Transcript Term Structure of Interest Rates

Chapter 3
The Level and Structure of
Interest Rates
Historical Interest
Rate Patterns
Over the last three decades interest rates have often
followed patterns of persistent increases or persistent
decreases with fluctuations around these trends.
• In the 1970s and early 1980s the U.S.’s inflation led to
increasing interest rates during that period. This period of
increasing rates was particularly acute from the late 1970s
through early 1980s when the U.S. Federal Reserve
changed the direction of monetary policy by raising
discount rates, increasing reserve requirements, and
lowering monetary growth.
Historical Interest
Rate Patterns
• This period of increasing rates was followed by a
period of declining rates from the early 1980s to
the late 1980s, then a period of gradually
increasing rates for most of the 1990s, and finally
a period of decreasing rates from 2000 through
2003.
• The different interest rates levels observed since
the 1970s can be explained by such factors as
economic growth, monetary and fiscal policy, and
inflation.
Years
2001
2000
1998
1996
1995
1993
1991
1990
1988
16
1986
1985
1983
1981
1980
1978
1976
1975
1973
1971
1970
T-bill Rates
Historical Interest Rate Patterns
18
TREASURY BILL RATES, 1970-2003
14
12
10
8
6
4
2
0
Historical Interest Rate Spreads
• In addition to the observed fluctuations in interest rate
levels, there have also been observed spreads between the
interest rates on bonds of different categories and terms to
maturity over this same period.
• For example, the spread between yields on Baa and AAA
bonds is greater in the late 1980s and early 1990s when the
U.S. economy was in recession compared to the differences
in the mid to late 1990s when the U.S. economy was
growing.
• In general, spreads can be explained by differences in each
bond’s characteristics: risk, liquidity, and taxability.
Historical Interest Rate Spreads
20
TREASURY BOND, Aaa CORPORATE,
Baa CORPORATE, AND MORTAGE RATES, 1970-2002
18
16
12
10
8
6
4
2
AAA
BBB
10-Yr T-Bond
30-yr Mortgage Rate
2002
2000
1999
1998
1996
1995
1994
1992
1991
1990
1988
1987
1986
1984
1983
1982
1980
1979
1978
1976
1975
1974
1972
1971
0
1970
Rates (%)
14
Historical Interest Rate Spreads
• Interest rate differences can be observed between similar
bonds with different maturities. The figures on the next
slide shows two plots of the YTM on U.S. government
bonds with different maturities for early 2002 and early
1981.
• The graphs are known as yield curves and they illustrate
what is referred to as the term structure of interest rates.
– The lower graph shows a positively-sloped yield curve
in early 2002 with rates on short-term government
securities lower than intermediate-term and long-term
ones.
– In contrast, the upper graph shows a negatively sloped
curve in early 1981 with short-term rates higher than
intermediate- and long-term ones.
Historical Interest Rate Spreads
Yield Curves
18
16
14
Rates (%)
12
10
8
6
4
2
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Years to Maturity
January, 1981
January, 2001
Objective
• Understanding what determines both the overall level
and structure of interest rates is an important subject in
financial economics. Here, we examine the factors that
are important in explaining the level and differences in
interest rates.
– Examining the behavior of overall interest rates using basic
supply and demand analysis
– Looking at how risk, liquidity, and taxes explain the
differences in the rates on bonds of different categories.
– Looking at four well-known theories that explain the term
structure on interest rates.
Supply and Demand Analysis
• One of the best ways to understand how market
forces determine interest rates is to use
fundamental supply and demand analysis.
• In determining the supply and demand for bonds,
let us treat different bonds as being alike and
simply assume the bond in question is a oneperiod, zero-coupon bond paying a principal of F
equal to 100 at maturity and priced at P0 to yield a
rate i.
• Given this type of bond, we want to determine the
important factors that determine its supply and
demand.
Bond Demand and Supply Analysis
Bond Demand Curve:
• Bond Demand Curve: The curve shows an
inverse relationship between, bond demand,
BD, and its price, P0, and a direct relation
between BD interest rate, i, given other factors
are constant.
• Bond demand curve is also called the supply of
loanable funds curve.
Bond Demand and Supply Analysis
Bond Demand Curve:
• The factors held constant include the overall wealth or
economic state of the economy, as measured by real
output, gdp, the bond’s risk relative to other assets, its
liquidity relative to other assets, expected future interest
rates, E(i) and inflation, and government policies:
BD f (i or P0 , gdp, E(i), E(Inflation ), risk , Liq , govt . policy )
Bond Demand and Supply Analysis
Bond Demand Curve:
• Bond demand is inversely related to its price and directly related
to interest rate.
• The bond demand curve showing bond demand and price relation
is negatively-sloped.
• This reflects the fundamental assumption that investors will
demand more bonds the lower the price or equivalently the greater
the interest rate.
• Changes in the economy, futures interest rate and inflation
expectations, risk, liquidity, and government policies lead to either
rightward or leftward shifts in the demand curve, reflecting greater
or less bond demand at each price or interest rate.
• Bond Demand Curve
Interest Rate
Bond Pr ice
( P )
B
0
D
i
P0
BD
(Supply of
Loanable Funds )
0
Quantity of Bonds
B  f (i or P0 , gdp, E(i), E( Inflation ), risk , Liq , govt . policy )
D
(i )
Bond Demand and Supply Analysis
Bond Supply Curve:
• The bond supply curve shows the quantity supplied of
bonds, BS, by corporations, governments, and
intermediaries is directly related to the bond’s price
and inversely related interest rate, given other factors
such as the state of the economy, government policy,
and expected future inflation are constant:
BS f (i or P0 , gdp, E(Inflation ), govt . policy )
• Bond supply curve is also called the demand of
loanable funds curve.
Bond Demand and Supply Analysis
Bond Supply Curve:
• The bond supply curve is positively sloped.
• The positively sloped curve reflects the fundamental
assumption that corporations, governments, and financial
intermediaries will sell more bonds the greater the bond’s
price or equivalently the lower the interest rate.
• The bond supply curve will shift in response to changes in the
state of the economy, government policy, and expected
inflation.
• Supply Curve for Bonds
Interest Rate
Bond Pr ice
S
( P )
B
B
D
0
(i )
i
P0
BS
(Demand of
Loanable Funds )
0
Quantity of Bonds
BS f (i or P0 , gdp, E(Inflation ), govt . policy )
Bond Demand and Supply Analysis
Equilibrium:
• The equilibrium rate, i* and price, P0*, are
graphically defined by the intersection of the
bond supply and bond demand curves.
• Supply and Demand for Bonds
Interest Rate
Bond Pr ice
S
( P )
B
B
D

P0*
BS
(Supply of
Loanable Funds )
0
Quantity of Bonds
B  f (i or P0 , gdp, E(i), E( Inflation ), risk , Liq , govt . policy )
BS  f (i or P0 , gdp, E(Inflation ), govt . policy )
D
(i )
i*
BD
(Demand of
Loanable Funds )
0
Bond Demand and Supply Analysis
Proof of Equilibrium:
• If the bond price were below this equilibrium
price (or equivalently the interest rate were
above the equilibrium rate), then investors
would want more bonds than issuers were
willing to sell.
• This excess demand would drive the price of
the bonds up, decreasing the demand and
increasing the supply until the excess was
eliminated.
Bond Demand and Supply Analysis
Proof of Equilibrium:
• If the price on bonds were higher than its
equilibrium (or interest rates lower that the
equilibrium rate), then bondholders would
want fewer bonds, while issuers would want
to sell more bonds.
• This excess supply in the market would lead
to lower prices and higher interest rates,
increasing bond demand and reducing bond
supply until the excess supply was
eliminated.
Bond Demand and Supply Analysis
Shifts in Bond Demand Cuve :
a. gdp   BD   BD BD Shifts right
Re ason : An increase in gdp may reflect an increase in wealth
that increases the demand for all assets , including bonds .
b. E(Inflation )  BD   BD BD Shifts left
Re ason : Investors will increase their purchase of consumptio n goods and decrease
their purchase of assets , including bonds .
c. E(i)  BD   BD BD Shifts right
Re ason : Expected lower rates means greater bond prices in the future and therefore greater
exp ected rates for some bonds .
Bond Demand and Supply Analysis
Shifts in Bond Demand Cuve :
d. Re lative Risk  BD   BD BD Shifts left
Re ason : If bonds become more risky relative to other sec urities or other sec urities become
less risky relative to bonds , then bond demand would decrease.
e. Re lative Liquidity   BD   BD BD Shifts right
f . Decrease in Re serve Re quirments  Bank loans   sup ply of funds   BD 
 BD BD Shifts right
g. Increase in Central Bank Discount Rate  Bank loans   sup ply of funds   BD 
 BD BD Shifts left
Bond Demand and Supply Analysis
Shifts in Bond Supply Cuve :
a. gdp   BS   BS BS Shifts right
Re ason : When the economy is growing producers will increase their
capital formation (inventory , accounts receivable , plant exp ansion , etc ).
To finance their capital exp ansion , they will sell more bonds.
b. E(Inflation )  BS   BS BS Shifts right
Re ason : If inf lation is exp ected in the future , then companies and
government s will be exp ecting greater borrowing needs in the future
given higher prices . They will therefore find it advantageo us to borrow more now .
c. Deficit   Treasury sells more bonds to finance shortfall  BS   BS BS Shifts right
Bond Demand and Supply Analysis
Shifts in Bond Supply Cuve :
d. Government Surplus  Treasury may buy existing government bonds  BS   BSBS Shifts left
e. Expansiona ry OMO  Central bank buys bonds  BS   BS BS Shifts left
f . Contractio nary OMO  Central bank sells its bonds  BS   BSBS Shifts right
Cases Using Demand and Supply Analysis
Expansionary Open Market Operation:
• Central Bank buys bonds, decreasing the bond
supply and shifting the bond supply curve to the
left.
• The impact would be an increase in bond prices
and a decrease in interest rates. Intuitively, as
the central bank buys bonds, they will push the
price of bond up and interest rate down.
Expansionary Open Market Operation
Interest Rate
Bond Pr ice
0
( P )
BS2
B1D
B1S

P1

P0
BS2
0
i**
B1S
i*
B1D
(i )
Cases Using Demand and Supply Analysis
•
Economic Recession:
In an economic recession, there is less capital formation and
therefore fewer bonds are sold.
•
This leads to a decrease in bond supply and a leftward shift in
the bond supply curve.
•
The recession also lowers bond demand, shifting the bond
demand curve to the left.
•
If the supply effect dominates the demand effect, then there
will be an increase in bond prices and a decrease in interest
rates.
Economic Recession
Interest Rate
Bond Pr ice
0
( P )
BS2
B1D
B D2
B1S

P1
P0
BS2
0
i**

i*
B1D
B1S
B D2
(i )
Cases Using Demand and Supply Analysis
Treasury Financing of a Deficit:
•
With a government deficit, the Treasury will have to
sell more bonds to finance the shortfall.
•
Their sale of bonds will increase the supply of
bonds, shifting the bond supply curve to the right,
initially creating an excess supply of bonds.
•
This excess supply will force bond prices down and
interest rates up.
Treasury Financing of Deficit
Interest Rate
Bond Pr ice
0
( P )
B1S
B1D
BS2

P0
i*

P1
B1S
0
(Tresury Issue )
i**
B1D
BS2
(i )
Cases Using Demand and Supply Analysis
•
Economic Expansion:
In a period of economic expansion, there is an increase in
capital formation and therefore more bonds are being sold to
finance the capital expansion.
•
This leads to an increase in bond supply and a rightward shift
in the bond supply curve.
•
The expansion also increases bond demand, shifting the bond
demand curve to the right.
•
If the supply effect dominates the demand effects, then there
will be a decrease in bond prices and an increase in interest
rates.
Economic Expansion
Interest Rate
Bond Pr ice
0
( P )
B
B1S
D
2
B1D
BS2

P0
P1

B1D
B1S
BS2
0
i*
i**
B D2
(i )
Risk and Risk Premium
•
Investment risk is the uncertainty that the actual
rate of return realized from a security will differ
from the expected rate.
•
In general, a riskier bond will trade in the market at
a price that yields a greater YTM than a less risky
bond.
•
The difference in the YTM of a risky bond and the
YTM of less risky or risk-free bond is referred to
as a risk spread or risk premium.
Risk and Risk Premium
•
The risk premium, RP, indicates how much
additional return investors must earn in order
to induce them to buy the riskier bond:
RP = YTM on Risky Bond - YTM on Risk-Free Bond
•
We can use the supply and demand model to
show how the risk premium is positive.
Risk and Risk Premium
•
Consider the equilibrium adjustment that
would occur for two identical bonds (C and
T) that are priced with the same yields, but
events occur that make one of the bonds
more risky.
Risk and Risk Premium
•
The increased riskiness on the one bond (Bond C)
would cause its demand to decrease, shifting its
bond demand curve to the left. That bond’s
riskiness would also make the other bond (Bond T)
more attractive, increasing its demand and shifting
its demand curve to the right.
•
At the new equilibriums, the riskier bond’s price is
lower and its rate greater than the other.
•
The different risk associated with bonds leads to a
market adjustment in which at the new equilibrium
there is a positive risk premium.
Risk Premium
Market for Bond C
P ()
D
1
Market for Bond T
i () P ()
S
B
B
B1D
B D2( Risk )

BS

C
0
C
1
i
i
B D2
BD2 (Risk in C )
B1D
Bond Qu
The riskiness of Bond C decreases its demand,
shifting its bond demand curve to the left.
Impact: A Higher Interest Rate on Bond C


i ()
BS
i1T
i T0
B D2
B1D
Bond Qu
The riskiness of Bond C increases the demand
for Bond T, shifting its bond demand curve to
the right.
Impact: A Lower Interest Rate on Bond T
Risk Premiums and Investors’
Return-Risk Premiums
•
The size of the risk premium depends on
investors’ attitudes toward risk.
•
To see this relation, suppose there are only
two bonds available in the market: a risk-free
bond and a risky bond.
Risk Premiums and Investors’
Return-Risk Premiums
•
Suppose the risk-free bond is a zero-coupon
bond promising to pay $1,000 at the end of one
year and currently is trading for $909.09 to
yield a one-year risk-free rate, Rf, of 10%:
$1,000
P0 
 $909.09
1.10
$1,000
Rf 
 1  .10
$909.09
Risk Premiums and Investors’
Return-Risk Premiums
•
Suppose the risky bond is a one-year zero coupon bond
with a principal of $1,000.
•
Suppose there is a .8 probability the bond would pay its
principal of $1,000 and a .2 probability it would pay
nothing.
•
The expected dollar return from the risky bond is
therefore $800:
E(Return) = .8($1,000) + .2(0) = $800
Risk Premiums and Investors’
Return-Risk Premiums
•
Given the choice of two securities, suppose that the market
were characterized by investors who were willing to pay
$727.27 for the risky bond, in turn yielding them an expected
rate of return of 10%:
E(R ) 
E(R ) 
•
•
E(Re turn )
1
P0
$800
 1  .10
$727.27
In this case, investors would be willing to receive an expected
return from the risky investment that is equal to the risk-free
rate of 10%, and the risk premium, E(R) - Rf, would be equal to
zero.
In finance terminology, such a market is described as risk
neutral.
RP = 0 → Risk-Neutral Market
Risk Premiums and Investors’
Return-Risk Premiums
•
Instead of paying $727.27, suppose investors like the chance of
obtaining returns greater than 10% (even though there is a
chance of losing their investment), and as a result are willing to
pay $750 for the risky bond. In this case, the expected return
on the bond would be 6.67% and the risk premium would be
negative:
$800
E(R ) 
 1  .0667
$750
RP  E(R )  R f  .0667  .10   .033
•
By definition, markets in which the risk premium is negative
are called risk loving.
RP < 0 → Risk-Loving Market
•
•
Risk Premiums and Investors’
Return-Risk Premiums
Risk loving markets can be described as ones
in which investors enjoy the excitement of the
gamble and are willing to pay for it by
accepting an expected return from the risky
investment that is less than the risk-free rate.
Even though there are some investors who are
risk loving, a risk loving market is an
aberration, with the exceptions being casinos,
sports gambling markets, lotteries, and
racetracks.
Risk Premiums and Investors’
Return-Risk Premiums
•
Suppose most of the investors making up our market were
unwilling to pay $727.27 or more for the risky bond.
•
In this case, if the price of the risky bond were $727.27 and the
price of the risk-free were $909.09, then there would be little
demand for the risky bond and a high demand for the risk-free
one.
•
Holders of the risky bonds who wanted to sell would therefore
have to lower their price, increasing the expected return. On the
other hand, the high demand for the risk-free bond would tend
to increase its price and lower its rate.
Risk Premiums and Investors’
Return-Risk Premiums
•
Suppose the markets cleared when the price of the
risky bond dropped to $701.75 to yield 14%, and the
price of the risk-free bond increased to $917.43 to
yield 9%:
$800
 1  .14
$701.75
$1,000
Rf 
 1  .09
$917.43
E(R ) 
•
In this case, the risk premium would be 5%:
RP  E(R)  R f  .14  .09  .05
•
Risk Premiums and Investors’
Return-Risk Premiums
By definition, markets in which the risk premium is
positive are called risk-averse markets.
RP > 0 → Risk-Averse Market
•
In a risk-averse market, investors require
compensation in the form of a positive risk
premium to pay them for the risk they are
assuming.
•
Risk-averse investors view risk as a disutility, not a
utility as risk-loving investors do.
Risk Premiums and Investors’
Return-Risk Premiums
•
Historically, security markets such as the stock and corporate
bond markets have generated rates of return that, on average,
have exceeded the rates on Treasury securities.
•
This would suggest that such markets are risk averse.
•
Since most markets are risk averse, a relevant question is the
degree of risk aversion.
•
The degree of risk aversion can be measured in terms of the
size of the risk premium. The greater investors’ risk
aversion, the greater the demand for risk-free securities and
the lower the demand for risky ones, and thus the larger the
risk premium.
Liquidity and Liquidity Premium
•
Liquid securities are those that can be easily
traded and in the short-run are absent of
risk.
•
In general, we can say that a less liquid
bond will trade in the market at a price that
yields a greater YTM than a more liquid
one.
Liquidity and Liquidity Premium
•
The difference in the YTM of a less liquid
bond and the YTM of a more liquid one is
defined as the liquidity premium, LP:
LP = YTM on Less Liquid Bond - YTM on More-Liquid Bond
Liquidity and Liquidity Premium
•
Consider the equilibrium adjustment that would occur for
two identical bonds that are priced with the same yields, but
events occur that make one of the bonds less liquid.
•
The decrease in liquidity on one of the bonds would cause its
demand to decrease, shifting its bond demand curve to the
left. The decrease in that bond’s liquidity would also make
the other bond relatively more liquid, increasing its demand
and shifting its demand curve to the right.
•
Once the markets adjust to the liquidity difference between
the bonds, then the less liquid bond’s price would be lower
and its yield greater than the relative more liquid bond.
•
Thus, the difference in liquidity between the bonds leads to a
market adjustment in which there is a difference between
rates due to their different liquidity features.
Liquidity Premium
Market for Bond C
P ()
D
1
B

C
0
C
1
i
i
B D2
BD2 (Liquidity  in C)
B1D
)

BS
i () P ()
S
B
B D2( Liquidity
Market for Bond T
B1D
Bond Qu
The decrease in liquidity of Bond C decreases
its demand, shifting its bond demand curve to
the left.
Impact: A Higher Interest Rate on Bond C


i ()
BS
i1T
i T0
B D2
B1D
Bond Qu
The decrease in liquidity of Bond C increases
the demand for Bond T, shifting its bond demand
curve to the right.
Impact: A Lower Interest Rate on Bond T
Taxability
•
An investor in a 40% income tax bracket who
purchased a fully-taxable 10% corporate bond at
par, would earn an after-tax yield, ATY, of 6%:
ATY = 10%(1-.4).
•
In general, the ATY can be found by solving for
that yield, ATY, that equates the bond’s price to
the present value of its after-tax cash flows:
CFt (1  tax rate )
P0  
t
(
1

ATY
)
t 1
M
Taxability and Pre-Tax Yield Spread
•
Bonds that have different tax treatments but otherwise are
identical will trade at different pre-tax YTM.
•
That is, the investor in the 40% tax bracket would be
indifferent between the 10% fully-taxable corporate bond and
a 6% tax-exempt municipal bond selling at par, if the two
bond were identical in all other respects.
•
The two bonds would therefore trade at equivalent after-tax
yields of 6%, but with a pre-tax yield spread of 4%:
Pr e  Tax Yield Spread  i C0  i 0M  10%  6%  4%
Taxability and Pre-Tax Yield Spread
•
In general, bonds whose cash flows are subject
to less taxes trade at a lower YTM than bonds
that are subject to more taxes.
•
Historically, taxability explains why U.S.
municipal bonds whose coupon interest is
exempt from federal income taxes, have traded
at yields below default-free U.S. Treasury
securities even though many municipals are
subject to default risk.
Term Structure of Interest Rates
• Term Structure examines the relationship
between YTM and maturity, M.
• Yield Curve: Plot of YTM against M for
bonds that are otherwise alike.
Term Structure of Interest Rates
• A yield curve can be constructed from current
observations. For example, one could take all outstanding
corporate bonds from a group in which the bonds are
almost identical in all respects except their maturities, then
generate the current yield curve.
• For investors who are more interested in long-run average
yields instead of current ones, the yield curve could be
generated by taking the average yields over a sample
period (e.g., 5-year averages) and plotting these averages
against their maturities.
• Finally, a widely-used approach is to generate a spot yield
curve from spot rates.
Term Structure of Interest Rates
Shapes: Yield curves have tended to take on one of the three
shapes:
They can be positively-sloped with long-term rates being
greater than shorter-term ones.
1.
•
2.
Yield curves can also be negatively-sloped, with short-term
rates greater than long-term ones.
•
3.
Such yield curves are called normal or upward sloping curves. They
are usually convex from below, with the YTM flattening out at higher
maturities.
These curves are known as inverted or downward sloping yield curves.
Like normal curves, these curves also tend to be convex, with the yields
flattening out at the higher maturities.
Yield curves can be relatively flat, with YTM being invariant to
maturity.
Term Structure of Interest Rates
YTM
Normal
Flat
Inverse
M
Theories of the Term
Structure of Interest Rates
The actual shape of the yield curve depends on:
•
The types of bonds under consideration (e.g., AAA
bond versus B bond)
•
Economic conditions (e.g., economic growth or
recession, tight monetary conditions, etc.)
•
The maturity preferences of investors and
borrowers
•
Investors' and borrowers' expectations about future
rates, inflation, and the state of economy.
Theories of the Term
Structure of Interest Rates
Four theories have evolved over the years to
try to explain the shapes of yield curves:
1. Market Segmentation Theory (MST)
2. Preferred Habitat Theory (PHT)
3. Liquidity Premium Theory (LPT)
4. Pure Expectation Theory (PET)
Market Segmentation Theory
• MST: Yield curve is determined by supply
and demand conditions unique to each
maturity segment.
• MST assumes that markets are segmented
by maturity.
Market Segmentation Theory
• Example: The yield curve for high quality
corporate bonds could be segmented into
two markets:
– short-term
– long-term
Market Segmentation Theory
Short-Term Market
• The supply of short-term corporate bonds, such as
commercial paper would depend on business demand for
short-term assets such as inventories, accounts receivables,
and the like
• The demand for short-term corporate bonds would emanate
from investors looking to invest their excess cash for short
periods.
• The demand for short-term bonds by investors and the
supply of such bonds by corporations would ultimately
determine the rate on short-term corporate bonds.
Market Segmentation Theory
Long-Term Market
• The supply of long-term bonds would come from
corporations trying to finance their long-term assets (plant
expansion, equipment purchases, acquisitions, etc.).
• The demand for such bonds would come from investors,
either directly or indirectly through institutions (e.g.,
pension funds, mutual funds, insurance companies, etc.),
who have long-term liabilities and horizon dates.
• The demand for long-term bonds by investors and the
supply of such bonds by corporations would ultimately
determine the rate on long-term corporate bonds.
Market Segmentation
Theory: Illustration
Yield Curve for corporate bonds with two maturity segments:
ST and LT
Short-Term Market
Supply : Financing of S  T assets :
Accounts receivable , inventorie s, etc.
ST Bonds
B
rST
Demand : Investors with S  T
horizon dates
Market Segmentation
Theory: Illustration
• Long-Term Market:
Supply : Financing of L  T assets :
Plants , equipment , acquisitio ns , etc.
LT Bonds
B
rLT
Demand : Investors with
L  T Horizon Date
Market Segmentation Theory
• Important to MST is the idea of unique or
independent markets.
• According to MST, the short-term bond
market is unaffected by rates determined in
the intermediate or long-term markets, and
vice versa.
• This independence assumption is based on the
premise that investors and borrowers have a
strong need to match the maturities of their
assets and liabilities.
MST: Supply and Demand Model
• One way to examine how market forces determine the shape of
yield curves is to examine MST using our supply and demand
analysis.
• Consider a simple world in which there are two types of
corporate and government treasury bonds:
– Corporate bonds: long-term (BcLT) and short-term (BcST)
– Treasury bonds: long-term (BTLT) and short-term (BTLT).
• Assumptions: The supplies and demands for each sector and
segment are based on the following assumptions:
MST: Supply and Demand Model
Assumption 1: Short-Term Bond Demand for Corporate
and Treasury
• The most important factors determining the demand for shortterm bonds (both corporate and Treasury) are the bond’s own
price or interest rate, government policy, liquidity, and risk.
• Short-term bond demand is assumed to be inversely related to its
price and directly related to its own rate (negatively sloped bond
demand curves); government actions that affect the supply of
loanable funds also can change bond demand (e.g., monetary
policy changing bank reserve requirements).
• The demand for the short-term bond in one sector is also assumed
to be an inverse function of the short-term rate in the other sector,
but not the long-term rate in either its sector or the other sector
given the assumption of segmented markets.
MST: Supply and Demand Model
• Assumption 1: Short-Term Bond Demand for
Corporate and Treasury
c
c
T
BD ST
 f (iST
, iST
, risk , liquidity , government policy )
T
T
c
BD ST
 f (iST
, iST
, risk , liquidity , government policy )
MST: Supply and Demand Model
Assumption 2: Long-Term Bond Demand for Corporate and
Treasury
• The most important factors determining the demand for long-term
bonds (both corporate and Treasury) are the bond’s own price or
interest rate, government policy such as monetary actions (e.g.,
change in bank reserve requirements), liquidity, and risk.
• Demand is assumed to be inversely related to its own price and
directly related to its own rate (negatively sloped bond demand
curves).
• In addition, the demand for the long-term bond in one sector is an
inverse function of the long-term rate in the other sector, but not a
function of short-term rates given the market segmentation
assumption.
MST: Supply and Demand Model
• Assumption 2: Long-Term Bond Demand for
Corporate and Treasury
BD cLT  f (i cLT , i TLT , risk , liquidity , government policy )
BD TLT  f (i TLT , i cLT , risk , liquidity , government policy )
MST: Supply and Demand Model
Assumption 3: Long-Term and Short-Term Bond
Supplies for Corporate
• The supplies of short-term and long-term corporate
bonds are directly related to their own prices and
inversely to their own interest rates (positively
sloped corporate bond supply curve) and directly
related to general economic conditions, increasing
in economic expansion and decreasing in
recession.
c
c
BSST
 f (iST
, gdp )
BS
c
LT
 f (i , gdp )
c
LT
MST: Supply and Demand Model
Assumption 4: Long-Term and Short-Term Bond
Supplies for Treasury
• The supplies of Treasury bonds depend only on government
actions (monetary and fiscal policy), and not on the
economic state or interest rates.
• This assumption says that the sale or purchase of Treasury
securities by the central bank or the Treasury is a policy
decision. The assumption that the supply of Treasury
securities depends on government actions and not interest
rates means that the bond supply curve is vertical.
T
BSST
 f (government policy )
BSTLT  f (government policy )
MST: Supply and Demand Model
• In the exhibit, the two equilibrium rates for
short-term and long-term corporate bonds are
plotted against their corresponding maturities
to generate the yield curve for corporate bonds.
• Similarly, the equilibrium rates for short-term
and long-term Treasury bonds are plotted
against their corresponding maturities to
generate the yield curve for Treasury bonds.
Market Segmentation Theory Model
Corporate Bond Market
P ()
Short Term
D ( i , risk , liquidity , government
B ST
T
ST
Treasury Bond Market
policy )

P ()
B SST ( gdp )
Long
Term
D (i TLT , risk , liquidity
BLT
i
i
i
C*
LT
C*
ST
P ()
Short Term
D (i , risk , liquidity
B ST
C
ST
Bond Qu
i ()
, government policy )
i ()
, government policy )
B SST ( Gov. Policy )

C*
i ST
P ()
T*
i ST
Bond Qu
Long
Term
D ( i , risk , liquidity , government policy )
BLT
BSLT ( Gov. Policy )
C
LT

i CLT*

BSLT
i ()
i ()
i TLT*
( gdp )
Bond Qu
Bond Qu
i
Yield Curve

ST
i TLT*
T*
i ST

LT
M
Yield Curve

ST

LT
M
MST: Supply and Demand Model
• These yield curves, in turn, capture an MST
world in which interest rates for each segment
are determined by the supply and demand for
that bond, with the rates on bonds in the other
maturity segments having no effect.
• In general, the positions and the shapes of the
yield curves depend on the factors that
determine the supply and demand for shortterm and long-term bonds.
MST: Cases Using S&D Model
Economic Expansion:
• When an economy moves into a period of economic
growth, business demand for short-term and long-term
assets increases.
• As a result, many companies issue more short-term
bonds to finance their larger inventories and accounts
receivables. They also issue more long-term bonds to
finance their increase in investments in plants,
equipment, and other long-term assets.
• In the bond market, these actions cause the short-term
and the long-term supplies of bonds to increase as the
economy grows.
MST: Cases Using S&D Model
Economic Expansion:
• At the initial interest rates, the increase in bonds
outstanding creates an excess supply. This drives bond
prices down and the YTM up.
• Using the supply and demand model, the economic
expansion shifts the corporate short-term and long-term
bond supply curves to the right, creating an excess
supply for short-term bonds at ic*ST and an excess
supply for long-term bonds at ic*LT.
• The excess causes corporate bond prices to fall and
rates to rise until a new equilibrium is reached (ic**ST
and ic**LT).
MST: Cases Using S&D Model
Economic Expansion:
• As the rates on short-term and long-term corporate
bonds increase, short-term and long-term Treasury
securities become relatively less attractive.
• As a result, the demands for short-term and long-term
Treasuries decrease, shifting the short-term and longterm Treasury bond demand curves to the left and
creating an excess supply in both Treasury markets at
their initial rates.
• Like the corporate bond markets, the excess supply in
the Treasury security markets will cause their prices
to decrease and their rates to rise until a new
equilibrium is attained.
MST: Cases Using S&D Model
Economic Expansion:
• Thus, the supply and demand analysis shows that a
recession has a tendency to increase both short-term
and long-term rates for corporate bonds, and by a
substitution effect, increase short-term and long-term
Treasury rates.
• Hence, an economic expansion causes the yield
curves for both sectors to shift up.
Economic Expansion
Corporate Bond Market
P ()
Short Term
i ()
B SST
Long Term
BDLT
i CLT**
iSTC**
i CLT*
iSTC*
BSLT
Yield Curve


S
( gdp
ST
i
i
i

B SST( gdp )
P ()
i CLT*


i ()
i TLT*
i TLT**
Yield Curve
i TLT**

iSTT**
T*
LT
T*
ST
i
i
M
Bond Qu
Bond Qu
i

LT


T*
i ST
T **
i ST
Long DTerm
S
B
LT
BLT
BDLT (i CLT )
C*
LT
Bond Qu

ST
C*
ST
C**
ST
)
Bond Qu
i ()
i ()
Short Term B S
D
ST
B ST
D
C
BST
(i ST
)
B

i
P ()
D
B ST

P ()
Treasury Bond Market


ST


LT
M
MST: Cases Using S&D Model
Government surplus in which the Treasury
buys existing long-term Treasury bonds:
• When the Treasury uses a surplus to buy long-term
Treasury securities there is a decrease in the supply
of long-term Treasuries (leftward shift in the
Treasury LT bond supply curve).
• The decrease in supply would push the price of the
long-term government securities up, resulting in a
lower long-term Treasury yield.
MST: Cases Using S&D Model
Government surplus in which the Treasury buys
existing long-term Treasury bonds:
• In the corporate bond market, the lower rates on long-term
government securities would lead to an increase in the
demand for long-term corporate securities (rightward shift in
the corporate LT bond demand curve), which, in turn, would
lead to an excess demand in that market.
• As bondholders try to buy long-term corporate bonds, the
prices on such bonds would increase, causing the yields on
long-term corporate bonds to fall until a new equilibrium is
reached.
MST: Cases Using S&D Model
Government surplus in which the Treasury buys existing
long-term Treasury bonds:
• Thus, the purchase of the long-term Treasury securities
decreases both long-term government and long-term corporate
rates.
• Since the long-term market is assumed to be independent of
short-term rates, the total adjustment to the Treasury’s
purchase of long-term securities would occur through the
decrease in long-term corporate and Treasury rates.
• If corporate and Treasury yield curves were initially flat, the
Treasury’s action would cause the yield curves to become
negatively sloped.
Government surplus in which the Treasury buys existing long-term Treasury bonds
Corporate Bond Market
P ()
Short Term
D
B ST
Treasury Bond Market
i ()
B SST

P ()
Bond
Long Term D T
BLT (i LT )
BDLT

i
C*
*
i ST
 iC
LT
i
i
i
i
BSLT
Yield Curve

C**
LT
ST
P ()
Long Term
BSLT

*
T*
iT
LT  i ST
Bond Qu
S
LT
B
i ()
T**
LT
T*
LT
i
i

Bond Qu
Yield Curve



ST
LT
iTLT**
M
T*
i ST
BDLT
i


LT

C*
LT
C*
LT
Bond Qu
i ()
Short Term B S
D
ST
B ST
C*
ST
Qu
i ()

P ()
M
MST: Cases Using S&D Model
Contractionary open market operation in which the
Central Bank sells some of it short-term Treasury
securities:
• A contractionary OMO in which the Fed sells shortterm Treasury securities would cause the price on
short-term Treasury securities to decrease and their
yield to increase. This would be reflected by a
rightward shift in the short-term Treasury bond supply
curve, as the Central Bank sells it securities to the
public.
• As the yield on short-term Treasuries increases, the
demand for short-term corporate would decrease
(demand curve shifting left), leading to lower prices
and higher yields on short-term corporate bonds.
MST: Cases Using S&D Model
Contractionary open market operation in which the
Central Bank sells some of it short-term Treasury
securities:
• Since the long-term market is assumed to be
independent of short-term rates, the total adjustment
to the Central bank’s sale of short-term securities to
the public would be in the short-term corporate and
Treasury markets with no impact on the long-term
markets.
• If both the Treasury and corporate yield curves were
initially flat, then the contractionary OMO would
result in new negatively sloped yield curves.
Contractionary Open Market Operation: Central Bank sells short-term Treasuries
Corporate Bond Market
Short Term
D
B ST
P ()
D
ST
B (i
T
ST
B SST
)
i ()
P ()
Short Term
Bond Qu
i ()
Long Term
BDLT
S
LT
P ()
B SST
B SST

Long Term
T*
i ST
D
B ST
T **
i ST
Bond Qu
i ()
S
LT
B
B

i ()

C*
i ST
C**
i ST


P ()
Treasury Bond Market

i CLT*
i TLT*
BDLT
i
C**
i ST
C*
*
i ST
 iC
LT
Bond Qu
Yield Curve
i
iSTT**



ST
LT
*
T*
iT
LT  i ST
M
Bond Qu
Yield Curve



ST
LT
M
MST: Outline of Cases Using S&D Model
Recession
• Outline: Decrease in capital formation (S-T
and L-T)  Fewer bonds sold (S-T and L-T)
 Excess demand for bonds (S-T and L-T)
 Bond prices increase and rates decrease. 
Downward shift in YC
MST: Outline of Cases Using S&D Model
Expansionary open market operation
in which the central bank buys shortterm Treasury securities
• Outline: Central bank buys S-T Treasuries (T-bills)
 T-bill prices increase and rates decrease 
Substitution effect in which the demand for S-T
corporate securities increase, causing their prices to
increase and their yields to decrease.  Tendency
for YC to become positively sloped.
MST: Outline of Cases Using S&D Model
Treasury Sale of long-term Treasury
bonds
• Outline: Treasury sells L-T Treasuries (TBonds)  T-Bond prices decrease and yields
increase  Substitution effect in which the
demand for L-T corporate securities
decrease, causing their prices to decrease and
their rates to increase.  Tendency for YC to
become positively sloped.
Preferred Habitat Theory (PHT)
• PHT assumes that investors and borrowers are
willing to give up their desired maturity segment
and assume market risk if rates are attractive.
• PHT asserts that investors and borrowers will be
induced to forego their perfect hedges and shift out
of their preferred maturity segments when supply
and demand conditions in different maturity markets
do not match.
Preferred Habitat Theory (PHT)
• PHT is a necessary extension of the MST:
– If an economy is poorly hedged (e.g., more
investors want ST investments and more borrowers
want to borrow LT), then the market will not be in
equilibrium.
– In such cases, ST and LT rates will change and the
markets will clear as investors and borrowers give
up their hedge.
Preferred Habitat Theory (PHT)
• To illustrate PHT, consider an economic world in which, on
the demand side, investors in corporate securities, on
average, prefer short-term to long-term instruments, while
on the supply side, corporations have a greater need to
finance long-term assets than short-term, and therefore
prefer to issue more long-term bonds than short-term.
• Combined, these relative preferences would cause an
excess demand for short-term bonds and an excess supply
for long-term claims and an equilibrium adjustment would
have to occur.
Preferred Habitat Theory (PHT)
• In the long-term market, the excess supply would force
issuers to lower their bond prices, thus increasing bond
yields and inducing some investors to change their shortterm investment demands.
• In the short-term market, the excess demand would cause
bond prices to increase and rates to fall, inducing some
corporations to finance their long-term assets by selling
short-term claims.
• Ultimately, equilibriums in both markets would be reached
with long-term rates higher than short-term rates, a
premium necessary to compensate investors and
borrowers/issuers for the market risk they've assumed.
Preferred Habitat Theory
• Poorly Hedged Economy: Investors, on average, prefer
ST investments; corporate borrowers, on average, prefer
to borrow LT (sell LT corporate bonds):
Investors
prefer ST
Borrower
prefer LT
r  attracts

B
, rST  ST
Excess Demand in ST PST
LT Borrowers
r  attracts
B
Excess Supply in LT PLT
, rLT   LT
ST Investors
Liquidity Preference Theory
• Long-term bonds are more price sensitive to
interest rate changes than short-term bonds. As a
result, the prices of long-term securities tend to be
more volatile and therefore more risky than shortterm securities.
• The Liquidity Premium Theory (LPT), also
referred to as the Risk Premium Theory (RPT),
posits that there is a liquidity premium for longterm bonds over short-term bonds.
Liquidity Preference Theory
• According to LPT, if investors were risk averse,
then they would require some additional return
(liquidity premium, LP) in order to hold long-term
bonds instead of short-term ones.
LP  rLT  rST  0
Liquidity Preference Theory
• Thus, if the yield curve were initially flat, but had no
risk premium factored in to compensate investors for
the additional volatility they assumed from buying
long-term bonds, then the demand for long-term
bonds would decrease and their rates increase until
risk-averse investors were compensated.
• In this case, the yield curve would become positively
sloped.
Pure Expectations Theory
• Expectation theories address the question of what
impact expectations have on the current yield curve.
• One of these theories is the Pure Expectations Theory
(PET); also referred to as the unbiased expectations
theory (UET).
• PET posits that the yield curve is governed by the
condition that the implied forward rate is equal to the
expected sport rate.
Pure Expectations Theory
To illustrate PET:
• Consider a market consisting of only two bonds: a risk-free
one-year zero-coupon bond and a risk-free two-year zerocoupon bond, both with principals of $100.
• Suppose that supply and demand conditions are such that both
the one-year and two-year bonds are trading at an 8% YTM.
• Suppose that the market expects the yield curve to shift up to
10% next year, but, as yet, has not factored that expectation into
its current investment decisions.
• Finally, assume the market is risk-neutral, such that investors do
not require a risk premium for investing in risky securities (i.e.,
they will accept an expected rate on a risky investment that is
equal to the risk-free rate).
Pure Expectations Theory
Question:
• What is the impact of the expectation
on the current yield curve?
Pure Expectations Theory
• Consider investors with HD = 2 years
• Alternatives:
– Buy 2-year bond at 8%
– Buy a series of 1-year bonds: 1-year bond today at
8% and 1-year bond one year later at E(r11) = 10%.
The expected return from the series would be 9%:
YTM 2:Series  (1.08)(1.10)
1/ 2
1  .09
• In a risk-neutral world, investors with HD = 2
years would prefer the series of 1-year bonds
over the 2-year bond.
Pure Expectations Theory
• Consider investors with HD = 1 year.
• Alternatives:
– Buy 1-year bond at 8%.
– Buy a 2-year bonds at 8% for P2 = 100/(1.08)2 =
85.734, then sell it one year later at an expected price
of E(P11) = 100/(1.10) = 90.91. The expected rate of
return would be 6%:
90.9185.734
E(r11 ) 
 .06
85.734
• In a risk-neutral world, investors with HD = 1 year would
prefer the 1-year bond over the 2-year bond.
Pure Expectations Theory
• Thus, in a risk-neutral market with an expectation of
higher rates next year, both investors with one-year
horizon dates and investors with two-year horizon
dates would purchase one-year instead of two-year
bonds
• If enough investors do this, an increase in the
demand for one-year bonds and a decrease in the
demand for two-year bonds would occur until the
average annual rate on the two-year bond is equal to
the equivalent annual rate from the series of one-year
investments (or the one-year bond's rate is equal to
the rate expected on the two-year bond held one
year).
Pure Expectations Theory
• Investors with HD of 2 years and those with HD
of 1 year would prefer one-year bonds over twoyear bonds.
• Market Response:
BD2   P2B  r2 
B1D   P1B  r1 
r
10%
8%




1 yr
2 yr
PET
M
YC becomes
Positively Sloped
Pure Expectations Theory
• In the example, if the price on a two-year bond fell
such that it traded at a YTM of 9% and the rate on
a one-year bond stayed at 8%, then investors with
two-year horizon dates would be indifferent
between a two-year bond yielding a certain 9%
and a series of one-year bonds yielding 10% and
8%, for an expected rate of 9%.
• Investors with one-year horizon dates would
likewise be indifferent between a one-year bond
yielding 8% and a two-year bond purchased at 9%
and sold one year later at 10%, for an expected
one-year rate of 8%.
Pure Expectations Theory
• Thus in this case, the impact of the market's
expectation of higher rates would be to push
2-year rates up to 9%.
• Note: With YTM2 = 9% and YTM1 = 8%,
the implied forward rate is f11 = 10% -- the
same rate as the expected rate E(r11).
Pure Expectations Theory
• Assume that the market response is one in which
only the demand for 2-year bonds is affected by
the expectations.
BD2   P2B  r2 
Note : When YTM 2  9%, YTM 1  8%,
r2  until r2  r2:Series  .09 then f11  10%  E(r11 ).
Thus , if PET holds , then f Mt  E(rMt )
r
10%
9%
8%


1 yr


2 yr
PET
M
Pure Expectations Theory
• In the above example, the yield curve is positively
sloped, reflecting expectations of higher rates.
• By contrast, if the yield curve were currently flat
at 10% and there was a market expectation that it
would shift down to 8% next year, then the
expectation of lower rates would cause the yield
curve to become negatively sloped.
Pure Expectations Theory
• That is, given a yield curve currently flat at 10% and a
market expectation that it would shift down to 8% next
year, an investor with a two-year horizon date would prefer
the two-year bond at 10% to a series of one-year bonds
yielding an expected rate of only 9% (E(R) =
[(1.10)(1.08)]1/2 -1 = .09).
• Similarly, an investor with a one-year horizon would also
prefer buying a two-year bond that has an expected rate of
return of 12% (P2 =100/(1.10)2 = 82.6446, E(P11) =
100/1.08 = 92.5926, E(R) = [92.5926-82.6446]/82.6446
= .12) to the one-year bond that yields only 10%.
Pure Expectations Theory
• In markets for both one-year and two-year
bonds, the expectations of lower rates
would cause the demand and price of the
two-year bond to increase, lowering its rate,
and the demand and price for the one-year
bond to decrease, increasing its rate.
Pure Expectations Theory
Market expects the yield curve to
shift down from 10% to 8%.
Investors with two-year horizon dates
would prefer the two-year bond at 10%
to a series of one-year bonds yielding
an expected rate of only 9%:
(E(R) = [(1.10)(1.08)]1/2 -1 = .09)
Investor with a one-year horizon would
prefer buying a two-year bond that has
an expected rate of return of 12% to the
one-year bond that yields only 10%:
P2 =100/(1.10)2 = 82.6446
E(P11) = 100/1.08 = 92.5926
E(R) = [92.5926-82.6446]/82.6446 = .12
Market Re sponse :
Market Re sponse :
BD2   P2B  r2 
r
B1D   P1B   r1 
10%
8%


1 yr


2 yr
PET
YC become
negatively sloped
M
Pure Expectations Theory
• The adjustments would continue until the rate on
the two-year bond equaled the average rate from
the series of one-year investments, or until the rate
on the one-year bond equaled the expected rate
from holding a two-year bond one year (or when
the implied forward rate is equal to expected spot
rates).
• In this case, if one-year rates stayed at 8%, then
the demand for the two-year bond would increase
until it was priced to yield 9% - the expected rate
from the series: [(1.10)(1.08)]1/2 -1 = .09
Pure Expectations Theory
• Assume that the market response is one in which
only the demand for 2-year bonds is affected by
the expectations.
BD2   P2B  r2 
r2  until r2  r2:Series  .09
r
10%
9%
8%


1 yr


2 yr
PET
M
Features of PET
1. One of the features of the PET is that in
equilibrium the yield curve reflects current
expectations about future rates. From our
preceding examples:
• When the equilibrium yield curve was
positively sloped, the market expected higher
rates in the future
• When the curve was negatively sloped, the
market expected lower rates.
Features of PET
2. PET intuitively captures what should be considered as
normal market behavior.
– For example, if long-term rates were expected to be higher
in the future, long-term investors would not want to
purchase long-term bonds now, given that next period they
would be expecting higher yields and lower prices on such
bonds. Instead, such investors would invest in short-term
securities now, reinvesting later at the expected higher
long-term rates.
– In contrast, borrowers/issuers wishing to borrow long-term
would want to sell long-term bonds now instead of later at
possibly higher rates.
– Combined, the decrease in demand for long-term bonds by
investors and the increase in the supply of long-term bonds
by borrowers would serve to lower long-term bond prices
and increase yields, leading to a positively-sloped yield
curve.
Features of PET
3. If PET strictly holds (i.e., we can accept all of the
model's assumptions), then the expected future
rates would be equal to the implied forward rates.
As a result, one could forecast futures rates and
future yield curves by simply calculating implied
forward rates from current rates.
Features of PET
• The last feature suggests that given a spot yield
curve, one could use PET to estimate next period's
spot yield curve by determining the implied forward
rates.
• The exhibit on the next slide shows spot rates on
bonds with maturities ranging from one year to five
years (Column 2). From these rates, expected spot
rates (St) are generated for bonds one year from the
present (Column 3) and two years from the present
(Column 4). The expected spot rates shown are equal
to their corresponding implied forward rates.
Features of PET
(1)
(2)
(3)
(4)
Maturity
Spot Rates
Expected Spot
Rates One year
from Present
Expected Spot
Rates Two Years
from Present
1
2
3
4
5
10.0%
10.5%
11.0%
11.5%
12.0%
f11 = 11.0%
f21 = 11.5%
f31 = 12.0%
f41 = 12.5%
f12 = 12.0%
f22 = 12.5%
f32 = 13.0%
Features of PET
f 12
f 32
S 3  [(1  S1 )(1  f 11 )(1  f 12 )]1 / 3  1
S5  [(1  S1 )(1  f11 )(1  f12 )(1  f13 )(1  f14 )]1 / 5  1
S 3  [(1  S 2 ) 2 (1  f 12 )]1 / 3  1
S5  [(1  S 2 ) 2 (1  f 32 ) 3 ]1 / 5  1
(1  S 3 ) 3
f 12 
1
2
(1  S 2 )
 (1  S5 ) 5 
f 32  
2
 (1  S 2 ) 
(1.11) 3
f 12 
 1  .12
(1.105) 2
 (1.12) 5 
f 32  
2
 (1.105) 
1/ 3
1
1/ 3
 1  .13
General Formula :
 (1  SM  t )
 
t
(
1

S
)
t

Mt
f Mt



1/ M
 1
Features of PET
• According to PET, if the market is risk-neutral,
then the implied forward rate is equal to the
expected spot rate, and in equilibrium, the
expected rate of return for holding any bond for
one year would be equal to the current spot rate
on one-year bonds.
Features of PET
•
For example, the expected rate of return from purchasing a
two-year zero-coupon bond at the spot rate of 10.5% and
selling it one year later at an expected one-year spot rate equal
to the implied forward rate of f11 = 11% is 10%. This is the
same rate obtained from investing in a one-year bond:
90.09  81.8984
E(R ) 
 .10
81.8984
100
E(P11 ) 
 90.09
1.11
100
P20 
 81.8984
2
(1.105)
Features of PET
•
Similarly, the expected rate of return from holding a threeyear bond for one year, then selling it at the implied
forward rate of f21 is also 10%. That is:
80.43596  73.1191
 .10
73.1191
100
E(P21 ) 
 80.43596
2
(1.115)
100
P30 
 73.1191
(1.11)3
E(R ) 
•
Any of the bonds with spot rates shown in the exhibit
would have expected rates for one year of 10% if the
implied forward rate were used as the estimated expected
rate.
Features of PET
• Similarly, any bond held for two years and sold at
its forward rate would earn the two-year spot rate
of 10.5%. For example, a four-year bond
purchased at the spot rate of 11.5% and expected
to be sold two years later at f22 = 12.5%, would
trade at an expected rate of 10.5% - the same as
the current two-year spot.
Features of PET
•
Analysts often refer to forward rates as hedgable rates.
•
The most practical use of forward rates or expected spot
yield curves generated from forward rates is that they
provide cut-off rates, useful in evaluating investment
decisions.
•
For example, an investor with a one-year horizon date should
only consider investing in the two-year bond in our above
example, if she expected one-year rates one year later to be
less than f11 = 11%; that is, assuming she is risk-averse and
wants an expected rate greater than 10%.
•
Thus, forward rates serve as a good cut-off rate for
evaluating investments.
Websites
• Historical interest rate data on different bonds can
be found at the Federal Reserve site
www.federalreserve.gov/releases/h15/data.htm
and www.research.stlouisfed.org/fred2
• For information on Federal Reserve policies go to
www.federalreserve.gov/policy.htm
• For information on European Central Banks go to
www.ecb.int
Websites
• Current and historical data on U.S.
government expenditures and revenues
can be found at www.gpo.gov/usbudget.
• Yield curves can be found at a number of
sites:www.ratecurve.com and
www.bloomberg.com