J. K. Dietrich - FBE Fall, 2005 Term Structure: Tests and Models Week 7 -- October 5, 2005

J. K. Dietrich - FBE Fall, 2005 Today’s Session u Focus on the term structure: the fundamental underlying basis for yields in the market u Three aspects discussed: –Tests of term structure theories –Models of term structure –Calibration of models to existing term structure u Goal is to gain a sense of how experts deal with important market phenomena

J. K. Dietrich - FBE Fall, 2005 Theories of Term Structure u Three basic theories reviewed last week: –Expectations hypothesis –Liquidity premium hypothesis –Market segmentation hypothesis u Expectations hypotheses posits that forward rates contain information about future spot rates u Liquidity premium posits that forward rates contain information about expected returns including a risk premium

J. K. Dietrich - FBE Fall, 2005 Forward Rate as Predictor u Use theories of term structure to analyze meaning of forward rates u 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 u Academic analysis must meet high standards, hence often difficult to read

J. K. Dietrich - FBE Fall, 2005 Some Technical Issues u We have used discrete compounding periods in all our examples: e.g. u Note that that since the price of a discount bond is: above expression includes ratios of prices.

J. K. Dietrich - FBE Fall, 2005 Technical Issues (continued) u Alternative is to use continuous compounding and natural logarithms: u For example, at 10%, discrete compounding yields price of.9101, continuous.9048 u Yield is:

J. K. Dietrich - FBE Fall, 2005 Technical Issues (continued) u Fama and Bliss use continuous compounding in their analysis u Their investigation is based on monthly yield and price date from 1964 to 1985 u Based on relations between prices, one- period spot rates, expected holding period yields, and implicit forward rates, they develop two estimating equations

J. K. Dietrich - FBE Fall, 2005 Fama and Bliss Estimations: I u 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

J. K. Dietrich - FBE Fall, 2005 Results of first regression u Example results for two-year and five-year bonds: u 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

J. K. Dietrich - FBE Fall, 2005 Fama and Bliss Estimations: II u 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

J. K. Dietrich - FBE Fall, 2005 Results of first regression u Example results for two-year and five-year bonds: u 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

J. K. Dietrich - FBE Fall, 2005 Summary of Fama-Bliss u Careful analysis of implications of theory with exact use of data can provide learning about determinants of term structure and information in forward rate u Term premiums seem to vary with short- rate and are not always positive u Forward rates fail to predict near-term interest-rate changes but are correlated with changes farther in the future

J. K. Dietrich - FBE Fall, 2005 Models of the Term Structure u Theoretical models attempt to explain how the term structure evolves u Theories can be described in terms behavior of interest rate changes u Two common models are Vasicek and Cox- Ingersoll-Ross (CIR) models u They both theorize about the process by which short-term rates change

J. K. Dietrich - FBE Fall, 2005 Vasicek Term-Structure Model u Vasicek (1977) assumes a random evolution of the short-rate in continuous time u 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

J. K. Dietrich - FBE Fall, Month Bill Rate

J. K. Dietrich - FBE Fall, 2005 Modelling 3-Month Bill Rate u For example, using 1950 to 2004 estimated  =.01 and standard deviation of change in rate of.46 starting with December 2003 level of.9%

J. K. Dietrich - FBE Fall, 2005 CIR Term-Structure Model u CIR (1985) assumes a random evolution of the short-rate in continuous time in a general equilibrium framework u 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

J. K. Dietrich - FBE Fall, 2005 Vasicek and CIR Models u To estimate these models, you need estimates of the parameters ( ,  and  ) and in CIR case,, a risk-aversion parameter u These models can explain a term structure in terms of the expected evolution of future short-term rates and their variability

J. K. Dietrich - FBE Fall, 2005 Black-Derman-Toy Model u 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 u Fitting a model to current observed variables is called calibration u Their model has practical significance in pricing interest-rate derivatives

J. K. Dietrich - FBE Fall, 2005 Binomial Process or Tree u A random variable changes at discrete time intervals to one of two new values with equal probability R 1,t R up 1,t R down 1,t R up 2,t R down or up 2,t R down 2,t

J. K. Dietrich - FBE Fall, 2005 BDT Model u Observe yields to maturity as of a given date u Assume or estimate variability of yields u 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

J. K. Dietrich - FBE Fall, 2005 BDT Solution for Future Rates u Rates can be solved for but have to use a search algorithm to find rates that fit u Equations are non-linear due to compounding of interest rates u For possible rates in one period, the problem is quadratic (squared terms only) u Can solve quadratic equations using quadratic formula:

J. K. Dietrich - FBE Fall, 2005 Rates using Quadratic Formula

J. K. Dietrich - FBE Fall, 2005 BDT Rates beyond One Year u Rates are unique and can be solved for but you need special mathematics u If you are patient, you can use a guess and revise approach u Once you have a tree of future rates, and you assume the binomial process is valid, you can price interest-rate derivatives

J. K. Dietrich - FBE Fall, 2005 Use of BDT Model u Model can be used to price contingent claims (like option contracts we discuss next week) u If you accept validity of model estimates of future possible outcome, it readily determines cash outflows in different states in the future

J. K. Dietrich - FBE Fall, 2005 Next time (October 12) u Midterm distributed; 90-minute examination is open book and open note; review old examinations and raise any questions about them in class u Read text Chapters 7 and 8 (focus on duration) and KMV reading on website for class on October 12