J. K. Dietrich - FBE 524 - Fall, 2005 Term Structure: Tests and Models Week 7 -- October 5, 2005

J. K. Dietrich - FBE 524 - Fall, 2005

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.

1)1(

)1(1

111

nnt

nnt

tnt R

Rr

nnt

t,n )R1(

1P


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


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

t1tt11nt111t1t1nt u)Rr(baRHPY


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


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


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


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

dzdt)r(dr


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


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


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


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

R1,t

Rup1,t

Rdown1,t

Rup2,t

Rdown or up2,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:

a

acbbx

2

42


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%


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

Documents

J. K. Dietrich - FBE 524 - Fall, 2005 Term Structure: Tests and Models Week 7 -- October 5, 2005