17,439 research outputs found
Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization
In this paper we investigate the expected terminal utility maximization
approach for a dynamic stochastic portfolio optimization problem. We solve it
numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which
is transformed by means of the Riccati transformation. We examine the
dependence of the results on the shape of a chosen utility function in regard
to the associated risk aversion level. We define the
Conditional value-at-risk deviation () based Sharpe ratio for
measuring risk-adjusted performance of a dynamic portfolio. We compute optimal
strategies for a portfolio investment problem motivated by the German DAX 30
Index and we evaluate and analyze the dependence of the -based Sharpe
ratio on the utility function and the associated risk aversion level
Multi-Period Trading via Convex Optimization
We consider a basic model of multi-period trading, which can be used to
evaluate the performance of a trading strategy. We describe a framework for
single-period optimization, where the trades in each period are found by
solving a convex optimization problem that trades off expected return, risk,
transaction cost and holding cost such as the borrowing cost for shorting
assets. We then describe a multi-period version of the trading method, where
optimization is used to plan a sequence of trades, with only the first one
executed, using estimates of future quantities that are unknown when the trades
are chosen. The single-period method traces back to Markowitz; the multi-period
methods trace back to model predictive control. Our contribution is to describe
the single-period and multi-period methods in one simple framework, giving a
clear description of the development and the approximations made. In this paper
we do not address a critical component in a trading algorithm, the predictions
or forecasts of future quantities. The methods we describe in this paper can be
thought of as good ways to exploit predictions, no matter how they are made. We
have also developed a companion open-source software library that implements
many of the ideas and methods described in the paper
On efficiency of mean-variance based portfolio selection in DC pension schemes
We consider the portfolio selection problem in the accumulation phase of a defined contribution (DC) pension scheme. We solve the mean-variance portfolio selection problem using the embedding technique pioneered by Zhou and Li (2000) and show that it is equivalent to a target-based optimization problem, consisting in the minimization of a quadratic loss function. We support the use of the target-based approach in DC pension funds for three reasons. Firstly, it transforms the difficult problem of selecting the individual's risk aversion coefficient into the easiest task of choosing an appropriate target. Secondly, it is intuitive, flexible and adaptable to the member's needs and preferences. Thirdly, it produces final portfolios that are efficient in the mean-variance setting. We address the issue of comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of expected utility (EU). The two natural notions of Variance Inefficiency and Mean Inefficiency are introduced, which measure the distance of an optimal inefficient portfolio from an efficient one, focusing on their variance and on their expected value, respectively. As a particular case, we investigate the quite popular classes of CARA and CRRA utility functions. In these cases, we prove the intuitive but not trivial results that the mean-variance inefficiency decreases with the risk aversion of the individual and increases with the time horizon and the Sharpe ratio of the risky asset. Numerical investigations stress the impact of the time horizon on the extent of mean-variance inefficiency of CARA and CRRA utility functions. While at instantaneous level EU-optimality and efficiency coincide (see Merton (1971)), we find that for short durations they do not differ significantly. However, for longer durations - that are typical in pension funds - the extent of inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, this result is a further element that supports the use of the target-based approach in DC pension schemes.Mean-variance approach; efficient frontier; expected utility maximization; defined contribution pension scheme; portfolio selection; risk aversion; Sharpe ratio
Tracking Error: a multistage portfolio model
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.
Mean-variance inefficiency of CRRA and CARA utility functions for portfolio selection in defined contribution pension schemes
We consider the portfolio selection problem in the accumulation phase of a defined contribution pension scheme in continuous time, and compare the mean-variance and the expected utility maximization approaches. Using the embedding technique pioneered by Zhou and Li (2000) we first find the efficient frontier of portfolios in the Black-Scholes financial market. Then, using standard stochastic optimal control we find the optimal portfolios derived via expected utility for popular utility functions. As a main result, we prove that the optimal portfolios derived with the CARA and CRRA utility functions are not mean-variance efficient. As a corollary, we prove that this holds also in the standard portfolio selection problem. We provide a natural measure of inefficiency based on the difference between optimal portfolio variance and minimal variance, and we show its dependence on risk aversion, Sharpe ratio of the risky asset, time horizon, initial wealth and contribution rate. Numerical examples illustrate the extent of inefficiency of CARA and CRRA utility functions in defined contribution pension schemes.Mean-variance approach, efficient frontier, expected utility maximization, defined contribution pension scheme, portfolio selection, risk aversion, Sharpe ratio
CAPM and APT-like models with risk measures.
The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk mea sures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as (strong) compatibility between prices and risks. Surprisingly, the lack of bounded optimal risk and/or return levels arises for important pricing models (Black and Scholes) and risk measures (VaR, CVaR, absolute deviation, etc.). Bounded problems present a Market Price of Risk and generate a pair of benchmarks. From these bench marks we introduce APT and CAPM like analyses, in the sense that the level of correlation between every available security and some economic factors explains the security expected return. The risk level non correlated with these factors has no influence on any return, despite the fact that we are dealing with risk functions beyond the standard deviation.Risk measure; Compatibility between prices and risks; Efficient portfolio; APT and CAPM-like models;
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A review of portfolio planning: Models and systems
In this chapter, we first provide an overview of a number of portfolio planning models
which have been proposed and investigated over the last forty years. We revisit the
mean-variance (M-V) model of Markowitz and the construction of the risk-return
efficient frontier. A piecewise linear approximation of the problem through a
reformulation involving diagonalisation of the quadratic form into a variable
separable function is also considered. A few other models, such as, the Mean
Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the
Minimax (MM) model which use alternative metrics for risk are also introduced,
compared and contrasted. Recently asymmetric measures of risk have gained in
importance; we consider a generic representation and a number of alternative
symmetric and asymmetric measures of risk which find use in the evaluation of
portfolios. There are a number of modelling and computational considerations which
have been introduced into practical portfolio planning problems. These include: (a)
buy-in thresholds for assets, (b) restriction on the number of assets (cardinality
constraints), (c) transaction roundlot restrictions. Practical portfolio models may also
include (d) dedication of cashflow streams, and, (e) immunization which involves
duration matching and convexity constraints. The modelling issues in respect of these
features are discussed. Many of these features lead to discrete restrictions involving
zero-one and general integer variables which make the resulting model a quadratic
mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the
algorithms and solution methods for this class of problems are also discussed. The
issues of preparing the analytic data (financial datamarts) for this family of portfolio
planning problems are examined. We finally present computational results which
provide some indication of the state-of-the-art in the solution of portfolio optimisation
problems
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