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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

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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


3-Month Bill Rate 1950 - 2004


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:


Rates 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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