1. The Risk Structure and Term Structure of Interest Rates
Chapter 6

2. Risk Structure of Interest Rates
Bonds with the same maturity have different interest rates due to:
Default risk
Liquidity
Tax considerations

3. Risk Structure of Interest Rates
Default risk: probability that the issuer of a bond is unable or unwilling to make interest payments or pay off the face value
U.S. Treasury bonds are considered default free (government can raise taxes).
Risk premium: the spread between the interest rates on bonds with default risk and the interest rates on (same maturity) Treasury bonds 3

4. Response to an Increase in Default Risk on Corporate Bonds – Supply/Demand Application4

5. Russian Default

6. Risk Structure of Interest Rates
Liquidity: the relative ease with which an asset can be converted into cash
Cost of selling a bond
Number of buyers/sellers in a bond market
Income tax considerations
Interest payments on municipal bonds are exempt from federal income taxes.

7. Interest Rates on Municipal and Treasury Bonds7

8. Taxes and Bond Prices
Coupon payments on municipal bonds are exempt from federal Income taxes
For 28% tax bracket:
After tax yield = (taxable yield) x (1 – tax rate)
3.60% = 5% x (1 – 0.28)
Tax equivalent yield =

9. Risk Structure of Long-Term Bonds in the United States

10. Bond (credit) Ratings and Risk
Bond Ratings -
Moody’s and Standard and Poor’s
Ratings Groups
Investment Grade
Non-Investment – Speculative Grade
Highly Speculative

11. Bond (credit) ratings S&P What it means AAA Aaa
Moody’s
What it means
AAA
Aaa
Highest quality and creditworthiness AA Aa
Slightly less likely to pay principal + interest A
Strong capacity to make payments, upper medium grade
BBB
Baa
Medium grade, adequate capacity to make payments BB Ba
Moderate ability to pay, speculative element, vulnerable B
Not desirable investment, long term payment doubtful
CCC
Caa
Poor standing, known vulnerabilities, doubtful payment CC Ca
Highly speculative, high default likelihood, known reasons C
Lowest rated class, most unlikely to reach investment grade D
Already defaulted on payments NR
No public rating has been requested
+ or -
& 1,2,3
Within-class refinement of AA to CCC ratings

12. Credit rating & historic default frequencies
Moody’s
Rating
1985
1990
1995
2000
2006
2008
2009
2010
Aaa 0% Aa A
1.201%
0.36%
Baa1
0.29%
0.271%
1.144%
Baa2
0.794%
0.74%
Baa3
0.98%
0.321%
0.70%
Ba1
2.67%
0.91%
2.27%
Ba2
1.63%
2.82%
0.66%
0.51%
0.60%
Ba3
3.77%
3.92%
1.72%
0.99%
2.715%
4.01% B1
4.38%
8.59%
4.35%
3.63%
1.783%
4.10%
0.85% B2
7.41%
22.09%
6.36%
3.84%
0.50%
0.825%
8.68% B3
13.86%
28.93%
11.72%
1.93%
3.198%
8.52%
0.56%

13. Default Risk – Price and YTM
Suppose risk-free rate is 4%
Suppose there is a company called FlimFlam that issues one-year, 4% coupon bond, FV=$100.
If risk free, the price of the FlimFlam bond is

14. Default Risk
Suppose 5% probability FlimFlam goes bankrupt – you get nothing
Expected Value of FlimFlam bond payment
Possibilities
Payoff
Probability
Payoff x Probability
Full Payment
$104
.95
$98.80
Default $0
.05
Expect to receive $98.80 one-year from now.
Discount at risk-free rate =
P = $95

15. Default Risk Premium Yield = ($104 / $95) -1 = .0947 or 9.47%
Default Risk Premium = rate – risk-free rate
= 9.47% - 4% = 5.47%

16. Default Risk Premium
We can calculate the probability of repayment from the interest rates.
Let 1+k be the return on a one-year corporate debt and 1+ i be the return on a one-year default risk-free treasury.
The probability of repayment is
the probability of default is 1 – p
The probability of repayment:

17. Default Risk
Suppose 10% probability FlimFlam goes bankrupt – you get nothing
Expected Value of FlimFlam bond payment
Possibilities
Payoff
Probability
Payoff x Probabilities
Full Payment
$104
.90
$93.60
Default $0
.10
Expect to receive $93.60 one-year from now.
Discount at risk-free rate =
Yield = ($104 / $90) -1 = or 15.55%
Default risk premium = 15.55% - 4% = 11.55%.

18. Bond Ratings and Risk Increased risk reduces bond demand.
The resulting shift to the left causes a decline in equilibrium price and an increase in the bond yield.
Bond Yield = U.S. Treasury Yield Default Risk Premium
Risk spread or default risk premium =
Bond Yield - U.S. Treasury Yield

19. Information Content of Interest Rates: Risk Structure
When the economy starts to slow, this puts a strain on private firms.
A slower economy means a higher default probability
Risk Spreads increase.

20. Information Content of Interest Rates: Risk Structure
Risk spread = Baa Corporate minus 10-year Treasury

21. Term Structure of Interest Rates
Definition of the Term Structure: The relationship among bonds with the same risk, liquidity and tax characteristics but different maturities is called the term structure of interest rates.
Yield Curve:
A plot of the term structure, with the yield to maturity on the vertical axis and the time to maturity on the horizontal axis.

22. Term Structure of Interest Rates

23. Term Structure of Interest Rates

24. Term Structure of Interest Rates: Facts to Explain
Interest rates (Yields) on different maturities tend to move together
Yields on short-term bond are more volatile than yields on long-term bonds
Long-term yields tend to be higher than short-term yields.
Also want to explain the fact that yield curves can be inverted.

25. Movements over Time of Interest Rates on U. S
Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities
Sources: Federal Reserve: 25

26. Three Theories to Explain the Three Facts
Pure Expectations Theory explains the first two facts but not the third
Segmented Markets Theory explains fact three but not the first two
Liquidity Premium Theory combines the two theories to explain all three facts

27. Pure Expectations Theory
The interest rate on a long-term bond will equal an average of the short-term interest rates that people expect to occur over the life of the long-term bond
Key Assumption: Buyers of bonds do not prefer bonds of one maturity over another.
Bonds of different maturities are considered to be perfect substitutes

28. Expectations Theory Notation
interest rate on 1-year bond today (t).
interest rate on 2-year bond today (t).
interest rate on n-year bond today (t).
interest rate on 1-year bond, 1-year from today (t+1).
Expected interest rate on 1-year bond, 1-year from today (t+1).
Expected interest rate on 1-year bond, n-years from today (t+n).

29. A Note on Averages
Geometric average of and =
Arithmetic average =

30. Expectations Theory:
Let the current interest rate on one-year bond (i1t) be 6%.
You expect the interest rate on a one-year bond next year ( ) to be 9%.
Then the expected return from buying 2 one-year bonds averages (6% + 9%)/2 = 7.5%.
Under the Expectations Theory the current interest rate on a two-year (i2t) bond must be 7.5% for you to be willing to purchase that bond.
Why?

31. Example: 2 year investment horizon
Strategy 1:
Invest $1,000 for 2-years at 8%:
Ending Balance = (1+0.08)2($1,000) = $1,166.40
Strategy 2:
Invest $1,000 1-year at 6% and expect 9% one year later:
Ending Balance = ( )(1+0.09)($1,000) = $1,155.40
Come out $11 ahead with Strategy 1.
What happens to S and D?

32. Expectations Theory ( Math)
1. Return from a 2-year bond over 2 years
2. Return from a 1-yr bond and then another 1-yr bond
3. If one and two year bonds are perfect substitutes, then:

33. Term Structure of Interest Rates: Expectations Theory
From:
We can derive the following arithmetic approximation:
Which says the long-term interest rate = average of current and expected future short-term interest rates.

34. Here is how we get the approximation:

35. Here is how we get the approximation:

36. Expectations Theory

37. Actual math: No Approximation
This is a geometric average

38. Expectations Hypothesis - Arithmetic Average
In words: The interest rate on a bond with n years to maturity at time t is the average of the n expected future one-year rates.
Numerical example:
One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%:
Interest rate on a two-year bond:
(5% + 6%)/2 = 5.5%
Interest rate for a five-year bond:
(5% + 6% + 7% + 8% + 9%)/5 = 7%
Interest rate for one, two, three, four and five-year bonds are:
5%, 5.5%, 6%, 6.5% and 7%.
This is the only interest rate that is known at time t

39. Expectations Hypothesis
Another example:
One-year interest rate over the next five years 7%, 6%, 5%, 4% and 3%:
Interest rate on a two-year bond:
(7% + 6%)/2 = 6.5%
Interest rate for a five-year bond:
(7% + 6% + 5% + 4% + 3%)/5 = 5%
Interest rate for one, two, three, four and five-year bonds:
7%, 6.5%, 6%, 5.5% and 5%.

40. Recall the Fisher Equation: i = r + πe
Holding r constant:
If inflation is expected to rise in the future, expected one-year interest rates will rise and the yield curve will slope upward.
If inflation is expected to fall in the future, expected one-year interest rates will fall and the yield curve will slope downward.
If inflation is expected to remain the same in the future, expected one-year interest rates will remain the same and the yield curve will be flat.

41. Term Structure of Interest Rates: Expectations Theory

42. Using the Pure Expectations Theory to Solve for Expected 1-year (forward) Interest rates
From the formula for the yield on a 2-year bond:
From the formula for the yield on a 3-year bond:
In general:

43. Actual math: No Approximation

44. Term Structure Facts and the Expectations Theory
Expectations Theory Explains:
Interest Rates of different maturities tend to move together
- long term interest rates are averages of expected future short-term interest rates.
Yields on short-term bond are more volatile than yields on long-term bonds –
But Expectations Theory does not explain:
3. Long-term yields tend to be higher than short-term yields.

45. Segmented Market Theory
Bonds of different maturities are not perfect substitutes for each other.

46. Segmented Markets Hypothesis
Assumptions:
Investors have specific preferences about the maturity or term of a security.
Investors do not stray from their preferred maturity.

47. Segmented Markets Hypothesis
The slope of the yield curve is explained by different demand and supply conditions for bonds of different maturities.
If the yield curve slopes up, it does so because the demand for short term bonds is relatively greater than the demand for long term bonds.
Short term bonds have a higher price and a lower yield as a result of the relatively greater demand. So the yield curve slopes upward.

48. Segmented Markets Hypothesis
Price
Price S S
P2s
P1s
P1l
P2l
D2s
D1l
D1s
D2l
Quantity of Short-term Bonds
Quantity of Long-term Bonds
Upward Sloping Yield Curve

49. Segmented Markets Hypothesis
The segmented markets hypothesis explains why….
Yield curves typically slope upward.
On average, investors prefer bonds with shorter maturities that have less interest rate risk.
Therefore, the demand for short term bonds is relatively greater than the demand for long-term bonds

50. Segmented Markets Hypothesis
But, the segmented markets hypothesis does not explain why…
Interest rates on different maturities move together.
The segmented markets hypothesis assumes that short and long markets are completely segmented.

51. Liquidity Premium Theory of the Term Structure of Interest Rates
Yield curve upward slope is explained by the fact that long-term bonds are riskier than short-term bonds.
Bondholders face both inflation risk and interest rate risk.
The longer the term of the bond, the greater both types of risk.
Investors need to be compensated for the greater risk.

52. Term Structure of Interest Rates
Liquidity Premium Theory
Liquidity or Risk Premium
(explains fact 3)
Pure Expectations Theory:
average of expected future short-term rates (explains facts 1&2)

53. Numerical Example Term in years (n) 1 2 3 4 5 5% 6% 7% 8% 9% 0% 0.25%
One year interest rate expectations 5% 6% 7% 8% 9%
Liquidity premium 0%
0.25%
0.5%
0.75%
1.0%
Pure expectations predicted n-year bond interest rates
5.5%
6.5%
Actual n-year bond interest rates, accounting for liquidity preference
5.75%
7.25%

54. Relationship Between the Liquidity Premium and Expectations Theories
(if short term interest rates are
expected to remain constant)

55. Information Content of Interest Rates: Term Structure
When the yield curve slopes down, it is called inverted
An inverted yield curve is a very valuable forecasting tool
It signals an economic downturn

56. Information Content of Interest Rates:10-year T-bond compared to 3-month T- bill

57. Market Predictions of Future Short Rates

58. The actual math is a lot more interesting
The actual math is a lot more interesting. Refer to the note on “Term Structure and Forward Interest Rates.”