Term Structure of Interest Rates

1
Lecture 8

• Term Structure of Interest Rates
• Default Structure of Interest Rates

2
Yield Curve

• The relationship between the default free
  interest rates and the bonds maturity is called
  the term structure of interest rates or yield
  curve.
• The yield curve is upwardly sloped in most cases,
  although it could be falling, hump-shaped, or
  flat.

3
Different Shapes of Yield Curve
4
Terminology

• The interest rate for a given time interval is
  called the short interest rate.
• The yield to maturity on zero-coupon bonds that
  prevails today is called the spot interest rate.
• The future short rate inferred from the yield
  curve is called the forward interest rate, which
  may not be the interest rate that actually will
  prevail at the future date.

5
Example 1 From Interest Rates Forecast to the
Term Structure
Expected One-Year Rates (short rates) in Coming
Years (Table 15.1) Year Interest Rate 0
(today) 8 1 10 2 11 3 11
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Pricing of Bonds Using Expected Rates
PV Present Value of 1 in n periods r1
One-year rate for period 1 r2 One-year rate for
period 2 rn One-year rate for period n
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Long-Term Rates and Bond Prices Using Expected
Short Rates
Time to Maturity Price of Zero Yield to
Maturity 1 925.93 8.00 2
841.75 8.995 3 758.33 9.660 4
683.18 9.993 1,000 Par value zero
8
Forward Rates from Observed Long-Term Rates
fn one-year forward rate for period n yn
yield for a security with a maturity of n
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Forward Rates Calculation
4 yr 9.993 3yr 9.660 fn ? (1.0993)4
(1.0966)3 (1fn) (1.46373) / (1.31870)
(1fn) fn .10998 or 11
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Example 2 From Zero Coupon Rates to Forward Rates
Zero-Coupon Rates Bond Maturity 12 1 11.7
5 2 11.25 3 10.00 4 9.25 5
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Forward Rates for Downward Sloping Yield Curve

• 1yr Forward Rates
• 1yr (1.1175)2 / 1.12 - 1 0.115006
• 2yrs (1.1125)3 / (1.1175)2 - 1 0.102567
• 3yrs (1.1)4 / (1.1125)3 - 1 0.063336
• 4yrs (1.0925)5 / (1.1)4 - 1 0.063008

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Constructing the Term Structure

• It is best to be constructed from the Treasury
  zero-coupon bonds. e.g. the U.S. Treasury STRIP
  program.
• Only 30 year (long) bonds and 10 year notes are
  eligible for stripping.
• Lots of databases now offer the yield curve
  information. You can get the yield curve for the
  Australian market from the Datastream.

13
Prices for Coupon Bonds a short-cut

• Denote b(0,1) the price of the Treasury zero
  maturing in one year b(0,2) the price of of the
  Treasury zero maturing in two years.
• Deriving the price for a two year coupon bond
  through the no-arbitrage condition
  
• p c b(0,1) (100c)b(0,2)
• This arbitrage condition sets the pricing
  relationship between coupon bonds and zeros.

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Theories of Term Structure

• Expectations
• Liquidity Preference
• Upward bias over expectations
• Market Segmentation
• Preferred Habitat

15
Expectations Theory

• Observed long-term rate is a function of todays
  short-term rate and expected future short-term
  rates
• Long-term and short-term securities are perfect
  substitutes
• If forward rates are available, then the term
  structure could be determined.

16
Spot Rates and Forward Rates

• (Repeat!) The following two approaches should
  yield the same return if there is no arbitrage.
• (1) buy a 3-yr bond today
• (2) buy a 2-yr bond and then buy a 1-yr bond in
  two years time. Then,

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Unbiased Expectations Hypothesis

• This version of the expectations hypothesis
  assumed that the forward rate is an unbiased
  estimator of the future one-period spot rate.
• Therefore, this version is referred to as the
  unbiased expectations hypothesis.

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Liquidity Premium Theory

• Long-term bonds are more risky
• Investors will demand a premium for the risk
  associated with long-term bonds
• Yield curve has an upward bias built into the
  long-term rates because of the risk premium
• Forward rates contain a liquidity premium and are
  not equal to expected future short-term rates

19
Market Segmentation and Preferred Habitat

• Short- and long-term bonds are traded in distinct
  markets
• Trading in the distinct segments determines the
  various rates
• Observed rates are not directly influenced by
  expectations
• Preferred Habitat
• Modification of market segmentation
• Investors will switch out of preferred maturity
  segments if premiums are adequate

20
The Default Risk Structure of Interest Rates

• Based on a bonds default risk, we need to derive
  a new set of discount rates to calculate risky
  bonds.
• Bond ratings give us the guide to adjust for the
  risk premium required. For example, check
  www.bondsonline.com.
• Distinguish promised yield-to-maturity and
  expected return.

21
Determinants of Bond Safety

• The key ration to evaluate bond safety are
• Coverage ratios ratios of company earnings to
  fixed costs.
• Leverage ratio debt/equity ratio.
• Liquidity ratios current ratio (CA/CL), and
  quick ratio ((CA-inventories)/CL).
• Profitability ratios ROE or ROA.
• Cashflow-to-debt ratio.

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Z-score

• Combinations of financial ratios are possible
  predictors of default.
• Z-score below 1.80 distress zone. Below 0.75
  foretells financial difficulty, above 3 safe.
• Real world application of the Z-Score
  successfully predicted 72 of corporate
  bankruptcies two years prior to these companies
  filing for Chapter 7 in the U.S. sample.

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Z-Score Example Telstra

• Market Cap 30,665m EBIT 6,692m
• Sales 20,906m total assets 34,993m total
  liabilities 19,632m
• Working capital current assets current
  liabilities 5,327m - 7,576m -2,249m
• Retained earnings profits dividends 4,154m
  - 3,284m 870m
• Z-score 2.1228 ? Its OK, but not in excellent
  shape.
• Data Source Dat Analysis, Oct. 2004.

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Z-Score Example Telstra (One year later)

• Market Cap 25,068m EBIT 6,893m
• Sales 22,657m total assets 36,310m total
  liabilities 21,429m
• Working capital current assets current
  liabilities 6,177m - 6,382m -205m
• Retained earnings profits dividends 4,447m
  - 4,131m 316m
• Z-score 1.96 ? worse than a year ago.
• Data Source Dat Analysis, Oct. 2005.

25
Evidence on Defaults

• Moodys defines default as a situation where an
  issuer misses or delays a contracted interest or
  principal payments. This includes
• The issuer offers a package of new securities
  that has a diminished financial obligations.
• The exchange has the purpose of helping the
  issuer to avoid default.
• Delays in payments within the grace period
  provided in the indenture.

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Estimated Cumulative Default Risks for Individual
Companies () One study

• Rating Year 1 Year 2 Year 3 Year 4 Year
  5
• AAA 0.00 0.00 0.00 0.03
  0.04
• AA 0.13 0.27 0.41 0.49
  0.62
• A 0.21 0.43 0.67 0.85
  1.10
• BBB 0.26 0.55 0.86 1.15
  1.48
• BB 0.31 0.66 1.06 1.44
  1.89
• up to B 1.26 4.18 6.62 10.03
  13.27

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Time to Default S P Study
28
Lower-Grade Bond Portfolio

• Even though lower-grade bonds (also called
  high-yield fixed income securities) are
  individually risky, much of this bond-specific
  risk can be eliminated in a diversified
  portfolio. Its adjusted st.dev is lower than
  other asset indexes.
• Duration of a lower-grade bond is lower than
  duration of a higher-grade bond.
• Lower-grade bonds are less sensitive to
  unexpected interest rate movements.

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Duration of Low-Grade Bonds

• High coupon rates
• Less call protection
• May be called even if there were no change in
  interest rates -- the issuer may refinance at a
  lower yield because its credit quality has
  improved
• More likely to default -- reduce the maturity.