Investment strategies and compensation of a mean-variance optimizing fund manager

This paper introduces a general continuous-time mathematical framework for solution of dynamic mean–variance control problems. We obtain theoretical results for two classes of functionals: the first one depends on the whole trajectory of the controlled process and the second one is based on its terminal-time value. These results enable the development of numerical methods for mean–variance problems for a pre-determined risk-aversion coefficient. We apply them to study optimal trading strategies pursued by fund managers in response to various types of compensation schemes. In particular, we examine the effects of continuous monitoring and scheme’s symmetry on trading behavior and fund performance

Black-Litterman model for continuous distributions

The Black–Litterman methodology of portfolio optimization, developed at the turn of the 1990s, combines statistical information on asset returns with investor’s views within the Markowitz mean-variance framework. The main assumption underlying the Black–Litterman model is that asset returns and investor’s views are multivariate normally distributed. However, empirical research demonstrates that the distribution of asset returns has fat tails and is asymmetric, which contradicts normality. Recent advances in risk measurement advocate replacing the variance by risk measures that take account of tail behavior of the portfolio return distribution. This paper extends the Black–Litterman model into general continuous distributions and deviation measures of risk. Using ideas from the Black–Litterman methodology, we design numerical methods (with variance reduction techniques) for the inverse portfolio optimization that extracts statistical information from historical data in a stable way. We introduce a quantitative model for stating investor’s views and blending them consistently with the market information. The theory is complemented by efficient numerical methods with the implementation distributed in the form of publicly available R packages. We conduct practical tests, which demonstrate significant impact of the choice of distributions on optimal portfolio weights to the extent that the classical Black–Litterman procedure cannot be viewed as an adequate approximation

Theoretical and empirical estimates of mean-variance portfolio sensitivity

This paper studies properties of an estimator of mean-variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher

Interest Rate Volatility and Risk Management: Evidence from CBOE Treasury Options

This paper investigates US Treasury market volatility and develops new ways of dealing with the underlying interest rate volatility risk. We adopt an innovative approach which is based on a class of model-free interest rate volatility (VXI) indices we derive from options traded on the CBOE. The empirical analysis indicates substantial interest rate volatility risk for medium-term instruments which declines to the levels of the equity market only as the tenor increases to 30 years. We show that this risk appears to be priced in the market and has a significant time-varying relationship with equity volatility risk. US Treasury market volatility is appealing from an investment diversification perspective since the VXI indices are negatively correlated with the levels of interest rates and of equity market implied volatility indices, respectively. Although VXI indices are affected by macroeconomic and monetary news, they are only partially spanned by information contained in the yield curve. Motivated by our results on the magnitude and the nature of interest rate volatility risk and by the phenomenal recent growth of the equity volatility derivative market, we propose the use of our VXI indices as benchmarks for monitoring, securitizing, managing and trading interest rate volatility risk. As a first step in this direction, we describe a framework of one-factor equilibrium models for pricing VXI futures and options on the basis of empirically favored mean-reverting jump-diffusions