Chapter 3
Structure of Interest Rates
Introduction
• The annual interest rate offered at any given time varies among debt
securities.
• Individual and institutional investors must understand why quoted
yields vary so that they can determine whether the extra yield on a
given security outweighs any unfavorable characteristics
Why Debt security Yields vary
• Debt securities offer different yields because they exhibit different
characteristics that influence the yield to be offered.
• The yields on debt securities are affected by the following
characteristics:
Credit (default) risk
Liquidity
Tax status
Term to maturity
• In general, securities with unfavorable characteristics must offer
higher yields to entice investors to buy them.
1- Credit (Default) Risk
• Because most securities are subject to the risk of default, investors
must consider the creditworthiness of the security issuer.
• Thus, if all other characteristics besides credit risk are equal, securities
with a higher degree of credit risk must offer a credit risk premium
(higher yield above the Treasury bond yield) if they are to attract
investors.
Use of Ratings Agencies to Assess Credit Risk
• Investors can personally assess the creditworthiness of corporations
that issue bonds, but they may prefer to rely on bond ratings provided
by rating agencies.
• These ratings are based on a financial assessment of the issuing
corporation, with a focus on whether the corporation will receive
sufficient cash flows over time to cover its payments to bondholders.
• The higher the rating on the bond, the lower the perceived credit
risk is.
Use of Ratings Agencies to Assess Credit Risk
• The most popular rating agencies are Moody’s Investors Service and
Standard & Poor’s Corporation (shown in Exhibit 3.1).
• The ratings issued by Moody’s range from Aaa for the highest
quality to C for the lowest quality, and those issued by Standard &
Poor’s range from AAA to D.
• Commercial banks typically invest only in investment-grade bonds,
which are bonds rated as Baa or better by Moody’s and as BBB or
better by Standard & Poor’s.
Use of Ratings Agencies to Assess Credit Risk
• At a given point in time, the credit risk premium offered on a
corporate bond is higher for bonds that are rated lower and, therefore,
are more likely to default.
• The credit risk premium might be 1percent on highly rated corporate
bonds, 2.5percent on medium-quality corporate bonds, and 5percent
on low-quality bonds.
Investment Grade Bonds
Exhibit 3.1 Rating Classification by Rating Agencies
8
DESCRIPTION OF
SECURIT Y
RATINGS ASSIGNED
BY: MOODY ’S
RATINGS ASSIGNED
BY: STANDARD &
POOR’S
Highest quality
Aaa
AAA
High quality
Aa
AA
High–medium quality
A
A
Medium quality
Baa
BBB
Medium–low quality
Ba
BB
Low quality (speculative)
B
B
Poor quality
Caa
CCC
Very poor quality
Ca
CC
Lowest quality (in default)
C
DDD, D
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Credit Ratings and Risk Premiums over Time
• Rating agencies can change bond ratings over time in response to
changes in the issuing firm’s financial condition.
• During weak economic conditions, corporations generate reduced
cash flows and may struggle to cover their debt obligations.
• The credit risk premium required by investors to invest in corporate
bonds then rises because of concerns that the firms’ credit risk is
rising.
Credit Ratings and Risk Premiums over Time
Example:
Medium-quality bond yields might contain a credit risk premium of 2.5
percent during normal economic conditions, but contain a credit risk
premium of 5 percent or higher during a recession.
2- Liquidity
• Investors prefer securities that are liquid, meaning that they can be
easily converted to cash without a loss in value.
• Thus, if all other characteristics are equal, securities with less liquidity
must offer a higher yield to attract investors.
• Debt securities with a short-term maturity or an active secondary
market have greater liquidity.
• Investors who need a high degree of liquidity (because they may
need to sell their securities for cash at any moment) prefer liquid
securities, even if it means accepting a lower return on their
investment.
• Investors who will not need their funds until the securities mature are
more willing to invest in securities with less liquidity so that they can
earn a slightly higher return.
3- Tax Status
• Investors are more concerned with after-tax income than with beforetax income earned on securities.
• If all other characteristics are similar, taxable securities must offer a
higher before-tax yield than do tax-exempt securities.
• The extra compensation required for taxable securities depends on the
tax rates of individual and institutional investors.
• Investors in high tax brackets benefit most from tax-exempt securities.
3- Tax Status
• When assessing the expected yields of various securities with similar
risk and maturity, it is common to convert them into an after-tax
form, as follows:
3- Tax Status
Example
Consider a taxable security that offers a before-tax yield of 8percent.
When converted to after-tax terms, the yield will be reduced by the tax
percentage. The precise after-tax yield depends on the tax rate T. If the
investor’s tax rate is 20percent, then the after-tax yield will be
3- Tax Status
• Exhibit 3.2 presents after-tax yields based on a variety of tax rates and
before-tax yields.
• For example, a taxable security with a before-tax yield of 4percent will
generate an after-tax yield of 3.60percent to an investor in the
10percent tax bracket, 3.12percent to an investor in the 22percent tax
bracket, and so on.
• This exhibit shows why investors in high tax brackets are attracted to
tax-exempt securities.
Exhibit 3.2 After-Tax Yields Based on Various Tax Rates and
Before-Tax Yields
BEFORE-TAX BEFORE-TAX BEFORE-TAX BEFORE-TAX BEFORE-TAX
YIELD TAX
YIELD 2%
YIELD 4%
YIELD 6%
YIELD 8%
RATE
16
10%
1.80%
3.60%
5.40%
7.20%
15
1.70
3.40
5.10
6.80
25
1.50
3.00
4.50
6.00
28
1.44
2.88
4.32
5.76
35
1.30
2.60
3.90
5.20
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Computing the Equivalent Before-Tax Yield
• In some cases, investors wish to determine the before-tax yield
necessary to match the after-tax yield of a tax-exempt security with a
similar risk and maturity.
• This can be done by rearranging the terms of the previous equation:
Computing the Equivalent Before-Tax Yield
• For instance, suppose that a firm in the 20percent tax bracket is aware
of a tax-exempt security that is paying a yield of 8percent.
• To match this after-tax yield, taxable securities must offer a before-tax
yield of
4- Term to Maturity
• Maturity dates will differ between debt securities.
• The term structure of interest rates defines the relationship between
possible terms to maturity and the annualized yield for a debt security
at a specific moment in time while holding other factors constant.
Modeling the Yield to be Offered on a Debt Security
When a company wants to issue debt, it needs to consider all the
characteristics just described so that it can determine the appropriate
yield to offer that will entice investors to buy its debt securities. The
following model incorporates the key characteristics for determining the
appropriate yield to be offered on a debt security:
Yn = Rf,n + DP + LP + TA
where:
Yn = yield of an n-day debt security
Rf,n = yield of an n-day Treasury (risk-free) security
DP = default premium to compensate for credit risk
LP = liquidity premium to compensate for less liquidity
TA = adjustment due to difference in tax status
20
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Example
Assume that Elizabeth Co. plans to issue 10-year bonds. It wants to
determine the yield that it must offer to successfully sell its debt
securities. First, it checks the annualized yield on a risk-free (Treasury)
bond with the same 10-year term to maturity, which is presently
6percent. Next, Elizabeth Co. must consider how its characteristics
affect the premiums that it needs to offer (above the prevailing risk-free
Treasury security with the same maturity) to sell its debt securities.
Assume Elizabeth Co. believes that a 2.1percent credit risk premium, a
0.4percent liquidity premium, and a 0.2percent tax adjustment are
necessary to persuade investors to purchase its bonds. The appropriate
yield to be offered on the bonds is
The Term Structure of Interest Rates
• The term-to-maturity of a loan is the length of time until the
principal amount is payable.
• The relationship between a security’s yield-to-maturity and term-to
maturity is known as the term structure of interest rates.
• We can view the term structure visually by graphically plotting yieldto-maturity and maturity for equivalent-grade securities at a point in
time.
• The plots are then connected in a smooth line called the yield curve.
The Expectation Theory
• The expectation theory holds that the shape of the yield curve is
determined by the investors’ expectations of future interest rate
movements and that changes in these expectations change the shape of
the yield curve.
• it assumes that investors are profit maximizers and that they have no
preference between holding a long-term security and holding a
series of short-term securities
• To see how changing expectations of interest rate movements can alter
the slope of the yield curve, let’s look at an example.
Case 1
• Suppose that an investor has a 2-year investment horizon and that
only 1-year bonds and 2-year bonds are available for purchase.
• Because both types of securities currently yield 6 percent, the
prevailing term structure is flat, as indicated by the yield curve shown
below.
The Expectation Theory
• Now suppose that new economic information becomes available and
investors expect interest rates on 1-year securities to rise to 12
percent within a year (Forward Rate).
the interest rate that will exist 1 year in the future
• Under such circumstances, investors would want to buy 1-year bonds
and sell any 2-year bonds they might own.
• To invest over a 2-year investment horizon, a profit-maximizing
investor in our example would examine the alternatives of (1) buying
a 2-year bond or (2) buying two successive 1-year bonds.
The Expectation Theory
• The investor would then select the alternative with the highest yield
over the 2-year holding period.
• Specifically, if an investor buys a 1-year bond that currently yields 6
percent and at the end of the year buys another 1-year bond expected
to yield 12 percent, the average expected holding-period yield for the
2-year period is 9 percent .
• Alternatively, if the investor purchases a 2-year security, the 2-year
holding-period yield is only 6 percent.
The Expectation Theory
• The process of buying 1-year bonds and selling 2-year bonds
continues until any differential in expected returns over the 2-year
investment period is eliminated.
• That condition could occur when the yields on 1-year securities equal
4 percent, the 2-year securities yield 8 percent, and the 1-year
forward rate remains 12 percent.
• With this term structure, an investor who purchases a 2-year bond has
a 2-year holding-period yield of 8 percent.
• This is identical to the investor who purchases a 4 percent 1-year
security and then reinvests the proceeds at the end of the first year in
the bonds expected to yield 12 percent.
• The expected yield for this strategy is 8 percent .
The Term Structure Formula
• the example in the preceding section illustrates that investors can trade
among securities of different maturities and, if they are profit
maximizers, obtain an equilibrium return across the entire
spectrum of maturities.
• a formal relationship between long- and short-term interest rates.
Specifically, the long-term rate of interest is a geometric average of the
current short-term interest rate and a series of expected short-term
forward rates.
The Term Structure Formula
• More formally, the yield on a bond maturing n years from now is
The Term Structure Formula
• The postscript identifies the maturity (n) of the security, and the
prescript represents the time period in which the security originates (t).
• Thus, tR1 is the actual market rate of interest on a 1-year security
today (time t), also called the spot rate.
• tR10 is the current market rate of interest for a 10-year security.
• For the forward rates, the prescript still identifies the time period in
which the security originates, but now it represents the number of
years in the future; thus t+1f1 refers to the 1-year interest rate 1 year in
the future; likewise, t+2f1 is the 1-year interest rate 2 years from now,
and so on.
Example
• Suppose the current 1-year rate is 6 percent. Further- more, the market
expects the 1-year rate a year from now to be 8 percent and the 1-year
rate 2 years from now to be 10 percent. Using our notation,
Example
• Given the market’s expectation of future interest rates, we can
calculate the current 3-year rate of interest by applying the previuos
Equation:
• Notice that an investor with a 3-year investment horizon will be
indifferent about buying a 3-year security yielding 7.99 percent or
three successive 1-year securities that also yield, on average, 7.99
percent
Term Structure And Liquidity Premiums
• Investors know from experience that short-term securities provide
greater marketability (more active secondary markets) and have
smaller price fluctuations (price risk) than do long-term securities.
• As a result, borrowers who seek long term funds to finance capital
projects must pay lenders a liquidity premium to purchase riskier
long-term securities.
• Thus, the yield curve must have a liquidity premium added to it.
• The liquidity premium increases as maturity increases because the
longer the maturity of a security, the greater its price risk and the less
marketable the security.
The Market-segmentation Theory
• The market-segmentation theory maintains that market participants have
strong preferences for securities of a particular maturity, and that they buy and
sell securities consistent with these maturity preferences.
• As a result, the yield curve is determined by the supply of and the demand for
securities at or near a particular maturity.
• Assumes that both issuers and investors have a preference for securities with a
narrow maturity range.
• Changes in interest rates in one segment of the yield curve, therefore, have little
effect on interest rates in other maturities.
• Under the segmentation theory, discontinuities in the yield curve are possible.
• The segmentation theory assumes that certain investors are almost completely
risk averse, which means that they do not shift the maturity of their holdings in
exchange for higher yields.
The preferred-habitat Theory
• asserts that investors will not hold debt securities outside their preferred
habitat (maturity preference) without an additional reward in the form of a
risk premium.
• Unlike market-segmentation theory, the preferred-habitat theory does not
assume that investors are completely risk averse; instead, it allows
investors to reallocate their portfolios in response to expected yield
premiums.
• The preferred-habitat theory can explain humps or twists in the yield curve
but does not allow for discontinuities in the yield curve, as would be
possible under the segmentation theory.