1 / 26

Term Structure: Tests and Models

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

Download Presentation

Term Structure: Tests and Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related