Interest Rates and Bond Valuation
Chapter 6
Sections 6.1 to 6.3, 6.5
Omit: Section 6.4
2. VALUATION OF BONDS
Bond Markets
A bond is a long-term debt instrument (a loan contract)
Issuers such as governments and corporations sell bonds to raise money
BOND ISSUANCE, t=0
$PRICE
BUYER
BOND
BOND
ISSUER
3
BOND CASH FLOWS, t=1,…,T
BOND ISSUANCE, t=1,…,T
$COUPONS
BUYER
BOND
ISSUER
4
1,000
50
Coupon = 50
50
50
50
50
50
50
50
50
50
PMT = 50
Face Value or Future value = 1,000
5
PV =
Price of Bond
Coupon
Face Value
6
TERMINOLOGY
The price of a bond is the present value of all future coupon payments and the
final principal payment
The maturity of a bond is the length of time during which the owner will receive
interest payments on the investment.
The coupon rate determines the dollar interest that the buyer of the bond
receives annually from the issuer.
The coupon rate is expressed as a percentage of the face value (the principal
amount) of the bond.
Current Yield (CY) is defined as the ratio of the annual coupon as percentage of
current market price.
Current market Price = 900
Coupon = 90
Face Value = 1,000
Coupon rate = 90/1,000 = 9%
Current Yield = 90/900 = 10%
7
Current Yield vs Yield to maturity (YTM)
Current Yield (CY) is defined as the ratio of the annual coupon as percentage of
current market price.
Yield or Yield to maturity (YTM) is the overall interest rate earned by an
investor who buys a bond at the market price and holds it until maturity.
Mathematically, YTM is the discount rate at which the sum of all future cash
flows (from coupons and principal repayment) equals the price of the bond.
K = YTM = Discount rate = Market Interest rate
8
BOND CASH FLOWS
0
1
P0
C
2
C
3
T
C
FV + C
WHERE k = YIELD TO MATURITY ON THE BOND
Same formula
but in a different
way
9
Here’s a bond issued by the Yukon Steel Company. The face value of the bond
is $1000, and it has 8 years to maturity.
Other relevant info is summarized below.
Coupon rate
= 10%
Maturity
= 8 years
Yield
= 7%
What is the price of this bond?
FV = 1,000
PMT = 10% of 1,000 = 100
I/Y = 7%
P/Y= 1
C/Y = 1
PV = ??
CPT PV = 1,179.14
N=8
END
The bond is selling at a premium. 1,179.14 – 1,000 = 179.14
10
Now suppose interest rates go up and the yield on bonds goes up to 11%.
The face value of the bond is $1000, and it has 8 years to maturity.
Other relevant info is summarized below.
Coupon rate
= 10%
Maturity
= 8 years
Yield
= 11%
What effect will it have on the price of Yukon Steel bonds??
FV = 1,000
I/Y = 11%
PV = ??
CPT PV = 948.54
PMT = 10% of 1,000 = 100
P/Y= 1
C/Y = 1
N=8
END
The price fell because Yield (Competitive interest rate increased)
The bond is now selling at a discount. 1,000 – 948.54 = 51.46
11
Coupon rate = 10 % > YTM= 7%
Price 1,179.14 > Face value=1000
Rule : whenever CR > YTM Bond is issued at a premium
Coupon rate = 10 % < YTM= 11%
Price 948.54 < Face value= 1,000
Rule : whenever CR < YTM Bond is issued at a discount
Coupon rate = 10 % = YTM= 10% (Try it yourself)
Price 1,000 < Face value= 1,000
Rule : whenever CR = YTM Bond is issued at Par or face value
Par means Face Value
12
LO1
Bonds… Terminology Recap…
• Bond
• Par value (face value)
• Coupon rate
• Coupon payment
• Maturity date
• Yield or Yield to maturity
13
For Canadian Bonds
▪ Semi-annual pay bonds form the overwhelming standard
for borrowing in the Canadian capital market.
▪ Bonds generally pay a fixed annual coupon rate of interest
in two equal semi-annual payments. Most bonds have a
fixed maturity date.
14
LO1
The Bond Indenture
• Contract between the bond issuer and the bondholders. It
includes:
•
•
•
•
•
•
The basic terms of the bonds
The total amount of bonds issued
A description of property used as security, if applicable
Sinking fund provisions
Call provisions
Details of protective covenants
15
Registered versus Bearer
Registered Bond:
A registered bond has its owner's name and contact information
recorded with the issuing entity, ensuring coupon payments are
correctly distributed.
Bearer bonds:
Which don't record the owner's info, are the opposite of registered
bonds.
Bearer bonds are gone if lost
Holders of bearer bonds will not be notified of changes in corporate policies
However, bearer bonds have lower transaction costs
16
Different Types of Bonds
• Mortgage bond: A mortgage bond is secured by a mortgage, or a pool of
mortgages, that are typically backed by real estate holdings and real property.
• Debenture: A debenture is a type of bond or other debt instrument that is not
secured by collateral
• Senior Debt: A loan and obligations which are prioritized for repayment in the
case of bankruptcy
• Subordinate (“Junior Debt”): A loan secured by collateral (assets) that are to be
paid if a company goes into default—but only after higher-priority debts (senior
debts) are settled
17
Different Types of Bonds
• Callable: Also known as a redeemable bond, is a bond that the issuer may redeem before it
reaches the stated maturity date. A callable bond allows the issuing company to pay off
their debt early.
• Extendable: A long-term debt security that includes an option which allows the
bondholder to extend its initial maturity to a later date.
• Retractable: Also known as variable-rate demand note, is a debt security that features a
put option which allows the holder to force the issuer to redeem the bond before
maturity at it's face value.
• Convertible: a type of debt security that provides an investor with a right or an obligation
to exchange the bond for a predetermined number of shares in the issuing company
18
Different Types of Bonds
• Zero coupon: A bond that pays no coupon
• Floating rate: A bond that has a variable interest rate
• Income Bonds: a bond that pays interest only if the issuing entity has
earned income.
19
• Bond Covenant: A bond covenant sets out certain activities that must
be undertaken, or what activities are forbidden.
• Negative (Restrictive ) Covenants: Activities that are forbidden
• Positive (Affirmative ) Covenants: Activities that must be undertaken
• Repayment Provisions: A contract clause that says how money
borrowed must be paid back.
• Sinking Fund: A type of fund that is created and set up purposely for repaying
debt. The owner of the account sets aside a certain amount of money regularly
and uses it only for a specific purpose.
20
• DBRS, S&P, Moody’s (These rating companies that charge for ratings)
• Rating reflects ability to pay or Likelihood of timely repayment
• BBB or better for investment grade
21
• Firms provide info, so only as good as what is disclosed.
• Bond price will fall when downgraded and rise when upgraded.
• Unrated debt (a.k.a. Junk Bonds):
• Junk bonds are bonds that carry a higher risk of default than
most bonds issued by corporations and governments.
• Junk bonds are riskier. They will be rated BB or lower by Standard &
Poor's and Ba or lower by Moody's. These lower-rated bonds pay a
higher yield to investors. Their buyers are getting a bigger reward for
taking a greater risk.
22
Good to know
• The price of a bond is more sensitive to changes in interest rates when
the YTM level is low.
• The price of a bond is more sensitive to changes in interest rates when
the maturity is high. (% change in price will be higher than for short
term)
• However, the higher the coupon rate the less sensitive the bond price
will be to changes in interest rates.
COMM308 - Tutorial Session #4
23
• What is the face value of a bond with coupon payments of $50 every
six months, a current value of $1,172.41, a remaining term of 10
years, and a risk-premium of 5%? The current risk-free rate is 3%.
Choose the closest answer. It’s a semi-annual bond.
PV = 1,174.41
K = Rf + Rp ,
PMT = 50
N= 2 x 10 = 20
Rf = Risk Free Rate, Rp = Risk premium
I/Y = (3 + 5)% = 8%
CPT FV = 1,080
P/Y = 2
C/Y = 2
Alternate Solution:
PV = 1,174.41
PMT = 50 N= 2 x 10 = 20
I/Y = (8/2)% = 4% (QR semi-annually to ESR)
CPT FV = 1,080
END
END
24
What is the face value of a bond with coupon payments of $60 every six
months, a current value of $1,300, a remaining term of 10 years, and a riskpremium of 5%? The current risk-free rate is 3%. Choose the closest answer.
PV =
I/Y =
CPT FV =
PMT = N=
P/Y =
C/Y =
BEG or END
Alternate Solution:
PV = 1,174.41 PMT = 50
N= 2 x 10 = 20
I/Y =
(QR semi-annually to ESR)
CPT FV =
BEG or END
25
What is the face value of a bond with coupon payments of $60 every six
months, a current value of $1,300, a remaining term of 10 years, and a riskpremium of 5%? The current risk-free rate is 3%. Choose the closest answer.
PV = 1,300
K = Rf + Rp ,
PMT = 60
N= 2 x 10 = 20
Rf = Risk Free Rate, Rp = Risk premium
I/Y = (3 + 5)% = 8%
CPT FV = 1,061.78
P/Y = 2 C/Y = 2
Alternate Solution:
PV = 1,174.41 PMT = 60
N= 2 x 10 = 20
I/Y = (8/2)% = 4%
(QR semi-annually to ESR)
CPT FV = 1,061.78
END
END
26
• Dominion Groceries Inc. has a 9-year, 6% annual coupon bond
outstanding with a $1,000 par value. Fresh Produce Inc. has a 10year, 5% annual coupon bond with a $1,000 par value. Both bonds
currently have a yield to maturity of 5.5%. Which of the following
statements is correct if the market yield increases to 5.75%?
• a. Both bonds would decrease in value by 2.50%.
• b. The Dominion bond will decrease in value by 1.89%.
• c. The Fresh bond will increase in value by $18.17.
• d. The Dominion bond will decrease in value by $17.57.
• e. The Fresh bond will increase in value by 1.70%.
27
• Dominion Groceries Inc. has a 9-year, 6% annual coupon bond
outstanding with a $1,000 par value. Fresh Produce Inc. has a 10year, 5% annual coupon bond with a $1,000 par value. Both bonds
currently have a yield to maturity of 5.5%. Which of the following
statements is correct if the market yield increases to 5.75%?
• Dominion Groceries:
N= 9
OLD I/Y = 5.5%
PMT = 60
FV = 1,000 P/Y = 1
C/Y = 1
New I/Y = 5.75%
CPT PV (5.5)= 1,034.76
CPT PV (5.75)= 1,017.19 Change = 1,017.19 – 1,034.76 = - 17.57
Fresh Produce:
N = 10
OLD I/Y= 5.5%
P/Y = 1
C/Y = 1
CPT PV (5.5)= 962.31
CPT PV (5.75)= 944.14
FV = 1,000
New I/Y = 5.75
PMT = 50
Change = 962.31 – 944.14 = - 18.17
28
• A zero-coupon bond with a face value of $1,000 is issued with an
initial price of $415.50. The bond matures in 10 years. What is the
implicit interest, in dollars, for the first year of the bond's life? Use
semiannual compounding.
PMT = 0
FV = 1,000
PV = 415.50
N=2 x 10 = 20
P/Y = 2
C/Y = 2
I/Y = ???
CPT = 8.978% , which is QR Semi-annual rate
ESR = (8.978/2) % = 4.489%
EAR = (1 + 4.489)2 – 1 = 9.179%
Interest in 1st year = 415.50 * 9.179% = 38.14 (Answer)
29