The premium of dynamic trading

It is well established that, in a market with inclusion of a risk-free asset, the single-period mean–variance efficient frontier is a straight line tangent to the risky region, a fact that is the very foundation of the classical CAPM. In this paper, it is shown that, in a continuous-time market where the risky prices are described by Itô processes and the investment opportunity set is deterministic (albeit time-varying), any efficient portfolio must involve allocation to the risk-free asset at any time. As a result, the dynamic mean–variance efficient frontier, although still a straight line, is strictly above the entire risky region. This in turn suggests a positive premium, in terms of the Sharpe ratio of the efficient frontier, arising from dynamic trading. Another implication is that the inclusion of a risk-free asset boosts the Sharpe ratio of the efficient frontier, which again contrasts sharply with the single-period case.     

The premium of dynamic trading in a discrete-time setting

Chiu and Zhou [Quant. Finance, 2011, 11, 115–123] show that the inclusion of a risk-free asset strictly boosts the Sharpe ratio in a continuous-time setting, which is in sharp contrast to the static single-period case. In this paper, we extend their work to a discrete-time setting. Specifically, we prove that the multi-period mean-variance efficient frontier generated by both risky and risk-free assets is strictly separated from that generated by only risky assets. As a result, we demonstrate that the inclusion of a risk-free asset strictly enhances the best Sharpe ratio of the efficient frontier in a multi-period discrete-time setting. Furthermore, we offer an explicit expression for the enhancement of the best Sharpe ratio, which was referred to as the premium of dynamic trading by Chiu and Zhou [op. cit.], although they do not present a computational formula for it. Our results further show that, in the case with a risk-free asset, if an investor can extract some money from his initial wealth at time 0, the efficient frontier with a risk-free asset can be tangent to that without a risk-free asset. Finally, based on real data from the American market, a numerical example is provided to illustrate the results obtained in this paper; a numerical comparison between the discrete-time case and the continuous-time case is also provided. Our numerical results reveal that the continuous-time model can be considered to be a limit of the discrete-time model.     

Comment on “Optimal portfolio selection in a value-at-risk framework”

This comment discusses some errors in [Journal of Banking and Finance 25 (2001) 1789]. Given the portfolio rate of return is normally distributed, the following can be inferred. First, taking expected portfolio return rate as the benchmark of value-at-risk (VaR), the risk–return ratio collapses to a multiple of the Sharpe index. However, using risk-free rate as the benchmark, then above inference does not hold. Second, whether the benchmark of VaR is expected portfolio return rate or the risk-free rate, the optimal asset allocations for maximizing the risk–return ratio and Sharpe index are identical.     

Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.     

Noise fit,estimation error and a Sharpe information criterion

When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjusted Sharpe ratio as model selection criterion analogously to the Akaike Information Criterion (AIC). Selecting a model with the highest adjusted Sharpe ratio selects the model with the highest estimated out-of-sample Sharpe ratio in the same way as selection by AIC does for the log-likelihood as a measure of fit.     

A subjective assessment of approximate probabilities with a portfolio application

This paper extends the assessment of approximate probabilities in two important directions. The first is to investigate some mathematical relations between the probability ranges and derives the most unbiased probability for the case when the limits are subjectively defined. The second is to suggest a simple method to determine the optimal solution which represents the optimal portfolio proportions of securities that possess the minimum risk measured by the maximum entropy measure. The paper considers the derivation of portfolio modeling under a fuzzy situation using probability theory, and provides various other (non-probabilistic) scenarios with their utility in risk modeling. A simple method for identification of mean-entropic frontier is proposed. Then, a comparison of mean-variance procedure with the discrete mean-entropic method is implemented by an example.     

Maximizing an equity portfolio excess growth rate: a new form of smart beta strategy?

It has been claimed that, for dynamic investment strategies, the simple act of rebalancing a portfolio can be a source of additional performance, sometimes referred to as the volatility pumping effect or the diversification bonus because volatility and diversification turn out to be key drivers of the portfolio performance. Stochastic portfolio theory suggests that the portfolio excess growth rate, defined as the difference between the portfolio expected growth rate and the weighted-average expected growth rate of the assets in the portfolio, is an important component of this additional performance (see Fernholz [Stochastic Portfolio Theory, 2002 (Springer)]). In this context, one might wonder whether maximizing a portfolio excess growth rate would lead to an improvement in the portfolio performance or risk-adjusted performance. This paper provides a thorough empirical analysis of the maximization of an equity portfolio excess growth rate in a portfolio construction context for individual stocks. In out-of-sample empirical tests conducted on individual stocks from 4 different regions (US, UK, Eurozone and Japan), we find that portfolios that maximize the excess growth rate are characterized by a strong negative exposure to the low volatility factor and a higher than 1 exposure to the market factor, implying that such portfolios are attractive alternatives to competing smart portfolios in markets where the low volatility anomaly does not hold (e.g. in the UK, or in rising interest rate scenarios) or in bull market environments.     

Robust portfolio optimization with a generalized expected utility model under ambiguity

This paper proposes a robust approach maximizing worst-case utility when both the distributions underlying the uncertain vector of returns are exactly unknown and the estimates of the structure of returns are unreliable. We introduce concave convex utility function measuring the utility of investors under model uncertainty and uncertainty structure describing the moments of returns and all possible distributions and show that the robust portfolio optimization problem corresponding to the uncertainty structure can be reformulated as a parametric quadratic programming problem, enabling to obtain explicit formula solutions, an efficient frontier and equilibrium price system. We would like to thank Prof. Zengjing Chen from School of Mathematics and System Sciences, Shandong University for helpful suggestions, and to thank the anonymous referee for valuable comments.     

Optimizing a portfolio of mean-reverting assets with transaction costs via a feedforward neural network

Optimizing a portfolio of mean-reverting assets under transaction costs and a finite horizon is severely constrained by the curse of high dimensionality. To overcome the exponential barrier, we develop an efficient, scalable algorithm by employing a feedforward neural network. A novel concept is to apply HJB equations as an advanced start for the neural network. Empirical tests with several practical examples, including a portfolio of 48 correlated pair trades over 50 time steps, show the advantages of the approach in a high-dimensional setting. We conjecture that other financial optimization problems are amenable to similar approaches.