J. K. Dietrich - FBE 524 - Fall, 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
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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.
1)1(
)1(1
111
nnt
nnt
tnt R
Rr
nnt
t,n )R1(
1P
J. K. Dietrich - FBE 524 - Fall, 2005
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:
)nRexp(ep t,nnR
t,nt,n
)pln(nR t,nt,n
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
t1tt11nt111t1t1nt u)Rr(baRHPY
J. K. Dietrich - FBE 524 - Fall, 2005
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
Dependent Variable a s(a) b s(b) R^2Net One-Year HPY -0.21 0.41 0.91 0.28 0.14Net Five-Year HPY -1.06 1.31 0.93 0.53 0.05
J. K. Dietrich - FBE 524 - Fall, 2005
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
t1tt11nt111t1ntnt u)Rr(baRR
J. K. Dietrich - FBE 524 - Fall, 2005
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
Dependent Variable a s(a) b s(b) R^2Change in Spot Rate in One Year 0.21 0.41 0.09 0.28 0Change in Spot Rate in Four Years 1.12 0.61 1.61 0.34 0.48
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
dzdt)r(dr
J. K. Dietrich - FBE 524 - Fall, 2005
3-Month Bill Rate 1950 - 2004Dependent Variable: DSTRMethod: Least SquaresDate: 10/03/05 Time: 17:20Sample (adjusted): 1950M02 2004M12Included observations: 659 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
DSTRMEAN 0.011824 0.006144 1.924443 0.0547
R-squared 0.005584 Mean dependent var 0.001684Adjusted R-squared0.005584 S.D. dependent var 0.459699S.E. of regression0.458414 Akaike info criterion 1.279427Sum squared resid138.2742 Schwarz criterion 1.286242Log l ikel ihood -420.5713 Durbin-Watson stat 1.429692
-8
-4
0
4
8
12
50 55 60 65 70 75 80 85 90 95 00
DSTR DSTRMEAN
Change in Short-Term Rate and Distance from Mean
J. K. Dietrich - FBE 524 - Fall, 2005
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%
-4
-2
0
2
4
6
8
10
2004M01 2004M04 2004M07 2004M10
ESTSTR FTB3
Project Random Rate and Actual Rate
J. K. Dietrich - FBE 524 - Fall, 2005
CIR Term-Structure Model
CIR (1985) assumes a random evolution of the short-rate in continuous time in a general equilibrium framework
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
dzrdt)r(dr
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
Binomial Process or Tree
A random variable changes at discrete time intervals to one of two new values with equal probability
R1,t
Rup1,t
Rdown1,t
Rup2,t
Rdown or up2,t
Rdown2,t
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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:
a
acbbx
2
42
J. K. Dietrich - FBE 524 - Fall, 2005
Rates using Quadratic FormulaMaturity 1 2
Yield 10.0% 11.0%Volatility 20.0% 19.0%
Price(s) at t=0 90.91$ 81.16$
beta1= 2*(1+r1)/(1+r2)^2 1.785569353beta2 = exp(2*vol) 1.462284589b = beta2-beta1*beta2-beta1-1 -1.934295312a = -beta1*beta2 -2.611010548c = 2 - beta1 0.214430647Using quadratic formula(1) r-d = 9.79%(2) r-d = (0.8387) r-u 14.32%
J. K. Dietrich - FBE 524 - Fall, 2005
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
J. K. Dietrich - FBE 524 - Fall, 2005
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
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