J. D. Jobson

Estimation for Markowitz Efficient Portfolios

Abstract Given a set of N assets a portfolio is determined by a set of weights xi, i = 1, 2, …, N; Σ N i=1 xi = 1 indicating the proportion of the value of the portfolio devoted to each asset. A Markowitz efficient portfolio is the vector of weights X m that minimizes the variance σ m 2 of the total return from the portfolio, subject to the condition that the portfolio mean premium return μ m has a certain value. The estimators for the N × 1 vector X m , the return premium μ m , and the variable σ m 2 require estimators for the mean premium return vector and for the covariance matrix Σ. The expectations, variances, and asymptotic distributions of the estimators of X m , μ m , and σ m 2 are derived under the assumption that returns are normally distributed. The use of these sampling properties for statistical inference is also discussed. The derived results are also compared with results obtained from a Monte Carlo simulation for a population of 20 stocks and several sample sizes.

Potential performance and tests of portfolio efficiency

Abstract The potential performance of an asset set may be obtained by choosing the portfolio proportions to maximize the Sharpe (1966) performance measure. If a portfolio has a Sharpe measure equivalent to the potential performance of the underlying set of assets, then it is efficient. Multivariate statistical procedures for comparing potential performance and testing portfolio efficiency are developed and then evaluated using simulations. Two likelihood ratio statistics are then used to compare stock and bond indices against sets of 20 and 40 portfolios. The procedures are also compared to the Gibbons (1982) methodology for testing financial models.

A Performance Interpretation of Multivariate Tests of Asset Set Intersection, Spanning, and Mean-Variance Efficiency

The purpose of this paper is to provide a link between the various multivariate tests of asset pricing and a performance measure for asset sets. The paper includes a unified summary of various F tests for mean-variance efficiency, intersection, and spanning for sets and subsets of financial assets. Both the risk-free asset and no risk-free asset environments are discussed. These tests are then related to the concept of potential performance for asset sets. The potential performance measure can be viewed as an extension of the Sharpe performance measure for single portfolios. The economic intuition behind the tests is that the multivariate tests of portfolio efficiency, intersection, and spanning are tests of zero potential performance at particular margins between the asset or portfolio subset and the full asset set.

Least Squares Estimation When the Covariance Matrix and Parameter Vector are Functionally Related

Abstract Estimation for the linear model y = Xβ + e with unknown diagonal covariance matrix G is considered. The diagonal elements of G are assumed to be known functions of the explanatory variables X and an unknown parameter vector Θ, where Θ is permitted to contain elements of β. A weighted joint least squares estimator is developed that is asymptotically equivalent to the maximum likelihood estimator. Asymptotic properties of the simple least squares estimator and of the weighted joint least squares estimator are obtained. A sampling experiment is used to compare the estimators.

Confidence regions for the mean-variance efficient set: An alternative approach to estimation risk

The mean-variance efficient set is used extensively in portfolio analysis and in addition underlies many of the models and tests of asset pricing. Despite this vital role, little direct attention has been paid to efficient frontier estimation. This article illustrates that an estimator for the efficient set hyperbola is composed of three mutually independent statistics whose distributions are known. This result is used to develop a confidence region for the efficient set hyperbola in (σ, μ) space. Two alternative approaches are used to define a confidence region. The first approach can be used to obtain an expression for a confidence region for σ given μ or for μ given σ. The second approach defines a confidence region in (σ, μ) space that contains the true hyperbola with a specified probability. In addition, anF test for mean-variance efficiency is used to generate a sample acceptance region. The sample acceptance region and the two confidence regions are compared graphically. A simulation experiment is used to examine the properties of the various procedures.

Statistical Inference in Two-Parameter Portfolio Theory with Multiple Regression Software

The purpose of this paper is to demonstrate how multiple regression software may be used for computing estimates of efficient set parameters and for performing tests of mean-standard deviation efficiency. Regression software also is shown to be useful for selecting, from a set of assets, a subset that maximizes performance and for comparing the performance of the set to the subset. The underlying multiple regression model fitted by the software has no relation to the analysis; the regression software is employed simply as a computing device. Since the multiple regression procedure is familiar to most finance researchers and since regression software is commonly available, the techniques presented here should be of wide interest.

A Coefficient of Equality for Questionnaire Items With Interval Scales

Given a sample of n responses to a pair of questionnaire items with interval scale values of V 1, V 2, ..., Vk it is sometimes of interest to know the degree to which respondents select the same response for both items. The coefficient of equality is designed to measure the departure from independence in the direction of equality. If the sample means and variances of the two items are equal this coefficient is equivalent to the Pearson product moment correlation coefficient. If the sample observations on the two items are such that the joint frequencies are the product of the marginals (independence) then the coefficient of equality is 0.

Towards the Development of a Canadian-American Scale: A Research Note

To many Canadians anti-Americanism is not only as old as Canada itself, but it also seems to have been and still is an integral part of the entire Canadian experience.1 To the historian Baker, anti-Americanism is a recurring theme in Canadian history and exists today as it did a century ago.2 However, anti-Americanism, to the best knowledge of the authors, has not been investigated through systematic behavioural empirical research. The development of a scale to measure anti-American attitudes is a necessary step in the empirical investigation of Canadian anti-Americanism. The purpose of this research note is to present some data that contribute to the empirical development of a measure of Canadian attitudes towards Americans which may be used as an antiAmerican scale.