Laurence Booth
Sean Cleary
6
Bond Valuation and Interest Rates
LEARNING OBJECTIVES
6.1
Describe the basic structure and the various features of different
types of bonds.
6.2
6.3
Explain how to value a bond given an appropriate discount rate.
Determine the discount rate or yield for a given market value of a
bond.
6.4
List and describe the factors, both domestic and global, that affect
interest rates.
6.5
List and describe the characteristics and pricing of other debt
instruments.
6.6
Explain how interest rate parity works.
6.1 THE BASIC STRUCTURE OF BONDS
• In the broadest sense, a bond is any debt instrument that
promises a fixed income stream to the holder
• Bonds usually have the following characteristics:
– A fixed face or par value, paid to the holder at maturity
– A fixed coupon, which specifies the interest payable over
the life of the bond
– A fixed maturity date
• Fixed income securities are often classified according to
maturity:
– Bills or paper have maturities less than one year
– Notes have maturities between one and seven years
– Bonds have maturities greater than seven years
• Bonds may be either bearer bonds or registered bonds
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
3
6.1 THE BASIC STRUCTURE OF BONDS
• The market price of a bond is the present value of the payments
promised by the bond
• The bond indenture is the contract between issuer and holder,
which specifies:
– Details regarding payment terms
– Collateral
– Positive or negative covenants
– Par value or face value (usually in increments of $1,000)
– Bond pricing, usually shown as the price per $100 of par
value which is equal to a percentage of the bond’s face value
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
4
6.1 THE BASIC STRUCTURE OF BONDS
• Term to maturity is the time remaining to the
maturity date
• Coupon rate is the annual percentage interest paid
on the bond’s face value
– Multiply the coupon rate by the bond’s face value
to calculate, and divide by two if the coupon is
paid semi-annually
– Example: A $1,000 bond with an 8% coupon rate
will have an $80 annual coupon or a $40 semiannual coupon
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
5
6.1 THE BASIC STRUCTURE OF BONDS
• Security and Protective Provisions
– Mortgage bonds are secured by real assets
– Debentures are either unsecured or secured, with
a floating charge over the firm’s assets
– Collateral trust bonds are secured by pledged
financial assets, such as common stock, other
bonds or Treasury bills
– Equipment trust certificates are secured by
pledged equipment, such as railway rolling stock
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.1 THE BASIC STRUCTURE OF BONDS
• Covenants are another type of protective provision
– Positive covenants specify actions the firm agrees
to do, such as supply periodic financial statements
and maintain certain ratios
– Negative covenants specify actions the firm agrees
to avoid, such as restrictions on the size of its debt
or acquiring or disposing of assets
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.1 THE BASIC STRUCTURE OF BONDS
Additional Bond Features
• Call features allow the issuer to redeem or pay off the
bond prior to maturity, usually at a premium
• Retractable bonds allow the holder to extend bonds
back to the issuer before maturity
• Extendible bonds allow the holder to extend the bond’s
maturity
• Sinking funds are funds set aside by the issuer to ensure
the firm is able to redeem the bond at maturity
• Convertible bonds can be converted into common stock
at a pre-determined conversion price
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.2 BOND VALUATION
• Equation 6-1 shows the price of a bond is a function of:
– Par or face value , F
– Term to maturity, n
– Interest or coupon payment, I
– The investor’s required rate of return (also known as
discount rate or yield to maturity), kb
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.2 BOND VALUATION
Annual Coupons
• Example: The market price of a 5% annual coupon-paying
Eurobond with 10 years to maturity, a face value of $1,000
and a yield-to-maturity of 6% is $926.40.
• TI BAII+ Calculator: (2nd) (CLRTVM) 1000 (FV) 50 (PMT) 10 (N)
6 (I/Y) (CPT) (PV) = 926.40
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
10
6.2 BOND VALUATION
Factors Affecting Bond Prices
• As Figure 6-3 shows, interest rates are inversely
related to bond prices
– Bond prices increase when interest rates decrease
– Bond prices decrease when interest rates increase
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
11
6.2 BOND VALUATION
Factors Affecting Bond Prices
• The relationship between the coupon rate and the
bond’s yield to maturity (YTM) determines if the
bond will sell at a premium, a discount, or at par
Coupon vs YTM
Price vs Face Value
Pricing
Coupon Rate < YTM
Price < Face Value
Discount
Coupon Rate = YTM
Price = Face Value
At Par
Coupon Rate > YTM
Price > Face Value
Premium
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
12
6.2 BOND VALUATION Semi-Annual Coupons
• Example: Find the price of a 4% semi-annual coupon Government of Canada
bond has five years to maturity, $1,000 in par value and a YTM of 6%.
• Since coupons are paid semi-annually, we must make three changes:
• Divide the coupon payment by 2
• Multiply the number of years by 2 (since we have twice as many semi-annual
periods
• Divide the yield to maturity by 2
• TI BAII+ Calculator: (2nd) (CLRTVM) 1000 (FV) 20 (PMT) 10 (N) 3 (I/Y) (CPT)
(PV) = 914.70
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
13
6.2 BOND VALUATION
Factors Affecting Bond Prices
• Yield to maturity, time to maturity and size of coupon affect the price
volatility of a bond
• Yield to Maturity
– Bond prices increase when the YTM decreases
– Bond prices decrease when the YTM increases
– The price-YTM relationship is convex to the origin (see Figure 6-2)
– Because of convexity, the price rise due to a fall in YTM is greater than
the price decline due to a rise in YTM given an identical change in YTM
– For a given change in YTM, bond prices will change more when
interest rates are low than when they are high
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.2 BOND VALUATION
Factors Affecting Bond Prices
• Time to Maturity
– Long-maturity bonds have greater price volatility than short-maturity bonds
– The longer the bond, the longer the period for which the cash flows are fixed
• Size of Coupon
– Low coupon bonds have greater price volatility than high coupon bonds
– High coupon bonds act like a stabilizing device, since a greater proportion of
the bond’s total cash flows occur closer to today and are therefore their
present value is less affected by a change in YTM
• Interest Rate Risk and Duration
– The sensitivity of a bond’s price to change in interest rates is a measure of
the bond’s interest rate risk. Interest rate risk is affected by: yield to
maturity, term to maturity and size of coupon
– These impact of interest rate risk is measured using duration
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.2 BOND VALUATION
Factors Affecting Bond Prices
• Duration measures of interest rate risk as a change in price for a
given change in interest rates
• The higher a bond’s duration, the more sensitive its price is to
changes in interest rates
• A bond’s duration will be higher if its:
– YTM is lower
– Term to maturity is longer
– Coupon is lower
• Bond Quotations
– Example: This quotation shows a 5.000% CIBC bond maturing on
September 10, 2010. Its YTM is 2.75% and its price is 106.97% of
face value.
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.2 BOND VALUATION
Cash Prices versus Quoted Prices
• The quoted price is the price reported by the media
• The cash price is the price paid by an investor, and includes both the
quoted price plus any interest that has accrued since the last coupon
payment date
• Example: Suppose you want to purchase a $1,000 bond with a 5% coupon,
paid semi-annually. Today is July 15th and the last coupon was paid on
June 30th. If the quoted price is $902, how much is the cash price?
– The cash price will be equal to the quoted price ($902) plus 15 days of
accrued interest (between June 30th and July 15th)
Cash Price  Quoted Price  Accrued Interest
 15 
 $902  $1,000  0.05

 365 
 $902  $2.05  $904.05
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.3 BOND YIELDS
Yield to Maturity (YTM/kb)
• The yield to maturity (YTM) is the discount rate used for bond
valuation
• YTM is the yield an investor would earn if:
– She purchases the bond at the current market price
– She holds the bond to maturity
– She reinvests all of the coupons paid by the bond at the YTM
• YTM is, therefore, the bond’s internal rate of return (IRR)
• YTM is, also, the discount rate that causes the present value
of the bond’s future cash flows to equal its current price
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
18
6.3 BOND YIELDS
Yield to Maturity (YTM/kb)
• There is no closed-form algebraic solution to the following equation
for YTM, so we must use a computer or calculator to solve YTM
• Example: What is the YTM on a 6% semi-annual coupon bond with
20 years to maturity that is selling for $1,030?
• The result is 2.87% each semi-annual period, or 5.74% per year.
Remember to express YTMs as annual rates, not semi-annual rates.
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
19
6.3 BOND YIELDS
Yield to Maturity (YTM/kb)
• Example: What is the YTM on a 6% semi-annual coupon bond
with 20 years to maturity that is selling for $1,030?
• Excel: = RATE(nper,pmt,pv,fv,type,guess)
= RATE(40,30,-1030,1000,,)
= 2.87% (multiply by 2 to annualize)
• TI BAII+ Calculator: (2nd) (CLRTVM) 30 (PMT) -1030 (PV) 1000
(FV) 40 (N) (CPT) (I/Y) = 2.87, then multiply by 2 to annualize
– Always enter the price as a negative number since it is a cash
outflow to purchase the bond, and enter the coupon and
principal payments as positive numbers since these are cash
inflows
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
20
6.3 BOND YIELDS
Yield to Call (YTC/kc)
• If a bond has a call feature, the issuer can call (or force the
investor to sell the bond back to it) before the maturity stated
in the bond indenture
• Callable bonds are initially protected from call for a period of
a few years (e.g., five, seven or ten), after which the issuer
may call the bond
• Replace the maturity date with the first call date, the face
value with the call premium (CP), and the time to maturity (n)
with the number of periods until expected call (c) to calculate
the yield to call (YTC), as in Equation 6-2:
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
21
6.3 BOND YIELDS
Yield to Call (YTC/kc)
• Example: What is the YTC on a 6% semi-annual coupon bond
with 20 years to maturity that is selling for $1,030? This bond
is callable in five years with a call price of $1,050.
• Excel: = RATE(nper,pmt,pv,fv,type,guess)
= RATE(10,30,-1030,1050,,)
= 3.08 % (multiply by 2 to annualize) = 6.16%
• TI BAII+ Calculator: (2nd) (CLRTVM) 30 (PMT) -1030 (PV) 1050
(FV) 10 (N) (CPT) (I/Y) = 3.08 then multiply by 2 to annualize =
6.16%
– Always use the call premium as the face value and the
expected call date as the time to maturity to find the yield
to earliest call
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
22
6.3 BOND YIELDS
Current Yield
• A bond’s current yield (CY) is the yield on the bond’s
current market price provided by the annual coupon
• Since current yield does not consider potential
capital gains or losses, it’s not a true measure of
return to the bondholder
• Example: What is the current yield of a 5.5% coupon
bond with a current market price of $1,050?
• Using Equation 6-3:
Annual Interest
$55
CY 

 5.24%
B
$1,050
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.4 INTEREST RATE DETERMINANTS
Base Interest Rates
• Interest is the “price” of borrowed money
• Interest is measured in basis points, or 1/100th of 1% (e.g., 250
basis points is 2.5%)
• Interest rates rise when the demand for loanable funds rises
• Interest rates fall when the demand for loanable funds falls
• The Risk-Free Interest Rate
– The risk-free rate is an abstract concept, and usually the yield on
short-term government treasury bills is used as a proxy for
practical purposes
– The risk-free rate is comprised of two components
• The real rate, which is compensation for deferring consumption
• The expected inflation rate, which is compensation for the expected
loss of purchasing power over the term of the short-term T-bill
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
24
6.4 INTEREST RATE DETERMINANTS
Base Interest Rates
• Figure 6-3 shows there is, generally, a relationship
where interest rates respond to the rate of inflation
Figure 6-4
Interest Rates
and Inflation
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.4 INTEREST RATE DETERMINANTS
Global Influences on Interest Rates
• Canadian interest rates are heavily influenced by
changes in interest rates in other countries
• Macroeconomic factors, both domestic and global,
also play an important role
• Interest rate parity theory (IRP) states that foreign
exchange forward rates will be established that
equalize the yield an investor can earn, whether
investing domestically or in a foreign jurisdiction
• Example: A country with both high inflation and high
interest rates will have a depreciating currency
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
26
6.4 INTEREST RATE DETERMINANTS
The Term Structure of Interest Rates
• The term structure of interest rates is the set of rates (YTM) for all
maturities of a given risk-class of debt securities (e.g., Government of
Canada bonds) at a given point in time
• When plotted on a graph the result is called a yield curve
• There are three typical shapes for the yield curve: upward sloping or
normal (1994), downward sloping or inverted (1990) and flat (1998)
Figure 6-5 Historical Yield
Curves 1990, 1994, 1998,
2004, and 2012
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.4 INTEREST RATE DETERMINANTS
The Term Structure of Interest Rates
Three Theories:
• Liquidity preference theory posits that investors must be paid
a liquidity premium in order to be compensated for the
interest rate risk inherent in holding less liquid, longer-term
debt
• Expectations theory suggests that longer-term interest rates
are the result of expectations of future short-term interest
rates. Or, in other words, the interest rates of various
maturities are dependent on each other
• Market segmentation theory suggests that different markets
exist for securities of different maturities that therefore the
two ends of the yield curve can have different factors
affecting them
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.4 INTEREST RATE DETERMINANTS
Risk Premiums
• More risky bonds (i.e., BBB-rated corporate bonds) will have
their own yield curve and it will plot at higher YTM at every
maturity than government bonds because of the additional
default risk that BBBs carry
• The yield spread is the difference between the YTM on a BBBrated corporate bond and a Government of Canada bond of
the same maturity and it represents the default risk premium
investors demand for investing in the more risky corporate
bond
• Yield spreads widen during recessions and narrow during
times of economic expansion
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
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6.4 INTEREST RATE DETERMINANTS
Risk Premiums
• Equation 6-5 can be used to determine the YTM on a
corporate bond:
k b  RF  Maturity yield differenti al  Spread
– RF is the risk-free rate
– The maturity yield differential is the extra yield
required for taking on a longer maturity
– The spread is the additional yield required for default
risk
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
30
6.4 INTEREST RATE DETERMINANTS
Risk Premiums
• Debt ratings – rating agencies, such as the Dominion Bond
Rating Service (DBRS), Standard & Poors (S&P), and Moody’s
assign all publicly traded bonds a risk rating
Figure 6-6 DebtRating Categories
for Standard &
Poor’s, Dominion
Bond Rating
Service, and
Moody’s
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
31
6.4 INTEREST RATE DETERMINANTS
Risk, Liquidity, and Bond Features:
• The greater the default risk, the higher the required
YTM
• The less liquid the bond, the higher the required YTM
• Call features generally increase the required YTM
• Extendable bonds generally have lower required
YTMs
• Retractable bonds generally have lower required
YTMs
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
32
6.5 OTHER TYPES OF BONDS/DEBT
INSTRUMENTS Treasury Bills
• Treasury bills are short-term obligations of the government with an initial
term to maturity of one year or less
• Issued at a discount to face value with face value being paid at maturity
• The difference between the discounted issue price and the face value is
treated as interest income
• Equation 6-7 can be used to value a Treasury bill:
where,
P = the market price of the T-bill
F = the face value of the T-bill
kBEY = the bond equivalent yield
n = the number of days until maturity
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
33
6.5 OTHER TYPES OF BONDS/DEBT
INSTRUMENTS Treasury Bills
• Example: What is the price of a $1 million Canadian treasury
bill with 80 days until maturity and a bond-equivalent yield of
4.5%?
PT-bill
F
$1,000,000


 $990,233.32
 n 
 80 
1  k BEY 
1

0
.
045



 365 
 365 
• Example: What is the yield on a $100,000 Treasury bill with
180 days until maturity and a market price of $98,200?
k BEY
F  P  365  $100,000  $98,200  365 




  3.72%
P  n 
$98,200
 180 
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
34
6.5 OTHER TYPES OF BONDS/DEBT
INSTRUMENTS Zero Coupon Bonds
• Zero coupon bonds are bonds issued at a discount which pay
no coupons and mature at par or face value
• Since no coupons are paid, there is no reinvestment rate risk
• Equation 6-9 shows the price of a zero-coupon bond is the
present value of the face value of the bond:
• Example: What is the market price of a $50,000 zero coupon
bond with 25 years to maturity that is currently yielding 6%?
B
Booth • Cleary – 3rd Edition
F
$50,000

 $11,649.93
n
25
(1  k b )
(1.06)
© John Wiley & Sons Canada, Ltd.
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6.5 OTHER TYPES OF BONDS/DEBT INSTRUMENTS
Floating Rate, Real Return, and Canada Savings Bonds
• Floating rate bonds have coupon rates that float with some
reference rate, such as the yield on Treasury bills
– Since the coupon rate floats, or is variable, the market price will
typically be close to the bond’s face value
• Real return bonds are issued by the Government of Canada to
protect investors against unexpected inflation
– Each period, the face value of the real return bond is grossed up by the
inflation rate. The coupon is then paid on the grossed up face value
• Canada Savings Bonds (CSBs) are issued by the Government of
Canada as either regular interest bonds (interest paid annually) or
compound interest bonds (interest compounds over the life of the
bond)
– There is no secondary market for Canada Savings Bonds; instead, they
are redeemable at any chartered bank in Canada at their face value
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
36
6.6 INTEREST RATE PARITY
• The interest rate parity (IRP) theory demonstrates why
differences in interest rates between countries should be
offset by forward exchange rates
• If no arbitrage opportunities exist, Equation 6A-1 should hold:
F 1  k domestic

S 1  k foreign
where,
F = current forward exchange rate (domestic units for foreign
units)
S = current spot exchange rate
kdomestic = domestic interest rate
kforeign = foreign interest rate
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
37
6.6 INTEREST RATE PARITY
• Example: Assume U.S interest rates are presently 1.25%
on one-year American T-Bills, that the U.S. dollar is
quoted at USD$1 = CAD$1.0500, and that the interest
rate on one-year Canadian T-Bills is 1.00%. Find the oneyear forward exchange rate.
 1  k domestic 
 1.01 


FS
 1.0500
  $1.0474
 1 k

 1.0125 
foreign 

Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
38
6.6 INTEREST RATE PARITY
• Example continued: Can an investor benefit from higher U.S.
interest rates without assuming risk? Suppose there are two
Canadian investors, both with $1,000 to invest. Investor A
invests domestically while investor B invests in the U.S. and
eliminates foreign exchange risk with a forward contract.
• Investor A’s ending wealth: $1,000(1.01) = $1,010
• Investor B’s ending wealth:


 $1,000 
1.04741.0125
  $1,010
 1.05 

• Notice that if $1.0474 is the prevailing forward rate, both
investors earn the same amount
Booth • Cleary – 3rd Edition
© John Wiley & Sons Canada, Ltd.
39
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