Linear beta pricing with efficient/inefficient benchmarks and short-selling restrictions

We provide two security pricing models that can be used when short sales of risky securities are not permitted. The first model uses a benchmark located on the expected return-standard deviation efficient frontier without short sales and presents security expected returns as a weighted linear function of two betas, one induced by the benchmark and the other adjusting for the short-sale constraints. The second model uses a benchmark that is inefficient relative to the efficient frontier, does not allow short sales, and expresses security returns as a weighted linear function of three betas:one induced by the inefficient benchmark, and the other two adjusting simultaneously for the short-sales restrictions and the benchmark's inefficiency.These results complement the linear pricing models that link expected returns and betas by allowing, for efficient or inefficient expected return-standard deviation, benchmarks with restricted short sales.     

Efficient Post-Retirement Asset Allocation

Abstract

To examine post-retirement asset allocation, an extension to the classic Markowitz risk-return framework is suggested. Assuming that retirees make constant (real dollar) annual withdrawals from their portfolios, reward and risk measures are defined to be the mean and standard deviation of wealth remaining at end of life. Asset returns and time of death are both treated as random variables. Assuming constant lifetime asset allocation, the risk and reward measures can be evaluated analytically, and an efficient frontier can be determined. Life annuities can be used to extend the left-hand (low-risk) side of the efficient frontier. The desired level of wealth at end of life can be used to choose a desirable portfolio on the efficient frontier. The desirable portfolio strongly depends on the withdrawal rate. It is suggested (although not proven) that asset allocations strategies that vary with age do not add efficiency in this model, and asset allocation strategies that vary with wealth can add efficiency.     

Short Sales Restrictions and Kinks on the Mean Variance Frontier

With a short sales restriction, there may be switching points along the mean variance frontier corresponding to changes in the set of assets held. Traditional wisdom holds that each switching point corresponds to a kink, while Ross has claimed that kinks never occur. This paper shows that the truth lies between the two views, since the efficient frontier may or may not be kinked at a switching point. There is some indication that kinks are rare, since a kink corresponds to a portfolio in which all assets have the same expected return.     

THE MEAN‐GINI EFFICIENT PORTFOLIO FRONTIER

A main advantage of the mean‐variance (MV) portfolio frontier is its simplicity and ease of derivation. A major shortcoming, however, lies in its familiar restrictions, such as the quadraticity of preferences or the normality of distributions. As a workable alternative to MV, we present the mean‐Gini (MG) efficient portfolio frontier. Using an optimization algorithm, we compute MG and mean‐extended Gini (MEG) efficient frontiers and compare the results with the MV frontier. MEG allows for the explicit introduction of risk aversion in building the efficient frontier. For U.S. classes of assets, MG and MEG efficient portfolios constructed using Ibbotson (2000) monthly returns appear to be more diversified than MV portfolios. When short sales are allowed, distinct investor risk aversions lead to different patterns of portfolio diversification, a result that is less obvious when short sales are foreclosed. Furthermore, we derive analytically the MG efficient portfolio frontier by restricting asset distributions. The MG frontier derivation is identical in structure to that of the MV efficient frontier derivation. The penalty paid for simplifying the search for the MG efficient frontier is the loss of some information about the distribution of assets.     

On the Cross-sectional Relation between Expected Returns and Betas

There is an exact linear relation between expected returns and true “betas” when the market portfolio is on the ex ante mean-variance efficient frontier, but empirical research has found little relation between sample mean returns and estimated betas. A possible explanation is that market portfolio proxies are mean-variance inefficient. We categorize proxies that produce particular relations between expected returns and true betas. For the special case of a zero relation, a market portfolio proxy must lie inside the efficient frontier, but it may be close to the frontier.     

A Benchmark Approach to Portfolio Optimization under Partial Information

This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.      

Linear programing models for portfolio optimization using a benchmark

We consider the problem of constructing a perturbed portfolio by utilizing a benchmark portfolio. We propose two computationally efficient portfolio optimization models, the mean-absolute deviation risk and the Dantzig-type, which can be solved using linear programing. These portfolio models push the existing benchmark toward the efficient frontier through sparse and stable asset selection. We implement these models on two benchmarks, a market index and the equally-weighted portfolio. We carry out an extensive out-of-sample analysis with 11 empirical datasets and simulated data. The proposed portfolios outperform the benchmark portfolio in various performance measures, including the mean return and Sharpe ratio.     

A dynamic balance sheet management model for a Canadian chartered bank

This paper presents a linear programming (LP) model based on Markowitz portfolio theory for solving the balance sheet management problem for the domestic assets and liabilities of a Canadian chartered bank. Given the bank's initial position, its economic forecasts, and the constraints under which it operates, the model will determine the current and expected future balance sheet adjustments which will meet the bank's expected profits goal with the minimum possible risk. By parametrically varying the expected profits goal, the model will generate the set of risk-return efficient decisions. Bankers need examine only the set of efficient decisions to choose their optimal solution.     

Simple Construction of the Efficient Frontier

We provide simple methods of constructing known results. At the core of our methods is the identification of a simple concise basis that spans the Capital Market Line (CML). We show that a portfolio whose risky assets weights are the product of the inverse variance‐covariance matrix of (nonredundant) security rates of return times the vector of the excess expected rates of return over the risk‐free rate is a CML portfolio. This portfolio and the risk‐free security span the CML. In addition, with this basis, there is immediate construction of the efficient frontier of risky assets (the 'hyperbola'), 'tangency' portfolios, 'reflection' portfolios, and a CAPM relationship. Our method is quick and simple. It is easy to derive, teach, implement, interpret, and remember.     

An accounting analysis of the risk-return relationship in bull and bear markets

This study provides evidence that accounting beta (earnings and cash flow-based) provides information consistent with the risk-return relationship in up- and down-markets. We are not able, however, to provide similar evidence using market beta. Considering that investors' ability to assess the risk-return trade-off in up- and down-markets is central to efficient portfolio formation, the results suggest that accounting data can provide appropriate measures of portfolio upside and downside risk.     

The econometrics of efficient portfolios

It is well known that the standard mean variance approach can be inappropriate when return distributions feature skewness, fat tails or multimodes. This is typically the situation for portfolios including derivatives. In this case, it can be necessary to come back to the basic expected utility approach. In this paper, an efficient portfolio maximizes the expected utility of future wealth. This paper presents an analysis of the efficiency frontier, formed by a set of efficient portfolios corresponding to a parameterized class of utility functions. First, we discuss the estimation of an efficient portfolio and introduce several tests of the efficiency hypothesis, depending on what is known about the utility function and the budget level. Next we analyse the shape of the frontier and develop a procedure for testing the separability of the efficiency frontier into K independent funds. The inference is semi-nonparametric because the return distribution is left unspecified. We illustrate our approach by an application to portfolios including derivatives.     

Portfolio frontiers with restrictions to tracking error volatility and value at risk

Asset managers are often given the task of restricting their activity by keeping both the value at risk (VaR) and the tracking error volatility (TEV) under control. However, these constraints may be impossible to satisfy simultaneously because VaR is independent of the benchmark portfolio. The management of these restrictions is likely to affect portfolio performance and produces a wide variety of scenarios in the risk-return space. The aim of this paper is to analyse various interactions between portfolio frontiers when risk managers impose joint restrictions upon TEV and VaR. Specifically, we provide analytical solutions for all the intersections and we propose simple numerical methods when such solutions are not available. Finally, we introduce a new portfolio frontier.     

Macroeconomic influences on optimal asset allocation

We develop a tactical asset allocation strategy that incorporates the effects of macroeconomic variables. The joint distribution of financial asset returns and the macroeconomic variables is modelled using a VAR with a multivariate GARCH (M-GARCH) error structure. As a result, the portfolio frontier is time varying and subject to contagion from the macroeconomic variable. Optimal asset allocation requires that this be taken into account. We illustrate how to do this using three risky UK assets and inflation as a macroeconomic factor. Taking account of inflation generates portfolio frontiers that lie closer to the origin and offers investors superior risk-return combinations.     

Fixed Versus Variable Rate Loans

This paper discusses the nature of fixed and variable loan contracts and derives the conditions which determine the optimal quantity of each. The results indicate that the payoff functions are quite different and dependent upon the project financed. The appropriate conditions for the allocation of loan terms to a set of borrowers are then developed. Finally, the analysis derives the optimal portfolio frontier and risk-return trade-off for the banking firm. Here, it is demonstrated that the solution is unlikely to be at a point of zero interest rate risk.     

Variable Rate Debt Instruments and Corporate Debt Policy

Sustained high rate of inflation has led to the creation of debt instruments with variable interest rate. The availability of these debt instruments presents management with the problem of the choice of the optimal debt portfolio. This paper deals with this problem assuming a given, and optimal, debt to equity ratio. Given expected monetary value maximization, an efficient frontier is derived in terms of the expected net income and probability of bankruptcy, where net income is defined as operating income minus debt repayment. This efficient frontier is shown to be also mean-variance efficient. It is also shown that in most cases the optimal debt portfolio includes more than one debt instrument. In other words, the firm will avoid the policy of minimizing the expected cost of its debt repayments or the policy of minimizing the costs of bankruptcy. The optimal solution itself is affected by market variables like the relative expected cost of different debt instruments and by firm specific variables like the variability of its operating income stream, and the covariance between the operating income and the debt repayments.     

Portfolio construction incorporating asymmetric dependence structures: a user's guide

We outline a method of portfolio selection incorporating asymmetric dependency structures using copula functions. Assuming normally distributed marginal returns, we illustrate how asymmetric return correlations affect the efficient frontier and subsequent portfolio performance under a dynamic rebalancing framework. Implementing this methodology within the context of tactically allocating a small set of market indices, we demonstrate several key findings. First, we establish the manner by which the efficient frontier constructed under asymmetric dependence differs from a mean‐variance frontier. By establishing a paper portfolio based on these differences, we find that asymmetric correlation structures do have real economic value. The primary source of this economic value is the ability to better protect portfolio value and reduce the size of any erosion in return relative to the normal portfolio when asymmetric return correlations are accounted for.     

Portfolio selection based on the mean–VaR efficient frontier

Value-at-Risk (VaR) has become one of the standard measures for assessing risk not only in the financial industry but also for asset allocations of individual investors. The traditional mean–variance framework for portfolio selection should, however, be revised when the investor's concern is the VaR instead of the standard deviation. This is especially true when asset returns are not normal. In this paper, we incorporate VaR in portfolio selection, and we propose a mean–VaR efficient frontier. Due to the two-objective optimization problem that is associated with the mean–VaR framework, an evolutionary multi-objective approach is required to construct the mean–VaR efficient frontier. Specifically, we consider the elitist non-dominated sorting Genetic Algorithm (NSGA-II). From our empirical analysis, we conclude that the risk-averse investor might inefficiently allocate his/her wealth if his/her decision is based on the mean–variance framework.     

On the Inverse of the Covariance Matrix in Portfolio Analysis

The goal of this paper is the derivation and application of a direct characterization of the inverse of the covariance matrix central to portfolio analysis. Such a characterization, in terms of a few primitive constructs, provides the basis for new and illuminating expressions for key concepts as the optimal holding of a given risky asset and the slope of the risk-return efficiency frontier faced by the individual investor. The building blocks of the inverse turn out to be the regression coefficients and residual variance obtained by regressing the asset's excess return on the set of excess returns for all other risky assets.     

Portfolio selection under institutional procedures for short selling: Normative and market-equilibrium considerations

In view of the acceptance of short selling of stocks as an investment tool in the portfolio context by a growing number of institutional investors in recent years, the present study considers both normative and market-equilibrium aspects of portfolio selection with short selling. Under the full-information covariance structure of security returns, the study accurately captures institutional procedures for short selling without sacrificing analytical tractability. While short selling enhances the portfolio's risk-return trade-off from a normative perspective, the equilibrium analysis reveals that there is a continuum of market-clearing prices within two boundaries for each security. Economic implications of the equilibrium pricing relationship are also explored in the study.     

Markowitz efficiency and size effect: evidence from the UK stock market

Academics and practitioners have frequently debated the relationship between market capitalization and expected return. We apply the Markowitz efficient frontier approach to develop a portfolio performance measure that compares the return of a portfolio to its optimal return, using data from the UK stock market over the period 1985–2012. Our results show that there is a negative relationship between portfolio size and portfolio return during the period under study. When comparing actual portfolio return with achievable return for the same level of risk, we find that as the portfolio size expands, underperformance of the portfolio increases, i.e. the larger the portfolio size, the greater the underperformance. This indicates that Markowitz efficiency is difficult to achieve, particularly in large portfolios. Changing model parameters leads to alternative efficient frontiers that impact upon the measurement of performance. However, the use of alternative efficient frontiers does not affect our result of the size effect on the relative performance of portfolios. Our study shows that the size effect is present over the full period. Our findings also suggest that the excess returns found in small portfolios are likely to be associated with higher levels of diversifiable risk in comparison with larger portfolios. Furthermore, in contrast to other studies, we find no evidence to support the size reversal effect in the data.