14,000 research outputs found
Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems
We address the statistical estimation of composite functionals which may be
nonlinear in the probability measure. Our study is motivated by the need to
estimate coherent measures of risk, which become increasingly popular in
finance, insurance, and other areas associated with optimization under
uncertainty and risk. We establish central limit formulae for composite risk
functionals. Furthermore, we discuss the asymptotic behavior of optimization
problems whose objectives are composite risk functionals and we establish a
central limit formula of their optimal values when an estimator of the risk
functional is used. While the mathematical structures accommodate commonly used
coherent measures of risk, they have more general character, which may be of
independent interest
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
Coherent measurement of factor risks
We propose a new procedure for the risk measurement of large portfolios. It
employs the following objects as the building blocks: - coherent risk measures
introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures
introduced in this paper, which assess the risks driven by particular factors
like the price of oil, S&P500 index, or the credit spread; - risk contributions
and factor risk contributions, which provide a coherent alternative to the
sensitivity coefficients.
We also propose two particular classes of coherent risk measures called Alpha
V@R and Beta V@R, for which all the objects described above admit an extremely
simple empirical estimation procedure. This procedure uses no model assumptions
on the structure of the price evolution.
Moreover, we consider the problem of the risk management on a firm's level.
It is shown that if the risk limits are imposed on the risk contributions of
the desks to the overall risk of the firm (rather than on their outstanding
risks) and the desks are allowed to trade these limits within a firm, then the
desks automatically find the globally optimal portfolio
Model Uncertainty and its Impact on Derivative Pricing
Financial derivatives written on an underlying can normally be priced and hedged accurately only after a suitable mathematical model for the underlying has been determined. This chapter explains the difficulties in finding a (unique) realistic model \u2014 model uncertainty. If the wrong model is chosen for pricing and hedging, unexpected and unwelcome financial consequences may occur. By wrong model we mean either the wrong model type (specification\ud
uncertainty) or the wrong model parameter (parameter uncertainty). In both cases, the impact of model uncertainty on pricing and hedging is significant. A variety of measures are introduced to value the model uncertainty of derivatives and a numerical example again confirms that these values are a significant proportion of the derivative price
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