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### Term Structure: Tests and Models

Week 7 -- October 5, 2005

Today’s 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

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

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

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 is:above expression includes ratios of prices.

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:

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

Fama and Bliss Estimations: I

- First equation examines relation between forward rate and 1-year expected HPYs for Treasuries of maturities 2 to 5 years:or, 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

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

Fama and Bliss Estimations: II

- Second equation examines relation between forward rate and expected future spot rates for Treasuries of maturities 2 to 5 years:or, 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

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

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

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

Vasicek Term-Structure Model

- Vasicek (1977) assumes a random evolution of the short-rate in continuous time
- Vasicek models change in short-rate, dr:where 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

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%

CIR Term-Structure Model

- CIR (1985) assumes a random evolution of the short-rate in continuous time in ageneral equilibriumframework
- CIR models change in short-rate, dr:where variables are defined as before but the variability of the rate change is a function of the level of the short-term rate

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

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

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

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

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:

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

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

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

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