Diana Barro

Tracking error: a multistage portfolio model

Abstract We study multistage tracking error problems. Different tracking error measures, commonly used in static models, are discussed as well as some problems which arise when we move from static to dynamic models. We are interested in dynamically replicating a benchmark using only a small subset of assets, considering transaction costs due to rebalancing and introducing a liquidity component in the portfolio. We formulate and solve a multistage tracking error model in a stochastic programming framework. We numerically test our model by dynamically replicating the MSCI Euro index. We consider an increasing number of scenarios and assets and show the superior performance of the dynamically optimized tracking portfolio over static strategies.

Dynamic portfolio optimization: Time decomposition using the Maximum Principle with a scenario approach

We study a dynamic portfolio management problem over a finite horizon with transaction costs and a risk averse objective function. We assume that the uncertainty faced by the investor can be modelled or approximated using discrete probability distributions via a scenario approach. To solve the resulting optimization problem we use stochastic programming techniques; in particular a scenario decomposition approach. To take advantage of the structure of the portfolio problem we propose a further decomposition obtained by means of a discrete version of the Maximum Principle. The result is a double decomposition of the original problem: The first, given by the scenario approach, focuses on the stochastic aspect of the problem while the second, using the discrete Maximum Principle, concerns the dynamics over time. Applying the double decomposition to our portfolio problem yields a simpler and more direct solution approach which we illustrate with examples.

Credit contagion in a network of firms with spatial interaction

This contribution studies the effects of credit contagion on the credit risk of a portfolio of bank loans. To this aim we introduce a model that takes into account the counterparty risk in a network of interdependent firms that describes the presence of business relations among different firms. The location of the firms is simulated with probabilities computed using an entropy spatial interaction model. By means of a wide simulation analysis we investigate the behavior of the model proposed and study the effects of default contagion on the loss distribution of a portfolio of bank loans.

Downside Risk in Multiperiod Tracking Error Models

The recent crisis made it evident that replicating the performance of a benchmark is not a sufficient goal to meet the expectations of usually risk-averse investors. The manager should also consider that the investors are seeking downside protection when the benchmark performs poorly and thus they should integrate a form of downside risk control. We propose a multiperiod double tracking error portfolio model which combines these two goals and provides enough flexibility. In particular, the control of the downside risk is carried out through the presence of a floor benchmark with respect to which we can accept different levels of shortfall. The choice of a proper measure for downside risk leads to different problem formulations and investment strategies which can reflect different attitudes towards risk. The proposed model is tested through a set of out-of-sample rolling simulations in different market conditions.

Combining stochastic programming and optimal control to decompose multistage stochastic optimization problems

The paper suggests a possible cooperation between stochastic programming and optimal control for the solution of multistage stochastic optimization problems. We propose a decomposition approach for a class of multistage stochastic programming problems in arborescent form (i.e. formulated with implicit non-anticipativity constraints on a scenario tree). The objective function of the problem can be either linear or nonlinear, while we require that the constraints are linear and involve only variables from two adjacent periods (current and lag 1). The approach is built on the following steps. First, reformulate the stochastic programming problem into an optimal control one. Second, apply a discrete version of Pontryagin maximum principle to obtain optimality conditions. Third, discuss and rearrange these conditions to obtain a decomposition that acts both at a time stage level and at a nodal level. To obtain the solution of the original problem we aggregate the solutions of subproblems through an enhanced mean valued fixed point iterative scheme.

Tracking error with minimum guarantee constraints

In recent years the popularity of indexing has greatly increased in financial markets and many different families of products have been introduced. Often these products also have a minimum guarantee in the form of a minimum rate of return at specified dates or a minimum level of wealth at the end of the horizon. Period of declining stock market returns together with low interest rate levels on Treasury bonds make it more difficult to meet these liabilities. We formulate a dynamic asset allocation problem which takes into account the conflicting objectives of a minimum guaranteed return and of an upside capture of the risky asset returns. To combine these goals we formulate a double tracking error problem using asymmetric tracking error measures in the multistage stochastic programming framework.

Integration of Non-financial Criteria in Equity Investment

In recent years the awareness of social, environmental and governance issues associated with investments have drawn relevant interest in the investment industry. Investors are more careful in considering investments that comply with their ethical and moral values, as well as with social impact. Hence, the ethical and social responsibility of investments (SRI) is becoming more popular in the academic literature due to the fact that socially responsible investment provide profitability and social commitment together. In this contribution we discuss the main issues that arise when integrating socially responsible criteria into a financial decision problem.

Volatility vs. Downside Risk: Optimally Protecting Against Drawdowns and Maintaining Portfolio Performance

As a consequence of recent market conditions an increasing number of investors are realizing the importance of controlling tail risk to reduce drawdowns thus increasing possibilities of achieving long-term objectives. Recently, so called volatility control strategies and volatility target approaches to investment have gained a lot of interest as strategies able to mitigate tail risk and produce better risk-adjusted returns. Essentially these are rule-based backward looking strategies in which no optimization is considered. In this contribution we focus on the role of volatility in downside risk reduction and, in particular, in tail risk reduction. The first contribution of our paper is to provide a viable way to integrate a target volatility approach, into a multiperiod portfolio optimization model, through the introduction of a local volatility control approach. Our optimized volatility control is contrasted with existing rule-based target volatility strategies, in an out-of sample simulation on real data, to assess the improvement that can be obtained from the optimization process. A second contribution of this work is to study the interaction between volatility control and downside risk control. We show that combining the two tools we can enhance the possibility of achieving the desired performance objectives and, simultaneously, we reduce the cost of hedging. The multiperiod portfolio optimization problem is formulated in a stochastic programming framework that provides the necessary flexibility for dealing with different constraints and multiple sources of risk.

Dynamic Tracking Error with Shortfall Control Using Stochastic Programming

In this contribution we propose a dynamic tracking error problem and we consider the problem of monitoring at discrete point the shortfall of the portfolio below a set of given reference levels of wealth. We formulate and solve the resulting dynamic optimization problem using stochastic programming. The resulting problem allows for a great flexibility in the combination of a tracking goal and a downside risk protection through a discrete monitoring of the shortfalls. We provide the results of a out-of-sample simulation experiments, on real data, for different portfolio configurations and different market conditions.