Loss-Based Risk Measures

Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative robustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions. Finally, we provide examples of statistically robust estimators for loss-based risk measures.Comment: 40 page

Sensitivity analysis in HMMs with application to likelihood maximization

International audienceThis paper considers a sensitivity analysis in Hidden Markov Models with continuous state and observation spaces. We propose an Infinitesimal Perturbation Analysis (IPA) on the filtering distribution with respect to some parameters of the model. We describe a methodology for using any algorithm that estimates the filtering density, such as Sequential Monte Carlo methods, to design an algorithm that estimates its gradient. The resulting IPA estimator is proven to be asymptotically unbiased, consistent and has computational complexity linear in the number of particles. We consider an application of this analysis to the problem of identifying unknown parameters of the model given a sequence of observations. We derive an IPA estimator for the gradient of the log-likelihood, which may be used in a gradient method for the purpose of likelihood maximization. We illustrate the method with several numerical experiments

Particle filter-based policy gradient for pomdps

International audienceOur setting is a Partially Observable Markov Decision Process with continuous state, observation and action spaces. Decisions are based on a Particle Filter for estimating the belief state given past observations. We consider a policy gradient approach for parameterized policy optimization. For that purpose, we investigate sensitivity analysis of the performance measure with respect to the parameters of the policy, focusing on Finite Difference (FD) techniques. We show that the naive FD is subject to variance explosion because of the non-smoothness of the resampling procedure. We propose a more sophisticated FD method which overcomes this problem and establish its consistency

Numerical methods for sensitivity analysis of Feynman-Kac models

The aim of this work is to provide efficient numerical methods to estimate the gradient of a Feynman-Kac flow with respect to a parameter of the model. The underlying idea is to view a Feynman-Kac flow as an expectation of a product of potential functions along a canonical Markov chain, and to use usual techniques of gradient estimation in Markov chains. Combining this idea with the use of interacting particle methods enables us to obtain two new algorithms that provide tight estimations of the sensitivity of a Feynman-Kac flow. Each algorithm has a linear computational complexity in the number of particles and is demonstrated to be asymptotically consistent. We also carefully analyze the differences between these new algorithms and existing ones. We provide numerical experiments to assess the practical efficiency of the proposed methods and explain how to use them to solve a parameter estimation problem in Hidden Markov Models. To conclude we can say that these algorithms outperform the existing ones in terms of trade-off between computational complexity and estimation quality

Incertitude de modèle en finance (mesures de risque et calibration de modèles)

PALAISEAU-Polytechnique (914772301) / SudocSudocFranceF

Sensitivity analysis in HMMs with application to likelihood maximization

This paper considers a sensitivity analysis in Hidden Markov Models with continuous state and observation spaces. We propose an Infinitesimal Perturbation Analysis (IPA) on the filtering distribution with respect to some parameters of the model. We describe a methodology for using any algorithm that estimates the filtering density, such as Sequential Monte Carlo methods, to design an algorithm that estimates its gradient. The resulting IPA estimator is proven to be asymptotically unbiased, consistent and has computational complexity linear in the number of particles. We consider an application of this analysis to the problem of identifying unknown parameters of the model given a sequence of observations. We derive an IPA estimator for the gradient of the log-likelihood, which may be used in a gradient method for the purpose of likelihood maximization. We illustrate the method with several numerical experiments.

Risk Parity and Beyond—From Asset Allocation to Risk Allocation Decisions

International audienceIn this article, the authors define the number of uncorrelated bets embedded within a given portfolio of N assets as the exponential of the entropy of the portfolio exposure to N uncorrelated factors. They present a set of formal results regarding the existence and uniqueness of portfolios designed to achieve the maximum effective number of bets. They also provide empirical evidence that incorporating constraints or target levels in a portfolio’s effective number of bets generates an improvement in out-of-sample risk-adjusted performance with respect to standard mean–variance analysis

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a 'risk measure' that summarizes the risk of the portfolio. We define the notion of 'risk measurement procedure', which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.Risk management, Risk measurement, Coherent risk measures, Law invariant risk measures, Value-at-Risk, Expected shortfall,

Goal-based Investing: Theory and Practice

International audienceGoal-based investing is a new paradigm that is expected to have a profound and long-lasting impact on the wealth management industry. This book presents the concept in detail and introduces a general operational framework that can be used by financial advisors to help individual investors optimally allocate their wealth by identifying performance-seeking assets and hedging assets. Grounded in the principles of asset pricing and portfolio optimisation, the goal-based investing approach leads to the design of investment solutions that truly respond to investors' problems, which can most often be summarized as follows: secure essential goals with the highest confidence level and maximize the chances to reach aspirational goals.A series of case studies guides the reader through the implementation of goal-based investing, illustrates the efficiency of this paradigm and explains how one can accommodate a variety of implementation features such as taxes, short-sales constraints, parameter estimation risk, as well as limited customisation