Romain Deguest

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.

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.

Default Intensities Implied by CDO Spreads: Inversion Formula and Model Calibration

We propose a simple computational method for constructing an arbitrage-free collateralized debt obligation (CDO) pricing model which matches a prespecified set of CDO tranche spreads. The key ingredient of the method is an inversion formula for computing the aggregate default rate in a portfolio, as a function of the number of defaults, from its expected tranche notionals. This formula can be seen as an analogue of the Dupire formula for portfolio credit derivatives. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient nonparametric method for calibrating CDO pricing models. Contrarily to the base correlation method, our method yields an arbitrage-free model. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. On the other hand, comparing the local intensity functions implied by different credit portfolio models reveals that apparently different models, such as the static Student-t copula models and the reduced-form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads.

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

While it is often argued that allocation decisions can be best expressed in terms of exposure to rewarded risk factors, as opposed to somewhat arbitrary asset class decompositions, the practical implications of this paradigm shift for the optimal design of the policy portfolio still remain largely unexplored. This paper aims at analyzing whether the use of uncorrelated underlying risk factors, as opposed to correlated asset returns, can lead to a more efficient framework for measuring and managing portfolio diversification. Following Meucci (2009), we use the entropy of the factor exposure distribution as the number of uncorrelated bets (also known as the effective number of bets, or ENB in short), implicitly embedded within a given asset allocation decision. We present a set of formal results regarding the existence and unicity of portfolios designed to achieve the maximum effective number of bets. We also provide empirical evidence that incorporating constraints, or target levels, on a portfolio effective number of bets generates an improvement in out-of-sample risk-adjusted performance with respect to standard mean-variance analysis.

Equity Correlations Implied by Index Options: Estimation and Model Uncertainty Analysis

We propose a method for constructing an arbitrage-free multiasset pricing model which is consistent with a set of observed single- and multiasset derivative prices. The pricing model is constructed as a random mixture of N reference models, where the distribution of mixture weights is obtained by solving a well-posed convex optimization problem. Application of this method to equity and index options shows that, whereas multivariate diffusion models with constant correlation fail to match the prices of index and component options simultaneously, a jump-diffusion model with a common jump component affecting all stocks enables to do so. Furthermore, we show that even within a parametric model class, there is a wide range of correlation patterns compatible with observed prices of index options. Our method allows, as a by product, to quantify this model uncertainty with no further computational effort and propose static hedging strategies for reducing the exposure of multiasset derivatives to model uncertainty.

Bond Portfolio Optimization in the Presence of Duration Constraints

Although there exists an abundant literature on the benefits and limits of scientific diversification in the equity universe, little is known about the out-of-sample performance of portfolio optimization models in the fixed-income universe. In this article, the authors address two key challenges that are specific to bond portfolio optimization, namely, the presence of duration constraints and the presence of no-arbitrage restrictions on risk parameter estimates, for which no equivalent exists in the equity universe. In an application to sovereign bonds in the eurozone, they find that the use of portfolio optimization techniques based on robust estimators for risk parameters generates an improvement in investor welfare compared with the use of ad hoc bond benchmarks such as equally weighted or cap-weighted portfolios. These results are robust with respect to changes in the number of constituents in the portfolio and the rebalancing period, and in the presence of duration or weight constraints.

A Reinterpretation of the Optimal Demand for Risky Assets in Fund Separation Theorems

In a continuous-time portfolio selection model with N risky assets and K state variables driving their risk and return parameters, we derive simple expressions for the allocation to each asset in the K + 1 risky funds of the (K + 2)-fund separation theorem. We show that the allocation to any given risky asset in each fund can be written in terms of the parameters of a regression of the excess returns of this asset on those of the N − 1 remaining assets. We also use these parameters to provide quantitative measures of the increase in Sharpe ratio of the speculative demand, or in the maximum correlation of each hedging demand with respect to the corresponding risk factor, associated with the introduction of a new asset in the investment universe. Finally, we show that in a multiperiod setting, an asset is “spanned” by others if and only if it improves neither the maximum Sharpe ratio of the speculative demand nor the maximum correlations of the hedging demands with the risk factors. This paper was accepted ...

Designing Multi-Factor Equity Portfolios

This chapter reviews efficient index design methods for factor indices referred to as smart factor investing. It then uses such smart factor indices as building blocks to design suitable allocation strategies to address specific risk/return objectives.