Econometric methods of analysis
and forecasting of financial markets
Lecture 9. Interest rates theory
This lecture helps to understand:
• The main hypotheses explaining the term
structure of interest rates
• Empirical estimation of interest rates
hypotheses
Contents
•
•
•
•
•
•
•
•
Useful definitions
Term structure of interest rate
Yield curve
Pure expectations theory
Liquidity preference theory
Market segmentation theory
Preferred habitat theory
Empirical estimation of term structure
hypotheses
Useful definitions
• Bond: debt security with the payment in nominal
(monetary units) or real (for example, by the index
of consumer prices) values. 2 types:
-discount bond: security with the income calculated as
the difference (discount) between the buying price of
bond and its par value under maturity.
-coupon bond: security with the income calculated as
the sum of coupon payments over the period of bond
circulation and, probably, positive or negative discount
between the buying price of bond and its par value
under maturity.
We consider only discount bonds.
Useful definitions
• Maturity: the date of payment of the par value set under the bond issue,
T.
• Term, time to maturity: time interval from the current date to the date
of maturity, m=T-t.
• Duration: weighted average of time intervals to all coupon payments for
the period before the maturity, where the weights are coupon rates, D.
For the discount bonds D=m.
• Market price p(t,T) or p(t,m)
• Bond price′ in every period of time is defined from:𝑃 𝑡 ′ , 𝑇 =
𝑝(𝑡, 𝑇)𝑒 𝑡 −𝑡 𝑟(𝑡,𝑇) , where r(t,T) is the interest rate
• In maturity 𝑝 𝑡 ′ , 𝑇 = 1:
ln(𝑝 𝑡, 𝑇 )
𝑟 𝑡, 𝑇 = −
𝑇−𝑡
In this form 𝑟 𝑡, 𝑇 is also called spot-rate or continuously compounded
rate to maturity or instantaneous compound interest
• Term structure of interest rates: 𝑟 𝑡, 𝑇 = 𝐹(𝑡, 𝑚)
ln(𝑝 𝑡, 𝑚 )
𝐹 𝑡, 𝑚 = −
𝑚
Term structure of interest rates
• 𝐹 𝑡, 𝑚 =
ln(𝑝 𝑡,𝑚 )
−
𝑚
• The term structure of interest rates in every
period of time is defined as the set of bond
prices with different maturity dates. If maturity
date is relatively small, then spot-rate is called
short-term rate, otherwise: long-term rate.
Yield curve
• Yield curve is the graph which presents the relation
between yields with different maturities and terms
(times to maturity). Types of yield curves:
Useful definitions
• Yield spread: difference between the yield with
maturity m and yield of the bond repayable at
time t+1: 𝑠 𝑚, 𝑡 = 𝑟 𝑡, 𝑚 − 𝑟 𝑡, 1
• Forward rate: implicit rate defined on the basis of
observable term structure of interest rates:
′
′
𝑇
−
𝑡
𝑟
𝑡,
𝑇
−
𝑡
−
𝑡
𝑟(𝑡,
𝑡
)
′
𝑓 𝑡, 𝑡 , 𝑇 =
𝑇 − 𝑡′
Instantaneous forward rate: 𝑓 𝑡, 𝑇 = 𝑟 𝑡, 𝑇 +
(𝑇 −
𝜕𝑟(𝑡,𝑇)
𝑡)
𝜕𝑇
or 𝑓 𝑡, 𝑚 = 𝑟 𝑡, 𝑚 +
𝜕𝑟(𝑡,𝑚)
𝑚
𝜕𝑚
Useful definitions
• Holding period rate: yield of bond purchased before
(time t’, price p(t’,T), bought before by the price
p(t,T)):
𝑇 − 𝑡 𝑟 𝑡, 𝑇 − 𝑇 − 𝑡′ 𝑟 𝑡′, 𝑇
′
ℎ 𝑡, 𝑡 , 𝑇 =
𝑡′ − 𝑡
• Forward term premium, liquidity premium:
difference between the forward rate and conditional
expectation of the future rate:
Φ𝑡 𝑡, 𝑡 ′ , 𝑇 = 𝑓 𝑡, 𝑡 ′ , 𝑇 − 𝐸𝑡 𝑟 𝑡 ′ , 𝑇 , 𝑡 < 𝑡 ′ < 𝑇
• Holding period term premium: difference between
yield conditional expectation and spot-rate:
Φℎ 𝑡, 𝑡 ′ , 𝑇 = 𝐸𝑡 ℎ 𝑡, 𝑡 ′ , 𝑇 − 𝑟 𝑡, 𝑡′ , 𝑡 < 𝑡 ′ < 𝑇
Expectations hypothesis
• Long-term interest rates represent expectations of
short-term rates.
• Fisher(1930): relationship between the real and
nominal interest rates
• Pure expectations hypothesis: long-term interest rates
equal average of expected short-term rates. This is
equivalent to the following statements:
1. 𝐸𝑡 ℎ 𝑡, 𝑡 + 𝑛, 𝑚 − 𝑟 𝑡, 𝑛 = 0 ∀𝑚, 𝑛 ≤
𝑚 𝑜𝑟 Φℎ 𝑡, 𝑡 + 𝑛, 𝑚 = 0
2. 𝑟 𝑡, 𝑛 = 𝐸𝑡 ℎ 𝑡, 𝑡 + 𝑛, 𝑚 , 𝑛 < 𝑚
𝑇−1
𝑡=0 𝑟(𝑡,1)
3. 𝑟 𝑡, 𝑇 = 𝑇−𝑡
4. Φ𝑓 𝑡, 𝑡′, 𝑚 = 0 ∀𝑚 𝑜𝑟 𝑓 𝑡, 𝑡 + 1, 𝑇 = 𝐸𝑡 𝑟 𝑡 + 1, 𝑇
Expectations hypothesis
• (4) follows that 1-period forward rate in investment
made over time n in future, should follow condition:
𝑓 𝑡, 𝑡 + 𝑛, 𝑡 + 𝑛 + 1 = 𝐸𝑡 𝑟 𝑡 + 𝑛, 𝑡 + 𝑛 + 1 =
= 𝐸𝑡 𝐸𝑡+1 𝑟 𝑡 + 𝑛, 𝑡 + 𝑛 + 1 = 𝐸𝑡 𝑓(𝑡 + 1, 𝑡 + 𝑛 +
1, 𝑡 + 𝑛 + 2)
• Jensen’s inequality failure => development of the
rational expectations theory:
𝐸𝑡 ℎ 𝑡, 𝑡 + 1, 𝑚 − 𝑟 𝑡, 1 = Φ ∀𝑚
• Forward premium over the period is constant and equal
for all terms to maturity:
Φ𝑓 𝑡, 𝑡 ′ , 𝑚 = Φ ∀𝑚
Liquidity preference theory
• Forward premium over the period is constant in
time but depends on the time to maturity:
Φ𝑓 𝑡, 𝑡 ′ , 𝑚 = Φ(𝑚)
• Bonds with higher time to maturity are
considered as more risky than short-term bonds.
With the maturity time growth liquidity premium
and expected rate for the bond holding period
increase:
𝐸𝑡 ℎ 𝑡, 𝑡 + 1, 𝑚 − 𝑟 𝑡, 1 = Φ 𝑚 , Φ 𝑚
>Φ 𝑚−1 >Φ 𝑚−2 >⋯
Market segmentation theory
• Different investors could have different preferences
on the desirable investment periods or they are
obliged legally to make investment in bonds with the
defined terms to maturity. There are some separated
markets for securities with different terms to
maturity, and bonds prices are defined according to
demand and supply at these markets. Arbitrage is
not possible.
• Forward premium for the period depends on
demand and supply on bonds with each of the terms
to maturity:
𝐸𝑡 ℎ 𝑡, 𝑡 + 1, 𝑚 − 𝑟 𝑡, 1 = Φ 𝑠 , 𝑠 = 𝑠(𝑡, 𝑚)
• 𝑠 𝑡, 𝑚 is the relative attraction of bonds with time
to maturity m in the overall value of all issued bonds.
Preferred habitat theory
• Denies macroeconomic fundamentals of forward
premium definition. It’s assumed that investor is not
a professional, he has his own horizon of investment
and prefers to buy bonds not outside its limits.
• Market term structure of yield securities is the result
of agents independent decision makings. There are
own demand and supply in every such “habitat” that
leads to any sign and change in forward premium.
• Only bonds with close terms to maturity could be
considered as substitutes and could have the same
forward premium.
Empirical estimation of term structure
hypotheses
• weak empirical support for the expectations hypothesis
• ARCH, GARCH, Engle 1982, Bollerslev 1986, Engle, Ng, Rotshild,
1990: introduction to the model conditional dispersion of
forecast errors presenting strong fluctuations of time-series
forward rates.
• Non-stationarity, unit-root testing of time series of yields with
different terms to maturity
• Cointegration of time-series: long-term relationship between
the rates of different terms, while their short-term fluctuations
could be considered as “random walk”
• Efcectiveness of the hypothesis of term structure: analysis of
the possibikity of interest rates with different terms of
maturity to forecast future inflation changes, i.e. Fisher
hypothesis testing about the term structure.
• Estimation of variant in time forward premium
Conclusions
• We’ve learned the main hypotheses
explaining the term structure of interest rates
• We covered approaches to empirical
estimation of interest rates hypotheses
References
• Brooks C. Introductory Econometrics for Finance. Cambridge University
Press. 2008.
• Cuthbertson K., Nitzsche D. Quantitative Financial Economics. Wiley. 2004.
• Tsay R.S. Analysis of Financial Time Series, Wiley, 2005.
• Y. Ait-Sahalia, L. P. Hansen. Handbook of Financial Econometrics: Tools and
Techniques. Vol. 1, 1st Edition. 2010.
• Alexander C. Market Models: A Guide to Financial Data Analysis. Wiley.
2001.
• Cameron A. and Trivedi P.. Microeconometrics. Methods and Applications.
2005.
• Lai T. L., Xing H. Statistical Models and Methods for Financial Markets.
Springer. 2008.
• Poon S-H. A practical guide for forecasting financial market volatility.
Wiley, 2005.
• Rachev S.T. et al. Financial Econometrics: From Basics to Advanced
Modeling Techniques, Wiley, 2007.