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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

Term Structure: Tests and Models

Week 7 -- October 5, 2005

today s session
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
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
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
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
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 continued7
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
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
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
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 regression11
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
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
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 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
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 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
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
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
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
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
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
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
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
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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