On the robustness of short-term interest rate models

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Accounting and Finance 43 (2003) 87–121
© AFAANZ, 2003. Published by Blackwell Publishing.

Blackwell Publishing Ltd108 Cowley Road, Oxford OX4 1JF and 350 Main Street, Malden MA 02148, USA.ACFI Accounting & Finance0810-5391© The Accounting Association of Australia and New Zeland, 200243 1 March 2003024 S. Treepongkaruna, S. Gray / Accounting and Finance 43 (2003) xxxx–xxxxS. Treepongkaruna, S. Gray / Accounting and Finance 43 (2003) xxxx–xxxx??? Original Article 0 0 Graphicraft Limited, Hong Kong
On the robustness of short-term interest rate models
Sirimon Treepongkaruna
a
, Stephen Gray
b
a
School of Finance and Applied Statistics, Australian National University,
Canberra ACT 0200, Australia
UQ Business School, University of Queensland, Brisbane 4072, Australia and Fuqua
School of Business, Duke University, Towerview Drive, Durham NC 90120, USA
b
Abstract
This paper investigates the robustness of a range of short-term interest rate
models. We examine the robustness of these models over different data sets,
time periods, sampling frequencies, and estimation techniques. We examine a
range of popular one-factor models that allow the conditional mean (drift) and
conditional variance (diffusion) to be functions of the current short rate. We
find that parameter estimates are highly sensitive to all of these factors in the
eight countries that we examine. Since parameter estimates are not robust, these
models should be used with caution in practice.
Key words
: Short-term interest rates; Mean Reversion; Conditional Volatility
JEL classification
: C52, E43, G15
1. Introduction
The short-term interest rate is one of the key financial variables in any eco-
nomy. It is a target instrument that central banks use to implement monetary
policy and an important economic indicator for regulators and governments. It
is also a key variable for business since it forms the basis of floating rate loans
and most of the financial instruments that can be used to manage interest rate
risk. Moreover, longer-term interest rates reflect, at least in part, the expected
values of future short rates. The short rate is also a relevant input in determining
the required return on any asset. For all of these reasons, a good model of the
short-term interest rate is of great practical importance.
Accordingly, the past few decades have seen the development of a large
theoretical and empirical literature that deals with various interest rate models.
We are grateful to Frank Finn, Philip Gray, Peter Clarkson, Terry O’Keefe, and Ron Weber
for helpful comments and suggestions. We also thank Ponladesh Poomimars of Bangkok
Bank, Piyajit Bhirombhakdi of Siam Commercial Bank, and Jack Pai of Industrial Bank of
Taiwan for their extensive help in providing data.

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88S. Treepongkaruna, S. Gray / Accounting and Finance 43 (2003) 87–121
© AFAANZ, 2003
The two major classes of interest rate models can be described as “no-arbitrage
term structure models” and “short rate models”. No-arbitrage term structure
models develop a time-series process for the short rate that is consistent with
the current term structure of interest rates. That is, a short rate process is de-
veloped to “fit” the current term structure. The advantage of this class of models
is that arbitrage against the current term structure of interest rates is impossible.
The disadvantage is that the modeled short rate process might not reflect some
of the key features of the empirically observed time series properties of interest
rates. Examples of this type of model include Ho and Lee (1986), Hull and
White (1990), Black and Karasinski (1991), Black
(1992).
Short rate models, on the other hand, propose an empirical time series pro-
cess for the short-term interest rate. The parameters of these models are usually
estimated by fitting the model to historical data. In some cases, the entire term
structure of interest rates can be inferred from the short rate process (see for
example, Vasicek, 1977; Cox
et al.
, 1985). In other cases, the short rate process
is too complicated to allow for a closed-form solution for the term structure
(see for example, Chan
et al.
, 1992; Aït-Sahalia, 1996). Within this class of
models there is a tradeoff between having the model simple enough to allow for
a closed-form solution for the term structure of interest rates, and having the
model complex enough to accurately capture the key empirical features of the
time series. Most of the popular short rate models incorporate two key features:
mean reversion and non-constant volatility. In particular, most models allow for
the short rate to revert to a long-run mean so that if the current rate is above
(below) the long-run mean it is expected to decrease (increase) towards the
long-run mean in the future. The volatility of interest rates is often made to be
dependent on the level of interest rates so that when rates are higher, they are
more volatile. These two features are common in short rate models because
they occur in historical US data and the vast majority of work on short rate
models is based on US data sets.
The seminal work of Vasicek (1977) and Cox
two of the most popular short rate models. Other variations of these models
have been developed, but until recently relatively little was known about
the empirical performance of different short rate models. Chan
(CKLS) develop a model that nests several short rate models and provides
a framework to empirically compare different models. In particular, the
CKLS model nests both the Vasicek and CIR models. In all three models, the
conditional mean (drift) and conditional variance (volatility) of the short
rate are functions of the current level of the short rate. All of the models incor-
porate linear mean reversion in the drift but use different diffusion (volatility)
processes.
The existing literature contains a number of papers that seek to estimate the
parameters of these short rate models. Different authors use different data sets,
time periods, sampling frequencies, and empirical methodologies. In this paper,
et al.
, (1990), and Heath
et al.
,
et al.
, (1985) (CIR) provide
et al.
, (1992)

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S. Treepongkaruna, S. Gray / Accounting and Finance 43 (2003) 87–121 89
© AFAANZ, 2003
our focus is on the robustness of parameter estimates to these choices. In particular,
we estimate the Vasicek, CIR, and CKLS models in eight different countries. We
compare and contrast the parameter estimates, linking differences to institutional
features and country-specific events. We find that the results from other countries
do not simply mirror those from the US.
We also estimate the parameters using two different empirical methodologies;
quasi-maximum likelihood estimation (QMLE) and the generalised method of
moments (GMM) technique. We find that, in general, the parameter estimates
are sensitive to the choice of empirical methodology. We also highlight the
circumstances in which the different methodologies are most likely to generate
parameter estimates that differ in an economically significant sense.
To examine the sensitivity of parameter estimates to the time period of data
chosen, we divide all of our data sets into two halves and estimate parameters
in each sub-period. We are able to almost uniformly reject the notion that
parameter estimates are equal across sub-periods. This suggests that we may
need a more flexible class of models that allows parameter estimates to change
over time or to switch between different regimes. We also estimate parameters
using data observed at a daily and weekly frequency. After annualising both
sets of parameter estimates, we demonstrate that the results are sensitive to the
sampling interval that is chosen.
The balance of the paper is organised as follows. Section 2 describes the
competing short rate models and Section 3 describes the data. The econometric
methods we use are developed in Section 4. Section 5 summarises the results of
the estimation of various models with particular emphasis on the implications
for the conditional mean and variance. Section 6 contains checks for robustness
to different data sets, time periods, sampling frequencies, and estimation techniques.
Section 7 concludes.
2. Short rate models
2.1. General specification
The assumption underlying the popular one-factor short rate models that we
examine is that the only factor driving the term structure of interest rates is the
short rate itself. Specifically, we examine three models that have the following
general continuous-time representation:
dr

=

µ
(
r
,
θ
)
dt

+

σ
(
r
,
θ
)
dW
(1)
where
the conditional variance or diffusion function,
estimated,
t
represents time, and
peting models differ in the way that the functions
r
is the short rate,
µ
(.) is the conditional mean or drift function,
θ
is a vector of parameters to be
W
is a standard Wiener process. The three com-
µ
σ
2
(.) is
(.) and
σ
2
(.) are defined.

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© AFAANZ, 2003
2.2. Vasicek model
The first model we examine is that of Vasicek (1977). Under the Vasicek model,
the continuous time representation of the short rate process is:
dr

=
(
α

+

β
r
)
dt

+

σ
dW
(2)
where
the long run mean of the short-term interest rate, and
standard deviation of innovations in the short rate. The Vasicek model effectively
assumes that the short rate follows a conditionally Gaussian diffusion process with
a mean-reverting drift and a constant variance. That is, the short rate tends to revert
to its long-run mean value of
−α
/
β
at the rate of
responds more quickly to any deviation from the long-run mean. In addition to this
mean-reverting drift is a normally distributed stochastic term with constant variance
To find a discrete-time analog to this continuous-time process, we use a simple
first-order Euler approximation:
−β
represents the speed of adjustment (or mean reversion),
−α
/
β
represents
σ
represents the constant
−
β
. For larger values of
β
, the short rate
σ
2
.
(2a)
where the approximation will be more accurate if
η
t
+
1
represents a draw from a standard normal distribution.
∆
t
is small. In this setting,
2.3. CIR model (CIR)
The model of Cox
Vasicek model, but uses a different diffusion process. The CIR model posits that
the short rate follows a square root diffusion process, which has the following
continuous-time representation:
et al.
, (1985) has the same linear mean reversion as the
(3)
where
the long run mean of the short-term interest rate, and
model, both the drift and the volatility change with the level of the short rate. It has
the same mean reversion feature as the Vasicek model, however the stochastic term
has a standard deviation proportional to the square root of the current short rate. This
implies that as the short rate increases, its standard deviation increases. Moreover,
the CIR model has the conceptual advantage that the short rate will be strictly non-
negative. As the short rate falls and approaches zero, the diffusion term (which
contains the square root of the short rate) also approaches zero. In this case, the
mean-reverting drift term dominates the diffusion term and pulls the short rate
back towards its long-run mean. This prevents the short rate from falling below zero.
As for the Vasicek model, we find a discrete-time analog to this continuous-
time process by using a simple first-order Euler approximation:
−β
represents the speed of adjustment (or mean reversion),
−α
/
β
represents
σ
is a constant. Under this
∆∆∆
rrtt
ttt
++
=++
11
(
α
)
βση
drr dt
)
β
rdW
(
=α++σ

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S. Treepongkaruna, S. Gray / Accounting and Finance 43 (2003) 87–12191
© AFAANZ, 2003
(3a)
2.4. Chan-Karolyi-Longstaff-Sanders model (CKLS)
The model of Chan
the Vasicek and CIR models, but has a more general diffusion process. The
continuous-time representation for this model is:
et al.
, (1992) has the same linear mean reversion as in
dr

=
(
α

+

β
r
)
dr

+

σ
r
γ
dW
(4)
where
the long run mean of the short-term interest rate,
the volatility of the short rate to depend on a power of the level of the short rate.
This model nests both the Vasicek (1977) (
models. It also nests the Merton (1973) model (
model (α = β = 0, γ = 1), the Black and Scholes (1973) Geometric Brownian
Motion (α = 0, γ = 1), the Brennan-Schwartz model (1980) (γ = 1), and the con-
stant elasticity of variance model of Cox (1975) (α = 0).
Chan et al., (1992) demonstrate that, within their data set, models with γ ≥ 1
out-perform those with γ < 1. They assert that the specification of γ ≥ 1 allows
the conditional variance to be nonlinear but monotonically increasing and hence
highly sensitive to the level of the short rate.
As for the Vasicek and CIR models, we find a discrete-time analog to this
continuous-time process by using a simple first-order Euler approximation:
−β
represents the speed of adjustment (or mean reversion),
−α
/
β
represents
γ
allows
σ
is a fixed constant, and
γ

=
0) and the CIR (1985) (
β
= γ = 0), the Dothan (1978)
γ

=
0.5)
(4a)
2.5. Summary
All three of the models we examine specify the same conditional mean (drift)
with linear mean reversion. They differ in their specifications of the conditional
variance (diffusion) as functions of the current level of the short rate. The para-
metric specifications are summarised in Table 1.
∆∆∆
rrtrt
tttt
++
=++
11
(
α
)
βση
∆∆∆
rrtrt
tttt
++
=++
11
(
α
) .
βσ η
γ
Table 1
Summary of competing short rate models
Model
Drift Function
[µ(.)]
Diffusion Function
[σ2(.)]
Vasicek (1977)
CIR (1985)
CKLS (1992)
α + βrt
α + βrt
α + βrt
σ2
σ2rt
σ
All models are of the general form
from N(0,1).
, ηt+1 denotes a random element drawn
γ
2 2rt
∆∆∆
rrtrt
tttt
++
=+
11
( , )
µ
( , )
σθθ η