Chapter 6

Q.1. What are premium, discount, and par bonds?

Ans.1. Premium (par, discount) bonds are bonds that sell for more than (the
same as, less than) their face or par value.

Q.2. In the United States, what is the normal face value for corporate and
U.S. government bonds? How are coupons calculated? How often are
coupons paid?

Ans.2. The face value is normally SR1,000 per bond. The coupon is expressed
as a percentage of face value (the coupon rate), so the annual riyal coupon is
calculated by multiplying the coupon rate by SR1,000. Coupons are normally
paid semi-annually; the semi-annual coupon is equal to the annual coupon
divided by two.

Q.3. What are the coupon rate and current yield on a bond? What happens to
these if a bond’s price rises?

Ans.3. The coupon rate is the annual riyal coupon expressed as a percentage of
face value. The current yield is the annual riyal coupon divided by the current
price. If a bond’s price rises, the coupon rate won’t change, but the current
yield will fall.

Q.4. What is interest rate risk? What are the roles of a bond’s coupon and
maturity in determining its level of interest rate risk?

Ans.4. Interest rate risk refers to the fact that bond prices fluctuate as interest
rates change. Lower coupon and longer maturity bonds have greater interest
rate risk.

1
Q.5. For a premium bond, which is greater, the coupon rate or the yield to
maturity? Why? For a discount bond? Why?

Ans.5. For a premium bond, the coupon rate is higher than the yield. The
reason is simply that the bonds sell at a premium because it offers a coupon
rate that is high relative to current market required yields. The reverse is true
for a discount bond: it sells at a discount because its coupon rate is too low.

Q.6. What is the difference between a bond’s promised yield and its realized
yield? Which is more relevant? When we calculate a bond’s yield to maturity,
which of these are we calculating?

Ans.6. A bond’s promised yield is an indicator of what an investor can expect

to earn if (1) all of the bond’s promised payments are made and (2) market
conditions do not change. The realized yield is the actual, after-the-fact return
the investor receives. The realized yield is more relevant, of course, but it is not
knowable ahead of time. A bond’s calculated yield to maturity is the promised
yield.

Q.7. Is the yield to maturity (YTM) on a bond the same thing as the required
return? Is YTM the same thing as the coupon rate? Suppose that today a 10
percent coupon bond sells at par. Two years from now, the required return
on the same bond is 8 percent. What is the coupon rate on the bond now?
The YTM?

Ans.7. The yield to maturity is the required rate of return on a bond expressed
as a nominal annual interest rate. For noncallable bonds, the yield to maturity
and required rate of return are interchangeable terms. Unlike YTM and
required return, the coupon rate is not used as the interest rate in bond cash
flow valuation, but is a fixed percentage of par over the life of the bond used to

2
set the coupon payment amount. For the example given, the coupon rate on
the bond is still 10 percent, and the YTM is 8 percent.

Q.8. Suppose you buy a 9 percent coupon, 15-year bond today when it’s first
issued. If interest rates suddenly rise to 15 percent, what happens to the
value of your bond? Why?

Ans.8. Since the yield increased, the price of the bond will decrease. This can
be explained in two ways. First, any new bonds will have a 15 percent coupon
rate in order to sell at par since that is the market interest rate. Investors will
pay less for a 9 percent coupon bond since they can buy a bond with a 15
percent coupon rate. Second, the decrease in price is a function of the time
value of money. The price of the bond is the present value of the coupon
payments plus the present value of the principal. In any present value
calculation, the present value declines when the interest rate increases.

Q.9. (a) What is the relationship between the price of a bond and its YTM? (b)
Explain why some bonds sell at a premium to par value, and other bonds sell
at a discount. What do you know about the relationship between the coupon
rate and the YTM for premium bonds? What about discount bonds? For
bonds selling at par value? (c) What is the relationship between the current
yield and YTM for premium bonds? For discount bonds? For bonds selling at
par value?

Ans.9. a. Bond price is the present value term when valuing the cash flows
from a bond; YTM is the interest rate used in valuing the cash flows from a
bond. They have an inverse relationship.

b. If the coupon rate is higher than the required return on a bond, the bond
will sell at a premium, since it provides periodic income in the form of coupon

3
payments in excess of that required by investors on other similar bonds. If the
coupon rate is lower than the required return on a bond, the bond will sell at a
discount, since it provides insufficient coupon payments compared to that
required by investors on other similar bonds. For premium bonds, the coupon
rate exceeds the YTM; and for discount bonds, the YTM exceeds the coupon
rate. For bonds selling at par, the YTM is equal to the coupon rate.

c. Current yield is defined as the annual coupon payment divided by the

current bond price. For premium bonds, the current yield exceeds the YTM; for
discount bonds the current yield is less than the YTM; and for bonds selling at
par value, the current yield is equal to the YTM. In all cases, the current yield
plus the expected one-period capital gains yield of the bond must be equal to
the required return.

Q.10. For callable bonds, the financial press generally reports either the yield
to maturity or the yield to call. Often yield to call is reported for premium
bonds, and yield to maturity is reported for discount bonds. What is the
reasoning behind this convention?

Ans.10. A premium bond is one with a relatively high coupon, and, in

particular, a coupon that is higher than current market yields. These are
precisely the bonds that the issuer would like to call, so a yield to call is
probably a better indicator of what is likely to happen than the yield to
maturity (the opposite is true for discount bonds). It is also the case that the
yield to call is likely to be lower than the yield to maturity for a premium bond,
but this can depend on the call price. A better convention would be to report
the yield to maturity or yield to call, whichever is smaller.

4
Multiple Choice Questions

1. The yield to maturity on a bond is:

a. below the coupon rate when the bond sells at a discount and above the
coupon rate when the bond sells at a premium
b. the interest rate that makes the present value of the payments equal to the
bond price
c. based on the assumption that all future payments received are reinvested at
the coupon rate
d. based on the assumption that all future payments received are reinvested
at future market rates

2. In which one of the following cases is the bond selling at a discount?

a. coupon rate is greater than current yield, which is greater than yield-to-
maturity
b. coupon rate, current yield and yield-to-maturity are all the same
c. coupon rate is less than current yield, which is less than yield-to-maturity
d. coupon rate is less than current yield, which is greater than yield-to-maturity

3. When are yield-to-maturity and current yield on a bond equal?

a. when market interest rates begin to level off
b. if the bond sells at a price in excess of its par value
c. when the expected holding period is greater than one year
d. if the coupon and market interest rate are equal

4. Consider a five-year bond with a 10 percent coupon that is presently trading

at a yield-to-maturity of 8 percent. If market interest rates do not change, one
year from now the price of this bond
a. will be higher
b. will be lower
c. will be the same
d. cannot be determined

5. Using semi-annual compounding, what would the price of a 15-year, zero

coupon bond that has a par value of SAR1,000 and a required return of 8
percent be?
a. SAR308
b. SAR315
c. SAR464
d. SAR555
5
6. Another term for bond duration is:
a. actual maturity
b. effective maturity
c. calculated maturity
d. near-term maturity

7. Which statement is true for the Macaulay duration of a zero-coupon bond?

a. it is equal to the bond’s maturity in years
b. it is equal to one-half the bond’s maturity in years
c.it is equal to the bond’s maturity in years divided by its yield to maturity
d. it cannot be calculated because of the lack of coupons

8. Which one of the following bonds has the shortest duration?

a. zero coupon, 10-year maturity
b. zero coupon, 13-year maturity
c. 8 percent coupon, 10-year maturity
d. 8 percent coupon, 13-year maturity

9. Identify the bond that has the longest duration (no calculations necessary).
a. 20-year maturity with an 8 percent coupon
b. 20-year maturity with a 12 percent coupon
c. 15-year maturity with a 0 percent coupon
d. 10-year maturity with a 15 percent coupon

10. Which bond has the longest duration?

a. 8-year maturity, 6 percent coupon
b. 8-year maturity, 11 percent coupon
c. 15-year maturity, 6 percent coupon
d. 15-year maturity, 11 percent coupon

11. The duration of a bond normally increases with an increase in:

a. term-to-maturity
b. yield-to-maturity
c. coupon rate
d. all of the above

12. When interest rates decline, what happens to the duration of a 30-year
bond selling at a premium?
a. it increases
b. it decreases

6
c. it remains the same
d. it increases at first, then declines

13. Mary just purchased a bond which pays SAR60 a year in interest. What is
this SAR60 called? 
A. coupon
B. face value
C. discount
D. call premium
E. yield

14. Bert owns a bond that will pay him SAR75 each year in interest plus a
SAR1,000 principal payment at maturity. What is the SAR1,000 called? 
A. coupon
B. face value
C. discount
D. yield
E. dirty price

15. A bond's coupon rate is equal to the annual interest divided by which one
of the following? 
A. call price
B. current price
C. face value
D. clean price
E. dirty price

16. The specified date on which the principal amount of a bond is payable is

referred to as which one of the following? 
A. coupon date
B. yield date
C. maturity
D. dirty date
E. clean date

7
17. The current yield is defined as the annual interest on a bond divided by
which one of the following? 
A. coupon
B. face value
C. market price
D. call price
E. dirty price

18. A bond that is payable to whomever has physical possession of the bond is
said to be in: 
A. new-issue condition.
B. registered form.
C. bearer form.
D. debenture status.
E. collateral status.

19. The Leeward Company just issued 15-year, 8 percent, unsecured bonds at

par. These bonds fit the definition of which one of the following terms? 
A. note
B. discounted
C. zero-coupon
D. callable
E. debenture

20. A bond that can be paid off early at the issuer's discretion is referred to as
being which one of the following? 
A. zero coupon
B. callable
C. senior
D. collateralized
E. unsecured

21. A SAR1,000 face value bond can be redeemed early at the issuer's
discretion for SAR1,030, plus any accrued interest. The additional SAR30 is
called which one of the following? 
A. dirty price
B. redemption value
C. call premium
D. original-issue discount
E. redemption discount

8
22. A call-protected bond is a bond that: 
A. is guaranteed to be called.
B. can never be called.
C. is currently being called.
D. is callable at any time.
E. cannot be called during a certain period of time.

9
23. A bond that has only one payment, which occurs at maturity, defines which
one of the following? 
A. debenture
B. callable
C. floating-rate
D. junk
E. zero coupon

24. You want to buy a bond from a dealer. Which one of the following prices
will you pay? 
A. call price
B. auction price
C. bid price
D. asked price
E. bid-ask spread

25. The difference between the price that a dealer is willing to pay and the
price at which he or she will sell is called the: 
A. equilibrium.
B. premium.
C. discount.
D. call price.
E. spread.

26. A bond is quoted at a price of SAR989. This price is referred to as which one
of the following? 
A. call price
B. face value
C. clean price
D. dirty price
E. wholesale price

27. Real rates are defined as nominal rates that have been adjusted for which
of the following? 
A. inflation
B. default risk
C. accrued interest
D. interest rate risk
E. both inflation and interest rate risk

10
28. The pure time value of money is known as the: 
A. liquidity effect.
B. Fisher effect.
C. term structure of interest rates.
D. inflation factor.
E. interest rate factor.
29. Callable bonds generally:
A. grant the bondholder the option to call the bond any time after the
deferment period.
B. are callable at par as soon as the call-protection period ends.
C. are called when market interest rates increase.
D. are called within the first three years after issuance.
E. have a sinking fund provision.

30. Bonds issued by the U.S. government: 

A. are considered to be free of interest rate risk.
B. generally have higher coupons than those issued by an individual state.
C. are considered to be free of default risk.
D. pay interest that is exempt from federal income taxes.
E. are called "munis".

31. Treasury bonds are: 

A. issued by any governmental agency in the U.S.
B. issued only on the first day of each fiscal year by the U.S. Department of
Treasury.
C. bonds that offer the best tax benefits of any bonds currently available.
D. generally issued as semi-annual coupon bonds.
E. totally risk-free.

 32. Municipal bonds: 
A. are totally risk-free.
B. generally have higher coupon rates than corporate bonds.
C. pay interest that is federally tax-free.
D. are rarely callable.
E. are free of default-risk.

11
33. A zero coupon bond: 
A. is sold at a large premium.
B. pays interest that is tax deductible to the issuer when paid.
C. can only be issued by the U.S. Treasury.
D. has more interest rate risk than a comparable coupon bond.
E. provides no taxable income to the bondholder until the bond matures.

Questions for practice

Example.1. Suppose a SR1,000 par value bond pays semi-annual coupons of

SR40. The annual coupon is then SR80, and, stated as a percentage of par
value, the bond’s coupon rate is:

Ans.1. Coupon rate = Annual coupon/Par value

SR80/SR1,000 = 8%.

Example.2. Suppose a SR1,000 par value bond paying an SR80 annual coupon
has a price of SR1,032.25. The current yield is:

Ans.2. Current yield = Annual coupon/Bond Price

SR80/SR1,032.25 = 7.75%.

Similarly, a price of SR969.75 implies a current yield of SR80/SR969.75 = 8.25%.

Notice that whenever there is a change in the bond’s price, the coupon rate
remains constant. However, a bond’s current yield is inversely related to its
price, and it changes whenever the bond’s price changes.

Example.3. Suppose a bond has a SR1,000 face value, 20 years to maturity, an

8 percent coupon rate, and a yield of 9 percent. What’s the price? Using the
straight bond pricing formula, the price of this bond is calculated as follows:

12
Ans.3. Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M
where: C = Annual coupon, the sum of two semi-annual coupons
FV = Face Value
M = Maturity in years
YTM = Yield to maturity

1. Present value of semi-annual coupons:

= SR80/.09 {1-1/(1.045)40} = SR736.06337

2. Present value of SR1,000 principal:

= 1000/(1.045)40 = SR171.92871
The price of the bond is the sum of the present values of coupons and
principal:
Bond price = SR736.06 + SR171.93 = SR907.99
So, this bond sells for SR907.99.

Example.4. Suppose a bond has 20 years to maturity and a coupon rate of 8

percent. The bond’s yield to maturity is 7 percent. What’s the price?

Ans.4. In this case, the coupon rate is 8 percent and the face value is SR1,000,
so the annual coupon is SR80. The bond’s price is calculated as follows:

Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M

1. Present value of semi-annual coupons:

= SR80/.07 {1-1/(1.035)40} = SR854.20289

2. Present value of SR1,000 principal:

= 1000/(1.035)40 = SR252.57247

The bond’s price is the sum of coupon and principal present values:
Bond price = SR854.20 + SR252.57 = SR1,106.77
This bond sells for SR1,106.77.

Example.5. Consider two bonds, both with eight years to maturity and a 7
percent coupon. One bond has a yield to maturity of 5 percent while the other
has a yield to maturity of 9 percent. Which of these bonds is selling at a

13
premium and which is selling at a discount? Verify your answer by calculating
each bond’s price.

Ans.5.
For the bond with a 9 percent yield to maturity, the coupon rate of 7 percent is
less than the yield, indicating a discount bond. The bond’s price is calculated as
follows:
Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M
= SR70/.09 {1-1/(1.045)16} + 1000/(1.045)16 = SR887.66
For the bond with a 5 percent yield to maturity, the coupon rate of 7 percent is
greater than the yield, indicating a premium bond. The bond’s price is
calculated as follows:
= SR70/.05 {1-1/(1.025)16} + 1000/(1.025)16 = SR1,130.55
Example.6. Consider two bonds, both with a 9 percent coupon rate and the
same yield to maturity of 7 percent, but with different maturities of 5 and 10
years. Which has the higher price? Verify your answer by calculating the prices.

Ans.6.
First, because both bonds have a 9 percent coupon and a 7 percent yield, both
bonds sell at a premium. Based on what we know, the one with the longer
maturity will have a higher price. We can check these conclusions by
calculating the prices as follows:
Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M
5-year maturity premium bond price
= SR90/.07 {1-1/(1.035)10} + 1000/(1.035)10 = SR1,083.17
10-year maturity premium bond price:
= SR90/.07 {1-1/(1.035)20} + 1000/(1.035)20 = SR1,142.12

14
Notice that the longer maturity premium bond has a higher price, as we
predicted.
Example.7. Consider two bonds, both with a 9 percent coupon rate and the
same yield to maturity of 11 percent, but with different maturities of 5 and 10
years. Which has the higher price? Verify your answer by calculating the prices.
These are both discount bonds. (Why?) The one with the shorter maturity will
have a higher price.
Ans.7. To check, the prices can be calculated as follows:
Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M
5-year maturity discount bond price:
= SR90/.11 {1-1/(1.055)10} + 1000/(1.055)10 = SR924.62
10-year maturity discount bond price:
= SR90/.11 {1-1/(1.055)20} + 1000/(1.055)20 = SR880.50
In this case, the shorter maturity discount bond has the higher price.
Example.8. Suppose we have a 6 percent bond with 10 years to maturity. Its
price is 90, meaning 90 percent of face value. Assuming a SR1,000 face value,
the price is SR900 and the coupon is SR60 per year. What’s the yield?

Ans.8. To find out, all we can do is try different yields until we come across the
one that produces a price of SR900. However, we can speed things up quite a
bit by making an educated guess using what we know about bond prices and
yields. We know the yield on this bond is greater than its 6 percent coupon
rate because it is a discount bond. So let’s first try 8 percent in the straight
bond pricing formula:
Bond price = C/YTM {1- 1/(1+ YTM/2)2M } + FV/(1+ YTM/2)2M
= SR60/.08 {1-1/(1.04)20} + 1000/(1.04)20 = SR864.10
The price with an 8 percent yield is SR864.10, which is somewhat less than the
SR900 price, but not too far off.
15
To finish, we need to ask whether the 8 percent we used was too high or too
low. We know that the higher the yield, the lower is the price, thus 8 percent is
a little too high. So let’s try 7.5 percent:
= SR60/.075 {1-1/(1.0375)20} + 1000/(1.0375)20 = SR895.78
Now we’re very close. We’re still a little too high on the yield (since the price is
a little low). If you try 7.4 percent, you’ll see that the resulting price is
SR902.29, so the yield is between 7.4 and 7.5 percent (it’s actually 7.435
percent).
Example.9. Suppose a 20-year bond has a coupon of 8 percent, a price of 98,
and is callable in 10 years. The call price is 105. What are its yield to maturity
and yield to call?
Ans.9. Callable Bond price = C/YTC {1- 1/(1+ YTC/2)2T } + CP/(1+ YTC/2)2T
where: C = Constant annual coupon
CP = Call price of the bond
T =Time in years until earliest possible call date
YTC = Yield to call assuming semi-annual coupons

we know the yield to maturity is slightly bigger than the coupon rate. (Why?)
After some calculation, we find it to be 8.2 percent.

To find the bond’s yield to call, we pretend it has a face value of 105 instead of
100 (SR1,050 versus SR1,000) and will mature in 10 years. With these two
changes, the procedure is exactly the same. We can try 8.5 percent, for
example:
= S80/.085 {1-1/(1.0425)20} + 1050/(1.0425)20 = SR988.51
Because $988.51 is a little too high, the yield to call is slightly bigger than 8.5
percent. If we try 8.6, we find that the price is $981.83, so the yield to call is
about 8.6 percent (it’s 8.6276 percent).

16
Example.10. Suppose a bond has a Macaulay duration of six years, and its
yield decreases from 10 percent to 9.5 percent. Calculate the resulting
percentage change in the price of the bond.

Ans.10.
Percentage change in bond price≈ - Duration * Change in YTM/ (1 + YTM/2)
-6*(.095-.10)/1.05 = 2.86%

Example.11. A bond has a Macaulay duration of 11 years, and its yield

increases from 8 percent to 8.5 percent. What will happen to the price of the
bond?
Ans.11.
The resulting percentage change in the price of the bond can be calculated as
follows:
Percentage change in bond price≈ - Duration * Change in YTM/ (1 + YTM/2)
-11*(.085-.08)/1.04 = - 5.29%

17