Chapter 6 The Risk and Term Structure of Interest Rate
In the previous section, we have generalized our discussion of the influence of various factors on the behavior in interest rate by examining only a particular type of bonds: namely, the 1-year zero coupon bond. However, there are many types of bonds: bonds with different maturity, bonds issued by different parties (i.e. government vs. corporate), etc. As a result, there is a different interest rate for each type of bond.
We will look at the behavior of interest rates of two groups of bonds: (1) Bonds with the same features but are issued by different agency. In other words, we want to look at the risk structure of interest rates. (2) Bonds issued by the same agency but have different term to maturity (i.e. life of the bond). In other words, we want to look at the term structure of interest rates.
1. Risk structure of interest rate
As we have discussed in the previous section, the (relative) risk level of an asset affects its demands according to the theory of asset demand. The higher the relative risk level, the lower the demand of that asset. According to the theory of asset demand, this leads to an increase in interest rate. In other words, investors need to be compensated with a higher return (in the form of higher interest rate) in order to induce them to hold the assets. There are a number of factors that affect the risk level of a bond. In this section, we will focus on only 3 of them: default risk, liquidity, and tax consideration.
(i) Default risk
Default risk represents the probability (or chance) that the issuer of the bonds will not be able to pay the coupon payments on time and the principal on the maturity date. The bond issuer will have to declare bankruptcy if it defaults on its bond issues. In general, Treasury securities (i.e. Tbills, T-notes and T-bonds) are considered to have very low or negligible level of default risk relative to other types of debt instruments. As a result, Treasury securities are considered to be risk-free assets (or default-free bonds) and are often used as a benchmark to compare the interest rates of other debt instruments. The interest rates difference between a Treasury security and a
Chapter 6-113
non-Treasury security is the risk premium. The risk premium represents the additional compensation an investor needed to hold the non-Treasury security.
Price
Interest SB SB
DB Treasury bonds
Risk premium
DB Corporate bonds
From the above diagram, we see that at the equilibrium point, the interest rate for Treasury securities are lower than the interest rate of corporate debt instruments. This is because corporate debt instruments are considered to be “riskier” than Treasury securities because they have a higher level of default risk. As a result, investors needed additional compensation (in addition to the interest rate of Treasury securities) to induce them to hold corporate debt instruments. Using the loanable fund framework, we know that as the default risk of the corporate bonds increases, the demand for corporate bonds (or supply of loanable funds to the business sector) decreases, and the demand for Treasury bonds (or supply of loanable funds to the public sector) increases. The default risk of corporate bonds could increase due to a number of reasons: major profit losses, drop in market shares, etc. As a result, the risk premium of the corporate bonds increases as shown in the diagram below.
Price
Interest SB Risk premium SB
DB Treasury bonds
DB Corporate bonds
There are a few rating companies (such as the Standard & Poor and Moody’s) that provide ratings for municipal (i.e. state and local government debt instruments) and corporate bonds Chapter 6-213
based on their default risk levels. The ratings assigned by those companies significantly affect the interest rates of these bonds. A drop in the rating usually sends a signal to the investors that a bond’s default risk has increased. This will lead to an increase in the interest rate of the bonds because investors will be seeking a higher risk premium to compensate them of the higher risk level.
(ii) Liquidity
According to the theory of asset demand, if everything remains the same, the more liquid an asset, the higher the demand for that asset. Treasury securities are considered to be the most liquid assets, and hence they are used as the benchmark.
Suppose a Treasury bond and a corporate bond both have the same features, i.e. same level of risk, maturity, etc. As a result, the two types of bonds have the same price,
Price Interest SB iT iC Liquidity premium Corporate bonds Interest SB iT iC Liquidity premium Corporate bonds SB SB
DB Treasury bonds Price
DB
(and hence the same level of
DB
DB
interest rates,
Treasury bonds
). Suppose the
corporate bond becomes less liquid than the Treasury bond. In this case, the demand for corporate bonds decreases while the demand for Treasury bonds increases. As a result, the interest rate of corporate bonds
Chapter 6-313
Price
Interest SB iT iC Liquidity premium Corporate bonds Price Interest SB iT iC Liquidity premium Corporate bonds SB SB
DB
DB
(
Treasury bonds
) is higher than the interest
DB
DB
rate of Treasury bonds (
Treasury bonds
). The
difference between the two interest rates represents the liquidity premium (since the two bonds have the same level of default risk).
Price
Interest SB iT iC Liquidity premium Corporate bonds SB
DB Treasury bonds
DB
(iii) Income tax consideration
Since individuals have to pay taxes on their capital gains, it is important to compare the after-tax returns of assets rather than the before-tax returns. This will not be a big issue if you are comparing returns of taxable assets. However, this will be very important when we are comparing a taxable with a non-taxable asset.
Chapter 6-413
Example: Suppose the following two types of assets are available to John and Mary: Asset A has a taxable return of 10%, and asset B has a non-taxable return of 7%. Which asset should John and Mary choose if John faces a tax rate of 35% while Mary faces a tax rate of 25%?
The tax benefit of non-taxable asset makes the asset even more attractive comparing to a taxable asset. As a result, the tax benefits of non-taxable assets make them more attractive, and hence the demand for such asset increases. In the United States, municipal bonds (i.e. bonds issued by state and local governments) are exempt from federal taxes and state and local taxes for state residents.
Suppose the treasury and municipal bonds both have the same features to begin with, i.e. same risk level, maturity, tax status, etc. In this case, the two types of bonds have the same price,
Price Interest SB iB SB
iT DB Treasury bonds Price
"Tax" premium Municipal bonds Interest SB iB
DB
(and hence the same level of
SB
iT DB
"Tax" premium Municipal bonds
DB
interest rates,
Treasury bonds
). Suppose the
municipal bonds are granted the tax-free status. As a result, the municipal bonds become more attractive than the treasury bonds. The demand for municipal bonds increases while the demand for treasury bonds decreases. In this case, the interest rate of municipal bonds
Chapter 6-513
Price
Interest SB iB SB
iT DB
"Tax" premium Municipal bonds
DB
(
Treasury bonds Price
) is lower than the interest
SB
Interest SB iB
iT DB
"Tax" premium Municipal bonds SB
DB
rate of Treasury bonds (
Price
Treasury bonds Interest SB iB
).
iT DB Treasury bonds
"Tax" premium Municipal bonds
DB
2. Term structure of interest rate
In this section, we will focus solely on how the yield of a bond is affected by its term to maturity. The relationship between the yield to maturity of a bond and its term to maturity is known as the terms structure of interest rates, and it is represented graphically by the yield curve. You can look up the yield curve daily in the Credit section of the Wall Street Journal. It is important to know that the yield curve assumes all the bonds have the same risk, liquidity and tax status.
The yield curve can be of any of the following four shapes:
Chapter 6-613
1. Normal yield curve: The short-term yield is lower than YTM the long-term yield. In other words, it is cheaper to borrow short-term than it is to borrow long-term.
Time to maturity
2. Inverted yield curve: The short-term yield is higher than YTM the long-term yield. In other words, it is more expensive to borrow short-term than it is to borrow long-term.
Time to maturity
3. Flat yield curve: The short-term yield is the same as the YTM long-term yield. In other words, the short-term cost of borrowing is the same as the long-term cost of borrowing.
Time to maturity
4. Humped yield curve: The intermediate yield is higher YTM than both the short-term and long-term yields. In other words, it is cheaper to borrow short-term or long-term than it is to borrow intermediate-term.
Time to maturity
There are 3 different theories that can help explain the shape of a yield curve: (i) pure expectation theory, (ii) liquidity preference theory, (iii) market segmentation theory, and (iv) preferred habitat theory.
Chapter 6-713
(i) Pure expectation theory
This theory claims that the term structure of the interest rate is based on the current expectations of future short-term interest rates. In other words, long-term interest rates are simply the (geometric) mean of the short-term interest rate in the same time period.
There are a few assumptions that are important to the pure expectation theory. It is assumed that there is no transaction cost and investors form similar expectations regarding future interest rate. The main assumption behind this theory is that investors do not prefer bonds of one maturity to bonds of another maturity (as long as they can maximize their holding period returns).
For example, if an investor wants to invest his/her money for a period of two years, he/she is indifferent between the following two options: (i) Buys a 1-year bond and when it matures, reinvests the money in another 1-year bond. (ii) Buys a 2-year bond and holds it until it matures.
Since the investor is indifferent between the two options, the return from the two options should be identical. To simplify our analysis, we will assume the investor only has $1 to invest. As a result, we know that the returns of the two options are as follow:
(i) Rolling over 1-year bonds
YTM
Term premium
Time to maturity
(ii) Buying a 2-year bond
YTM
Term premium
Time to maturity
Chapter 6-813
YTM
Term premium
where
YTM
Time to maturity current 1-year interest rate
Term premium
Time to maturity 1-year interest rate 1 year from now YTM
Term premium
Time to maturity current 2-year interest rate
Since we know the returns of the two strategies are identical, we know the following must be true:
YTM
Term premium
Time to maturity
We can rewrite the above equation as follows:
Chapter 6-913
YTM
Term premium
Time to maturity
The above equation indicates that the long-term interest rate is simply the geometric mean of the short-term interest rates.
We can also look at the relationship between the long-term interest rates and short-term interest rates in a slightly different manner. By expanding the original equation, we know the above relationship can be rewritten as:
YTM
Term premium
Time to maturity
YTM
YTM
Term premium
Term premium
In general,
Time to maturity and
Time to maturity are so small that
they are negligible. As a result, we can rewrite the above relationship as:
YTM
Term premium
Time to maturity
Chapter 6-1013
In this case, we can see that the interest rate for the two-year bond is simply the arithmetic mean of the interest rate of the 1-year for this period and the expected interest rate of a 1-year bond for next period.
It is important to note that the geometric mean represents a more accurate relationship between the short-term interest rates and the long-term interest rates. However, the arithmetic mean is a lot easier to calculate.
Example: If the 1-year rate this year is 10% and it is expected to be 11% the next year, according to the expectation hypothesis, the 2-year rate this year should be:
(i) Using geometric mean
YTM
Term premium
Time to maturity
(ii) Using arithmetic mean
YTM
Term premium
Time to maturity
It is important to note that if the relationship between the long-term interest rates and short-term rates do not follow the one dictated by the pure expectation theory, it is possible for an investor to profit through arbitraging (i.e. making money out of nothing).
Chapter 6-1113
Example: Suppose the current 1-year and expected 1-year interest rates are 10% and 11%, respectively. According to the pure expectation theory, the current two-year interest rate should be 10.5%. What happen if the current 2-year interest rate is 10.7%?
In this particular scenario, it is possible for investors to profit through arbitraging. To simplify our illustration, we will use the arithmetic mean representation of the relationship between longterm and short-term interest rates. What kind of strategy can investor adopt to make money out of zero initial investment?
Strategy: Borrow $1000 in the short-term market (i.e. 1 year at 10%) and loan it out in the longterm market (i.e. 2 years at 10.7% a year).
1. Money borrowed After 1 year, the $1000 borrowed comes due and the investor owes a total of $1100
YTM
Term premium
(=
Time to maturity), and it will be rolled forward with another 1-year loan at an
interest rate of 11%. As a result, the total amount due at the end of the second year will be $1221
YTM
Term premium
(=
Time to maturity).
2. Money loaned
Chapter 6-1213
The investor has loaned out the $1000 borrowed at 10.7% a year for two years. At the end of the two-year
YTM
period,
the
investor
will
be
able
to
collect
an
amount
of
$1225.45
Term premium
(
Time to maturity ).
In this particular scenario, the investor owed $1221 for the $1000 borrowed, but was able to collect $1225.45 for the $1000 loaned. In other words, he/she is able to make a profit of $4.45 based on a zero investment. This might be a small amount, but it will grow as the amount borrowed/loaned gets bigger.
We
YTM
can
easily
generalize
the
relationship
between
longer-term
interest
rate
Term premium
(
Time to maturity ) and short-term interest rates as follows:
(i) Using geometric mean
YTM
Term premium
Time to maturity
(ii) Using arithmetic mean
Chapter 6-1313
YTM
Term premium
Time to maturity
According to the pure expectation theory, if investors expect short-term interest rate to: (1) Rise in the future, the yield curve would slope upward. (2) Remain constant, the yield curve would be flat. (3) Fall in the future, the yield curve would slope downward.
(ii) Liquidity preference (or liquidity premium) theory
The liquidity preference theory is very similar to the pure expectation theory, with one modification. This theory claims that long-term interest rate should be higher than short-term interest rate for the following reasons:
1. Savers have to be compensated for giving up cash (i.e. liquidity). And the longer the period of time they have to give up, the more they need to be compensated. 2. Long-term bonds are more sensitive to interest rate changes than short-term bonds. Hence, the return for a longer-term bond needs to be higher than a shorter-term bond. In other words, returns of long-term bonds need to include a liquidity premium to induce investors to buy them.
As a result, investors (or savers) need a positive liquidity (or term) premium to induce them to give up their money for a period of time. The longer the period of time they have to give up their money, the larger the term premium.
By incorporating the term premium, we can alter the following shapes of different forms of yield curve as predicted by the pure expectation theory:
Chapter 6-1413
YTM
YTM
Term premium
Time to maturity
Time to maturity
YTM
YTM
Time to maturity
Time to maturity
(iii) Market segmentation theory
The market segmentation theory assumes that bonds of different maturity are not substitutes from an investor’s point of view. This differs from the expectation theory that investors are indifferent to bonds of different maturity. As a result, the market segmentation theory assumes that there are different demands and supplies for bonds of different maturity. In other words, the short-term interest rate is determined by the demand and supply of short-term bonds, while the long-term interest rate is determined by the demand and supply of long-term bonds.
Using the market segmentation theory, we know that we will have a normal yield curve if there is a lower demand for (or higher supply of) short-term bonds relative to long-term bonds. On the other hand, we know that we will have an inverted yield curve if there is a higher demand for (or lower supply of) short-term bonds relative to long-term bonds.
(iv) Preferred habitat theory
The preferred habitat theory is a combination of the expectation theory and the liquidity preference theory. In other words, long-term interest rates are determined by investors expected future short-term interest rates and the habitat premium demanded. In other words, this theory assumes that bonds of different maturity are substitutes but investors have preference for bonds
Chapter 6-1513
of one maturity over bonds of another maturity (hence, the name preferred habitat). In this case, investors invest mostly in bonds of their preferred maturity, and invest in bonds of other maturity (usually longer maturity) only if they provide a high enough return (in the form of a habitat premium) to induce them to do so.
Example: Suppose investors expect 1-year interest rate to be declining from the current 10% to 9% to 8% to 7% to 6%. In addition, the habitat premium for 1-year to 5-year bonds are 0%, 0.2%, 0.4%, 0.6% and 0.8%. What are the current interest rates for 2-year, 3-year, 4-year and 5year bonds (using the arithmetic mean relationship)?
Interest Rate Peak
Trough
2-year bond:
Time Interest Rate Peak
Trough
3-year bond:
Time
Chapter 6-1613
Interest Rate Peak
Trough
4-year bond:
Time Interest Rate Peak
Trough
5-year bond:
Time
From the above example, we see that despite the investors requesting an increasingly positive habitat premium for longer-term bonds, it is possible to have a downward sloping yield curve if the investors expect a sharp decrease in expected future short-term interest rates.
How well do the theories explain the shape of the yield curve in the real world? So far, we have looked at four different theories (or models) that attempt to explain the shape of the yield curve. The question is how well do those theories explain some of the common facts regarding the shape of the yield curve in the real world. There are few common facts regarding the shape of the yield curve that we will look at more carefully to determine how the four theories can explain them.
Fact 1: The yield curve is upward sloping most of the time. Fact 2: The yield curve typically shifts rather than rotates. Fact 3: Short-term and long-term interest rates exhibit procyclical pattern, and short-term rates demonstrate larger amplitude in changes.
Chapter 6-1713
1. Explaining Fact 1: Yield curve is upward sloping 2. The pure expectation theory is not very good at explaining why the yield curve is upward sloping most of the time. According to the pure expectation theory, individuals are rational, which means that the individuals are equally likely to expect interest rates to rise as they are to fall. In other words, individuals expect the yield curve to be equally upward and downward sloping. 3. 4. The liquidity premium theory does a better job of explaining why we tend to see an upward sloping yield curve because of the increasing term premium. Since the preferred habitat theory is very similar to the liquidity preference theory, it is adequate in explaining why it is more common to see an upward sloping yield curve. 5. 6. Explaining Fact 2: Yield curve typically shifts rather than rotates. 7. With the pure expectation theory, assets with different maturities are assumed to be perfect substitutes. As a result, an increase in short-term interest rate will translate to an increase in long-term interest rate as individuals adjust their investment strategies. For example, the Federal Reserve System flooded the market with T-bills. As a result, the price of the T-bills will fall, which leads to an increase in short-term interest rate. Investors would like to take advantage of this increase in short-term interest rate and they sell longer-term Tbonds to do so. In this case, the price of the T-bonds drops, which leads to an increase in long-term interest rate. Hence, we see that an increase in short-term interest rate will lead to an increase in long-term interest rate. In other words, the yield curve shifts. Since the liquidity preference and preferred habitat theories are modification of the pure expectation theory, they are able to explain why yield curve generally shifts rather than rotates. 8. 9. The market segmentation theory is not adequate in explaining why the yield curve shifts and not rotates. That is because this theory assumes that assets of different maturities are not good substitutes for one another. Hence, an increase in the sale of T-bills by the Federal Reserve System will have an impact on the short-term interest rate but not on the long-term interest rate. As a result, we will see the yield curve rotating but not shifting. 10. 11. Explaining Fact 3: Short-term and long-term interest rates exhibit procyclical pattern, and short-term rates demonstrate larger amplitude in changes.
Chapter 6-1813
Short-term and long-term interest rates demonstrate procyclical pattern. In other words, they rise during an economic expansion and fall during an economic recession. We know that during an economic expansion, the interest rates increase as the economy picks up steam. Rational investors know that the interest rates are going to start falling as soon as the economy weakens (or slows down).
Interest Rate Peak
Trough
Time
According to the pure expectation theory (and its modifications), as the economy reaches its peak, investors know that interest rates will start falling. They will start selling short-term assets and use the funds to buy long-term assets (so that they can lock in the higher rate). As a result, the prices of the short-term assets fall and the prices of the long-term assets rise. This translates into an increase in short-term interest rate and a decrease in long-term interest rate. What this means is that the yield curve will become flatten or inverted. We can easily explain the situation when the economy reaches its trough.
What about the market segmentation theory? How well does it explain this particular fact? During economic expansion, the demand for loan increases and the banks will be force to sell off its holdings of short-term securities to free up funds to make the loans. As a result, the short-term interest rate will rise. Similarly, we can explain with this theory how the short-term interest rate is affected in an economic recession.
Chapter 6-1913