Term Structure: Tests and Models

1. Term Structure: Tests and Models Week 7 -- October 5, 2005

2. 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

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 is:above 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 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

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 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

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, 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

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 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

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