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Chapter Six
Chapter 6
Bonds, Bond Prices, and the
Determination of Interest Rates
Goals of the Chapter
1. Present value and bond pricing
2. Relationship of bond prices, interest rates
(yields), and returns.
3. Key drivers of bond prices - supply and
demand in the bond market determine bond
prices .
4. Why bonds are risky - default, inflation, and
interest changes.
Bond Prices
• A standard bond specifies the fixed
amounts to be paid and the exact dates of
the payments.
• We will examine four basic types.
Bond Prices
1. Zero-coupon or discount bond
– Promise a single payment on a future date
– Example: Treasury bill
2. Fixed-payment loan (just skim)
– Sequence of fixed payments
– Example: Mortgage or car loan
3. Coupon bond
– periodic interest payments + principal repayment at maturity
– Example: U.S. Treasury Bonds and most corporate bonds
4. Consol
– periodic interest payments forever, principal never repaid
– Example: U.K. government has some outstanding
Zero-Coupon Bonds
• U.S. Treasury bills (T-bills) are money market
instruments - mature is less than a year.
• Straightforward type of bond.
– Each T-bill represents a promise by the U.S.
government to pay $1000 on a fixed future date.
– No coupon payments - zero-coupon bonds
– Also called pure discount bonds (or discount bonds)
since the price is less than face value - they sell at a
discount.
• How to price a $1000 face value zero-coupon
bond $1000
P
(1  i )
n
Zero-Coupon Bonds
Assume i = 5%
Price of a One-Year Treasury Bill
1000

 $952.38
(1  0.05)
Price of a Six-Month Treasury Bill
1000

 $975.90
1/ 2
(1  0.05)
Remember units must match: annual interest rate, therefore n must
be annual – in the case ½ of a year.
Zero-Coupon Bonds
• When the price moves, the interest rate moves
with it in the opposite direction.
• We can compute the interest rate from the price
using the present value formula.
– Suppose the price of a one-year $1000 face value Tbill is $950.
i = ($1000/$950) - 1 = 0.0526 = 5.26%
Fixed-Payment Loans
Conventional home mortgages and car loans are
fixed-payment loans.
– They promise a fixed number of equal payments at
regular intervals.
– Amortized loans - the borrower pays off part of the
principal along with the interest for the life of the
loan.
• Value of a Fixed Payment Loan =
FixedPayment FixedPayment


2
(1 i)
(1 i)
FixedPayment

(1 i) n
• The sum of the present value of the payments.

Coupon Bond Price
A simple contract-
Price of a Coupon Bond
 C
C
C
C  FV
PB  


 ...... 

1
2
3
n
n
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)


Present Value of Coupon Payments
Present Value of
Principal Payment
Price of Coupon Bond (PB) =
Present Value of yearly coupon payments (C)
+ Present Value of the Face Value (FV),
where: i = interest rate and n = time to maturity
Example: Price of a 10%, n-year Coupon Bond
Coupon Payment (C) =$100, Face value (FV) = $1,000,
and n = time to maturity
 $100
$100
$100  $1000
PB  

 ...... 

1
2
n
n
(1  i)  (1  i)
 (1  i) (1  i)
C is contractually fixed.
Given values for i and n, we can determine the bond
price PB
Definition: Coupon Rate = Coupon Payment / Face Value
Price of a 10-year Coupon Bond
If n = 10, i = 0.10, C = $100 and FV = $1000.
 $100
$100
$100  $1000
PB  $1000  

 ...... 

1
2
10 
10
(
1

.
1
)
(
1

.
1
)
(
1

.
1
)
(
1

.
1
)


What’s the Coupon Rate?
Excel Example
Price of a 10-year Coupon Bond
If n = 10, i = 0.12, C = $100 and FV = $1000.
 $100
$100
$100 
$1000
PB  $887  

 ...... 

1
2
10 
10
(
1

.
12
)
(
1

.
12
)
(
1

.
12
)
(
1

.
12
)


i goes up and PB goes down
What’s the Coupon Rate for this bond?
Yield to Maturity -YTM
• yield to maturity:
– The yield bondholders receive if they hold the
bond to its maturity when the final principal
payment is made.
– Simple example:
$5
$100 $105


Price of 1-yr, 5% Coupon Bond =
(1  i) (1  i) (1  i)
• The value of i that solves the equation is the
yield to maturity (YTM).
Current Yield
• Current yield is the measure of the proceeds the
bondholder receives for making a loan.
Yearly Coupon Payment
Current Yield 
Price Paid
• The current yield measures that part of the
 return from buying the bond that arises solely
from the coupon payments.
Current Yield and YTM
Example:
1 year, $100 FV, 5% coupon bond selling for $99
Current Yield = 5  0.0505 , or 5.05%
99
Yield to maturity for this bond is 6.06 percent
found as the solution to:
$5
$100 $105


 $99
(1  i) (1  i) (1  i)
 $105 
(1  i )  
  1.0606  i  1.0606  1  .0606
 $99 
Yield to Maturity - YTM
• This relationship should make sense.
• If you pay $99 for a $100 face value bond, you
will receive both the interest payments and
the increase in value from $99 to $100.
• This rise in value is referred to as a capital gain
and is part of the return on your investment.
• When the price of a bond is higher than face
value, the bondholder incurs a capital loss.
YTM: 10-year Coupon Bond
Suppose n = 10, PB = $950, C = $100 and FV = $1000.
 $100 $100
$100  $1000
PB  $950  

 ...... 

1
2
10 
10
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)


What is the Coupon Rate?
What is the Current Yield?
What’s the YTM? - more complicated

YTM = .1085 or 10.85%
FV  P
C
n
Approx.YTM 
FV  P
2
C = the coupon payment
FV = Face Value
P = Price
n = years to maturity
Using the Approximation Formula
Previous example:
n = 10, PB = $950, C = $100 and FV = $1000.
Yearly Coupon Payment
Current Yield 
Price Paid
For our 10-year, $1000 FV bond with a $100
coupon payment selling at $950:
Coupon rate = 10%
Current Yield = 10.52%
YTM = 10.85%
Approx. YTM = 10.77%
• When the coupon bond is priced at its face value, the
yield to maturity equals the coupon rate
• The price of a coupon bond and the yield to maturity are
negatively related
• The yield to maturity is greater than the coupon rate when
the bond price is below its face value
More on Zero Coupon or Discount Bonds
•
Definition: A discount bond is sold at some price P, and pays
a larger amount (FV) after t years. There is no periodic
interest payment.
Let PB = price of the bond, i= interest rate, n = years to
maturity, and FV = Face Value (the value at maturity):
PB 
FVn
(1 i ) n
Zero Coupon Bonds - Price
Price of a One-Year Treasury Bill at 4% and FV = $1,000:
1000
PB 
 $961.53
(1  0.04)
Price of a Six-Month Treasury Bill at 4% and FV = $1,000:
1000
PB 
 $980.58
1/ 2
(1  0.04)
Price of a 20-Year zero coupon bond at 8% and FV =
$20,000:
$20000
PB 
 $4,290.96
20
(1  0.08)
YTM - Zero Coupon Bonds
P
 FV
(1  i )  
 P
n
FV
(1 i ) n

 FV 
 1 i  


 P 
(1 / n )
 FV 
i

 P 
(1 / n )
1
Zero Coupon Bonds - YTM
• For a discount bond with FV = $15,000 and P = $4,200,
and n = 20, the interest rate (or yield to maturity) would
be:
(1 / n )
FV


i
1

 P 
 15,000 
i

 4,200 
(1/ 20)
1
i = 1.0657 -1=> i =6.57%
Note: This is the formula for compound annual rate of growth
Zero Coupon Bonds - YTM
• For a discount bond with FV = $10,000 and P = $6,491,
and n = 7, the interest rate (or yield to maturity) would be:
 10,000 
i

 6,491 
(1 / 7 )
1
i = 1.06368 – 1 = .06368 or 6.368%
http://online.wsj.com/mdc/public/p
age/2_3020-tstrips.html
From a Coupon Bond to Zero Coupon Bonds Zero
Coupon Bonds are called Strips – here’s why.
 C
C
C
C  FV
PB  


 ...... 

1
2
3
n
n
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)


Create n+1 discount bonds
http://www.treasurydirect.gov/instit
/marketables/strips/strips.htm
Consols
• Consols or perpetuities, are like coupon bonds
whose coupon payments last forever.
• The borrower pays only interest, never repaying
the principal.
• The U.S. government sold consols once in 1900,
but the Treasury has bought them all back.
• The price of a consol is the present value of all
future coupon payments.
PConsol 
Yearly Coupon Payment
i
Yearly Coupon Payment
i  YTM 
Pconsol
Holding Period Return
• The holding period return is the return to holding
a bond and selling it before maturity.
• The holding period return can differ from the
yield to maturity.
One Year Holding Period Return
Example:
– 10 year bond
– 6% coupon rate
– Purchase at face value, $100
– Hold for one year and then sell it
Holding Period Return
What if market interest rates at the time of
the sale fall to 5%?
1-yr Holding Period Return =
$6 $107.11  $100 $13.11


 0.1311
$100
$100
$100
Current Yield + Capital Gain
The investor earned $13.11 on a $100 investment.
The 1-yr Holding Period Return = 13.11%
Holding Period Return
What if market interest rates at the time of the
sale rise to 7%?
1yr Holding Period Return =
$6 $93.48  $100  $0.52


 0.0052
$100
$100
$100
1-yr Holding Period Return = - 0.52%
Holding Period Return
• The one-year holding period return is the sum of the
yearly coupon payment divided by the price paid for the
bond and the change in the price divided by the price
paid.
Yearly Coupon Payment

Price Paid
Change in Price of the Bond

Price of the Bond
=
+ Capital Gain (as a %)
Current Yield
Holding Period Return
• You purchase coupon bond and sell one year later.
RET( t t 1)
C ( Pt 1  Pt ) C  Pt 1  Pt
 

Pt
Pt
Pt
Current
Yield
Capital Gain
Holding Period Return
You purchase a 10-year, 10% coupon bond,
face value = $1,000. If the interest rate one
year later is the same at 10%:
One year holding period return =
$100 $1000  $1000 $100


 .10
$1000
$1000
$1000
or 10.0%
Holding Period Return
If the interest rate one year later is lower, say at
8%:
One year holding period return =
$100 $1125  $1000 $225


 .225
$1000
$1000
$1000
or 22.5%
• Where did the $1125 come from?
• In the previous example, where did the
$107.11 come from?
Holding Period Returns
If the interest rate in one year is higher at 12%:
One year holding period return =
$100 $893  $1000
 $7


 .007
$1000
$1000
$1000
or -.70%
Where did the $893 come from. You need to know how to
calculate the $1125 and $893
Key Conclusions From Table
• The return equals the yield to maturity (YTM)
only if the holding period equals the time to
maturity.
• A rise in interest rates is associated with a fall in
bond prices, resulting in a capital loss if the
holding period is less than the time to maturity.
• The greater the percentage price change
associated with an interest-rate change, the more
distant the maturity.
Interest-Rate Risk
• Change in bond price due to change in
interest rate
• Prices and returns for long-term
bonds are more volatile than those for
shorter-term bonds
• There is no interest-rate risk for a bond
whose time to maturity matches the
holding period
Reinvestment (interest rate) Risk
• If
investor’s holding period exceeds the term to
maturity
 proceeds
from sale of bond are reinvested
at new interest rate
 the
investor is exposed to
reinvestment risk
• The
investor benefits from rising interest rates,
and suffers from falling interest rates
Bond Supply, Bond Demand and Equilibrium in the
Bond Market
• Supply and demand determine bond prices
and bond yields.
• The bond supply curve is the relationship
between the price and the quantity of bonds
investors, corporations and governments are
willing to sell, all else equal.
• The higher the price of a bond, the larger the
quantity supplied.
– The bond supply curve slopes upward.
• Why?
Bond Supply
1. Investors:
– The higher the price, the more tempting it is to sell a
bond they currently hold – quantity supplier
increases.
2. Companies seeking to finance projects and that
finance spending:
– The higher the price at which they can sell bonds,
the cheaper it is to borrow - quantity supplied
increases.
• For a $100 one-year zero-coupon bond, the
quantity supplied will be higher at $95 than it
will be at $90, all other things being equal.
Bond Demand
• The bond demand curve is the relationship
between the price and the quantity of bonds
that investors demand, all else equal.
• The bond demand curve slopes downward.
– Why?
• Investors:
– The lower the price bond buyers must pay for a
fixed-dollar payment on a future date, the more
likely they are to buy a bond.
• For a $100 one-year zero-coupon bond, the
quantity demanded will be higher at $90 than it
will be at $95, all other things being equal.
Bond Supply, Bond Demand and Equilibrium
in the Bond Market
Equilibrium is
the point at
which supply
equals
demand.
Bond Supply, Bond Demand and Equilibrium in the Bond
Market
• If bond prices are above equilibrium?
– Quantity supplied > quantity demanded.
– Excess supply puts downward pressure on
the price until supply equals demand.
• If the price is below equilibrium?
– Quantity demanded > quantity supplied.
– Excess demand puts upward pressure on
the price until supply equals demand.
Bond Supply, Bond Demand and Equilibrium in the Bond
Market
• To understand bond prices, need to
understand what determines supply and
demand.
• Remember the distinction between moving
along a curve (change in quantity supplied or
demanded) versus a shift in the curve (change
in supply or demand).
• A shift in either supply or demand changes the
price of bonds, so it changes yields as well.
Factors That Shift Bond Supply
1. Changes in Government Borrowing
– Any increase in the government’s borrowing needs
increases the quantity of bonds outstanding,
shifting the bond supply curve to the right.
2. Change in General Business Conditions
– As business conditions improve, the bond supply
curve shifts to the right.
3. Changes in Expected Inflation
– When expected inflation rises, the cost of
borrowing in real terms falls, shifting the bond
supply curve to the right.
Factors That Shift Bond Supply
When borrowers’ desire for funds increases, the supply curve shifts to the
right. This lowers bond prices, thereby raising interest rates.
Factors That Shift Bond Demand
1. Wealth
– Increase in wealth increases bond demand, shifts the
demand for bonds to the right.
2. Expected Inflation
– Declining inflation means promised payments have
higher value - bond demand shifts right.
3. Expected Returns and Expected Interest Rates
– If the expected return on bonds rises relative to the
return on alternative investments, bond demand will
shift right.
– When interest rates are expected to fall, price prices
are expected to rise shifting bond demand to the right.
Factors That Shift Bond Demand
4. Risk Relative to Alternatives
– If bonds become less risky relative to alternative
investments, demand for bonds shifts right.
5. Liquidity Relative to Alternatives
– Investors like liquidity: the more liquid the bond,
the higher the demand.
– If bonds become less risky relative to alternative
investments, demand for bonds shifts right.
Factors That Shift Bond Demand
When bonds
become more
attractive for
investors, the
demand curve
shifts to the right.
This raises bond
prices, lowering
interest rates.
Case Study: Increase in Expected Rate of
Inflation
• Recall expected inflation affects both bond supply
and bond demand.
• Increasing expected inflation
– Reduces the real cost of borrowing shifting bond
supply to the right.
– But, lowers real return on lending (also possible
capital loss), shifting bond demand to the left
• What happens to bond prices and interest rates?
Response to an Increase in the Expected Rate of
Inflation – the Fisher Effect
6-58
Expected Inflation and Interest Rates (Three-Month
Treasury Bills), 1953–2011
6-59
Case Study: Interest Rates and the Business Cycle
• A business cycle downturn (recession):
– Reduces business investment opportunities
shifting bond supply to the left.
– Reduces wealth, shifting bond demand to the left
also.
• In theory, both shifts could give an ambiguous
answer.
• The data is not ambiguous. In recessions,
interest rates tend to fall meaning that bond
prices rise.
FIGURE 7 Business Cycle and Interest Rates (Three-Month Treasury Bills),
1951–2008
6-61
Understanding Changes in Equilibrium Bond
Business Cycle Downturn
S1
1) Shift S left
2) Shift D left
Price per Bond
S0
E1
E0
D0
D1
Quantity of Bonds
We know price
increases so
supply shifts
more than
demand.
Bonds Are Risky!!!
Bonds are a promise to pay a certain amount
in the future. How can that be risky?
1. Default risk - the chance the bond’s issuer
may not make payment.
2. Inflation risk - investor cannot be aware of
the real of the payments made.
3. Interest rate risk - a rise in interest rates
before bond is sold could mean a capital
loss.
6-63
Case Study: What Happened Here?
• Investors’ concerns about
risk affect their demand for
U.S. Treasuries.
• 1998, Russia defaulted and
people lost confidence in
emerging market countries.
• Safest assets are U.S.
Treasuries.
• Demand increased, price
increased, and yields
declined.
Why Bonds are Risky
• Risk arises because an investment has many
possible payoffs during the holding horizon.
• We need to look at the risk the bondholder
faces, what are the possible payoffs, and how
likely each is to occur.
• We will compare the risk on a bond’s return
relative to the risk-free rate.
Default Risk
• Although there is little to no default risk with U.S.
Treasury bonds, there is with other government
and corporate bonds.
• We can look at an example of a corporate bond.
• Assume the one-year risk-free interest rate is 5
percent.
• A company has issued a 5 percent coupon bond
with a face value of $100.
Default Risk
• If this bond was considered default risk-free, the
price of the bond would be the present value of
the $105 payment.
Price of risk free bond = ($100 + $5)/$1.05 = $100
• Suppose, however, there is a 10% probability that
the company will go bankrupt before paying back
the loan.
– Assume the outcome is either $105 or $0.
– Expected value equals $94.50
Price of bond = $94.50/1.05 = $90
Default Risk
Suppose 10% probability firm goes bankrupt – you get nothing
Expected Value of bond payment
Possibilities
Payoff
Probability
Payoff x
Probability
Full Payment
$105
$0
.90
.10
$94.50
$0
Default
•Expect to receive $94.50 one-year from now.
•Discount at risk-free rate = $94.50  $90
1.05
•P = $90
Default Risk
• If the price of the bond is $90, what yield to
maturity does this price imply?
YTM = $105/$90 - 1 = 0.1667
• Default risk premium is the promised yield to
maturity minus the risk-free rate:
Risk Premium = 16.67 percent - 5 percent
= 11.67 percent.
• The higher the default risk, the higher the
yield and risk premium.
Default Risk Premium
• We can calculate the probability of repayment
from the interest rates.
• Let 1+k be the return on a one-year corporate
debt and 1+ i be the return on a one-year default
risk-free treasury.
• At equilibrium: 1+i = p(1+k)
1 i
• The probability of repayment is
p
1 k
• the probability of default is 1 – p
• The probability of repayment: 1.05  0.90
1.1667
6-70
Yields on 9/20/16
3mo
6mo
9mo
1yr
2yr
3yr
5yr
10yr
20yr
30yr+
BONDS
U.S. Treasury
0.29% 0.51% 0.60% 0.68% 0.82% 0.94% 1.23% 1.67% 2.01% 2.40%
U.S. Treasury
Zeros
--
Agency/GSE
0.71% 0.81% 0.96% 1.01% 1.13% 1.42% 1.62% 2.42% 3.13% 3.20%
Corporate
(Aaa/AAA)
--
--
--
0.39% --
0.56% 0.73% 0.89% 1.25% 1.82% 2.32% --
0.73% 1.05% 1.40% 1.59% 2.45% 3.35% 3.91%
Corporate (Aa/AA) 0.74% 1.15% 1.08% 1.22% 1.44% 1.62% 1.96% 2.75% 3.60% 4.49%
Corporate (A/A)
0.93% 1.25% 1.32% 1.42% 1.84% 1.89% 2.65% 3.44% 4.25% 4.32%
Corporate
(Baa/BBB)
1.21% 1.65% 2.15% 2.15% 2.54% 4.48% 4.88% 5.13% 6.08% 6.42%
Municipal
(Aaa/AAA)
0.60% 0.60% 0.77% 0.75% 0.97% 1.12% 1.43% 2.15% 2.85% 2.93%
Municipal (Aa/AA)
0.86% 0.75% 0.85% 0.88% 1.63% 1.61% 1.65% 2.59% 3.07% 3.28%
Municipal (A/A)
0.75% 0.80% 1.00% 0.99% 1.20% 1.68% 2.52% 3.45% 3.50% 3.45%
Taxable Municipal* 0.73% 1.11%
1.05% 1.10% 1.72% 2.05% 2.09% 4.30% 3.94% 3.36%
6-71
• Inflation-indexed bonds are structured so that
the government promises to pay you a fixed
interest rate plus the change in the consumer
price index (CPI).
• The U.S. Treasury sells two types of bonds that
are adjusted for inflation.
1. Series I savings bonds
2. Treasury Inflation Protected Securities (TIPS)
Inflation Risk
• With few exceptions, bonds promise to make
fixed-dollar payments.
• Remember that we care about the purchasing
power of our money, not the number of
dollars.
– This means bondholders care about the real
interest rate.
• How does inflation risk affect the interest
rate?
Inflation Risk
• Think of the interest rate having three
components:
1. The real interest rate
2. Expected inflation, and
3. Compensation for inflation risk.
• Example:
– Real interest rate is 3 percent.
– Inflation could be either 1 percent or 3 percent.
– Expected inflation is 2 percent, with a standard
deviation of 1.0 percent.
Inflation Risk
• Nominal interest rate should equal:
i = 3 percent real interest rate +
2 percent expected inflation +
Compensation for inflation risk
• The greater the inflation risk, the larger the
compensation for it.
Interest-Rate Risk
• Interest-rate risk arises from the fact that
investors don’t know the holding period return of
a long-term bond.
– The longer the term of the bond, the larger the price
change for a given change in the interest rate.
• For investors with holding periods shorter than
the maturity of the bond, the potential for a
change in interest rates creates risk.
– The more likely the interest rates are to change during
the bondholder’s investment horizon, the larger the
risk of holding a bond.