Term Structure: Tests and Models

1
Term Structure Tests and Models

• Week 7 -- October 5, 2005

2
Todays Session

• Focus on the term structure the fundamental
  underlying basis for yields in the market
• Three aspects discussed
• Tests of term structure theories
• Models of term structure
• Calibration of models to existing term structure
• Goal is to gain a sense of how experts deal with
  important market phenomena

3
Theories of Term Structure

• Three basic theories reviewed last week
• Expectations hypothesis
• Liquidity premium hypothesis
• Market segmentation hypothesis
• Expectations hypotheses posits that forward rates
  contain information about future spot rates
• Liquidity premium posits that forward rates
  contain information about expected returns
  including a risk premium

4
Forward Rate as Predictor

• Use theories of term structure to analyze meaning
  of forward rates
• Many investigations of these issues have been
  published, we are discussing Eugene F. Fama and
  Robert R. Bliss, The Information in Long-Maturity
  Forward Rates, American Economic Review, 1987
• Academic analysis must meet high standards, hence
  often difficult to read

5
Some Technical Issues

• We have used discrete compounding periods in all
  our examples e.g.
• Note that that since the price of a discount bond
  isabove expression includes ratios of prices.

6
Technical Issues (continued)

• Alternative is to use continuous compounding and
  natural logarithms
• For example, at 10, discrete compounding yields
  price of .9101, continuous .9048
• Yield is

7
Technical Issues (continued)

• Fama and Bliss use continuous compounding in
  their analysis
• Their investigation is based on monthly yield and
  price date from 1964 to 1985
• Based on relations between prices, one-period
  spot rates, expected holding period yields, and
  implicit forward rates, they develop two
  estimating equations

8
Fama and Bliss Estimations I

• First equation examines relation between forward
  rate and 1-year expected HPYs for Treasuries of
  maturities 2 to 5 yearsor, in words, regress
  excess of n-year bond holding period yield over
  one-year spot rate on the forward rate for n-year
  bond in n-1 years over one-year spot rate

9
Results of first regression

• Example results for two-year and five-year
  bonds
• Authors interpret these results to mean
• Term premiums vary over time (with changes in
  forward rates and one-year rates)
• Average premium is close to zero
• Term premium has patterns related to one-year rate

10
Fama and Bliss Estimations II

• Second equation examines relation between forward
  rate and expected future spot rates for
  Treasuries of maturities 2 to 5 yearsor, in
  words, regress change in one-year spot rate in n
  years on the forward rate for n-year bond in n-1
  years over one-year spot rate

11
Results of first regression

• Example results for two-year and five-year
  bonds
• Authors interpret these results to mean
• One-year out forecasts in forward rate have no
  explanatory power
• Four year ahead forecasts explain 48 of change
• Evidence of mean reversion

12
Summary of Fama-Bliss

• Careful analysis of implications of theory with
  exact use of data can provide learning about
  determinants of term structure and information in
  forward rate
• Term premiums seem to vary with short-rate and
  are not always positive
• Forward rates fail to predict near-term
  interest-rate changes but are correlated with
  changes farther in the future

13
Models of the Term Structure

• Theoretical models attempt to explain how the
  term structure evolves
• Theories can be described in terms behavior of
  interest rate changes
• Two common models are Vasicek and
  Cox-Ingersoll-Ross (CIR) models
• They both theorize about the process by which
  short-term rates change

14
Vasicek Term-Structure Model

• Vasicek (1977) assumes a random evolution of the
  short-rate in continuous time
• Vasicek models change in short-rate, drwhere
  r is short-term rate, ? is long-run mean of
  short-term rate, ? is an adjustment speed, and ?
  is variability measure. Time evolved in small
  increments, d, and z is a random variable with
  mean zero and standard deviation of one

15
3-Month Bill Rate 1950 - 2004
16
Modelling 3-Month Bill Rate

• For example, using 1950 to 2004 estimated ? .01
  and standard deviation of change in rate of
  .46starting withDecember 2003level of .9

17
CIR Term-Structure Model

• CIR (1985) assumes a random evolution of the
  short-rate in continuous time in a general
  equilibrium framework
• CIR models change in short-rate, drwhere
  variables are defined as before but the
  variability of the rate change is a function of
  the level of the short-term rate

18
Vasicek and CIR Models

• To estimate these models, you need estimates of
  the parameters (?, ? and ? ) and in CIR case, ?,
  a risk-aversion parameter
• These models can explain a term structure in
  terms of the expected evolution of future
  short-term rates and their variability

19
Black-Derman-Toy Model

• Rather than estimate a model for interest-rate
  changes, Black-Derman-Toy (BDT) assume a binomial
  process (to be defined) and use current observed
  rates to estimate future expected possible
  outcomes
• Fitting a model to current observed variables is
  called calibration
• Their model has practical significance in pricing
  interest-rate derivatives

20
Binomial Process or Tree

• A random variable changes at discrete time
  intervals to one of two new values with equal
  probability

Rup2,t
Rup1,t
Rdown or up2,t
R1,t
Rdown1,t
Rdown2,t
21
BDT Model

• Observe yields to maturity as of a given date
• Assume or estimate variability of yields
• Fit a sequence of possible up and down moves in
  the short-term rate that would produce
• The observed multi-period yields
• Produce the assumed variability in yields

22
BDT Solution for Future Rates

• Rates can be solved for but have to use a search
  algorithm to find rates that fit
• Equations are non-linear due to compounding of
  interest rates
• For possible rates in one period, the problem is
  quadratic (squared terms only)
• Can solve quadratic equations using quadratic
  formula

23
Rates using Quadratic Formula
24
BDT Rates beyond One Year

• Rates are unique and can be solved for but you
  need special mathematics
• If you are patient, you can use a guess and
  revise approach
• Once you have a tree of future rates, and you
  assume the binomial process is valid, you can
  price interest-rate derivatives

25
Use of BDT Model

• Model can be used to price contingent claims
  (like option contracts we discuss next week)
• If you accept validity of model estimates of
  future possible outcome, it readily determines
  cash outflows in different states in the future

26
Next time (October 12)

• Midterm distributed 90-minute examination is
  open book and open note review old examinations
  and raise any questions about them in class
• Read text Chapters 7 and 8 (focus on duration)
  and KMV reading on website for class on October 12