Transcript Chapter 5

Chapter 5
Pages 94-115 only
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
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)
Then we move to the spread or risk premium.
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 October 7, 2005:
Maturity
1-Mo
3-Mo
6-Mo
1-Yr
2-Yr
3-Yr
5-Yr
10-Yr
30-Yr
Yield (%)
3.33
3.61
3.99
4.17
4.19
4.21
4.23
4.36
4.57

If you want to invest in a 10-year bond you would
never accept less than a 4.36% yield.
Spread (Risk Premium)

Non-Treasury securities trade at a spread over a
similar maturity Treasury security.

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The spread is a risk premium that reflects the additional
risks an investor faces by buying a bond riskier than a
Treasury security.
How do we measure spreads?
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Basis points (difference in yields) [most important]

Relative yield spread (% difference between yields)
Yield ratio (one yield divided by another)

Example of Spreads
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The current 10-year US Treasury Note yields 3.60%.
A 10-year AAA rated corporate bond yields 4.96%.
What is the spread on the AAA rated bonds?
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Basis points: 4.96-3.60 = 1.36% or 136 bps
Relative yield:
4.96  3.60
 0.38
3.60

Yield ratio:
4.96
 1.38
3.60
Factors Affecting Yield Spreads

Type of Issuer (market segments):
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Different market segments (and sub-segments) have different ability
to satisfy contractual obligations.
e.g., Municipals, corporates, agencies, etc.
Credit Worthiness of Issuer:


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Spread between a Treasury and non-Treasury security that are
identical in all respects except for credit quality is call a credit
spread.
“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.
Factors Affecting Yield Spreads

Inclusion of Options (e.g., put and call provisions):

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Expected Liquidity of Issue:

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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. The reduced rate increases
the spread between on-the-run and off-the-run Treasuries.
Factors Affecting Yield Spreads
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Term to Maturity:
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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:
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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:
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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).
Answer:

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After-tax muni yield is 3.73%.
After-tax corporate yield is 3.42% [= 5.26  (1 – 0.35)]
Taxability of Bonds
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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

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Is 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)

C
Face Value


N
(1  y)
(1  y) N
However, YTM is accurate for yield curve construction
only under the following conditions:
(1) Yield curve is flat.
(2) Coupons can be reinvested at a rate equal to YTM.

Otherwise YTM is incorrect. Why?

A bond is a portfolio of zero-coupon bonds.
Getting Yields for Yield Curve

A portfolio of zero-coupon bonds:
P


C
Face Value

(1  yN ) N
(1  yN ) N
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)
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
C
C


2
(1  y1 ) (1  y2 )
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.
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 yield 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
liquidity risk premium.
Tax treatment of strips is different from that of original
treasury securities. Accrued interest is taxed, creating a
negative coupon (strip yields reflect this tax disadvantage).
Where Else Can We Get Risk-Free
Zeros?
1.
2.
3.

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On-the-run Treasury issues.
On-the-run issues along with selected off-the-run
Treasury issues.
All Treasury coupon security coupons and bills.
Note: most of 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
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We need to extract 60 spot (zero) rates from couponbearing Treasury bonds.
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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 equals
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
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Suppose we have par yields two on-the-run Treasuries:
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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:

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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.

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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
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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
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
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:
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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%).
Comments On Bootstrapping
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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 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:
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The most common methodology is exponential spline fitting.
Revisiting the Theoretical Spot Curve

Let’s return to the base rate in the formula:
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.
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 one 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
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:
zt  [(1.0250)(1.0237)(1.0255)(1.0262)]1/4 1  0.0251 (or 5.02%)
Another Generalization


Suppose given spot rates we want to find the forward
rate.
We use the following formula:
(1  zt n )t n  (1  zt )t (1  ft ,n )n

Where:




zt = the annualized t-period spot rate.
zt+n = the annualized (t+n)-period spot rate.
ft,n = the an n-period forward rate beginning in period t.
Best to illustrate by example.
Another Example


Suppose we have the following
annualized semi-annual yields:
What is the two-year forward rate on a
three-year bond?
Year
Period Yield
0.50
1
3.0%
1.00
2
3.4
1.50
3
3.8
That is, in two years, what will be the
expected yield on a three-year bond?
We use the following formula:
2.00
4
4.2
2.50
5
4.8
3.00
6
5.4
3.50
7
5.8
4.00
8
6.4
4.50
9
6.8
5.00
10
7.2


(1  zt n )t n  (1  zt )t (1  ft ,n )n
(1.036)10  (1.021)4 (1  f4,6 )6
10
f 4,6
(1.036)

 1  4.61% (or 9.22%)
4
(1.021)
6