1. chapter 5 The Risk and Term Structure of Interest Rates
Instructor: Xiajing Dai

2. preview
In our supply and demand analysis of interest-rate behavior in Chapter 5, we examined the determination of just one interest rate. Yet we saw earlier that there are enormous numbers of bonds on which the interest rates can and do differ.
In this chapter, we complete the interest-rate picture by examining the relationship of the various interest rates to one another
Year
Economics

3. Figure 1 shows the yields to maturity for several categories of long-term bonds from 1919 to 2002.
It shows us two important features of interest-rate behavior for bonds of the same maturity:
(1)Interest rates on different categories of bonds differ from one another in any given year
(2)the spread (or difference) between the interest rates varies over time.
Year
Economics

4. Year
Economics

5. Default risk
One attribute of a bond that influences its interest rate is its risk of default, which occurs when the issuer of the bond is unable or unwilling to make interest payments when promised or pay off the face value when the bond matures.
By contrast, U.S. Treasury bonds have usually been considered to have no default risk because the federal government can always increase taxes to pay off its obligations. Bonds like these with no default risk are called default-free bonds.
Year
Economics

6. The spread between the interest rates on bonds with default risk and default-free bonds, called the risk premium, indicates how much additional interest people must earn in order to be willing to hold that risky bond
Year
Economics

7. Year
Economics

8. Conclusion
A bond with default risk will always have a positive risk premium, and an increase in its default risk will raise the risk premium.
Year
Economics

9. Because default risk is so important to the size of the risk premium, purchasers of bonds need to know whether a corporation is likely to default on its bonds.
Two major investment advisory firms, Moody’s Investors Service and Standard and Poor’s Corporation, provide default risk information by rating the quality of corporate and municipal bonds in terms of the probability of default
Year
Economics

10. Year
Economics

11. Liquidity
The differences between interest rates on corporate bonds and Treasury bonds (that is, the risk premiums) reflect not only the corporate bond’s default risk but its liquidity, too.
This is why a risk premium is more accurately a “risk and liquidity premium,” but convention dictates that it is called a risk premium.
Year
Economics

12. Income Tax Considerations
Why is it, then, that these bonds have had lower interest rates than U.S. Treasury bonds for at least 40 years, as indicated in Figure 1? The explanation lies in the fact that interest payments on municipal bonds are exempt from federal income taxes, a factor that has the same effect on the demand for municipal bonds as an increase in their expected return.
Year
Economics

13. Year
Economics

14. Summary
The risk structure of interest rates is explained by three factors: default risk, liquidity, and the income tax treatment of the bond’s interest payments.
As a bond’s default risk increases, the risk premium on that bond (the spread between its interest rate and the interest rate on a default-free Treasury bond) rises.
The greater liquidity of Treasury bonds also explains why their interest rates are lower than interest rates on less liquid bonds.
If a bond has a favorable tax treatment, its interest rate will be lower.
Year
Economics

15. Term Structure of Interest Rates
A plot of the yields on bonds with differing terms to maturity but the same risk, liquidity, and tax considerations is called a yield curve, and it describes the term structure of interest rates for particular types of bonds.
Yield curves can be classified as upward-sloping, flat, and downward-sloping (the last sort is often referred to as an inverted yield curve).
Year
Economics

16. Year
Economics

17. A good theory of the term structure of interest rates must explain the following three important empirical facts:
1. As we see in Figure 4, interest rates on bonds of different maturities move together over time.
2. When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term interest rates are high, yield curves are more likely to slope downward and be inverted.
3. Yield curves almost always slope upward, as in the “Following the Financial News” box.
Year
Economics

18. (1) the expectations theory (2) the segmented markets theory
Three theories have been put forward to explain the term structure of interest rates; that is, the relationship among interest rates on bonds of different maturities reflected in yield curve patterns:
(1) the expectations theory
(2) the segmented markets theory
(3) the liquidity premium theory
Year
Economics

19. Expectation Theory
The expectations theory of the term structure states the following commonsense proposition: The interest rate on a long-term bond will equal an average of short-term.
Year
Economics

20. The key assumption behind this theory is that buyers of bonds do not prefer bonds of one maturity over another, so they will not hold any quantity of a bond if its expected return is less than that of another bond with a different maturity. Bonds that have this characteristic are said to be perfect substitutes
Year
Economics

21. let us consider the following two investment strategies:
1. Purchase a one-year bond, and when it matures in one year, purchase another one-year bond.
2. Purchase a two-year bond and hold it until maturity
Year
Economics

22. Expected return from strategy 2
(1 + i2t)(1 + i2t) – (i2t) + (i2t)2 – 1 =
1 1
Since (i2t)2 is extremely small, expected return is approximately 2(i2t)
© 2006 Pearson Addison-Wesley. All rights reserved

23. (1 + it)(1 + iet+1) – 1 1 + it + iet+1 + it(iet+1) – 1 = 1 1
1 1
Since it(iet+1) is also extremely small, expected return is approximately
it + iet+1
From implication above expected returns of two strategies are equal: Therefore
2(i2t) = it + iet+1
Solving for i2t
i2t = 2
© 2006 Pearson Addison-Wesley. All rights reserved

24. More generally for n-period bond:
it + iet+1 + iet iet+(n–1)
int = n
In words: Interest rate on long bond = average short rates expected to occur over life of long bond
Numerical example:
One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%:
Interest rate on two-year bond:
(5% + 6%)/2 = 5.5%
Interest rate for five-year bond:
(5% + 6% + 7% + 8% + 9%)/5 = 7%
Interest rate for one to five year bonds:
5%, 5.5%, 6%, 6.5% and 7%.

25. 1. Short rate rises are persistent
Expectations Hypothesis explains Fact 1 that short and long rates move together
1. Short rate rises are persistent
2. If it  today, iet+1, iet+2 etc.   average of future rates   int 
3. Therefore: it   int , i.e., short and long rates move together
© 2006 Pearson Addison-Wesley. All rights reserved

26. Doesn’t explain Fact 3 that yield curve usually has upward slope
Explains Fact 2
1. When short rates are low, they are expected to rise to normal level, and long rate = average of future short rates will be well above today’s short rate: yield curve will have steep upward slope
2. When short rates are high, they will be expected to fall in future, and long rate will be below current short rate: yield curve will have downward slope
Doesn’t explain Fact 3 that yield curve usually has upward slope
Short rates as likely to fall in future as rise, so average of future short rates will not usually be higher than current short rate: therefore, yield curve will not usually slope upward
© 2006 Pearson Addison-Wesley. All rights reserved

27. Segmented Markets Theory
Key Assumption: Bonds of different maturities are not substitutes at all
Implication: Markets are completely segmented: interest rate at each maturity determined separately
Explains Fact 3 that yield curve is usually upward sloping
People typically prefer short holding periods and thus have higher demand for short-term bonds, which have lower interest rates than long bonds
Does not explain Fact 1 or Fact 2 because assumes long and short rates determined independently
© 2006 Pearson Addison-Wesley. All rights reserved

28. Liquidity Premium (Preferred Habitat) Theories
Key Assumption: Bonds of different maturities are substitutes, but are not perfect substitutes
Implication: Modifies Expectations Theory with features of Segmented Markets Theory
Investors prefer short rather than long bonds  must be paid positive liquidity (term) premium, lnt, to hold long-term bonds
Results in following modification of Expectations Theory
it + iet+1 + iet iet+(n–1)
int = lnt n
© 2006 Pearson Addison-Wesley. All rights reserved

29. Relationship Between the Liquidity Premium (Preferred Habitat) and Expectations Theories
© 2006 Pearson Addison-Wesley. All rights reserved

30. The implication of figure 5
The liquidity premium is always positive and typically grows as the term to maturity increases
© 2006 Pearson Addison-Wesley. All rights reserved

31. Numerical Example
1. One-year interest rate over the next five years: 5%, 6%, 7%, 8% and 9%
2. Investors’ preferences for holding short-term bonds, liquidity premiums for one to five-year bonds:
0%, 0.25%, 0.5%, 0.75% and 1.0%.
Interest rate on the two-year bond:
(5% + 6%)/ % = 5.75%
Interest rate on the five-year bond:
(5% + 6% + 7% + 8% + 9%)/ % = 8%
Interest rates on one to five-year bonds:
5%, 5.75%, 6.5%, 7.25% and 8%.
© 2006 Pearson Addison-Wesley. All rights reserved

32. Liquidity Premium (Preferred Habitat) Theories: Term Structure Facts
Explains all 3 Facts
Explains Fact 3 of usual upward sloped yield curve by investors’ preferences for short-term bonds
Explains Fact 1 and Fact 2 using same explanations as expectations hypothesis because it has average of future short rates as determinant of long rate
© 2006 Pearson Addison-Wesley. All rights reserved

33. Market Predictions of Future Short Rates
© 2006 Pearson Addison-Wesley. All rights reserved

34. Answers in brief
1. bond with a C rating should have a higher interest rate because it has a higher default risk, which reduces its demand and raises its interest rate relative to that on the Baa bond.
© 2006 Pearson Addison-Wesley. All rights reserved

35. 3. During business cycle booms, fewer corporations go bankrupt and there is less default risk on corporate bonds, which lowers their risk premium. Similarly, during recessions, default risk on corporate bonds increases and their risk premium increases.
© 2006 Pearson Addison-Wesley. All rights reserved

36. 5. If yield curves on average were flat, this would suggest that the risk premium on long‑term relative to short‑term bonds would equal zero and we would be more willing to accept the expectations hypothesis.
© 2006 Pearson Addison-Wesley. All rights reserved

37. (a) yield curve would be upward- and then downward‑sloping 7.
(a) yield curve would be upward- and then downward‑sloping
(b) yield curve would be downward- and then up ward‑sloping
If people prefer shorter-term bonds over longer-term bonds, the yield curve tend to be even more upward sloping because long‑term bonds would then have a positive risk premium.
© 2006 Pearson Addison-Wesley. All rights reserved

38. 9. The steep upward‑sloping yield curve at shorter maturities suggests that short‑term interest rates are expected to rise in the near future .
The downward slope for longer maturities indicates that short‑term interest rates are eventually expected to fall sharply.
Since interest rates and expected inflation move together, the yield curve suggests that the market expects inflation to rise moderately in the near future but fall later on.
© 2006 Pearson Addison-Wesley. All rights reserved

39. 11. The government guarantee will reduce the default risk on corporate bonds, making them more desirable relative to Treasury securities. The increased demand for corporate bonds and decreased demand for Treasury securities will lower interest rates on corporate bonds and raise them on Treasury bonds.
© 2006 Pearson Addison-Wesley. All rights reserved

40. 13. Abolishing the tax‑exempt feature of municipal bonds would make them less desirable relative to Treasury bonds. The resulting decline in the demand for municipal bonds and increase in demand for Treasury bonds would raise the interest rates on municipal bonds, while the interest rates on Treasury bonds would fall.
© 2006 Pearson Addison-Wesley. All rights reserved

41. 15. The slope of the yield curve would fall because the drop in expected future short rates means that the average of expected future short rates falls so that the long rate falls.
© 2006 Pearson Addison-Wesley. All rights reserved