Electronic trading of equities and other securities makes heavy use of “arrival price ” algorithms, that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, mean-variance optimal trading strategies are static: they do not modify the execution speed in response to price motions observed during trading. We show that substantial improvement is possible by using dynamic trading strategies, and that the improvement is larger for large initial positions. We develop a technique for computing optimal dynamic strategies to any desired degree of precision. The asset price process is observed on a discrete tree with a arbitrary number of levels. We introduce a novel dynamic programming technique in which the control variables are not only the shares traded at each time step, but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. The resulting adaptive strategies are “aggressive-in-the-money”: they accelerate the execution when the price moves in the trader’s favor, spending parts of the trading gains to reduce risk.

It is now known that the very impressive investment returns generated by Bernie Madoff were based on a sophisticated Ponzi scheme. Madoff claimed to use a splitstrike conversion strategy. This strategy consists of a long equity position plus a long put and a short call. In this paper we examine Madoff’s returns and compare his investment performance with what could have been obtained using a split-strike conversion strategy based on the historical data. We also analyze the split-strike strategy in general and derive expressions for the expected return, standard deviation, Sharpe ratio and correlation with the market of this strategy. We find that Madoff’s returns lie well outside their theoretical bounds and should have raised suspicions about Madoff’s performance.

We solve the dynamic mean-variance portfolio problem and derive its time-consistent solution using dynamic programming. Previous literature, in contrast, only determines either myopic or precommitment (committing to follow the initially optimal policy) solutions. We provide a fully analytical simple characterization of the dynamically optimal mean-variance portfolios within a general incomplete-market economy. We also identify a probability measure that incorporates intertemporal hedging demands and facilitates tractability. We illustrate this by easily computing portfolios explicitly under various stochastic investment opportunities. A calibration exercise shows that the mean– variance hedging demands are economically significant. (JEL G11, D81, C61) The mean-variance analysis of Markowitz (1952) has long been recognized as the cornerstone of modern portfolio theory. Its simplicity and intuitive appeal have led to its widespread use in both academia and industry. Orig-inally cast in a single-period framework, the mean-variance paradigm has no doubt also inspired the development of the multiperiod portfolio choice literature. To this day, the mean-variance criteria are employed in many multi-

1I would like to thank Anil Bera and Zhijie Xiao for their comments that helped earlier drafts of the paper. I would also like to acknowledge part of the data provided from JAE website for Diez de los Rios and Garcia, 2009. This project is funded by SMU O ¢ ce of Research Grant No: 09-C244-SMU-004. This is a preliminary version of the paper, please do not quote without authors permission. Copyright A. Ghosh, 2011. Traditional tests of …nancial risk for optimal portfolio choice based on Sharpe ratios are inherently ensconsed in the normality assumption of the return distribution be-sides independence. Such tests are not strictly valid for …nancial data that are known to be leptokurtic, and often show persistence in levels or volatility. We propose a smooth total moment risk measure with directional components that address the drawbacks of such procedures for practical implementation and inference. Our illus-tration of the proposed test on hedge fund indices with other existing measures show promising future for the new risk measure that has known tabulated distributions.