TESTING A STANDARD THEORY OF PORTFOLIO SELECTION*
| Date | 01 February 1980 |
| Author | PATRICK HONOHAN |
| DOI | http://doi.org/10.1111/j.1468-0084.1980.mp42001002.x |
| Published date | 01 February 1980 |
TESTING A STANDARD THEORY OF PORTFOLIO
SELECTION*
By PATRICK HONOHAN
1. INTRODUCTION AND SUMMARY
In applied analysis of portfolio selection, most authors still employ the mean-
and-variance approach. That is to say, they assume that investors form
expectations with regard to the probability of achieving various yields from each
portfolio, and that the investors then choose between portfolios on the basis of the
mean and variance of these probability distributions, having regard to their
preference as between risk and return.
It is the contention of this paper, however, that, when attempting to estimate
the parameters underlying this choice, most authors neglect valuable information
provided by the mean-variance theory, and by the data. Furthermore, they usually
provide only weak tests of the model which they are estimating. In what follows,
an econometric model imposing all the usual conditions of the mean-variance
approach is developed. An innovation is the endogenous estimation of the
subjective covariances of asset yields, along Bayesian lines. (Special attention is
given to the problem of time horizon in expectations formation.) This model is
nested in a more general monopolistic model against which it may be tested.
The theory leads to a non-linear system of equations which is not easy to
estimate. Here we apply the model to the investment portfolios of life assurance
companies in the United Kingdom. However, the model does not perform well,
possibly due to data deficiencies, and, when tested against the monopolistic model,
the standard mean-variance theory, with its usual assumption of price-taking
behaviour, is rejected.
A byproduct is a new quarterly series of insurance companies' assets, valued at
market prices.
PART A: THEORY
2. THE MODEL
The aim here is not to present a new model, but to develop for a model that is
quite standard in the literature, an operational version which tests more assumptions
than usual. The mean-variance theory is such a strong one that its rejection must
a priori be considered likely, especially at an aggregate level (albeit an aggregate
over a fairly homogeneous population) in view of the failure of much weaker
theories of demand to pass statistical tests.1 But it is a plausible and easily handled
theory, and one that will continue to be used in the theoretical literature in
preference to more cumbersome alternatives.
To test the model we must first specify a mechanism generating the decision-
* This paper draws on my LSE PhD thesis. I am indebted toC. McCarthy,J. Davies, M. Morishima,
B. Nolan, S. E. Pudney,J. R. S. Reveil and C. R. Wymer for helpful comments.
l A recent survey of empirical consumer demand theories is contained in Barten, 1977.
17
18 BULLETIN
maker's subjective probability distribution over the returns on different assets
from information that is available to him. This is done in Section 3. The decision-
maker's chosen portfolio will depend on the mean and variance of this subjective
distribution in a way which will be specified in Section 4. Finally (Section 5) we
must specify alternative hypotheses within which the standard theory is nested to
allow the usual likelihood ratio tests to be applied.
3. EXPECTATIONS FORMATION
3.1 We begin from the assumption, maintained throughout, that the insurance
companies attempt to maximize an objective function depending only on the mean
and variance of their subjective probability distribution of the yield on their
portfolio. That is to say, the companies are seen as forming expectations as to the
possible yield of the portfolio. These expectations take the form of a probability
attached to each conceivable outcome. The mean yield derived from these
probabilities is taken as a measure of the likely yield, and the variance as a measure
of the risk that the actual yield will deviate from the likely yield. The chosen
portfolio is the one which, according to their preferences as between risk and return
(as measured by the 'objective function'), provides the best balance between risk
and return that can be obtained.
This mean-variance assumption is almost universal in the empirical literature.
It can have the status of a primitive assumption, not relying on deeper axioms, as
in the vast literature on capital market efficiency, etc., or it can be derived from the
principle of expected utility maximization, with suitable supplementary assump-
tions on the functional form of the utility function and/or on the subjective
probability distribution. The latter course has the merit of resting on firm
foundations in decision theory, but the necessary restrictions on the form of utility
function or on the probability distribution are implausibly severe (cf. Baron
(1977)). Accordingly, we prefer to work directly with a mean-variance objective
function which may be regarded as an empirical approximation to the true
objective function.
3.2 In order to obtain estimates of the mean and covariance of returns used in
the decision-maker's maximization exercise, we need to model his process of
expectations formation. In the literature, this is usually done in a rough and ready
fashion, by taking sample means and covariance. However, in a time series study
we must take account of the fact that new information is available to the decision-
makers in each new period and that, accordingly, these expectations will be updated
in each period. We could do this by using rolling samples in calculating mean and
covariance. However, we will carry out the updating in a more systematic manner,
using a Bayesian methodology. Both procedures have a pedigree in the literature
so far as revision of means is concerned. In particular the Bayesian approach
which we adopt here has parallels in the literature on rational expectations.2
2See, for example, Friedman, 1975.
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