Solution Manual For Fundamentals of Corporate Finance 7th Canadian Edition Richard A Brealey

Solution Manual for Fundamentals of Corporate Finance 7th Canadian Edition Richard A.
Breale
Solution Manual for Fundamentals of Corporate

Finance 7th Canadian Edition Richard A. Brealey
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Brealey 7CE
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 bond’s 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]
Please note that financial calculators are programmed considering the concept of cash
inflows and cash outflows, that means if PV has a negative sign (you are paying now),
the PMT and FV will have a positive sign (you are receiving coupon payments and the
maturity price in future) and the vice-versa!
6-1
Copyright © 2020 McGraw-Hill Ryerson Limited
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
firm’s 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 1.389%. To buy the bond, investors pay the
ask price. The investor would pay 103.816 percent of par value. With $1,000 par
value, this means paying $1,038.16 to buy a bond.
9. Coupon payment = interest = .05 × 1000 = 50

Capital gain = 1100 – 1000 = 100
interest + capital gain 50 + 100

Rate of return = purchase price = 1000 = .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 interest + after-tax capital gain

After-tax rate of return = purchase price
35 + 85
= 1000 = .12 = 12%
11. Bond 1
year 1: PMT = 80, FV = 1000, I/Y = 10%, n = 10; Compute PV0 = $877.11
6-2
year 2: PMT = 80, FV = l000, I/Y = 10%, n = 9; Compute PV1 = $884.82
80 + (884.82 - 877.11)
Rate of return = 877.11 = .10 = 10%
Bond 2
year 1: PMT = 120, FV = 1000, I/Y = 10%, n = 10; Compute PV0 = $1122.89
year 2: PMT = 120, FV = l000, I/Y = 10%, n = 9; Compute PV1 =$1115.18
120 + (1115.18 - 1122.89)

Rate of return = 1122.89 = .10 = 10%
Both bonds provide the same rate of return.
12. Accrued interest =

number of days from last coupon to purchase date
Coupon payment × number of days in coupon period
= 22.5 × 122/184= $14.92
Dirty bond price= clean bond price + accrued interest = $990+ $14.92= $1004.92
The quoted clean price is $990. The bond pays semi-annual interest. The last $22.5
coupon was paid on March 1, 2019, and the next coupon will be paid on September 1,
2019. The number of days from the last coupon payment to the purchase date is 122
(from March 1 to July 1) and the total number of days in the coupon period is 184 (from
March 1 to September 1). The accrued interest is $14.92, and the total cost of buying one
bond is $1004.92.
13. a. If YTM = 8%, price will be $1000. This is a bond theorem: if the yield on bond is
equals to coupon rate, the value of bond will be equals to its par value!
interest + capital gain
b. Rate of return = original price
80 +(1000 − 1100 )
= = -0.0182 = -1.82%
1100
1 + nominal interest rate

c. Real return = 1 + inflation rate –1
0.9818
= – 1 = –.0468 = – 4.68%
1.03
6-3
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 ,
I/Y =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, I/Y = 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, I/Y = 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:
First question: PV = -300, PMT = 0, n = 30, FV = 1000, and compute I/Y.
6-4
Second question: PV = -300, PMT = 0, n = 15.64, FV = 1000, and compute I/Y.
Third question: PV = -385.54, PMT = 0, n = 10, FV = 1000, and compute I/Y.
18. Because current yield = .098375, bond price can be solved from: $90/Price = .098375,
which implies that price = $914.87.
On your financial calculator, you can now enter: I/Y = 10;

PV = -914.87; FV = 1000; PMT = 90, and solve for n to find that n =20 years.
19. Given that the yield to maturity of the bond is 7% and number of years remain to
maturity is 9 years, the semi-annual YTM is 7%/2 or 3.5% and number of semi-annuals
to maturity is 18. We can solve the following equation:
PMT × present value annuity factor(3.5%, 18 periods) + 1000/(1.035)18 = $1065.95
To solve, use a financial calculator to find the PMT that makes the PV of the bond cash
flows equal to $1065.95. You should find PMT = $40. This is semiannual coupon
amount; therefore, the annual coupon amount is $80. Therefore, the coupon rate is
80/1000 = 8%.
20. 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/Y =6.4941%.

[Calculator entries: N = 16; PV = -634.34; FV = 800; PMT = 35, compute I/Y]
On an annual basis, this is 2×6.49415% or 12.98830%
21. a. Today, at a price of 980 and maturity of 10 years, the bond’s yield to maturity is
8.30%
Financial calculator: n = 10, PV = -980, PMT = 80, FV = 1000, compute I/Y.
6-5
In one year, at a price of 1100 and remaining maturity of 9 years, the bond’s yield
to maturity is 6.4978%.
Financial calculator: n = 9, PV = -1100, PMT = 80, FV = 1000, compute I/Y.
80 + (1100 − 980 )
b. Rate of return = = 20.41%
980
22. Assume the bond pays an annual coupon. The answer is (provide following inputs in
your financial calculator to get the answer):
PV0 = $908.71 (n = 20, PMT = 80, FV = 1000, I/Y = 9)
PV1 = $832.70 (n = 19, PMT = 80, FV = 1000, I/Y = 10)
80 + 832 .70 − 908 .71

Rate of return = = 0.004393 or 0.4391%
908 .71
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 (provide following inputs in
your financial calculator to get the answer):
PV0 = $907.99
Financial calculator: n = 2 × 20 = 40, PMT = 80/2 = 40, FV = 1000, I/Y = 9/2 = 4.5,
compute PV
PV1 = $831.32
Financial calculator: n = 2 × 19 = 38, PMT = 80/2 = 40, FV = 1000, i = 10/2 = 5,
compute PV
Assuming that the yield doesn’t 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:
41 .80 + 40 + 831 .32 − 907 .99

Rate of return = = 0.005650 or 0.5650%
907 .99
23. The price of the bond at the end of the year depends on the interest rate at that time. With
6-6
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 = .03318 = 3.318%
24. The bond price is originally $627.73. (On your calculator, input n = 30, PMT =
40, FV =1000, and I/Y = 7%; and compute PV.) 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/Y = 8%; and compute PV.) The rate of return is
therefore [40 + (553.66 – 627.73)]/627.73 = –.054275 = –5.4275%.
25. a. Annual coupon = .08 × 1000 = $80.

Total coupons received after 5 years = 5 × 80 = $400
Total cash flows, after 5 years = 400 + 1000 = $1400

1/5
Rate of return = (1400
975 ) – 1 = .075 = 7.5%
b. Future value of coupons after 5 years

= 80 × future value annuity factor (1%, 5 years) = 408.08
Financial calculator inputs: n=5, I/Y=1, PMT=80, PV=0 compute FV
Total cash flows, after 5 years = 408.08 + 1000 = $1408.08

1/5
Rate of return = (1408.08
975 ) – 1 = .0763 = 7.63%
Financial calculator inputs: n=5, PV=-975, FV=1408.08, compute, I/Y
c. Future value of coupons after 5 years

= 80 × future value annuity factor(8.64%, 5 years) = 475.35
Financial calculator inputs: n=5, I/Y=8.64, PMT=80, PV=0, compute FV
Total cash flows, after 5 years = 475.35 + 1000 = $1475.35

1/5
975 ) – 1 = .0864 = 8.64%
Financial calculator inputs: n=5, PV=-975, FV=1475.35, compute, I/Y
6-7
26. 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 bond’s 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/Y.
You will find I/Y = 8.64%, the same answer we found in 25 (c).
27. a. False. Since a bond's coupon payments and principal are fixed, as interest rates
rise, the present value of the bond's 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 bond's 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 bond's 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 part a above with the 6% coupon bond in
c. When YTM rises from 6% to 7%, the 3% coupon bond's 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 sensitive 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%, 2×5) + 1000/(1 + .02)10 = $1,269.48
to:
Price = 50 × annuity factor(2.5%, 2×5) + 1000/(1 + .025)10 = $1,218.80
The wealth of the investor falls 4% (=$1,218.80 - $1,269.48/$1,269.48).
6-8
28. 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 aren’t 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
Yield spread Yield Spread
(Provincial (Corporate vs.
Date V122544 V122517 V122518 vs. Canada) 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
6-9
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 bonds：0.42%
Average Yield Spread of the corporate bonds over the Canada bonds： 1.09%
6-10
Yields Spread
7.5
7
6.5
YTM of long-term
YTM (percent)
6 Corporate Bonds
YTM of long-term
5.5
Provincial Bonds
5 YTM of long-term
Canada Bonds
4.5
4
3.5
07 12 05 10 03 08 01 06 11 04 09 02
02/ 02/ 03/ 03/ 04/ 04/ 05/ 05/ 05/ 06/ 06/ 07/
20 20 20 20 20 20 20 20 20 20 20 20
Date
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.
29. a. Strips pay no interest, only principal.
Bond Time to Maturity YTM Financial calculator

(Years, n) = (1000/Price)1/n- 1 Inputs
June 2020 2 = (1000/969.4)1/2 - 1 = PV= -969.4, FV=1000,
.0157 =1.57% n=2, compute I/Y
June 2022 4 = (1000/910.4)1/4 - 1 = PV= -910.4, FV=1000,
.0237 = 2.37% n=4, compute I/Y
June 2025 7 = (1000/805.8)1/7 - 1 = PV= -805.8, FV=1000,
.0313 = 3.13% n=7, compute I/Y
June 2029 11 = (1000/654.3)1/11 - 1 = PV= -654.3, FV=1000,
.0393 = 3.93% n=11, compute I/Y
June 2035 17 = (1000/457.5)1/17 - 1 = PV= -457.5, FV=1000,
.0471 = 4.71% n=17, compute I/Y
In
b. As maturity period increase, the yield on zero coupon bond increases, therefore, the
6-11
term structure (yield curve) is upward sloping.
30. 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
(Note: PVIFA refers to present value of annuity factor and PVIF refers to present value
interest factor)
OR:
Par value = $1000

Year Cash flows PV @5%
1 40 38.10
2 40 36.28
3 40 34.55
4 50 41.14
5 50 39.18
6 50 37.31
7 60 42.64
8 60 40.61
60+1000 =
9 1060 683.29
Total Present Value 993.09
Using financial calculator:

Inputs:
I/Y =5,
CF: enter cash flows for each period as follows:
CF0=0; CF1=40, CF2 = 40, CF3=40, …. CF9=1060
Compute: NPV. You will get 993.09.
31. 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
6-12
(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%.
32. The bond’s 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 = 2×10 = 20, PMT = 70/2 = 35, FV =1000, i = 3.6822)
Price at A rating = $954.90 (n = 2×10 = 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.
33. Internet: Credit spreads on corporate bonds
Note: While answering this question, visit the websites and use the most recent data. Here,
the answer is provided based on July 24, 2019 data.
On July 24, 2019, at https://fred.stlouisfed.org/categories/33446, the spread for a 10 year

Moody’s Seasoned Baa rated bond yield relative to 10-year Treasury bond was 2.15%,
while the yield on 10-year Treasury bond was 2.05% (from
https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView
.aspx?data=yield). That means, the estimated required rate of return on Baa rated corporate
bond is:
Required rate of return = US treasury bond yield to maturity + credit spread = 2.05% +
2.15% =4.20%
34. YTM = 4%
Real interest rate = 1 + nominal interest rate = 1.04 - 1 = .0196, or 1.96%
6-13
1 + expected rate of inflation 1.02
Real interest rate ≈ nominal interest rate - expected inflation rate = 4% - 2% = 2%
35. 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%
36. 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.
20 + 40 × 1.02 1.0608
a. Nominal return = 1000 = .0608 Real return = 1.02 – 1 = .04
40 + 40 × 1.04 1.0816
b. Nominal return = 1000 = .0816 Real return = 1.04 – 1 = .04
60 + 40 × 1.06 1.1024
c. Nominal return = 1000 = .1024 Real return = 1.06 – 1 = .04
80 + 40 × 1.08 1.1232
d. Nominal return = 1000 = .1232 Real return = 1.08 – 1 = .04
37. First year income Second year income

a. 40x1.02=$40.80 1040 x 1.022 = $1082.02
b. 40x1.04=$41.60 1040 x 1.042 = $1124.86
c. 40x1.06=$42.40 1040 x 1.062 = $1168.54
d. 40x1.08=$43.20 1040 x 1.082 = $1213.06
38. a. YTM = 5.76% (Financial calculator inputs: n=15, PV = -1048, PMT=62.5, FV=1000)
6-14
b. YTC = 6.33% (n=10, PV = -1048, PMT=62.5, FV=1100)
39. a. Current price = 1,112.38 (Inputs: n=6, I/Y=4.8%, PMT=70, FV=1000)
b. Current call price = 1,137.35 (Inputs: n=6, I/Y=4.35%, PMT=70, FV=1000)
40. a. YTM on ABC bond at issue = 5.5% (since sold at par, coupon rate = required rate of
return)
10-year Gov't of Canada bond yield at issue
= ABC bond YTM - credit spread = 5.5% - .25% = 5.25%
Required yield to meet Canada call:

= 10-year Gov't 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.
41. 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.
1000
The zero coupon bond will fall from an initial price of 1.0830 = $99.38 to a new
1000
price of 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 bond’s 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 zero–coupon bond, by
contrast, makes only one payment at the final maturity date.
6-15
42. 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
1/5
1255.625
Rate of return = (
975 ) – 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:
after-tax coupons + after-tax capital gain

Five-year rate of return = 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 annuity factor((1–.35)×1%, 5 years) = 263.4
Total cash flows, after 5 years = 263.4 + 1000 – 4.375 = $1259.025

1/5
975 ) – 1 = .0525 = 5.246%
c. Future value of coupons after 5 years

= (1 – .35) × 80 × future value annuity factor((1–.35)×8.64%, 5 years) = 290.89
Total cash flows, after 5 years = 290.89 + 1000 – 4.375 = $1286.5

1/5
975 ) – 1 = .057 = 5.7%
43. 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%.
6-16
44.
Expected results: Students should be able to see some evidence supporting the
difference in the bond ratings of these two companies. (As of Fiscal, 2018)
BCE, Inc. provides wire line and wireless communications services, Internet access,
data services, and video services in Canada. BCE has S&P rating of BBB+
Cameco Corporation is one of the world’s largest uranium producers. The company has
S&P rating of rating: BBB-
EBIT
BCE: Times interest earned= interest payment = 5.06
BCE: Debt/Equity = 1.2
EBIT
CCO: Times interest earned= interest payment = 1.36
CCO: Debt/Equity= 0.43
BCE has a higher times interest earned ratio of 5.06 while CCO’s times interest earned
is 0.99. Thus, BCE has greater ability to make its interest payment than CCO. However,
BCE’s indebtedness is higher than CCO because it has higher debt to equity ratio than
CCO. However, higher debt/equity ratio contradicts BCE’s higher credit rating. When
providing a credit rating to a firm, each company’s business risk is evaluated in addition
to the financial risks.
Appendix 6A Solutions
6A.1 a. Equation 6A.4:

(1 + rn)n = (1 + rn−1)n−1  (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 6A.4 to solve for the forward rate:

fn = (1 + rn)n -1
(1 + rn−1)n−1
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
6-17
(1.0247)3
(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
50 50+1000
Price today = 1.0126 + = 1,058.61
(1.02)2
(ii) 5%, 5-Year bond: Annual coupon payment = .05 x 1000 = $50
50 50 50 50 50+1000
Price today = 1.0126 + (1.02)2 + + (1.0279)4 + (1.0302)5 = 1,093.56
(1.0247)3
(iii) 10%, 5-Year bond: Annual coupon payment = .10 x 1000 = $100
100 100 100 100 100+1000
Price today = 1.0126 + (1.02)2 + + (1.0279)4 + = 1,325.34
(1.0247)3 (1.0302)5
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%
YTM (I/Y) = 2.96%
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 can’t be sure that the future rates are only due to expectation of
future interest rates.
6-18
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:
1
𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑛
YTM = ( 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑟𝑖𝑐𝑒 ) −1
The 2014 strip bond matures in 1 year from 2013 so n = 1

YTM on 2014 strip bond = (1000/988.53) -1 = .011603 = 1.16%
Financial Calculator: PV=-988.53, N=1, FV=1000, compute I/Y => 1.16%

YTM on 2015 strip bond = (1000/969.15)1/2 -1 = .015791= 1.58%

YTM on 2016 strip bond = (1000/945.5)1/3 -1 = .018856 = 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
6A.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. Here’s the
formula:
(1 + YTM2016)3 = (1 + YTM2015)2 x (1+ f2015)
So: f2015 = [(1 + YTM2016)3 / (1 + YTM2015)2] -1
= [(1.018856)3/(1.015791)2]-1 = .025013 = 2.50%
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
6-19
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:
50 50 50+1000
Price today = 1.0116 + (1.0158)2 + (1.0189)3 = 1,090.53
Please note that we are using four decimal places for rounding the numbers. If
you don’t round the numbers in intermediate calculation, the price will be
$1093.39.
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
50 1
by the 2013 interest rate: PV of 2014 interest payment = (1+ f ) x (1+ r )
2014 2013
50
= (1+ f ) x (1+ r )
2014 2013
But (1 + YTM2015)2 = (1+ r2013) x (1+ f2014)

50 50
So: = (1+ f ) x (1+ r )
(1.0158)2 2014 2013
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:
50+1000 1 1
PV of 2015 interest payment = (1+ f ) x (1+ f ) x (1+ r ) =
2015 2014 2013
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
6-20
Solution Manual for Fundamentals of Corporate Finance 7th Canadian Edition Richard A. Breale
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 aren’t 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.
6-21
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1. a. The coupon payments are fixed at $60 per year.

d. Current yield = coupon payment/bond price. As coupon payment remains the

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.

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

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

interest + capital gain 50 + 100

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

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

after-tax interest + after-tax capital gain

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

120 + (1115.18 - 1122.89)

Both bonds provide the same rate of return.

12. Accrued interest =

1 + nominal interest rate

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

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

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

Price= 1000/(1 + YTM)maturity

Price Maturity (years) YTM

Alternatively, the problem can be solved using a financial calculator:

On your financial calculator, you can now enter: I/Y = 10;

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

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

which can be solved on the calculator to show that I/Y =6.4941%.

Financial calculator: n = 10, PV = -980, PMT = 80, FV = 1000, compute I/Y.

Financial calculator: n = 9, PV = -1100, PMT = 80, FV = 1000, compute I/Y.

PV0 = $908.71 (n = 20, PMT = 80, FV = 1000, I/Y = 9)

PV1 = $832.70 (n = 19, PMT = 80, FV = 1000, I/Y = 10)

80 + 832 .70 − 908 .71

41 .80 + 40 + 831 .32 − 907 .99

a. Price = 1065/1.06 = $1004.72

b. Price = 1065/1.08 = $986.11

c. Price = 1065/1.10 = $968.18

25. a. Annual coupon = .08 × 1000 = $80.

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

b. Future value of coupons after 5 years

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

c. Future value of coupons after 5 years

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

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

Time Series: Low/High/Average

29. a. Strips pay no interest, only principal.

Bond Time to Maturity YTM Financial calculator

30. Price of bond today

Par value = $1000

Using financial calculator:

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

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

33. Internet: Credit spreads on corporate bonds

On July 24, 2019, at https://fred.stlouisfed.org/categories/33446, the spread for a 10 year

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