Chapter 25
Factors Affecting Bond Yields
Fabozzi:Investment Management
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Learning Objectives
• You will understand why the yield on a Treasury
security is the base interest rate.
• You will study the factors that affect the yield spread
between two bonds.
• You will be able to describe what a yield curve is.
• You will learn what a spot rate and a spot rate curve
mean.
• You will calculate the theoretical spot rates from the
Treasury yield curve.
Learning Objectives
• You will explore what the term structure of interest
rates is.
• You will study why the price of a Treasury bond should
be based on spot rates.
• You will learn what is meant by a forward rate and be
able to calculate a forward rate.
• You will explore how long-term rates are related to the
current short-term rate and short-term forward rates.
• You will understand the different theories about the
determinants of the shape of the term structure: pure
expectations theory, the liquidity theory, the preferred
habitat theory, and the market segmentation theory.
Introduction
Bond yields depend on a variety of factors, including
the type of issuer, the characteristics of the bond issue,
and the state of economy. In this chapter we look at the
following factors
•Investor’s minimum interest rate or yield on U.S.
Treasury securities
•The difference in yields between a non- U.S. and a
U.S. Treasury security
•Bond maturity
The base interest rate
U.S. Treasury securities which are backed by the full faith
and credit of the U.S. government, are known throughout
the financial world as risk-free securities. Because of this,
the interest rate on Treasuries is the key interest rate in
world markets and the Treasury market is the most liquid
in the world.
Benchmark interest rate or base interest rate
•Minimum interest rate that investors will accept
•The yield on the most recently issued (on-the-run)
•Treasury security with a comparable maturity to the
investor’s potential bond purchase.
Risk premium
Risk premium – the spread between the rate on non-Treasury
securities and a particular on-the-run Treasury security. This is
the difference between the risk-free rate and the additional risk
inherent in other securities.
Or, base interest rate + risk premium
Factors affecting the spread
1.type of issuer
2.issuer’s perceived creditworthiness
3.term or maturity of the instrument
4.provisions that grant either issuer or investors the option
to do something
5.taxability of the interest received
6.expected liquidity of the security
Types of issuers
Market sectors (classifications of issuers)
-U.S. government and its agencies
-Municipal governments
-Corporations (foreign and domestic)
-Foreign governments
Types of issuers
Each sector has different risk/reward ratios as each
issuer (except the Treasury) has differing abilities to
meet their contractural obligations
Intermarket sector spread- difference in interest rates
offered by different sectors on security with same
maturity
Intramarket sector spread – spread between two issues
within a market sector
Perceived creditworthiness of the
issuer
Default or credit risk
The risk that the issuer will be unable to make
timely interest or principal payments. This risk is
assessed by commercial rating companies
Quality spread or credit spread
Spread between Treasuries and non-Treasuries that
are identical except for quality
Inclusion of options
Embedded options
Call provision – issuer can retire debt, in full or
partially, prior to maturity
Effect on spread
Investors require a larger spread on bonds with
embedded options comparable to Treasuries if the
option benefits the issuer
Investors require a smaller spread (or a negative
spread!) on bonds with embedded options
comparable to Treasuries if the option benefits the
investor.
Taxability of interest
Interest income is taxable at the federal level, as well as
possible taxes at the state and local level. The interest
from municipal bonds is exempt, which makes their
yields lower than similar Treasuries. The benchmark for
calculating spreads on tax-exempt bonds is a generic
AAA G.O. bond with specified maturity.
Taxability of interest
After-tax yield for taxable bond issues
After tax-yield = pretax yield x (1- marginal tax rate)
Equivalent taxable yield – taxable issue yield = after-tax yield on taxexempt issue
Equivalent taxable yield = tax exempt yield
(1- marginal tax rate)
Municipal bonds
General obligations
Revenue
Housing
Power
Hospitals
Insured
Expected liquidity of an issue and
term to maturity
Expected liquidity of an issue
The greater the expected liquidity, the lower required yield.
Term to maturity
The longer the term to maturity, the greater the bond price volatility
Short-term bonds – maturity between 1 and 5 years
Intermediate-term bonds – maturity between 5 and 12 years
Long-term bonds – maturity > 12 years
Maturity spread – spread between two maturity sectors
Term structure of interest rates – the relationship between yields on
identical bonds with different maturities
Yield curve
Insert Figure 25-1
The Treasury yield curve is a benchmark for pricing
bonds and setting yields in other areas of the debt
market. While this method has been used for a long time
due to the liquidity and creditworthiness of the Treasury
market, it has been inexact. That is, securities with the
same maturity may have different yields. How then
should we price bonds?
Why the yield curve should not be
used to price a bond
Recall that the price of a bond = PV of its cash flows and that one
interest rate (derived from a Treasury security) is used to discount all
cash flows.
Problems
Different cash flow patterns require different discount rates
Each bond should be viewed as package of zero-coupon
instruments, with each coupon date being its own maturity.
Finding the spot rate - Take the yield on a zero-coupon Treasury for
each maturity
Spot rate curve – graphical description of the rate and its maturity
All zero-coupon Treasuries maturities < 1 year, so we find
the theoretical spot rate curve.
Constructing the theoretical spot
rate curve for Treasuries
This curve is constructed from the yield on the following
Treasury securities:
1.on-the-run Treasury issues
2.on-the-run Treasury issues and selected off-the-run
Treasury issues
3.all Treasury coupon securities and bills
4.Treasury coupon strips
The methodology depends on which securities are used.
Bootstrapping occurs when on-the-run Treasury issues with or
without selected off-the-run Treasury issues are used.
On-the-run Treasury issues
These include the most recently auctioned issues of a given
maturity with each having an observed yield. Estimated yield is
used in the analysis when the issue is not trading at par. The
resulting yield curve is the par coupon curve.
Here we construct a theoretical spot rate curve with 60
semiannual spot rates – 6 month rate to 30 year rate. Seven
maturity points are available, with the rest extrapolated. This
yield is found by
Yield at higher maturity – yield at lower maturity
Number of semiannual periods between the two maturity points
On-the-run Treasury issues
Then, the yield for intermediate points is found by adding
to the yield at the lower maturity the amount from the
computation above.
Problems with this method are:
•Large gap between maturity points resulting in
misleading yields for those maturities estimated
•Yields for on-the-run issues may be misleading since the
true yield tends to be > quoted yield.
Estimating the theoretical spot rate curve
Insert Table 25-6
Use the 6 month Treasury bill with an annualized yield of 5.25% = spot
rate and 1 year Treasury with yield of 5.5% as the 1 year spot rate.
Coupon rate for a 1.5 year Treasury is 5.75%. Compute the spot rate for
a theoretical 1.5 year zero-coupon Treasury.
$100 = par
Cash flows are: 0.5 year
0.575 x $100 x 0.5
= $ 2.875
1.0 year
0.075 x $100 x 0.5
= $ 2.875
1.5 years
0.575 x $100 x 0.5 + 100
=$102.875
and given:
z1= one-half the annualized 6 month theoretical spot rate
z2= one-half the 1 year theoretical spot rate
z3= one-half the annual value of the 1.5 year theoretical spot rate
Estimating the theoretical spot rate curve
We know that
z1
= 0.02625 (0.0525/2)
z2
= 0.0275 (0.055/2)
therefore,
100 = 2.801461 + 2.723166 + 102.875
(1 + z3)3
94.47537 = 102.875
(1 + z3)3
(1 + z3)3 = 1.028798, z3 = 0.028798
We double this yield to find the bond-equivalent yield of 5.76%.
This procedure can be used to find the theoretical spot rates for all
maturities. See Table 25-7. Insert Table 25-7
On-the-run Treasury issues and
selected off-the-run Treasury issues
To lessen the problem of gaps between maturities, some
dealers use selected off-the-run Treasury issues (i.e. 20
year and 25 year issues). The linear extrapolation method
is used to fill in the gaps and bootstrapping is used to
construct the theoretical spot rate curve.
Insert Table 25-8
All Treasury coupon securities
and bills
Even when using only on-the-run issues with a few off-therun issues, information is embodied in Treasury prices is lost
to the analysis.
It is more appropriae to used all Treasury coupon securities
and bills to construct the theoretical spot rate curve.
Using the theoretical spot rate curve
Insert Table 25-9
What forces a Treasury to be price based on the spot
rates? Arbitrage.
By viewing the bond as a package of zero-coupon
instruments and applying the appropriate interest
rates, we avoid underpricing the bond relative to
its theoretical value, creating an arbitrage
opportunity.
Base interest rate = theoretical Treasury spot rate for
that maturity
Forward rates
Alternative 1: An investor buying a one-year instrument will
realize the on-year spot rate, a certain amount.
Alternative 2: An investor buying a six-month instrument and
then replacing it with another at maturity, will realize an
unknown amount. The logic is that in six months rates may be
higher. The value of f denotes some rate on a six month
instrument six months from now that will make the investor
indifferent between alternatives 1 and 2.
Insert Figure 25-2
Double this to give the bond-equivalent yield for the 6 month
rate, 6 months from now.
Forward rates: an example
Given:
6 month spot rate
1 year spot rate
= 0.0525, z1 = 0.02625
= 0.0550, z2 = 0.02750
That is, the annual rate for f on a bond equivalent basis = 5.75%
If the 6 month rate 6 months from now < 5.75%, then the investor
should go with alternative one since the total value will be higher.
It is important to understand that expectations of future rates are
built into the rates offered on investments with different maturities.
Forward rate – future interest rate calculated from spot rates or the
yield curve
Relationship between six-month
forward rates and spot rates
Given a t-period spot rate, the current spot rate and six
month forward rates (ft), the formula for the relationship is
zt = [(1 + z1)(1 + f1)(1 + f2)(1+ f3)…(1 + ft-1)] 1/t -1
Other forward rates
We can calculate the forward rates for any time in the
future for any investment horizon by using spot rates.
But do forward rates do a good job at predicting future
interest rates?
While studies have concluded that forward rates do not
predict rates well, forward rates show how an investor’s
expectations must differ from the market consensus to
choose the correct alternative.
Determinants of the shape of the
term structure
Four typical term structure shapes
Insert Figure 25-3
Two theories have evolved to explain these structures:
1.Expectations theories
-Pure expectations
-Liquidity
-Preferred habitat
2.Market segmentation theory
The pure expectations theory
Forward rates exclusively represent the expected future rates.
The entire term structure reflects the market’s expectations of future
short-term rates. Therefore, a normal curve indicates a rising trend in
short-term rates for the near future, a flat term structure indicates
constant rates, and a falling structure indicates declining rates.
Looking at the normal structure, the following scenario could exist:
1.Long-term horizon investors would shy away from long-term bonds
since an increase in rates would lower bond prices. They would buy
short-term bonds.
2.Speculators would sell long-term bonds before prices fell and shortsell others. They would reinvest proceeds in short-term debt
instruments.
3.Borrowers would tend to borrow now, rather than wait, as lending
rates are expected to increase.
The pure expectations theory
Long-term bonds would be sold and demand for shortterm debt instruments would rise, making long-term
yields rise. That is, the behavior initiated by expectations
would affect the predicted result.
The pure expectations theory: a
problem
This theory ignores the inherent risks of bonds brought on
by the uncertainty of future interest rates. More
specifically, the risks are:
1.Price risk
Uncertainty about the price of a bond at the end
of an investment horizon due to dependence on
future interest rates; it may be lower than
expected.
2.Reinvestment risk
Uncertainty about the rate at which proceeds can
be reinvested
The pure expectations theory:
interpretations
1.Investors expect the return for any investment horizon to be the
same regardless of maturity strategy. However, due to the price risk
associated with investing in bonds with a maturity greater than the
investment horizon, expected returns will differ significantly.
2.Local expectations theory states that the returns on bonds of
different maturities will be the same over a short-term investment
horizon. This interpretation can be sustained given equilibrium.
3.The return-to-maturity expectations interpretation states that the
return an investor will realize by rolling over short-term bonds to
some investment horizon will be the same as holding zero-coupon
bond with a maturity equal to that investment horizon. There is much
doubt about this interpretation.
The liquidity theory
Investors will hold longer-term maturities if they are
offered a forward rate that reflects both interest rate
expectations and a liquidity (or risk) premium.
According to this liquidity theory of the term structure,
implied forward rates will not be an unbiased estimate of
expectations of future interest rates because there is a
liquidity premium built in. It also assumes that the risk
premium rises uniformly with maturity.
The preferred habitat theory
Given mismatched demand and supply for funds in a maturity
range, lenders and borrowers will need to shift to maturities
showing the opposite imbalances. They will require a risk
premium reflecting the extent of aversion to price or
reinvestment risk.
The shape of the yield curve, under this theory, is
determined by both expectations of future interest rates
and the risk premium that will coax investors to shift out
of their preferred habitat.
Determination of shape
Upward and downward sloping as well as humped are
possible.
Market segmentation theory
This theory states that the shape of the yield curve is
affected by the asset/liability constraints and/or creditors
(borrowers) restricting their lending (financing) to specific
maturity sectors.
Assumes neither investors nor borrowers are willing
to shift maturity sectors to take advantage of forward
rate or expectations opportunities.
Determination of yield curve shape
Supply and demand for securities within each
maturity sector