Chapter 6 Rate The Risk and Term Structure of Interest 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 6-1 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 based on their default risk levels. The ratings assigned by those companies significantly affect the Chapter 6-2 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, P * (and hence the same level of interest rates, i * ). 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 ( ic ) is higher than the interest rate of Treasury bonds ( iT ). 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 6-3
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, P * (and hence the same level of interest rates, i * ). Suppose the municipal bonds are granted the taxfree 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 ( i B ) is lower than the interest rate of Treasury bonds ( iT ). Price Interest SB iB iT DB Treasury bonds SB "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 6-4
1. Normal yield curve: The short-term yield is lower than the long-term yield. In other words, it is cheaper to borrow short-term than it is to borrow long-term. YTM Time to maturity 2. Inverted yield curve: The short-term yield is higher than the long-term yield. In other words, it is more expensive to borrow short-term than it is to borrow long-term. YTM Time to maturity 3. Flat yield curve: The short-term yield is the same as the long-term yield. In other words, the short-term cost of borrowing is the same as the long-term cost of borrowing. YTM Time to maturity 4. Humped yield curve: The intermediate yield is higher 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. YTM 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. (i) Pure expectation theory Chapter 6-5
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 Return = (1 + i1t )(1 + i1e,t +1 ) (ii) Buying a 2-year bond Return = (1 + i 2t )(1 + i 2t ) where i1t = current 1-year interest rate i1e,t +1 = 1-year interest rate 1 year from now i 2t = current 2-year interest rate Since we know the returns of the two strategies are identical, we know the following must be true: (1 + i1t )(1 + i1e,t +1 ) = (1 + i 2t )(1 + i 2t ) (1 + i 2t ) 2 We can rewrite the above equation as follows: 1 + i 2t = (1 + i1t )(1 + i1e,t +1 ) i 2t = (1 + i1t )(1 + i1e,t +1 ) - 1 Chapter 6-6
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: 1 + i1t + i1e,t +1 + i1t i1e,t +1 = 1 + 2i 2t + i 22t e 2 In general, i1t i1,t +1 and i2t are so small that they are negligible. As a result, we can rewrite the above relationship as: i2t = i1t + i1e,t +1 2 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 i 2t = 1.1 1.11 - 1 = 0.10498 10.5% (ii) Using arithmetic mean i 2t = 0.1 + 0.11 = 0.105 = 10.5% 2 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 6-7
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 (= 1000 1.1 ), 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 (= 1100 1.11 ). 2. Money loaned 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 ( = 1000 1.107 2 ). 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 ( int ) and short-term interest rates as follows: (i) Using geometric mean i nt = n (1 + i1t )(1 + i1e,t +1 )(1 + i1e,t + 2 )...(1 + i1e,t + n -1 ) - 1 (ii) Using arithmetic mean Chapter 6-8 int = i1t + i1e,t +1 + i1e,t + 2 + ... + i1e,t + n -1 n 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: YTM YTM Term premium Time to maturity Time to maturity Chapter 6-9
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 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%, Chapter 6-10
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)? 0.1 + 0.09 + 0.002 = 0.097 = 9.7% 2 0.1 + 0.09 + 0.08 + 0.004 = 0.094 = 9.4% 3 0.1 + 0.09 + 0.08 + 0.07 + 0.006 = 0.091 = 9.1% 4 0.1 + 0.09 + 0.08 + 0.07 + 0.06 + 0.008 = 0.088 = 8.8% 5 2-year bond: 3-year bond: 4-year bond: 5-year bond: i 2t = i 2t = i 2t = i 2t = 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 .