### Chapter 5

```Chapter 5
Factors Affecting Bond Yields and the Term
Structure of Interest Rates
Introduction



We have spent a lot of time discussing the
required yield (interest rate) on a bond.
However there is an IMPORTANT point to
remember: There is no single market yield.
Every bond has it’s own yield.
In this chapter, we learn about the factors that
affect bond yields.
Two Ways to Get Money
1. Buy it: Earn it in productive activities (equity).
2. Rent it: An interest rate is the price of renting
money (debt).

Like all other prices, interest rates reflect
supply and demand.


That is, the supply and demand for renting money.
There is an interest rate for each rental period.
Supply and Demand of Money
Interest
rate (r)
r
S\$
Why is D\$ downward sloping?
• If r - less money demanded
• If r - more money demanded
D\$
Quantity of money
U.S. Treasuries




Viewed as having zero default risk
Largest and most liquid market
Lowest rate (yield)
Rate (yield) is the base or benchmark rate
Treasury yields



Minimum yield investors will accept
Treasury yield
Examples of current U.S. Treasury rates
Bond Yield

All bond yields can be expressed as:
Bond yield = Base interest rate + Spread
Bond yield = Base interest rate + Risk premium


Our first task is to understand the base rate
(also called the benchmark rate)
Base Rate

Minimum rate an investor would ever accept for
investing in non-Treasury securities.


It is measured as the YTM of a comparable on-the-run Treasury
security.
Example: US Treasury yields on February 20, 2009:
Maturity
1-Mo
3-Mo
6-Mo
1-Yr
2-Yr
3-Yr
5-Yr
10-Yr
30-Yr
Yield (%)
0.19
0.27
0.48
0.64
0.96
1.30
1.81
2.78
3.56

If you want to invest in a 10-year bond you would
never accept less than a 2.78% yield.

similar maturity Treasury security.


risks an investor faces by buying a bond riskier than a
Treasury security.

Basis points (difference in yields) [most important]

Relative yield spread (% difference between yields)
Yield ratio (one yield divided by another)




The current 10-year US Treasury Note yields 2.78%.
A 10-year AAA rated corporate bond yields 4.96%.
What is the spread on the AAA rated bonds?


Basis points: 4.96-2.78 = 2.18% or 218 bps
Relative yield:
4.96  2.78
 0.78%
2.78

Yield ratio:
4.96
 1.78%
2.78

Type of Issuer (market segments):


Different market segments (and sub-segments) have different ability
to satisfy contractual obligations.
e.g., Municipals, corporates, agencies, etc.

Credit Worthiness of Issuer:



Spread between a Treasury and non-Treasury security that are
identical in all respects except for credit quality is call a credit
“Identical in all respects” means identical in terms of embedded
options, liquidity, taxability, etc.
Otherwise the spread reflects the value of items other than
default.

Inclusion of Options (e.g., put and call provisions):



Expected Liquidity of Issue:



Callable bonds will have ________ spreads over Treasury rates.
Putable bonds will have ________ spreads over Treasury rates.
Bonds with lower liquidity trade with higher yields because they are
more difficult to sell quickly for a fair price (hence are more risky).
Treasury securities have very high liquidity, although off-the-runs are
less liquid than on-the-runs.
Financeability:

Treasury bonds can be used as collateral for loans. The more
desirable a particular (usually on-the-run) T-bond is, the lower the
rate the lender will charge for the loan.

Term to Maturity:


Holding all other bond factors constant, the longer the
time until maturity of the bond, the more risky (volatile)
it will be when yields change.
Taxability of Interest:



Coupon payments are taxable at federal and state level.
Exception: Municipal bond interest is exempt at the
federal level (and state level in some cases).
This means municipal securities will pay lower coupon
rate (why?)
Taxability of Bonds

To compare the yields on municipals with yields on
taxable bonds, we need to look at after-tax yields:
after-tax yield = pretax yield  (1  marginal tax rate)

Example:



A 10-year A-rated corporate bond has a yield of 5.26% and a
10-year A-rated municipal bond has a yield of 3.73%.
If your federal tax rate is 35% which bond would you prefer?
(assuming all other features of the bonds are equal).


After-tax muni yield is 3.73%.
After-tax corporate yield is 3.42% [= 5.26  (1 – 0.35)]
Taxability of Bonds


We can also determine the yield that must be offered
on a taxable bond to give the same after-tax yield as a
tax-exempt issue.
This is called the equivalent taxable yield:
tax-exempt yield
equivalent taxable yield =
1  marginal tax rate

From the previous example, what taxable yield would
offer the same after-tax yield as 3.73% at 35% tax rate:
3.73
 5.74%
1  0.35
Term Structure of Interest Rates



The relationship between yield and maturity on
bonds that are identical in every way except
maturity.
The term structure is an important tool in
valuing bonds.
A graphical representation of the term structure
is called the yield curve.
Term Structure of Interest Rates
r
Flat
Notice:
(1) Term structure can change over time.
(2) Short-term rates more volatile than long-term rates.
Downward sloping
Upward sloping (most common)
Maturity (term)
Where Do We Get A Yield Curve?

It may seem a logical first step to use the YTM from
bonds of different maturities:
C
C
P


2
(1  y) (1  y)

However, YTM is accurate for yield curve construction
only if:


C
Face Value


N
(1  y)
(1  y) N
Coupons can be reinvested at a rate equal to YTM.
Otherwise YTM is incorrect.
Getting Yields for Yield Curve

A bond is a portfolio of zero-coupon bonds:
C
C
P


2
(1  y1 ) (1  y2 )

If each coupon is sold as a zero-coupon bond, then each
should be discounted at a different rate (reflecting the
maturity of the cash flow)




C
Face Value


N
(1  yN )
(1  yN ) N
1 year coupon should be discounted at 1-year rate.
2 year coupon should be discounted at 2-year rate.
…and so on.
Such rates are called spot rates.

These are rates on zero-coupon bonds.
Theoretical Spot Rate Curve

Treasury Bond valuation:





portfolios of zero coupon securities.
Each cash flow should be discounted by the
appropriate yield
Cannot use yields on coupon bonds.
Need a zero coupon yield curve.
There are no zero coupon Treasuries with
maturities > 1 year so a curve must be
constructed
Constructing the Spot Rate Curve

Which securities?





On the run Treasuries
On the run and selected off the run Treasuries
All Treasuries
Treasury coupon strips
Must use increasingly rigorous methods as
What Characteristics Should Spot
Rates Have?

Spot rates should reflect the required yield for a
single cash flow (i.e., a single maturity).
 Therefore,
bonds from which spot rates are determined
ideally should have no intermediate cash flows.

Be default risk free:
 Spot
rates should reflect the pure supply and demand of
loanable funds, not default risk.

Where do we get these rates?
 From
risk-free zero-coupon bonds.
 The spot rate curve comes from risk-free zero-coupon rates.
Where Do We Get Risk-Free Zeros?

Treasury coupon strips (logical starting point!):



Coupons are “stripped” from the bond and sold off
separately as individual zero-coupon bonds.
Resulting securities are called STRIPS (Separate Trading
of Registered Interest and Principal of Securities).
Problems with strips:


Liquidity of strips is less than the liquidity of the original
Treasury securities. Thus, yields on strips reflect a
Tax treatment of strips is different from that of original
treasury securities. Accrued interest on strips is taxed.
Where Else Can We Get Risk-Free
Zeros?
1.
2.
3.



On-the-run Treasury issues.
On-the-run issues along with selected off-the-run
Treasury issues.
All Treasury coupon securities and bills.
The securities above are coupon paying bonds, not
zero-coupon bonds!
So we will need some techniques to create the spot
(zero-coupon) yield curve from coupon-paying bonds.
The resulting yield curve is called the theoretical spot
rate curve.
On-The-Run Treasury Issues

We need to extract 60 spot (zero) rates from couponbearing Treasury bonds.



Why 60? There are 60 semi-annual coupon payments in 30 years.
Complication: There are usually only 6 or so on-the-run Treasuries
available.
How do we estimate the remaining 54 yields?
1.
2.
Construct par yield curve – yield curve constructed from 6 couponbearing Treasuries assuming the bonds are priced at par (yield equal
to the coupon rate). Use linear interpolation to fill in gaps.
“Convert” the par yield curve to theoretical zero coupon curve using a
technique called bootstrapping.
Par Yield Curve Construction

Suppose we have par yields for two on-the-run Treasuries:



2-year: 6.0%
5-year: 6.6%
From these par yields we can interpolate the 2½, 3, 3½, 4, and
4½ year par yields using the following formula:
6.6%  6.0%
Yield at longer maturity  Yield at shorter maturity

 0.10
6
Number of semi-annual periods between maturity points

Interpolated yields are:





2½ -year: 6.00% + 0.10 = 6.10%
3-year: 6.10% + 0.10 = 6.20%
3½-year: 6.20% + 0.10 = 6.30%
4-year: 6.30% + 0.10 = 6.40%
4½-year: 6.40% + 0.10 = 6.50%
Problems With Interpolation

There are large gaps between the 5-year and 10-year
bonds and 10-year and 30-year bonds.



Using such long distances between maturities can reduce the
accuracy of interpolation.
On-the-runs may be “special” in that they are desirable
as collateral for loans. This can distort yields.
Solution?


In addition to the on-the-runs, use some selected off-the-run
Treasuries to help fill in the gaps.
Usually the 20 and 25-year off-the-runs are used.
Bootstrapping



Bootstrapping enables us to take (par) yields from
coupon bonds and generate a spot yield curve.
Bootstrapping uses the concept that a coupon-bearing
bond is a portfolio of zero-coupon bonds and should
be priced accordingly.
Best illustrated by an example.
Example of Bootstrapping

Suppose we have the following on-the-run Treasury
securities (coupons paid semi-annually):
Bond
A
B
C
D

Maturity
(in years)
0.50
1.00
1.50
2.00
Yield/
Coupon Rate
5.25
5.50
5.75
6.00
All bonds have a
face value of \$100.
The first 2 bonds
are zero coupon
bonds (why?).
Our goal: Extract zero-coupon yields.
Bond
A
B
C
D
Solution


Yield/
Coupon Rate
5.25
5.50
5.75
6.00
Since the first 2 bonds are zero-coupon bonds (i.e., T-bills) their
par rates are spot rates.
The next spot rate we need is the 1½ year rate:


Since it has a 5.75% coupon rate, the coupon on this bond should be \$2.875
every six months.
The value of this bond is \$100 since it is based on par yields.
\$100 
\$100 

Maturity
(in years)
0.50
1.00
1.50
2.00
2.875
2.875 102.875


(1  z1 ) (1  z2 )2 (1  z3 )3
2.875
2.875
102.875


(1.02625) (1.0275) 2 (1  z3 )3
Solving for z3 we get: 0.028798

Double the yield to get an annual spot yield: 0.0576 (or 5.76%).
Bond
A
B
C
D
Solution

Yield/
Coupon Rate
5.25
5.50
5.76
6.00
Now we find the spot yield for the 2-year maturity:


Since it has a 6.0% coupon rate, the coupon on this bond should be \$3.00
every six months.
So, we can now find z4:
\$100 
\$100 

Maturity
(in years)
0.50
1.00
1.50
2.00
3.00
3.00
3.00
103.00



(1  z1 ) (1  z2 ) 2 (1  z3 )3 (1  z4 ) 4
3.00
3.00
3.00
103.00



(1.02625) (1.0275) 2 (1028798)3 (1  z4 ) 4
Solving for z4 we get: 0.030095

Double the yield to get an annual spot yield: 0.062 (or 6.02%).


This process continues until the entire spot
yield curve is constructed.
The bootstrapped yields are yields the market
would apply to zero-coupon Treasury bonds,
if such securities existed.
All Treasury Securities



Using only on-the-run issues (even with selected off-therun issues) fails to recognize all the information
contained in Treasury security prices.
Some argue that all Treasury securities and T-bills
should be used to construct the theoretical spot yield
curve.
If all securities are used, methodologies other than
bootstrapping must be used because there may be more
than one yield for each maturity:

The most common methodology is exponential spline fitting.
Revisiting the Theoretical Spot Curve

Bond yield = Base interest rate + Spread

Earlier, we said the base rate came from the YTM of
an on-the-run Treasury security.


We have to modify that explanation.
The base rate comes from the theoretical spot rate
curve that we just learned to construct.
Example using theoretical spot rates (exhibit 5-7)
Period
Year
Cash Flow
Spot Rate (%)
PV of \$1 at
PV of CF
1
0.5
5
5.25
0.974421
4.872107
2
1
5
5.5
0.947188
4.735942
3
1.5
5
5.76
0.918351
4.591756
4
2
5
6.02
0.888156
4.440782
5
2.5
5
6.28
0.856724
4.283619
6
3
5
6.55
0.824206
4.12103
7
3.5
5
6.82
0.790757
3.953783
8
4
5
6.87
0.763256
3.81628
9
4.5
5
7.09
0.730718
3.653589
10
5
5
7.2
0.701952
3.509758
11
5.5
5
7.26
0.675697
3.378483
12
6
5
7.31
0.650028
3.250138
13
6.5
5
7.43
0.622448
3.112238
14
7
5
7.48
0.597889
2.989446
15
7.5
5
7.54
0.573919
2.869594
16
8
5
7.67
0.547625
2.738125
17
8.5
5
7.8
0.521766
2.608831
18
9
5
7.79
0.502665
2.513325
19
9.5
5
7.93
0.477729
2.388643
20
10
105
8.07
0.453268
47.59317
Theoretical Value = 115.4206
Forward Interest Rates


Borrowers and lenders often enter into agreements to
make loans in the future.
This creates a need for a forward interest rate.


Forward rates are interest rates implied by current spot rates of
interest.
A forward rate is often viewed as the market’s consensus for future
interest rates.
Example of Forward Rates

Consider the following zero rates:
Period
(n)
1
2
3
4
5
Zero Rate
z1 = 5.25
z2 = 5.50
z3 = 5.76
z4 = 6.02
z5 = 6.28
zT = The current T-period spot (zero) interest rate (i.e., annual
interest rate that prevails from T0 to time T).
Forward Interest Rates


Spot Rate
z1 = 5.25
z2 = 5.50
z3 = 5.76
z4 = 6.02
z5 = 6.28
Consider the following two strategies:
(1) Invest \$100 at 2.75% for 1 year (i.e., 2 six-month periods:


Period (n)
1
2
3
4
5
\$100(1.0275)2 = \$105.5756.
(2) Invest \$100 at 2.625% for 6 months and then reinvest the
funds for 6 more months at the prevailing rate:


At the end of 6 months we would have: \$100(1.02625) = \$102.625
At the end of 1 year we would have: \$102.625(1 + f) = ?

We don’t know what f is (it’s a forward rate). However…

100(1.0275)2 = 100(1.02625)(1 + f):



Solving for f we get: f = 2.875%
To annualize, multiply: 5.75%
5.75% is the market’s consensus for the six-month rate six
months from now.
These
must be
equal
How to use the forward rate…

If the six-month rate six months from now is
less (greater) than 5.75%, then the total
dollars at the end of one year would be higher
(lower) by investing in the one-year
instrument.
Forward rates cont…

In general:
(1  zt 1 )t 1
ft 
1
t
(1  zt )

where ft is the 6-month forward rate beginning t 6-month periods from
now.
Generalizing Our Result…

The relationship between the t-period spot rate, current
six-month spot rate, and the six-month forward rates is:
(1  zt )t  (1  z1 )(1  f1 )(1  f2 )
zt  t (1  z1 )(1  f1 )(1  f 2 )
(1  ft 1 )
(1  ft 1 )  1
where ft is the 6-month forward rate beginning t 6-month periods from now.
t=0
t=1
z1
t=2
f1
t=3
f2
t=4
f3
Example

Suppose we have the following rates:





z1 = 0.0250
f1 = 0.0237
f2 = 0.0255
f3 = 0.0262
We can see how the 2-year spot rate is related to the various sixmonth forward rates:
z4  [(1.0250)(1.0237)(1.0255)(1.0262)]1/ 4 1  0.0251 (or 5.02%)
Calculating forward rates


Invest \$100 in each strategy
Alternative 1 payoff at end of year


Alternative 2 payoff at end of year


\$100(1 + z2)2
\$100(1 + z1)(1+f)
Investors are indifferent if:


\$100 (1 + z2)2 = \$100(1 + z1)(1+f)
f = ((1 + z2)2 /(1 + z1)) -1
Theories of the Term Structure


If we plot the yield curve, what will it look like?
Typical patterns are:




Upward sloping (called normal or positive yield curve)
Downward sloping (called inverted yield curve)
Flat
We want a theory that explains two things:


What explains the shape of the yield curve?
What explains two empirical regularities we observe:


1. Most of the time the yield curve is upward sloping.
2. Sometimes the yield curve is downward sloping
Expectations-Based Theories

Expectations Theories:




Three theories: (1) pure, (2) liquidity, (3) preferred habitat
Assume that forward rates are related to the market’s
The various theories differ in whether other factors also affect
forward rates.
Market Segmentation Theory.
Pure Expectations Theory


Under the Pure Expectations Theory, forward rates
equal expected future spot rates. [ft = E(zt)]
Thus, the yield curve reflects the market’s current
expectations of future short-term rates:




Rising yield curve → market expects ST rates to rise in the future.
Flat yield curve → market expects rates to be mostly constant.
Declining yield curve → market expects ST rates to decline.
The shape of the yield curve results from bond prices
level of interest rates.
Example of How Expectations
Affect Shape of Yield Curve


Suppose the yield curve is flat at 5% and investors
expect interest rates to increase over the next year to 7%.
An investor with a two-year horizon would weigh two
investing strategies:
1.
2.

Buy a two-year bond and hold it for two years (earning 5%)
Buy a one-year bond now and a 7% bond in one year (earning about
6% over two years)
Investors would choose alternative #2 and ignore #1:



1-year bond prices would increase, pushing down 1-year yields.
2-year bond prices would decline, pushing up 2-year yields.
The yield curve would go from flat to upward sloping.
Strengths and Weaknesses of PET

Strengths:


Weaknesses:



PET is consistent with the observation that yields tend to move
together in the yield curve.
PET does not explain the persistent upward slope of the yield curve.
Risk factors are ignored in PET, but seem to play a role in shaping the
yield curve.
What risk factors are ignored by PET?



Reinvestment risk
Price risk
Liquidity (term) risk.
Liquidity Theory



Investors are averse to risk.
σLTB > σSTB, so investors demand the less volatile ST
bonds unless compensated for buying LT bonds.
Therefore, liquidity theory says investors must be paid
a risk premium for investing in LT bonds:



And, the risk premium rises uniformly with maturity.
LT bond yields > ST bond yields and the yield curve will be upward
sloping.
Strength of Liquidity Theory:

Consistent with the persistent positive slope of the yield curve.
Preferred Habitat Theory

Suggests that different investors have a different
“maturity preference” for investing in bonds.




e.g., Insurance companies prefer to invest in long-term bonds whereas
banks prefer shorter-term instruments.
However, investors will leave their maturity habitat if
compensated (i.e., paid a risk premium) for doing so.
Firms issuing bonds in an “undesirable” maturity range
will have to offer a risk premium to attract investors.
Shape of yield curve is based on expectations as well as
risk premiums offered in different maturity habitats.
Market Segmentation Theory


Similar to the Preferred Habitat Theory except
investors are not willing to leave their
preferred maturity segment.
Each maturity segment of the yield curve has
it’s own supply and demand for funds that is
not related to any other segment.
Reality


What are the main influences of the shape of the
Treasury yield curve?
Research shows there are three main influences:



Expectations of future interest rate changes.
Bond risk premiums—demand for long-term bonds
pushes up prices and reduces yields relative to what we
would expect from expectations alone.
Convexity bias—attractive convexity properties of LT
bonds pushes LT bond prices up (and yields down)
relative to a pure expectations driven yield curve.
```