Bob Korkie

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.

Tests of Conditional Asset Pricing with Time-Varying Moments and Risk Prices

This paper tests conditional capital asset pricing models in a multivariate GARCH framework employing both constant and time-varying prices of market and bond risk. Depending on the interpretation of the market portfolio, the ICAPM with one hedge portfolio or the two-factor myopic CAPM are supported using weekly data from July 1983 to December 1989. In contrast, we reject the myopic single-factor CAPM under a constant price of market risk. We find that interest rate risk is highly significant, which suggests that previous rejections of the conditional CAPM using only stock market data may be due to omitted hedge terms or an incomplete market factor.

A Mean-Variance Analysis of Self-Financing Portfolios

This paper develops the analytics and geometry of the investment opportunity set IOS and the test statistics for self-financing portfolios. A self-financing portfolio is a set of long and short investments such that the sum of their investment weights, or net investment, is zero. This contrasts with a standard portfolio that has investment weights summing to one. Examples of self-financing portfolios are hedges, overlays, arbitrage portfolios, swaps, and long/short portfolios. A standard portfolio plus the IOS of self-financing portfolios form a restricted IOS hyperbola with restricted efficient set constants that differ from the usual constants. The restrictions affect statistical tests of portfolio efficiency, which are developed for the self-financing restrictions. As an application, we consider the self-financing portfolios formed by Fama and French 1992, 1993, 1995, based on market capitalization and value. In contrast to Fama and French 1992, 1993, 1995, we find that their restricted IOS is significantly different from the unrestricted IOS with the implication that the Fama-French tests are misspecified.

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.

Variance Spillover and Skewness in Financial Asset Returns

Bond and stock returns have been observed in the literature to exhibit unconditional skewness and temporal persistence in conditional skewness. We demonstrate that observed persistence in conditional third central moments can be due to the spillover of conditional variance dynamics. The confounding of true skewness and a variance spillover effect is problematic for financial modeling. Using market data, we empirically demonstrate that a simple standardization approach removes the variance-induced skewness persistence. An important implication is that more parsimonious return and asset pricing models result if skewness persistence need not be modeled.

Market Line Deviations and Market Anomalies with Reference to Small and Large Firms

Previous anomaly research may have misinterpreted corrected, for the market index, mean returns on small firms. Assuming mean-variance preferences, it is shown theoretically that corrected mean returns (i.e., market line deviations) are not indicative of the relative desirability of increasing the proportional investment in small firms. The correct improvement criterion is derived and estimated. Tests indicate that the value-weighted market index is not significantly improved with greater weight on small firms, in the average month. When seasonality is considered, the observed performance improvements due to small or large firms are significant in some months, but the required portfolio position is unclear. If a case exists for a small firm anomaly in January, it probably exists in other months and it also exists for large firms. It is doubtful whether such an anomalous stock market exists.

A Note on the Analytics and Geometry of Limiting Mean-Variance Investment Opportunity Sets

This paper extends the mathematics developed by Merton (1972) to the limiting investment opportunity set as smaller risk assets are added. Investment opportunity sets of risky assets are well-known to be described by hyperbolae in mean-standard deviation space. In practice, the asset classes in portfolios may vary from high risk common stocks to near cash assets. Low variability assets change the appearance of the investment opportunity set to the extent that a unique optimum risky asset portfolio disappears. The limiting result is similar to the investment opportunity set that arises when two assets are perfectly correlated. The location of the IOS is shown to mathematically depend upon the level of the riskless interest rate and one slope parameter. The slope parameter is estimable, using a finite number of assets, and represents a bound on market Sharpe ratios.

A contingent claim analysis of closed-end fund premia

Abstract We estimate contingent claims that replicate monthly net asset value (NAV) payoffs from closed-end funds. A claims theoretical value is obtained by martingale pricing methods. The resulting net present value (NPVS) sequence is the theoretical premia sequence that is compared to the actual market premia sequence. The theoretical premia, like actual premia, are uncorrelated with NAV returns and are positively autocorrelated due to autocorrelation in the pricing information. However, there is poor correspondence between the theoretical and actual premia that seems due to the markets systematic errors in estimating a funds management value. Risky arbitrage may be available to insiders.