Covariance complexity and rates of return on assets

This paper considers the estimation of the expected rate of return on a set of risky assets. The approach to estimation focuses on the covariance matrix for the returns. The structure in the covariance matrix determines shared information which is useful in estimating the mean return for each asset. An empirical Bayes estimator is developed using the covariance structure of the returns distribution. The estimator is an improvement on the maximum likelihood and Bayes–Stein estimators in terms of mean squared error. The effect of reduced estimation error on accumulated wealth is analyzed for the portfolio choice model with constant relative risk aversion utility.     

Econometrical analysis of the sample efficient frontier

The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.     

Market proxies and the conditional prediction of returns

Ex post efficient proxies for the market portfolio are tested against the equal weight proxy. The equal weight proxy outperforms the others when the criterion is squared error of conditional prediction of returns. The ex post efficient proxies use maximum likelihood estimates of return. Stein estimates of return will generally be different from the maximum likelihood estimates and they necessarily correspond to market proxies which are not efficient ex post. In other words, there generally exists a better, inefficient, proxy than an ex post efficient proxy when the criterion is squared error of conditional prediction of return.     

Dominance of a Class of Stein type Estimators for Optimal Portfolio Weights When the Covariance Matrix is Unknown

For the estimation problem of mean-variance optimal portfolio weights, several previous studies have proposed applying Stein type estimators. However, few studies have addressed this problem analytically. Since the form of the loss function used in this problem is not of the quadratic type commonly used in statistical studies, there have been some difficulties in showing analytically the general dominance results. However, dominance results are given here of a class of Stein type estimators for the mean-variance optimal portfolio weights when the covariance matrix is unknown and is estimated. The class of estimators is broader than the one given in a previous study. The results we have obtained enable us to clarify conditions for some previously proposed estimators in finance to have smaller risks than the estimator which we obtain by plugging in the sample estimates.     

The likelihood ratio test statistic of mean-variance efficiency without a riskless asset

The question whether a given porfolio is mean-variance efficient is a basic problem of investment analysis. Mean-variance efficiency is also the basis of the Capital Asset Pricing Model. This paper presents the explicit form of the likelihood ratio test of the hypothesis that a given portfolio, or a particular market index, is ex-ante mean-variance efficient in the case where there is no riskless asset. Geometric relations are illustrated to provide intuition about the constrained maximum likelihood estimators and the test statistic, and two simple economic interpretations of the test are given.     

Jump robust two time scale covariance estimation and realized volatility budgets

We estimate the daily integrated variance and covariance of stock returns using high-frequency data in the presence of jumps, market microstructure noise and non-synchronous trading. For this we propose jump robust two time scale (co)variance estimators and verify their reduced bias and mean square error in simulation studies. We use these estimators to construct the ex-post portfolio realized volatility (RV) budget, determining each portfolio component’s contribution to the RV of the portfolio return. These RV budgets provide insight into the risk concentration of a portfolio. Furthermore, the RV budgets can be directly used in a portfolio strategy, called the equal-risk-contribution allocation strategy. This yields both a higher average return and lower standard deviation out-of-sample than the equal-weight portfolio for the stocks in the Dow Jones Industrial Average over the period October 2007–May 2009.     

On the distribution and estimation of trading costs

This paper investigates the uncertainty about the trading costs associated with a given portfolio strategy. I derive accurate approximations of the ex ante probability distributions of proportional trading costs and portfolio turnover under the conventional assumption of normal asset returns. Based on these approximations, I express the expected trading costs as a function of asset and portfolio characteristics. All else equal, the expected trading costs increase with: i) the deviations of the expected asset returns from the expected portfolio return, ii) the assets' volatility and iii) the portfolio volatility. At the same time, they decrease with the covariance between the assets and the portfolio. Furthermore, I propose novel estimators of the expected turnover and trading costs and show that they offer small bias and low variance, even when the sample size is small. Finally, I incorporate my results into a portfolio selection framework to produce portfolios with low levels of risk and trading costs. Several experiments with real and simulated data confirm the practical value of the results.     

European capital market integration: An empirical study based on a European asset pricing model

This paper investigates the integration between the capital markets of 15 European countries, all of which are members of the European Union. Integration is tested under the joint hypothesis of a European multifactor asset pricing model. A European portfolio is constructed from which common factors are extracted using maximum likelihood factor analysis. Empirical tests are undertaken to determine whether these European factors are not only priced, but also equally priced across the European capital markets. The results show that a number of common factors are extracted from the European portfolio and a degree of capital market integration is shown to exist across the European capital markets.     

A note on the consistency of maximum likelihood estimators

Abstract

A sequence of maximum likelihood estimators based on a sequence of independent but not necessarily identically distributed random variables is shown to be consistent under certain assumptions. Some examples are given to show that these assumptions are easy to verify and not very restrictive.     

Estimating correlation and covariance matrices by weighting of market similarity

We discuss a weighted estimation of correlation and covariance matrices from historical financial data. To this end, we introduce a weighting scheme that accounts for the similarity of previous market conditions to the present situation. The resulting estimators are less biased and show lower variance than either unweighted or exponentially weighted estimators. The weighting scheme is based on a similarity measure that compares the current correlation structure of the market to the structures at past times. Similarity is then measured by the matrix 2-norm of the difference of probe correlation matrices estimated for two different points in time. The method is validated in a simulation study and tested empirically in the context of mean–variance portfolio optimization. In the latter case we find an enhanced realized portfolio return as well as a reduced portfolio risk compared with alternative approaches based on different strategies and estimators.     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.      

Implications of the Discreteness of Observed Stock Prices

Stock prices on the organized exchanges are restricted to be divisible by ?. Therefore, the “true” price usually differs from the observed price. This paper examines the biases resulting from the discreteness of observed stock prices. It is shown that the natural estimators of the variance and all of the higher order moments of the rate of returns are biased. An approximate set of correction factors is derived and a procedure is outlined to show how the correction can be made. The natural estimators of the “beta” and of the variance of the market portfolio, on the other hand, are “nearly” unbiased.     

The geometric chain-ladder

Abstract

The log normal reserving model is considered. The contribution of the paper is to derive explicit expressions for the maximum likelihood estimators. These are expressed in terms of development factors which are geometric averages. The distribution of the estimators is derived. It is shown that the analysis is invariant to traditional measures for exposure.     

(Non-)robustness of maximum likelihood estimators for operational risk severity distributions

The quality of operational risk data sets suffers from missing or contaminated data points. This may lead to implausible characteristics of the estimates. Outliers, especially, can make a modeler's task difficult and can result in arbitrarily large capital charges. Robust statistics provides ways to deal with these problems as well as measures for the reliability of estimators. We show that using maximum likelihood estimation can be misleading and unreliable assuming typical operational risk severity distributions. The robustness of the estimators for the Generalized Pareto distribution, and the Weibull and Lognormal distributions is measured considering both global and local reliability, which are represented by the breakdown point and the influence function of the estimate.     

Stock recommendation of an analyst who trades on own account

This article analyzes the provision of information acquisition and truthful reporting incentives to a financial analyst who can privately trade on own account. In a binary message and state space, I show that the analyst's reward scheme essentially provides him with a portfolio endowment traded in the market. Regardless of the true signal, the analyst issues the report that corresponds to the portfolio endowment with maximum market value, given security prices. The analyst's information acquisition incentive is driven only by private portfolio considerations: he acquires information only if he will be holding a large enough position in the stock he covers.     

Mean–variance and mean–semivariance portfolio selection: a multivariate nonparametric approach

While univariate nonparametric estimation methods have been developed for estimating returns in mean-downside risk portfolio optimization, the problem of handling possible cross-correlations in a vector of asset returns has not been addressed in portfolio selection. We present a novel multivariate nonparametric portfolio optimization procedure using kernel-based estimators of the conditional mean and the conditional median. The method accounts for the covariance structure information from the full set of returns. We also provide two computational algorithms to implement the estimators. Via the analysis of 24 French stock market returns, we evaluate the in-sample and out-of-sample performance of both portfolio selection algorithms against optimal portfolios selected by classical and univariate nonparametric methods for three highly different time periods and different levels of expected return. By allowing for cross-correlations among returns, our results suggest that the proposed multivariate nonparametric method is a useful extension of standard univariate nonparametric portfolio selection approaches.     

Discussions on the spurious hyperbolic memory in the conditional variance and a new model

This paper studies the spurious hyperbolic memory in the conditional variance caused by the Markov Regime-Switching GARCH (MRS-GARCH) process. We firstly propose an illustrative cause of this spuriousness and provide simulation evidence. An MRS Hyperbolic GARCH (MRS-HGARCH) model is then developed to successfully address it. Related statistical properties including the stationarity conditions and asymptotic behaviours of the maximum likelihood estimators of the MRS-HGARCH process are also investigated. An empirical study of the S&P 500 and TOPIX indexes returns is then conducted which demonstrates that our MRS-HGARCH model can provide a more reliable estimator of the hyperbolic-memory parameter and outperform both the HGARCH and MRS-GARCH models.     

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Abstract

Estimation of the tail index parameter of a single-parameter Pareto model has wide application in actuarial and other sciences. Here we examine various estimators from the standpoint of two competing criteria: efficiency and robustness against upper outliers. With the maximum likelihood estimator (MLE) being efficient but nonrobust, we desire alternative estimators that retain a relatively high degree of efficiency while also being adequately robust. A new generalized median type estimator is introduced and compared with the MLE and several well-established estimators associated with the methods of moments, trimming, least squares, quantiles, and percentile matching. The method of moments and least squares estimators are found to be relatively deficient with respect to both criteria and should become disfavored, while the trimmed mean and generalized median estimators tend to dominate the other competitors. The generalized median type performs best overall. These findings provide a basis for revision and updating of prevailing viewpoints. Other topics discussed are applications to robust estimation of upper quantiles, tail probabilities, and actuarial quantities, such as stop-loss and excess-of-loss reinsurance premiums that arise concerning solvency of portfolios. Robust parametric methods are compared with empirical nonparametric methods, which are typically nonrobust.     

Sampling error and double shrinkage estimation of minimum variance portfolios

Shrinkage estimators of the covariance matrix are known to improve the stability over time of the Global Minimum Variance Portfolio (GMVP), as they are less error-prone. However, the improvement over the empirical covariance matrix is not optimal for small values of n, the estimation sample size. For typical asset allocation problems, with n small, this paper aims at proposing a new method to further reduce sampling error by shrinking once again traditional shrinkage estimators of the GMVP. First, we show analytically that the weights of any GMVP can be shrunk – within the framework of the ridge regression – towards the ones of the equally-weighted portfolio in order to reduce sampling error. Second, Monte Carlo simulations and empirical applications show that applying our methodology to the GMVP based on shrinkage estimators of the covariance matrix, leads to more stable portfolio weights, sharp decreases in portfolio turnovers, and often statistically lower (resp. higher) out-of-sample variances (resp. Sharpe ratios). These results illustrate that double shrinkage estimation of the GMVP can be beneficial for realistic small estimation sample sizes.     

Sampling distribution of the relative risk aversion estimator: Theory and applications

Brown and Gibbons (1985) developed a theory of relative risk aversion estimation in terms of average market rates of return and the variance of market rates of return. However, the exact sampling distributions of the relative risk aversion estimators have not been derived. The main purpose of this paper is to derive the exact sampling distribution of an appropriate relative risk aversion estimator. First, we have derived theoretically the density of Brown and Gibbons' maximum likelihood estimator. It is shown that the centralt is not appropriate for testing the significance of estimated relative risk aversion distribution. Then we derived the minimum variance unbiased estimator by a linear transformation of the Brown and Gibbons' maximum likelihood estimator. The density function is neither a central nor a noncentralt distribution. The density function of this new distribution has been tabulated. There is an empirical example to illustrate the application of this new sampling distribution.