The impact of association measures within the portfolio dimensionality reduction problem

The dependency structure of random sources plays a crucial role in portfolio theory and in several pricing and risk management problems. In this paper, we discuss the possible usage of alternative association measures in portfolio problems. Among association measures, we highlight those that are consistent with the choices of risk-averse investors and we characterise semidefinite positive association measures. Additionally, we propose new portfolio selection problems that optimise the association between the portfolio and market benchmarks and follow a dimensionality reduction problem. Finally, by carrying out an empirical analysis, we show the impact of selected association measures within the portfolio problem. This analysis proves that the proper usage of both a risk measure and an association measure can increase the portfolio performance substantially

International Portfolio Management under Uncertainty

Although the consideration of foreign investments may have a positive impact on the overall market risk of the portfolio through diversi cation, it also adds a new source of uncertainty due to changes in the value of the currency. We investigate portfolio optimization models that account separately for the local asset returns and the currency returns, providing the investor with a full investment strategy. We tackle the uncertainty inherent to the estimation of the parameters with the aid of robust optimization techniques. We show how, by using appropriate assumptions regarding the formulation of the uncertainty sets, the original non-linear and non-convex models may be reformulated as second order cone or as semide nite programs. Additionally to the guarantees provided by robust optimization, we consider the use of hedging instruments such as forward contracts and options. The proposed hedging strategies are implemented from a portfolio perspective, and therefore do not depend on the individual value or behavior of any particular asset or currency. Hedging decisions are taken at the same time as investment decisions in a holistic approach to portfolio management. While dynamic decision making has traditionally been represented as scenario trees, these may become severely intractable and di cult to compute with an increasing number of time periods. We present an alternative approach to multiperiod international portfolio optimization based on an a ne dependence between the decision variables and the past returns. We add to our formulation the minimization of the worst case value-at-risk and show the close relationship with robust optimization. The proposed theoretical framework is supported by various numerical experiments with simulated and historical market data demonstrating its potential bene ts

Configuration model for correlation matrices preserving the node strength

Correlation matrices are a major type of multivariate data. To examine properties of a given correlation matrix, a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices, such as those derived from random matrix theory, that partially preserve properties of the original matrix. We propose a model to generate such reference correlation and covariance matrices for the given matrix. Correlation matrices are often analysed as networks, which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node. Given this background, the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes, akin to configuration models for conventional networks. Our algorithm is derived from the maximum entropy principle. We will apply the proposed algorithm to measurement of clustering coefficients and community detection, both of which require a null model to assess the statistical significance of the obtained results.Comment: 8 figures, 4 table

The Wishart short rate model

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves

Efficient Solution of Portfolio Optimization Problems via Dimension Reduction and Sparsification

The Markowitz mean-variance portfolio optimization model aims to balance expected return and risk when investing. However, there is a significant limitation when solving large portfolio optimization problems efficiently: the large and dense covariance matrix. Since portfolio performance can be potentially improved by considering a wider range of investments, it is imperative to be able to solve large portfolio optimization problems efficiently, typically in microseconds. We propose dimension reduction and increased sparsity as remedies for the covariance matrix. The size reduction is based on predictions from machine learning techniques and the solution to a linear programming problem. We find that using the efficient frontier from the linear formulation is much better at predicting the assets on the Markowitz efficient frontier, compared to the predictions from neural networks. Reducing the covariance matrix based on these predictions decreases both runtime and total iterations. We also present a technique to sparsify the covariance matrix such that it preserves positive semi-definiteness, which improves runtime per iteration. The methods we discuss all achieved similar portfolio expected risk and return as we would obtain from a full dense covariance matrix but with improved optimizer performance.Comment: 14 pages, 3 figure

Distributionally robust optimization with applications to risk management

Many decision problems can be formulated as mathematical optimization models. While deterministic optimization problems include only known parameters, real-life decision problems almost invariably involve parameters that are subject to uncertainty. Failure to take this uncertainty under consideration may yield decisions which can lead to unexpected or even catastrophic results if certain scenarios are realized. While stochastic programming is a sound approach to decision making under uncertainty, it assumes that the decision maker has complete knowledge about the probability distribution that governs the uncertain parameters. This assumption is usually unjustified as, for most realistic problems, the probability distribution must be estimated from historical data and is therefore itself uncertain. Failure to take this distributional modeling risk into account can result in unduly optimistic risk assessment and suboptimal decisions. Furthermore, for most distributions, stochastic programs involving chance constraints cannot be solved using polynomial-time algorithms. In contrast to stochastic programming, distributionally robust optimization explicitly accounts for distributional uncertainty. In this framework, it is assumed that the decision maker has access to only partial distributional information, such as the first- and second-order moments as well as the support. Subsequently, the problem is solved under the worst-case distribution that complies with this partial information. This worst-case approach effectively immunizes the problem against distributional modeling risk. The objective of this thesis is to investigate how robust optimization techniques can be used for quantitative risk management. In particular, we study how the risk of large-scale derivative portfolios can be computed as well as minimized, while making minimal assumptions about the probability distribution of the underlying asset returns. Our interest in derivative portfolios stems from the fact that careless investment in derivatives can yield large losses or even bankruptcy. We show that by employing robust optimization techniques we are able to capture the substantial risks involved in derivative investments. Furthermore, we investigate how distributionally robust chance constrained programs can be reformulated or approximated as tractable optimization problems. Throughout the thesis, we aim to derive tractable models that are scalable to industrial-size problems

PORTFOLIO OPTIMIZATION IN ELECTRICITY TRADING WITH LIMITED LIQUIDITY

In principle, portfolio optimization in electricity markets can make use of the standard mean-variance model going back to Markowitz. Yet a key restriction in most electricity markets is the limited liquidity. Therefore the standard model has to be adapted to cope with limited liquidity. An application of this model shows that the optimal hedging strategy for generation portfolios is strongly dependent on the size of the portfolio considered as well as on the variance-covariancematrix used and the liquidity function assumed.optimization; electricity, liquidity; electricity trading; mean-variance-model