Risk minimization and portfolio diversification

We consider the problem of minimizing capital at risk in the Black-Scholes setting. The portfolio problem is studied given the possibility that a correlation constraint between the portfolio and a financial index is imposed. The optimal portfolio is obtained in closed form. The effects of the correlation constraint are explored; it turns out that this portfolio constraint leads to a more diversified portfolio

Worst-Case Value-at-Risk of Non-Linear Portfolios

Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are further compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR over all return distributions of the derivative underliers with given first- and second-order moments. The derivative returns are modelled as convex piecewise linear or - by using a delta-gamma approximation - as (possibly non-convex) quadratic functions of the returns of the derivative underliers. These models lead to new Worst-Case Polyhedral VaR (WCPVaR) and Worst-Case Quadratic VaR (WCQVaR) approximations, respectively. WCPVaR is a suitable VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WCQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that WCPVaR and WCQVaR optimization can be formulated as tractable second-order cone and semidefinite programs, respectively, and reveal interesting connections to robust portfolio optimization. Numerical experiments demonstrate the benefits of incorporating non-linear relationships between the asset returns into a worst-case VaR model.Value-at-Risk, Derivatives, Robust Optimization, Second-Order Cone Programming, Semidefinite Programming

Robust Optimization of Currency Portfolios

We study a currency investment strategy, where we maximize the return on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques to maximize the return on the portfolio for the worst-case foreign exchange rate scenario. Currency portfolios differ from stock only portfolios in that a triangular relationship exists among foreign exchange rates to avoid arbitrage. Although the inclusion of such a constraint in the model would lead to a nonconvex problem, we show that by choosing appropriate uncertainty sets for the exchange and the cross exchange rates, we obtain a convex model that can be solved efficiently. Alongside robust optimization, an additional guarantee is explored by investing in currency options to cover the eventuality that foreign exchange rates materialize outside the specified uncertainty sets. We present numerical results that show the relationship between the size of the uncertainty sets and the distribution of the investment among currencies and options, and the overall performance of the model in a series of backtesting experiments.robust optimization, portfolio optimization, currency hedging, second-order cone programming

Robust Portfolio Optimization with Derivative Insurance Guarantees

Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust portfolio optimization model that provides additional strong performance guarantees for all possible realizations of the asset returns. This insurance is provided via optimally chosen derivatives on the assets in the portfolio. The resulting model constitutes a convex second- order cone program, which is amenable to efficient numerical solution. We evaluate the model using simulated and empirical backtests and conclude that it can out- perform standard robust portfolio optimization as well as classical mean-variance optimization.robust optimization, portfolio optimization, portfolio insurance, second order cone programming

Global Portfolio Optiomization Revisted: A Least Discrimination Alternative to Black-Litterman

Global portfolio optimization models rank among the proudest achievements of modern finance theory, but practitioners are still struggling to put them to work. In 1992, Black and Litterman put the problem down to difficulties portfolio managers have in extrapolating views about some expected asset returns into full probabilistic forecasts about all asset returns and proposed a method to alleviate this problem. We propose a more general method based on a least discrimination (LD) principle. It produces a probabilistic forecast that remains true to personal views but is otherwise as close as possible to the forecast implied by a reference portfolio. The LD method produces optimal portfolios for a variety of views, including views on volatility and correlation, in which case optimal portfolios include option-like pay-offs. It also justifies a simple linear interpolation between market and personal forecasts, should a compromise be reached.Global portfolio optimization, black-litterman model, least discrimination, utility theory, mean-variance analysis, relative entropy, generalized relative entropy, non-linear pay-offs

Relative Robust Portfolio Optimization

Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management

Ex Ante versus Ex Post Regulation of Bank Capital

The current debate on the new Basel Accord gives rise to a natural question about the appropriate form of capital regulation. We construct a general framework to study this issue. We show that ex ante regulation wastes the expertise of a bank in measuring its risk exposure, while an ex post regime makes full use of it. However, the latter is more vulnerable to the problem of unknown managerial risk preference. The results imply that the two regimes are complements, rather than substitutes. Further, under plausible conditions, an ex post regime emerges as the dominant element of the optimal combination. We use the results to shed light on current policy concerns.Ex Ante Regulation, Ex Post Regulation, Asymmetric Information, Safety Loss, Overportection, Loss, Safety Bias, Basel II

Ex Ante Versus Ex Post Regulation of Bank Capital

The current debate on the new Basel Accord gives rise to a natural question about the appropriate form of capital regulation.We construct a simple framework to analyze this issue. In our model the risk carried by a bank as well as managerial risk preference are a bank's private information. We show that ex ante constraints waste the superior risk information of a bank, while an ex post regime makes full use of it. However, the latter is more vulnerable to the problem of unknown managerial risk-aversion. The results imply that the two regimes are complements, rather than substitutes. Further, under plausible conditions, an ex post regime emerges as the dominant element of the optimal combination. We use the results to shed light on current policy concerns. In particular, our results provides theoretical underpinning for the inclusion of pillar 2 alongside pillar 1 in Basel II.Ex Ante Regulation, Ex Post Regulation, Asymmetric Information, Safety Loss, Overprotection Loss, Safety Bias, Basel II.