bond price can be solved from: 90/Price = .098375, which implies that price = $914.87. On your calculator, you can now enter: i = 10;

PV = (-)914.87; FV = 1000; PMT = 90, and solve for n to find that n =20 years.

20. Assume that the yield to maturity is a stated rate. Thus the per-period rate is 7%/2 or 3.5%. We must solve the following equation:

PMT annuity factor(3.5%, 18 periods) + 1000/(1.035)18 = $1065.95

To solve, use a calculator to find the PMT that makes the PV of the bond cash flows equal to $1065.95. You should find PMT = $40. The coupon rate is 240/1000 = 8%.

21. a. Since the bonds were issued at par value the coupon rate equaled the yield to maturity at issue. With a yield to maturity of 7% at issue, the coupon rate must be 7%. The semi-annual coupon payment is 0.07/2 $1,000 = $35. With 8 years left to maturity 16 payments of semiannual coupons will be made. Now that the current yield to maturity is 15% the per-period discount rate is .15/2 = .075

Now, the price is

35 Annuity factor(7.5%, 16 periods) + 1000/1.07516 = $634.34

b. The investors pay $634.34 for the bond. They expect to receive the promised coupons plus $800 at maturity. We calculate the yield to maturity based on these expectations:

35 Annuity factor(i, 16 periods) + 800/(1 + i)16 = $634.34

which can be solved on the calculator to show that i =6.4941%. On an annual basis, this 26.4941% or 12.9882%

[Calculator enteries: N = 16; PV = -634.34; FV = 800; PMT = 35]

22. a. Today, at a price of 980 and maturity of 10 years, the bonds yield to maturity is 8.3% (n = 10, PV = (-) 980, PMT = 80, FV = 1000).

In one year, at a price of 1100 and remaining maturity of 9 years, the bonds yield to maturity is 6.4978% (n = 9, PV = (-) 1100, PMT = 80, FV = 1000).

b. Rate of return = = 20.41%

23. Assume the bond pays an annual coupon. The answer is:

PV0 = $908.71 (n = 20, PMT = 80, FV = 1000, i = 9)

PV1 = $832.70 (n = 19, PMT = 80, FV = 1000, i = 10)

Rate of return = = .4391%

If the bond pays coupons semi-annually, the solution becomes more complex. First, decide if the yields are effective annual rates or APRs. Second, make an assumption regarding the rate at which the first (mid-year) coupon payment is reinvested for the second half of the year. Your assumptions will affect the calculated rate of return on the investment. Here is one possible solution:

Assume that the yields are APR and the yield changes from 9% to 10% at the end of the year. The bond prices today and one year from today are:

PV0 = $907.99 (n = 2 20 = 40, PMT = 80/2 = 40, FV = 1000, i = 9/2 = 4.5)

PV1 = $831.32 (n = 2 19 = 38, PMT = 80/2 = 40, FV = 1000, i = 10/2 = 5)

Assuming that the yield doesnt increase to 10% until the end of year, the $40 mid-year coupon payment is reinvested for half a year at 9%, compounded monthly. Its future value at the end of the year is: $40 (1.045) = $41.80 and the rate of return on the bond investment is:

Rate of return = = .56%

24. The price of the bond at the end of the year depends on the interest rate at that time. With one year until maturity, the bond price will be $ 1065/(1 + r).

a. Price = 1065/1.06 = $1004.72

Return = [65 + (1004.72 1000)]/1000 = .06972 = 6.972%

b. Price = 1065/1.08 = $986.11

Return = [65+ (986.11 1000)]/1000 = .05111 = 5.111%

c. Price = 1065/1.10 = $968.18

Return = [65 + (968.18 1000)]/1000 = .0332 = 3.32%

25. The bond price is originally $627.73. (On your calculator, input n = 30, PMT =

40, FV =1000, and i = 7%.) After one year, the maturity of the bond will be 29 years and its price will be $553.66. (On your calculator, input n = 29, PMT = 40, FV = 1000, and i = 8%.) The rate of return is therefore [40 + (553.66 627.73)]/627.73 = .054275 = 5.4275%.

26. a. Annual coupon = .08 1000 = $80.

Total coupons received after 5 years = 5 80 = $400

Total cash flows, after 5 years = 400 + 1000 = $1400

Rate of return = (1400 975 )1/5 1 = .075 = 7.5%

b. Future value of coupons after 5 years

= 80 future value factor(1%, 5 years) = 408.08

Total cash flows, after 5 years = 408.08 + 1000 = $1408.08

Rate of return = (1408.08 975 )1/5 1 = .0763 = 7.63%

c. Future value of coupons after 5 years

= 80 future value factor(8.64%, 5 years) = 475.35

Total cash flows, after 5 years = 475.35 + 1000 = $1475.35

Rate of return = (1475.35 975 )1/5 1 = .0864 = 8.64%

27. To solve for the rate of return using the YTM method, find the discount rate that makes the original price equal to the present value of the bonds cash flows:

975 = 80 annuity factor( YTM, 5 years ) + 1000/(1 + YTM)5

Using the calculator, enter PV = (-)975, n = 5, PMT = 80, FV = 1000 and compute i. You will find i = 8.64%, the same answer we found in 26 (c).

28. a. False. Since a bonds coupon payments and principal are fixed, as interest rates

rise, the present value of the bonds future cash flow falls. Hence, the bond price falls.

Example: Two-year bond 3% coupon, paid annual. Current YTM = 6%

Price = 30 annuity factor(6%, 2) + 1000/(1 + .06)2 = 945

If rate rises to 7%, the new price is:

Price = 30 annuity factor(7%, 2) + 1000/(1 + .07)2 = 927.68

b. False. If the bonds YMT is greater than its coupon rate, the bond must sell at a discount to make up for the lower coupon rate. For an example, see the bond in a. In both cases, the bonds coupon rate of 3% is less than its YTM and the bond sells for less than its $1,000 par value.

c. False. With a higher coupon rate, everything else equal, the bond pays more future cash flow and will sell for a higher price. Consider a bond identical to the one in a. but with a 6% coupon rate. With the YTM equal to 6%, the bond will sell for par value, $1,000. This is greater the $945 price of the otherwise identical bond with a 3% coupon rate.

d. False. Compare the 3% coupon bond in a with the 6% coupon bond in c. When YTM rises from 6% to 7%, the 3% coupon bonds price falls from $945 to $927.68, a -1.8328% decrease (= (927.68 945)/945). The otherwise identical 6% bonds price falls to 981.92 (= 60 annuity factor(7%, 2) + 1000/(1 + .07)2) when the YTM increases to 7%. This is a -1.808% decrease (= 981.92 1000/1000), which is slightly smaller. The prices of bonds with lower coupon rates are more sensitivity to changes in interest rates than bonds with higher coupon rates.

e. False. As interest rates rise, the value of bonds fall. A 10 percent, 5 year Canada bond pays $50 of interest semi-annually (= .10/2 $1,000). If the interest rate is assumed to be compounded semi-annually, the per period rate of 2% (= 4%/2) rises to 2.5% (=5%/2). The bond price changes from:

Price = 50 annuity factor(2%, 25) + 1000/(1 + .02)10 = $1,269.48

to:

Price = 50 annuity factor(2.5%, 25) + 1000/(1 + .025)10 = $1,218.80

The wealth of the investor falls 4% (=$1,218.80 $1,269.48/$1,269.48).

29. Internet: Using historical yield-to-maturity data from Bank of Canada

Tips: Students will need to read the instructions on how to put the data into a spreadsheet. They will want to save the data in CSV format so that it will be easily moved into the spreadsheet. The data will be automatically put into Excel if you access the website with Internet Explorer. Watch that the headings for the columns of data in your spreadsheet arent out of line (we found that the Government of Canada bond yield heading took two columns, displacing the other two headings the data itself were in the correct columns).

Expected results: Long-term Government of Canada bonds have the lowest yield, followed by the yields for the provincial long bonds and then for the corporate bonds. The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time. For an even clearer picture, have the students pick data from 1990 onward.

Time Series: Low/High/Average (Accessed November 22, 2008)

Date Range: 2002/07 2007/06

V122544=Government of Canada benchmark bond yields long-term

V122517=Average weighted bond yields (Scotia Capital Inc.) Provincial long-term

V122518=Average weighted bond yields (Scotia Capital Inc.) All corporates long-term

Date V122544 V122517 V122518 Yield spread (Provincial vs. Canada) Yield Spread (Corporate vs. Canada)

2002/07 5.73 6.13 7.19 0.4 1.46

2002/08 5.58 6 6.99 0.42 1.41

2002/09 5.43 5.83 6.84 0.4 1.41

2002/10 5.63 6.05 7.17 0.42 1.54

2002/11 5.58 5.99 6.96 0.41 1.38

2002/12 5.42 5.81 6.73 0.39 1.31

2003/01 5.49 5.92 6.85 0.43 1.36

2003/02 5.46 5.88 6.81 0.42 1.35

2003/03 5.58 6.02 7.06 0.44 1.48

2003/04 5.41 5.82 6.7 0.41 1.29

2003/05 5.12 5.52 6.35 0.4 1.23

2003/06 5.03 5.41 6.22 0.38 1.19

2003/07 5.4 5.7 6.48 0.3 1.08

2003/08 5.44 5.79 6.54 0.35 1.1

2003/09 5.23 5.57 6.29 0.34 1.06

2003/10 5.38 5.73 6.39 0.35 1.01

2003/11 5.29 5.63 6.27 0.34 0.98

2003/12 5.2 5.52 6.07 0.32 0.87

2004/01 5.23 5.5 6.03 0.27 0.8

2004/02 5.09 5.37 5.87 0.28 0.78

2004/03 5.04 5.38 5.85 0.34 0.81

2004/04 5.31 5.66 6.15 0.35 0.84

2004/05 5.32 5.71 6.25 0.39 0.93

2004/06 5.33 5.78 6.36 0.45 1.03

2004/07 5.29 5.76 6.34 0.47 1.05

2004/08 5.15 5.58 6.17 0.43 1.02

2004/09 5.04 5.44 6.05 0.4 1.01

2004/10 5 5.39 5.99 0.39 0.99

2004/11 4.9 5.29 5.88 0.39 0.98

2004/12 4.92 5.3 5.82 0.38 0.9

2005/01 4.74 5.14 5.66 0.4 0.92

2005/02 4.76 5.11 5.62 0.35 0.86

2005/03 4.77 5.21 5.73 0.44 0.96

2005/04 4.59 5.04 5.58 0.45 0.99

2005/05 4.46 4.89 5.46 0.43 1

2005/06 4.29 4.69 5.2 0.4 0.91

2005/07 4.31 4.72 5.25 0.41 0.94

2005/08 4.12 4.52 5.04 0.4 0.92

2005/09 4.21 4.64 5.15 0.43 0.94

2005/10 4.37 4.82 5.34 0.45 0.97

2005/11 4.18 4.67 5.24 0.49 1.06

2005/12 4.02 4.54 5.09 0.52 1.07

2006/01 4.2 4.71 5.3 0.51 1.1

2006/02 4.15 4.67 5.27 0.52 1.12

2006/03 4.23 4.78 5.37 0.55 1.14

2006/04 4.57 5.07 5.67 0.5 1.1

2006/05 4.5 5.01 5.6 0.51 1.1

2006/06 4.67 5.18 5.81 0.51 1.14

2006/07 4.45 4.96 5.6 0.51 1.15

2006/08 4.2 4.69 5.33 0.49 1.13

2006/09 4.07 4.55 5.18 0.48 1.11

2006/10 4.24 4.7 5.33 0.46 1.09

2006/11 4.02 4.47 5.11 0.45 1.09

2006/12 4.1 4.56 5.18 0.46 1.08

2007/01 4.22 4.66 5.28 0.44 1.06

2007/02 4.09 4.53 5.15 0.44 1.06

2007/03 4.21 4.64 5.27 0.43 1.06

2007/04 4.2 4.64 5.38 0.44 1.18

2007/05 4.39 4.84 5.63 0.45 1.24

2007/06 4.56 5.07 5.82 0.51 1.26

Average Yield Spread of the provincial bonds over the Canada bonds0.42%

Average Yield Spread of the corporate bonds over the Canada bonds 1.09%

We can see that long-term Government of Canada bonds have the lowest yield over time, followed by the yields for long-term provincial long bonds and then for the corporate bonds. The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time. The result makes sense because YTM of long-term Canada bonds has the lowest risk premium of the three, followed by YTM of the provincial bonds. YTM of long-term corporate bonds has larger spreads over Canada bonds because it has much higher default and liquidity risk than Canada Bonds.

30. a. Strips pay no interest, only principal. Assume each bond pays $100 principal on the maturity date

Bond Time to Maturity (Years) YTM = (100/Price)1/time to maturity 1

June 2014 1.583 = (100/96.94)1/1.583 1 = .0198

June 2016 3.583 = (100/91.04)1/3.583 1 = .02655

June 2019 6.583 = (100/80.58)1/6.583 1 = .03334

June 2023 10.583 = (100/65.43)1/10.583 1 = .04090

June 2029 16.583 = (100/45.75)1/16.583 1 = .04828

b. The term structure (yield curve) is upward sloping.

31. Price of bond today

= 40 PVIFA(5%, 3) + 50 PVIFA(5%,3) PVIF(5%,3)

+ 60 PVIFA(5%,3)PVIF(5%,6) + 1000 PVIF(5%, 9)

= 108.93 + 117.62 + 121.93 + 644.61 = $993.09

32. a., b. Price of each bond at different yields to maturity

Maturity of bond

4 years 8 years 30 years

Yield (%)

7 1033.87 1059.71 1124.09

8 1000.00 1000.00 1000.00

9 967.60 944.65 897.26

Difference between prices

(YTM=7% vs YTM=9%) 66.27 115.06 226.83

c. The table shows that prices of longer-term bonds respond with more sensitivity to changes in interest rates. This can be illustrated in a variety of ways. In the table we compare the prices of the bonds at 7 percent and 9 percent yields. When the yield falls from 9 to 7%, the price of the 30-year bond increases $226.83 but the price of the 4-year bond only increases $66.27. Another way to compare the bonds sensitivity to changes in the yield is to look at the percentage change in the prices. For example, with an increase in the yield from 8 to 9%, the price of the 4-year bond falls (967.6/1000) 1, or 3.24% but the 30-year bond price falls (897.26/1000) 1, or 10.27%.

33. The bonds yield to maturity will increase from 7.5%, effective annual interest (EAR) to 7.8%, EAR, when the perceived default risk increases.

6 month interest rate equivalent to 7.5% EAR = (1.075)1/2 1 = .036822

6 month interest rate equivalent to 7.8% EAR = (1.078)1/2 1 = .038268

Price at AA rating = $974.53 (n = 210 = 20, PMT = 70/2 = 35, FV =1000, i = 3.6822)

Price at A rating = $954.90 (n = 210 = 20, PMT = 70/2 = 35, FV =1000, i = 3.8268)

The price falls by $19.63 dollars due to the drop in the bond rating and the increase in the required rate of return.

34. Internet: Credit spreads on corporate bonds

At www.bondsonline.com/Todays_Market/Corporate_Bond_Spreads.php, the spread for a 10 year A2/A-rated bond was reported to be 95, meaning 95 basis points (bp) or .95%. The spread for a 10 year B2/B-rated bond was 405 bp or 4.05%. As of August 1,2011 the yield to maturity on a 10 year US Treasury bond was 2.78%. The estimated required rate of return on each corporate bond is:

Required rate of return = US treasury bond yield to maturity + credit spread

10 year A2/A-rated bond required rate of return = 2.78% + .95% = 3.73%

10 year B2/B-rated bond required rate of return = 2.78% + 4.05% = 6.83%

35. Internet: Canadian corporate bond yields (As of August 1, 2011)

Tips: If you click on Bond Type it will sort the bonds by type, making it easier to find a set of corporate bonds. Alternatively, the data in the table can be copied and pasted into Excel and sorted there. If you sort by type and maturity, it is easier to get a group of corporate bonds with similar maturity dates. At www.dbrs.com, type the company name into the search box. If the company is rated, it will be listed. Click on the name and pick the rating of the subordinated debt (or just the lowest rating). Find a Government of Canada bond (CANADA FEDGOV) with a similar maturity date in the bond list. Calculate the yield spread: corporate bond yield government bond yield and compare the yields and spread with the different ratings.

Heres sample of data taken from the globeinvestor.com bond table and assembled into a table in Excel. Spread in the final column is calculated as the difference between the corporate bond yield and the corresponding Government of Canada bond. Federal government bonds could not be found with exactly the same maturity date for all corporate bonds. So the Federal bond with closest maturity date was chosen. The BBB rate bonds have the largest spread, between 94 and 132 basis points. By contrast the two AA corporate bonds (Bank of Montreal and Bank of Nova Scotia) are substantially smaller, only about 78 to 91 basis points.

CORPORATE BONDS CANADA FEDGOV

DBRS Rating Coupon Rate Coupon Freq. Maturity Price Yield Maturity Yield Spread

CANADIAN NATURAL RESOURCES BBB(high) 4.95 S 06/01/2015 108.73 2.53 06/01/2015 1.59 0.94

METRO INC BBB 4.98 S 10/15/2015 108.82 2.65 06/01/2015 1.59 1.06

ROGERS COMM BBB 5.8 S 05/26/2016 111.76 3.14 06/01/2016 1.82 1.32

BANK OF MTL AA(low) 4.55 S 08/01/2017 107.96 2.95 06/01/2017 2.04 0.91

BANK OF NOVA SCOTIA AA(low) 3.93 S 04/27/2015 105.43 2.37 06/01/2015 1.59 0.78

36. YTM = 4%

Real interest rate = 1 + nominal interest rate = 1.04 1 = .0196, or 1.96%

1 + expected rate of inflation 1.02

Real interest rate nominal interest rate expected inflation rate = 4% 2% = 2%

37. The nominal return is 1060/1000, or 6%. The real return is 1.06/(1 + inflation) 1.

a. 1.06/1.02 1 = .0392 = 3.92%

b. 1.06/1.04 1 = .0192 = 1.92%

c. 1.06/1.06 1 = 0%

d. 1.06/1.08 1 = .0185 = 1.85%

38. The principal value of the bond will increase by the inflation rate, and since the coupon is 4% of the principal, it too will rise along with the general level of prices. The total cash flow provided by the bond will be

1000 (1 + inflation rate) + coupon rate 1000 (1 + inflation rate).

Since the bond is purchased for par value, or $1000, total dollar nominal return is therefore the increase in the principal due to the inflation indexing, plus coupon income:

Income = 1000 inflation rate + coupon rate 1000 (1 + inflation rate)

Finally, the nominal rate of return = income/1000.

a. Nominal return = 20 + 40 1.02 1000 = .0608 Real return = 1.0608 1.02 1 = .04

b. Nominal return = 40 + 40 1.04 1000 = .0816 Real return = 1.0816 1.04 1 = .04

c. Nominal return = 60 + 40 1.06 1000 = .1024 Real return = 1.1024 1.06 1 = .04

d. Nominal return = 80 + 40 1.08 1000 = .1232 Real return = 1.1232 1.08 1 = .04

39. First year income Second year income

a. 401.02=$40.80 1040 x 1.022 = $1082.02

b. 401.04=$41.60 1040 x 1.042 = $1124.86

c. 401.06=$42.40 1040 x 1.062 = $1168.54

d. 401.08=$43.20 1040 x 1.082 = $1213.06

40. a. YTM = 5.76% (n=15, PV = (-)1048, PMT=62.5, FV=1000)

b. YTC = 6.33% (n=10, PV = (-)1048, PMT=62.5, FV=1100)

41. a. Current price = 1,112.38 (n=6, i=4.8%, PMT=70, FV=1000)

b. Current call price = 1,137.35 (n=6, i=4.35%, PMT=70, FV=1000)

42. a. YTM on ABC bond at issue = 5.5% (since sold at par, coupon rate = required rate of

return)

10-year Govt of Canada bond yield at issue

= ABC bond YTM credit spread = 5.5% .25% = 5.25%

Required yield to meet Canada call:

= 10-year Govt of Canada bond yield + .15% = 5.25 + .15% = 5.4%

Call price at issue = 1,007.57 (n=10, i=5.4%, PMT=55, FV=1000)

b. Required yield to call bond = 4.9% + .15% = 5.05%

Call price now, 5 years later = 1,019.46 (n=5, i=5.05%, PMT=55, FV=1000)

c. Based on new interest rates, the bond price is:

Price now, 5 years later = 1,021.65 (n=5, i=5%, PMT=55, FV=1000)

Now the current price is greater than the call price. The company can call bonds and

reduce its cost of debt.

43. The coupon bond will fall from an initial price of $1000 (when yield to maturity = 8%) to a new price of $897.26 when YTM immediately rises to 9%. This is a 10.27% decline in the bond price.

The zero coupon bond will fall from an initial price of 1000 1.0830 = $99.38 to a new

price of 1000 1.0930 = $75.37. This is a price decline of 24.16%, far greater than that of

the coupon bond.

The price of the coupon bond is much less sensitive to the change in yield. It seems to act like a shorter maturity bond. This makes sense: the 8% bond makes many coupon payments, most of which come years before the bonds maturity date. Each payment may be considered to have its own maturity date which suggests that the effective maturity of the bond should be measured as some sort of average of the maturities of all the cash flows paid out by the bond. The zerocoupon bond, by contrast, makes only one payment at the final maturity date.

44. a. Annual after-tax coupon = (1 .35) .08 1000 = $52.

Total coupons received after 5 years = 5 52 = $260

Capital gains tax = .5 .35 (1000 975) = 4.375

After-tax capital gains = 1000 975 4.375 = 20.625

Total cash flows, after 5 years = 260 + 1000 4.375 = $ 1255.625

Rate of return = (1255.625 975 )1/5 1 = .05189, or 5.189%

Note: This can also be answered by first calculating the five-year rate of return and then converting it into a one-year rate of return. This way students can continue to use the coupons + capital gains/original investment approach:

Five-year rate of return =after-tax coupons + after-tax capital gain original investment

= 260 + 20.625 975 = .28782

The one-year rate of return equivalent to the five-year rate of return is:

(1 + .28782) 1/5 1 = .05189, or 5.189%.

b. Future value of coupons after 5 years

= (1 .35) 80 future value factor((1.35)1%, 5 years) = 263.4

Total cash flows, after 5 years = 263.4 + 1000 4.375 = $1259.025

Rate of return = (1259.025 975 )1/5 1 = .0525 = 5.246%

c. Future value of coupons after 5 years

= (1 .35) 80 future value factor((1.35)8.64%, 5 years) = 290.89

Total cash flows, after 5 years = 290.89 + 1000 4.375 = $1286.5

Rate of return = (1286.5 975 )1/5 1 = .057 = 5.7%

45. The new bonds must be priced to have a yield to maturity of 5% + 1.5% = 6.5%. To sell at par, the coupon rate on the new bonds must be set at 6.5%.

46.

Expected results: Students should be able to see some evidence supporting the

difference in the bond ratings of these two companies.

BCE, Inc. provides wire line and wireless communications services, Internet access, data services, and video services in Canada. BCE has DBRS rating of BBB(high)

Agrium, Inc. produces and markets agricultural nutrients, industrial products, and specialty products worldwide. The company has DBRS Issuer credit rating: BBB

BCE: Times interest earned= EBIT interest payment = 5.67

BCE: Debt/Equity = 61.49%

AGU: Times interest earned=EBIT interest payment = 10.31

AGU: Debt/Equity= 39.61%

Agrium has a higher times interest earned ratio of 10.31 while BCEs times interest earned is 5.67. Thus, Agrium has greater ability to make its interest payment than BCE. BCEs indebtedness is higher than AGU because it has higher debt to equity ratio than AGU. However, both ratios contradicts BCEs higher credit rating. When providing a credit rating to a firm, each companys business risk is evaluated in addition to the financial risks. Evidently, Agriums higher business risks resulted in a slightly lower credit rating than BCE even though its times interest earned is higher and indebtedness is lower.

Appendix 6A Solutions

6A.1 a. Equation 6B.4:

(1 + rn)n = (1 + rn1)n1 (1 + fn)

rn = spot interest rate for n year investment

rn-1 = spot interest rate for n-1 year investment

fn = forward interest rate for year n

Rearrange equation 6B.4 to solve for the forward rate:

fn = (1 + rn)n 1

(1 + rn1)n1

Year 2 forward rate = (1+.02)2 -1 = 0.2745 = 2.745%

(1.0126)

Year 3 forward rate = (1+.0247)3 -1 = .03417 = 3.417%

(1.02)2

Year 4 forward rate = (1+.0279)4 -1 = .03756 = 3.756%

(1.0247)3

Year 5 forward rate = (1+.0302)5 -1 = .03945 = 3.945%

(1.0279)4

b. To calculate the bond prices use the yield to maturity that corresponds to the payment date for the bonds:

(i) 5%, 2-Year bond: Annual coupon payment = .05 x 1000 = $50

Price today = = 1,058.61

(ii) 5%, 5-Year bond: Annual coupon payment = .05 x 1000 = $50

Price today = = 1,093.56

(iii) 10%, 5-Year bond: Annual coupon payment = .10 x 1000 = $100

Price today = = 1,325.34

c. Using calculator to calculate yield to maturity:

5%, 2-Year bond: PMT = 50, N= 2, FV = 1000, PV = -1,058.61

YTM (I/Y) = 1.98%

5%, 5-Year bond: PMT = 50, N= 5, FV = 1000, PV = -1,093.56

YTM (I/Y) = 2.96%

10%, 5-Year bond: PMT = 100, N= 5, FV = 1000, PV = -1,325.34

YTM (I/Y) = 2.91%

d. The 10% 5-year bond yield to maturity is slightly lower than the 5% 5-year coupon bond. The purchase price of the 10% coupon 5-year bond is higher than the purchase price of the 5% coupon 5-year bond. Although both bond prices were calculated using the same discount rates you have to pay more to buy the bond that pays higher coupon payments each year. But the bond that pays the higher coupon payment has more payments earlier. Since the yield to maturity is effectively an average of the future interest rates that fact that the 10% bond pays more earlier, when rates were lower, the effective interest rate on the investment (the yield to maturity) is slightly lower for the 10% bond!

6A.2 a. The forward rates are higher each year. If the expectations theory is correct, the forward rates are also the expected future interest rates. The expected future interest rates indicate that interest rates are expected to increase over time!

b. With liquidity-preference, longer-term bonds earn higher return to compensate investors for the liquidity risk. So the spot rate on a longer term bond includes both expectation of future interest rates and also liquidity risk premium. So, unfortunately we cant be sure that the future rates are only due to expectation of future interest rates.

6A.3 a. Since each bond pays zero coupons (strip bonds) the yield to maturity for a bond maturing in n years from today can be calculated as:

YTM = maturity payment 1/n

current price

The 2014 strip bond matures in 1 year from 2013 so n = 1

YTM on 2014 strip bond = (1000/988.53) -1 = .0116 = 1.16%

The 2015 strip bond matures in 2 year from 2013 so n = 2

YTM on 2015 strip bond = (1000/969.15)1/2 -1 = .0158 = 1.58%

The 2016 strip bond matures in 3 year from 2013 so n = 3

YTM on 2015 strip bond = (1000/945.5)1/3 -1 = .0189 = 1.89%

b. The yield to maturity for the 2014 bond, 1.16%, is the one year interest rate as of June 2013, r2013. Now the yield to maturity for the 2015 bond (YTM2015), 1.58%, reflects the 2013 interest rate and also the forward interest rate as of June 2014, f2014.

To calculate the June 2014 forward rate you can use this version of Equation 6B.2:

(1 + YTM2015)2 = (1+ r2013) x (1+ f2014)

So: f2014 = [(1 + YTM2015)2 / (1+ r2013)] -1 = [(1.0158)2 /(1.0116)] 1 = .0200 = 2.00%

The forward interest rate as of June 2015, f2015, is calculated with the yield to maturity on the 2015 bond (YTM2015), and the yield to maturity on the 2016 bond (YTM2016). Buying the 2016 bond is making a 3 year investment. Heres the formula:

(1 + YTM2016)3 = (1 + YTM2015)2 x (1+ f2015)

So: f2015 = [(1 + YTM2016)3 / (1 + YTM2015)2] -1

= [(1.0189)3/(1.0158)2]-1 = .0251 = 2.51%

Note: Because there is no bond maturing in 2017 the forward rate for June 2016 cannot be calculated.

c. Assume that this is a bond with $1000 face value and pays annual coupons in June of each year. So the annual coupon is .05 x 1000 = $50. If you discount at the yield to maturity for bonds maturing at each of payment dates the price of the bond at June 2013 is:

Price today = = 1,090.53

Note: if you use the forward rates to discount the coupons and principal and carry all the decimal places you would get the same answer. Rather than doing the calculation we will show you that the formula for discounting using the forward rates is equivalent to discounting using the yields to maturity.

The present value of the $50 payment at June 2015 was calculated above as 50/(1+ YMT2015)2 = 50/(1.0158)2

It could also be calculated by first discounting by the 2014 forward rate and then by the 2013 interest rate: PV of 2014 interest payment = 50 (1+ f2014) x 1 (1+ r2013) = 50 (1+ f2014) x (1+ r2013)

But (1 + YTM2015)2 = (1+ r2013) x (1+ f2014) =

So: = 50 (1+ f2014) x (1+ r2013)

Also if the payment at the end of 2015 were discounted by the forward rates and the 2013 interest rate the answer would be the same as discounting using the 2015 yield to maturity:

PV of 2015 interest payment = 50+1000 (1+ f2015) x1 (1+ f2014) x1 (1+ r2013) = 50 (1+ f2015)x (1+ f2014) x (1+ r2013)

But (1 + YTM2016)3 = (1 + YTM2015)2 x (1+ f2015)

and (1 + YTM2015)2 = (1+ r2013) x (1+ f2014)

So: (1 + YTM2016)3 = (1+ r2013) x (1+ f2014)x (1+ f2015)

So the price of the bond if all of the cash flows are discounted using all of the forward rates will be exactly the same as discounting using the yield to maturities. You can do the calculations but you must carry of the decimal places to make the numbers exactly the same.

6A.4 Assuming the expectations theory the upward sloping yield curve implies that future annual interest rates will be higher than the current interest rate. If the company only needs money to borrow money for a short period, then it will be cheaper to borrow short term than long term. Now, the other issue is that liquidity premium. So if you borrow long term you must pay the liquidity risk premium. So even if future rates arent much higher than today if liquidity is relevant, borrowing short term again and again can be cheaper than borrowing long term. However, there is one other issue for a company. If they need to borrow for a long period of time but borrow short term, they need to repeatedly negotiate loans, every time the loan matures. Then if they get into financial trouble or there is an unanticipated financial crisis, they might not be able to get the new loan in the future. So, borrowing long term can be less risky but possibly more expensive.

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