Constant proportion portfolio insurance" (CPPI) is nowadays one of the most popular techniques for portfolio insurance strategies. It simply consists of reallocating the risky part of a portfolio with respect to market conditions, via a leverage parameter-called the multiple-guaranteeing a predetermined floor. We propose to introduce a conditional time-varying multiple as an alternative to the standard unconditional CPPI method, directly linked to actual risk management problematics. This ex ante approach for the conditional multiple aims to diversify the risk model associated, for example, with the expected shortfall (ES) or extreme risk measure estimations. First, we recall the portfolio insurance principles, and main properties of the CPPI strategy, including the time-invariant portfolio protection (TIPP) strategy, as introduced by Estep and Kritzman (1988). We emphasize the existence of an upper bound on the multiple, for example to hedge against sudden drops in the market. Then, we provide the main properties of the conditional multiples for well-known financial models including the discrete-time portfolio rebalancing case and Lévy processes to describe the risky asset dynamics. For this purpose, we precisely define and evaluate different gap risks, in both conditional and unconditional frameworks. As a by-product, the introduction of discrete or random time portfolio rebalancing allows us to determine and/or estimate the density of durations between rebalancements. Finally, from a more practical and statistical point of view due to trading restrictions, we present the class of Dynamic AutoRegressive Expectile (DARE) models for estimating the conditional multiple. This latter approach provides useful complementary information about the risk and performance associated with probabilistic approaches to the conditional multiple. D'une stratégie d'assurance de portefeuille dynamique fondée sur un modèle autorégressif de quantile Résumé « L'assurance de portefeuille à proportion constante" (CPPI en anglais) est aujourd'hui l'une des techniques les plus populaires pour les stratégies d'assurance de portefeuille. Elle consiste simplement à réallouer la partie risquée d'un portefeuille en fonction des conditions de marché qui déterminent le paramètre de l'effet de levier-appelé multiple-de façon à garantir un plancher prédéterminé de performance. Nous proposons dans cet article d'introduire un multiple variant dans le temps conditionnellement aux turbulences de marché, comme alternative à la méthode CPPI inconditionnelle classique, en utilisant une approche standard de gestion des risques. Le multiple est ainsi modélisé à partir de la moyenne prévue des pertes potentielles (ES), dans le cadre d'un modèle autorégressif dynamique de quantiles, qui présente, au final, l'avantage de conférer à la méthode d'assurance de portefeuille proposée, une certaine flexibilité adaptative de l'exposition au risque.

Abstract"Constant proportion portfolio insurance" (CPPI) is nowadays one of the most popular techniques for portfolio insurance strategies. It simply consists of reallocating the risky part of a portfolio with respect to market conditions, via a leverage parameter -called the multiple -guaranteeing a predetermined floor. We propose to introduce a conditional time-varying multiple as an alternative to the standard unconditional CPPI method, directly linked to actual risk management problematics. This ex ante approach for the conditional multiple aims to diversify the risk model associated, for example, with the expected shortfall (ES) or extreme risk measure estimations. First, we recall the portfolio insurance principles, and main properties of the CPPI strategy, including the time-invariant portfolio protection (TIPP) strategy, as introduced by Estep and Kritzman (1988). We emphasize the existence of an upper bound on the multiple, for example to hedge against sudden drops in the market. Then, we provide the main properties of the conditional multiples for well-known financial models including the discrete-time portfolio rebalancing case and Lévy processes to describe the risky asset dynamics. For this purpose, we precisely define and evaluate different gap risks, in both conditional and unconditional frameworks. As a by-product, the introduction of discrete or random time portfolio rebalancing allows us to determine and/or estimate the density of durations between rebalancements. Finally, from a more practical and statistical point of view due to trading restrictions, we present the class of Dynamic AutoRegressive Expectile (DARE) models for estimating the conditional multiple. This latter approach provides useful complementary information about the risk and performance associated with probabilistic approaches to the conditional multiple. Abstract"Constant proportion portfolio insurance" (CPPI) is nowadays one of the most popular techniques for portfolio insurance strategies. It simply consists of reallocating the risky part of a portfolio with respect to market conditions, via a leverage parameter -called the multiple -guaranteeing a predetermined floor. We propose to introduce a conditional time-varying multiple as an alternative to the standard unconditional CPPI method, directly linked to actual risk management problematics. This ex ante approach for the conditional multiple aims to diversify the risk model associated, for example, with the expected shortfall (ES) or extreme risk measure estimations. First, we recall the portfolio insurance principles, and main properties of the CPPI strategy, including the time-invariant portfolio protection (TIPP) strategy, as introduced by Estep and Kritzman (1988). We emphasize the existence of an upper bound on the multiple, for example to hedge against sudden drops in the market. Then, we provide the main properties of the conditional multiples for well-known financial models including the discrete-time portfolio rebalancing case and Lévy processes to describe the r...

Cet article introduit une nouvelle classe de modèles pour la Value-at-Risk ( VaR ) et l’ Expected Shortfall ( ES ), appelés modèles Dynamic AutoRegressive Expectiles ( DARE ). Notre approche est fondée sur une moyenne pondérée de modèles de VaR et d’ ES , calculés à partir des expectiles, i.e . les modèles Conditional Autoregressive Expectile ( CARE ) introduits par Taylor (2008a) et Kuan et al . (2009). Premièrement, nous recensons brièvement les principales approches non paramétriques, paramétriques et semi paramétriques d’estimation de la VaR et de l’ ES . Deuxièmement, nous détaillons l’approche DARE et montrons comment les expectiles peuvent être utilisés pour estimer ces mesures de risque. Troisièmement, nous utilisons différents tests de validation ( backtesting ) afin de comparer l’approche DARE à différentes méthodes alternatives de prévision de la VaR . Finalement, nous évaluons l’impact du choix des pondérations sur la qualité des prévisions et déterminons les poids optimaux dans le but de sélectionner de façon dynamique le modèle de prévision le plus adapté.

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Controlling and managing potential losses is one of the main objectives of the Risk Management. Following Ben Ameur and Prigent (2007) and Chen et al. (2008), and extending the first results by Hamidi et al. (2009) when adopting a risk management approach for defining insurance portfolio strategies, we analyze and illustrate a specific dynamic portfolio insurance strategy depending on the Value-at-Risk level of the covered portfolio on the French stock market. This dynamic approach is derived from the traditional and popular portfolio insurance strategy (Cf. Black and Jones, 1987;Black and Perold, 1992): the so-called "Constant Proportion Portfolio Insurance" (CPPI). However, financial results produced by this strategy crucially depend upon the leverage -called the multiple -likely guaranteeing a predetermined floor value whatever the plausible market evolutions. In other words, the unconditional multiple is defined once and for all in the traditional setting. The aim of this article is to further examine an alternative to the standard CPPI method, based on the determination of a conditional multiple. In this time-varying framework, the multiple is conditionally determined in order to remain the risk exposure constant, even if it also depends upon market conditions. Furthermore, we propose to define the multiple as a function of an extended Dynamic AutoRegressive Quantile model of the Value-at-Risk (DARQ-VaR). Using a French daily stock database (CAC40 and individual stocks in the period 1998-2008), we present the main performance and risk results of the proposed Dynamic Proportion Portfolio Insurance strategy, first on real market data and secondly on artificial bootstrapped and surrogate data. Our main conclusion strengthens the previous ones: the conditional Dynamic Strategy with Constant-risk exposure dominates most of the time the traditional Constant-asset exposure unconditional strategies.

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International audienceThe recent experience from the global financial crisis has raised serious questions about the accuracy of standard risk measures as a tool to quantify extreme downward risks. These standard risk measures, such as the var, emerge over the last decades as the industry standard for risk management and asset allocation (Basak and Shapiro [2001] ; Montfort [2008]). We estimate the riskiness of risk models and we evaluate its impact on optimal portfolios at various time horizons. Based on a long sample of u.s. data, we find an inverse U-shape relation between var model errors and the horizon that impacts the optimal asset allocation of the representative agent.Les récents épisodes de turbulence financière sont venus remettre en cause la précision des mesures classiques de risque pour évaluer les risques extrêmes. Ces mesures de risques, telles que la var, sont devenues incontournables dans la gestion des risques et l'allocation d'actifs (Basak et Shapiro [2001] ; Monfort [2008]). Nous estimons le risque de modèle des mesures de risque et nous évaluons son impact sur l'allocation d'actifs optimale à différents horizons. Nos résultats montrent, sur des données longues américaines, une relation en U inversé entre notre mesure du risque de modèle sur les var historiques et l'horizon de l'agent représentatif, qui impacte sensiblement son allocation optimale

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