Quadratic programming for portfolio optimization

Portfolio optimization is a procedure for generating a portfolio composition which yields the highest return for a given level of risk or a minimum risk for given level of return. The problem can be formulated as a quadratic programming problem. We shall present a new and efficient optimization procedure taking advantage of the special structure of the portfolio selection problem. An example of its application to the traditional mean-variance method will be shown. Formulation of the procedure shows that the solution of the problem is vector intensive and fits well with the advanced architecture of recent computers, namely the vector processor.     

Tracking models and the optimal regret distribution in asset allocation

We present a tracking model for asset allocation that tracks desired investment goals. The model is shown to be optimal with respect to an investor's ‘regret distribution’, the cumulative distribution of the difference between the revenue under perfect foresight and that possible without foresight. Relationships with Markowitz mean/variance models are also explored.     

Dynamic capital allocation with irreversible investments

Capital allocation models generally assume that the risk portfolio is constructed at a single point in time, when the underwriter has full information about available underwriting opportunities. However, in practice, opportunities are not all known at the beginning but instead arrive over time. Moreover, a commitment to an opportunity is not easy to change as time passes. Thus, to optimize a portfolio, the underwriter must make decisions on opportunities as they arrive while making use of assumptions about what will arrive in the future. This paper studies capital allocation rules in this setting, finding important differences from the static setting. The pricing of an opportunity is based on an expected future marginal cost of risk associated with that opportunity—one that will be fully understood only after the risk portfolio is finalized. The risk charge for today’s opportunity is thus a probability-weighted average of the product of the marginal value of capital in future states of the world and the amount of capital consumed by the opportunity in those future states. Our numerical examples illustrate how the marginal cost of risk for an opportunity is shaped by when it arrives in time, as well as what has arrived before it.     

Risk-adjusted probability measures in portfolio optimization with coherent measures of risk

We consider the problem of optimizing a portfolio of n assets, whose returns are described by a joint discrete distribution. We formulate the mean–risk model, using as risk functionals the semideviation, deviation from quantile, and spectral risk measures. Using the modern theory of measures of risk, we derive an equivalent representation of the portfolio problem as a zero-sum matrix game, and we provide ways to solve it by convex optimization techniques. In this way, we reconstruct new probability measures which constitute part of the saddle point of the game. These risk-adjusted measures always exist, irrespective of the completeness of the market. We provide an illustrative example, in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk-averse portfolios.     

Resource Allocation with Wobbly Functions

We consider resource allocation with separable objective functions defined over subranges of the integers. While it is well known that (the maximisation version of) this problem can be solved efficiently if the objective functions are concave, the general problem of resource allocation with functions that are not necessarily concave is difficult.In this article, we focus on a large class of problem instances, with objective functions that are close to a concave function or some other smooth function, but with small irregularities in their shape. It is described that these properties are important in many practical situations.The irregularities make it hard or impossible to use known, efficient resource allocation techniques. We show that, for this class of functions the optimal solution can be computed efficiently. We support our claims by experimental evidence. Our experiments show that our algorithm in hard and practically relevant cases runs up to 40–60 times faster than the standard method.     

Tractable Almost Stochastic Dominance

LL-Almost Stochastic Dominance (LL-ASD) is a relaxation of the Stochastic Dominance (SD) concept proposed by Leshno and Levy that explains more of realistic preferences observed in practice than SD alone does. Unfortunately, numerical applications of this concept, such as identifying if a given portfolio is efficient or determining a marketed portfolio that dominates a given benchmark, are computationally prohibitive due to the structure of LL-ASD. We propose a new Almost Stochastic Dominance (ASD) concept that is computationally tractable. For instance, a marketed dominating portfolio can be identified by solving a simple linear programming problem. Moreover, the new concept performs well on all the intuitive examples from the literature, and in some cases leads to more realistic predictions than the earlier concept. We develop some properties of ASD, formulate efficient optimization models, and apply the concept to analyzing investors’ preferences between bonds and stocks for the long run.     

Fair resource allocation for demands with sharp lower tail inequalities

We consider a fairness problem in resource allocation where multiple groups demand resources from a common source with the total fixed amount. We show that for many common demand distributions satisfying sharp lower-tail inequalities, a natural allocation that provides resources proportional to each group's average demand performs very well. More specifically, this natural allocation is approximately fair and efficient (i.e., it provides near maximum utilization). We also show that, when a small amount of unfairness is allowed, the Price of Fairness (PoF), in this case, is close to 1.     

Risk measurement in the presence of background risk

A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolio’s re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measure’s aggregation properties are obtained.     

Iterative approaches for a dynamic memory allocation problem in embedded systems

Memory allocation has a significant impact on energy consumption in embedded systems. In this paper, we are interested in dynamic memory allocation for embedded systems with a special emphasis on time performance. We propose two mid-term iterative approaches which are compared with existing long-term and short-term approaches, and with an ILP formulation as well. These approaches rely on solving a static version of the allocation problem and they take advantage of previous works for addressing the static problem. A statistic analysis is carried out for showing that the mid-term approach is the best one in terms of solution quality.     

The optimal portfolio problem with coherent risk measure constraints

One of the basic problems of applied finance is the optimal selection of stocks, with the aim of maximizing future returns and constraining risks by an appropriate measure. Here, the problem is formulated by finding the portfolio that maximizes the expected return, with risks constrained by the worst conditional expectation. This model is a straightforward extension of the classic Markovitz mean–variance approach, where the original risk measure, variance, is replaced by the worst conditional expectation.The worst conditional expectation with a threshold α of a risk X, in brief WCE_α(X), is a function that belongs to the class of coherent risk measures. These are measures that satisfy a set of properties, such as subadditivity and monotonicity, that are introduced to prevent some of the drawbacks that affect some other common measures.This paper shows that the optimal portfolio selection problem can be formulated as a linear programming instance, but with an exponential number of constraints. It can be solved efficiently by an appropriate generation constraint subroutine, so that only a small number of inequalities are actually needed.This method is applied to the optimal selection of stocks in the Italian financial market and some computational results suggest that the optimal portfolios are better than the market index.     

Scenario optimization asset and liability modelling for individual investors

We develop a scenario optimization model for asset and liability management of individual investors. The individual has a given level of initial wealth and a target goal to be reached within some time horizon. The individual must determine an asset allocation strategy so that the portfolio growth rate will be sufficient to reach the target. A scenario optimization model is formulated which maximizes the upside potential of the portfolio, with limits on the downside risk. Both upside and downside are measured vis-à-vis the goal. The stochastic behavior of asset returns is captured through bootstrap simulation, and the simulation is embedded in the model to determine the optimal portfolio. Post-optimality analysis using out-of-sample scenarios measures the probability of success of a given portfolio. It also allows us to estimate the required increase in the initial endowment so that the probability of success is improved.     

A cost allocation problem arising in hub–spoke network systems

This paper studies a cost allocation problem arising from hub–spoke network systems. When a large-scale network is to be constructed jointly by several agents, both the optimal network design and the fair allocation of its cost are essential issues. We formulate this problem as a cooperative game and analyze the core allocation, which is a widely used solution concept. The core of this game is not necessarily non-empty as shown by an example. A reasonable scheme is to allocate the cost proportional to the flow that an agent generates. We show that, if the demand across the system has a block structure and the fixed cost is high, this cost allocation scheme belongs to the core. Numerical experiments are given with real telecommunication traffic data in order to illustrate the usefulness of our analytical findings.     

长期投资者的战略资产配置选择

在股票期望收益率服从一个均值回复过程的假设下,推导出具有幂效用函数的投资者的资产配置函数,着重分析了投资期限对投资者资产配置结果的影响,发现长期投资者比短期投资者在股票上配置更大的资产比例.虽然不同投资期限的投资者具有相同的短视配置,但是战略配置随着投资期限的增大而增大.     

商业银行激励费用分配的数学模型

研究商业银行激励费用分配问题，建立了激励费用分配的数学模型，并对模型进行了分析和求解。目前对激励费用分配问题尚缺少科学的系统研究，本文给出了处理该问题的一种理论依据，其方法和结论也可用于其它类似的问题。     

On the convergence of the Weiszfeld algorithm for continuous single facility location–allocation problems

A general family of single facility continuous location–allocation problems is introduced, which includes the decreasingly weighted ordered median problem, the single facility Weber problem with supply surplus, and Weber problems with alternative fast transportation network. We show in this paper that the extension of the well known Weiszfeld iterative decrease method for solving the corresponding location problems with fixed allocation yields an always convergent scheme for the location allocation problems. In a generic way, from each starting point, the limit point will be a locally minimal solution, whereas for each possible exceptional situation, a possible solution is indicated. Some computational results are presented, comparing this method with an alternating location–allocation approach. The research of the second author was partially supported by the grant of the Algerian Ministry of High Education 001BIS/PNE/ENSEIGNANTS/BELGIQUE.     

Allocation strategies in hub networks

In this paper, we study allocation strategies and their effects on total routing costs in hub networks. Given a set of nodes with pairwise traffic demands, the p-hub median problem is the problem of choosing p nodes as hub locations and routing traffic through these hubs at minimum cost. This problem has two versions; in single allocation problems, each node can send and receive traffic through a single hub, whereas in multiple allocation problems, there is no such restriction and a node may send and receive its traffic through all p hubs. This results in high fixed costs and complicated networks. In this study, we introduce the r-allocation p-hub median problem, where each node can be connected to at most r hubs. This new problem generalizes the two versions of the p-hub median problem. We derive mixed-integer programming formulations for this problem and perform a computational study using well-known datasets. For these datasets, we conclude that single allocation solutions are considerably more expensive than multiple allocation solutions, but significant savings can be achieved by allowing nodes to be allocated to two or three hubs rather than one. We also present models for variations of this problem with service quality considerations, flow thresholds, and non-stop service.     

Dynamic asset allocation for varied financial markets under regime switching framework

Asset allocation among diverse financial markets is essential for investors especially under situations such as the financial crisis of 2008. Portfolio optimization is the most developed method to examine the optimal decision for asset allocation. We employ the hidden Markov model to identify regimes in varied financial markets; a regime switching model gives multiple distributions and this information can convert the static mean–variance model into an optimization problem under uncertainty, which is the case for unobservable market regimes. We construct a stochastic program to optimize portfolios under the regime switching framework and use scenario generation to mathematically formulate the optimization problem. In addition, we build a simple example for a pension fund and examine the behavior of the optimal solution over time by using a rolling-horizon simulation. We conclude that the regime information helps portfolios avoid risk during left-tail events.     

On the combinatorial structure of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, supplying wireless access to voice/data communication networks for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee.As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.     

Optimal capital allocation in a hierarchical corporate structure

We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization.