3,656 research outputs found
On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory
In the paper, we consider three quadratic optimization problems which are
frequently applied in portfolio theory, i.e, the Markowitz mean-variance
problem as well as the problems based on the mean-variance utility function and
the quadratic utility.Conditions are derived under which the solutions of these
three optimization procedures coincide and are lying on the efficient frontier,
the set of mean-variance optimal portfolios. It is shown that the solutions of
the Markowitz optimization problem and the quadratic utility problem are not
always mean-variance efficient. The conditions for the mean-variance efficiency
of the solutions depend on the unknown parameters of the asset returns. We deal
with the problem of parameter uncertainty in detail and derive the
probabilities that the estimated solutions of the Markowitz problem and the
quadratic utility problem are mean-variance efficient. Because these
probabilities deviate from one the above mentioned quadratic optimization
problems are not stochastically equivalent. The obtained results are
illustrated by an empirical study.Comment: Revised preprint. To appear in European Journal of Operational
Research. Contains 18 pages, 6 figure
Theory and Applications of Robust Optimization
In this paper we survey the primary research, both theoretical and applied,
in the area of Robust Optimization (RO). Our focus is on the computational
attractiveness of RO approaches, as well as the modeling power and broad
applicability of the methodology. In addition to surveying prominent
theoretical results of RO, we also present some recent results linking RO to
adaptable models for multi-stage decision-making problems. Finally, we
highlight applications of RO across a wide spectrum of domains, including
finance, statistics, learning, and various areas of engineering.Comment: 50 page
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
Risk measures and their applications in asset management
Several approaches exist to model decision making under risk, where risk can be broadly defined as the effect of variability of random outcomes. One of the main approaches in the practice of decision making under risk uses mean-risk models; one such well-known is the classical Markowitz model, where variance is used as risk measure. Along this line, we consider a portfolio selection problem, where the asset returns have an elliptical distribution. We mainly focus on portfolio optimization models constructing portfolios with minimal risk, provided that a prescribed expected return level is attained. In particular, we model the risk by using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). After reviewing the main properties of VaR and CVaR, we present short proofs to some of the well-known results. Finally, we describe a computationally efficient solution algorithm and present numerical results.conditional value-at-risk;elliptical distributions;mean-risk;portfolio optimization;value-at-risk
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
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