A Rational Expectations Consistent Measure of Risk: Using Financial Market Data from a Middle Income Context*

Published date01 December 2010
DOIhttp://doi.org/10.1111/j.1468-0084.2010.00598.x
AuthorJohannes Fedderke,Neryvia Pillay
Date01 December 2010
769
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2010. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 72, 6 (2010) 0305-9049
doi: 10.1111/j.1468-0084.2010.00598.x
A Rational Expectations Consistent Measure of Risk:
Using Financial Market Data from a Middle Income
ContextÅ
Johannes Fedderke† and Neryvia Pillay
Pennsylvania State University, University Park, USA; University of Witwatersrand,
Johannesburg, South Africa; and Economic Research Southern Africa, Cape Town,
South Africa (e-mail: jwf15@sia.psu.edu)
School of Economics, University of Cape Town, Cape Town, South Africa
(e-mail: neryvia.pillay@uct.ac.za)
Abstract
Although economic theory assumes that risk is of central importance in nancial deci-
sion making, it is difcult to measure the uncertainty faced by investors. Commonly used
empirical proxies for risk (such as the moving standard deviation of the returns on an
asset) are not rmly grounded in economic theory. Risk measures have been developed
by other studies, but these are often based on subjective weights attaching to a range of
objective component indicators, are difcult to replicate and are not strictly consistent
with underlying theory. The contribution of this article is to develop a methodology to
construct rational expectations consistent empirical risk measures. It has the advantages of
being explicitly consistent with economic theory and easily replicable. We illustrate this
methodology by specic application to the South African context. The time-varying risk
measure developed in this article is consistent with a rational expectations application of
the expectations hypothesis. The constructed measure is a broad one (it includes political
risk and peso problems for instance) and reects investors’ perceptions of systematic risk.
I. Introduction
Although economic theory assumes that risk is important, it is difcult to construct a
measure that appropriately captures the risks faced by investors. The contribution of
this article is to develop a methodology to construct a rational expectations consistent
measure of risk. We illustrate this methodology by specic application to the South
African context.
The methodology developed here constructs a risk measure that is consistent with the
expectations hypothesis of the term structure of interest rates. The expectations hypothesis
is the simplest and most commonly used hypothesis for explaining the term structure of
ÅThe authors gratefully acknowledge nancial support from Economic Research Southern Africa.
JEL Classication numbers: G32, E43.
770 Bulletin
interest rates.1It holds that long-term rates on government bonds are related to expectations
of future short-term rates on government bonds. Thus, changes in the shape of the term
structure reect a changed outlook for future interest rates relative to current rates. This
framework enables us to extract the risk implied by the returns on government bonds.
Since government bonds are free of default risk, and their prices are determined in the
bond market, yields on government bonds can be assumed to reect the pure country risk
perceived by investors.
The empirical evidence on the expectations hypothesis is mixed at best.2The measure
of risk presented in this article identies the presence of a time-varying risk premium as an
explanation of the empirical failure of the expectations hypothesis, and hence as the foun-
dation for an estimation strategy for obtaining a measure of risk.3It does so by identifying
the structure that the time-varying risk premium would have to exhibit for the expectations
hypothesis to hold under standard tests of the hypothesis.4
Our approach therefore has the advantage of allowing for the generation of a measure
of the risk premium that varies over time, and is explicitly rational expectations consistent,
rather than relying on a proxy for the time-varying risk premium. Moreover, it allows us
to compare the structure of standard proxies of the time-varying risk premium used in
the literature with the structure of the time-varying risk premium that is required for the
expectations hypothesis to hold.
It should be noted at the outset that the objective is not to develop a formal measure of
risk,5nor to operationalize ‘coherent risk’ measures.6Instead, the objective is to develop
an operational measure of risk as manifested in bond prices. Crucially, the measure could
1There are three main alternative theories: the liquidity preference hypothesis developed by Hicks (1946), the mar-
ket segmentation hypothesis of Culbertson (1957) and arbitrage-free interest rate modelling. The liquidity preference
hypothesis also involves the expected values of future spot rates, but emphasizes the risk preferences of agents. The
risk premium is assumed to increase with the time to maturity of the bond. Under the market segmentation hypothesis,
agents have strong preferences for specic maturities. Thus, bond prices are determined in separate markets, and
bonds of neighbouring maturities are not substitutes. Heath, Jarrow and Morton (1992) specify a general framework
for all arbitrage-free interest rate models. These models can be calibrated to t the current term structure.
2Although some studies nd support for the expectations hypothesis, by way of example see Engle and Granger
(1987) and Campbell and Shiller (1987), there are many that reject it, for instance Fama (1976), Shiller (1979),
Shiller, Campbell and Schoenholtz (1983), Mankiw and Summers (1984), Mankiw (1986), Campbell and Shiller
(1991) and Hall, Anderson and Granger (1992).
3It should be noted that studies that have attempted to incorporate a time-varying risk premium using various
proxies into their tests of the expectations hypothesis also generate mixed results. Thus Shiller et al. (1983), Froot
(1989) and Simon (1989) do not nd support for the expectations hypothesis under a time-varying risk premium,
but Tzavalis and Wickens (1997) demur. One explanation for this is that all of these approaches have to rely on a
proxy for risk, which may be subject to signicant measurement error. Variables used to capture the risk premium
have included a moving standard deviation of short-term rates, a variable representing the relative volumes issued
of different maturities (Shiller et al., 1983), survey data on interest rate expectations (Froot, 1989), a risk premium
that is specied to be proportional to the volatility of excess returns using instrumental variables (Simon, 1989)
and the ex post excess holding period return of one maturity as a proxy for the term premium of another maturity
(Tzavalis and Wickens,1997). Harris (2001) uses panel data techniques with the risk premium captured by individual
and time-specicxed effects, which has the advantage of allowing a risk premium that varies with maturity and
over time without the need for a proxy that may not be based on economic theory, but the study still nds that the
expectations hypothesis is rejected. Other possible explanations for the failure of the expectations hypothesis include
small sample bias, measurement error and peso problems. See the more extensive explanation in section II.
4Since the risk premium is derived from the expectations hypothesis, it cannot be incorporated in a test of the
expectations hypothesis and this article is not intended as a test of the expectations hypothesis.
5See, for instance, the discussion in Gollier (2004) and Jorion (1997).
6See the discussion in Artzner et al. (1997, 1999).
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2010

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