Municipal Bond Interest Rates by MaryJeanMenintigar

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									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. T-
bills, 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




                                                Risk
                                        DB    premium                     DB

                       Treasury bonds                         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                           SB
                                              premium




                                        DB                                DB

                       Treasury bonds                         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                                             SB
                                   iT


                                            iC
                                   Liquidity
                            DB     premium                                 DB

           Treasury bonds                                Corporate bonds        (and hence the same level of
                  Price                     Interest
                                            SB                                           SB
                                                    iT


                                                             iC
                                                    Liquidity
                                            DB      premium                              DB

interest rates,           Treasury bonds                              Corporate 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                                              SB
                                     iT


                                              iC
                                     Liquidity
                             DB      premium                                 DB

(           Treasury bonds                                 Corporate bonds        ) is higher than the interest
                             Price                          Interest
                                                            SB                                       SB
                                                                 iT


                                                                           iC
                                                                  Liquidity
                                                            DB    premium                            DB

rate of Treasury bonds (                  Treasury bonds                           Corporate 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                                         SB
                                                      iT


                                                               iC
                                                      Liquidity
                                                 DB   premium                              DB

                          Treasury bonds                                 Corporate bonds




(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                                             SB
                                           iB


                                iT
                                  "Tax"
                           DB    premium                                  DB

        Treasury bonds                                  Municipal bonds        (and hence the same level of
                  Price                    Interest
                                            SB                                          SB
                                                              iB


                                                   iT
                                                     "Tax"
                                            DB      premium                             DB

interest rates,           Treasury bonds                             Municipal 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                                              SB
                                                iB


                                     iT
                                       "Tax"
                              DB      premium                                DB

(           Treasury bonds                                 Municipal bonds        ) is lower than the interest
                             Price                         Interest
                                                            SB                                       SB
                                                                        iB


                                                                 iT
                                                                   "Tax"
                                                            DB    premium                            DB

rate of Treasury bonds (                  Treasury bonds                           Municipal bonds        ).
                  Price                         Interest
                                                 SB                                        SB
                                                                 iB


                                                      iT
                                                        "Tax"
                                                 DB    premium                             DB

                          Treasury bonds                               Municipal bonds


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                 Time to maturity current 1-year interest rate
        YTM



                         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                                 Term
                                premium                              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 long-
term 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 long-
term 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        period,    the   investor    will    be   able     to   collect   an   amount   of   $1225.45
    YTM



                          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        can    easily     generalize      the     relationship    between       longer-term   interest   rate
    YTM



                          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 5-
year 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 T-
   bonds 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

								
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