CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles

TLDR

Value at Risk (VaR) is the standard market‑risk metric used by financial institutions, defined as the loss level that a portfolio will exceed with a specified probability over a given horizon, yet its estimation remains a difficult statistical problem with no satisfactory existing methods. This paper proposes a new quantile‑estimation approach that interprets VaR as the conditional quantile of future portfolio values, thereby avoiding the extreme assumptions of normality or i.i.d. returns used by prior methods. The authors develop the Conditional Autoregressive Value‑at‑Risk (CAViaR) model, which models the quantile’s time evolution with an autoregressive process, estimates parameters via regression quantiles, optimizes the non‑differentiable objective using a differential evolutionary genetic algorithm, and introduces the Dynamic Quantile test to assess model adequacy.

Abstract

Value at Risk (VaR) has become the standard measure of market risk employed by financial institutions for both internal and regulatory purposes. VaR is defined as the value that a portfolio will lose with a given probability, over a certain time horizon (usually one or ten days). Despite its conceptual simplicity, its measurement is a very challenging statistical problem and none of the methodologies developed so far give satisfactory solutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation which does not require any of the extreme assumptions invoked by existing methodologies (such as normality or i.i.d. returns). The Conditional Autoregressive Value-at-Risk or CAViaR model moves the focus of attention from the distribution of returns directly to the behavior of the quantile. We specify the evolution of the quantile over time using a special type of autoregressive process and use the regression quantile framework introduced by Koenker and Bassett to determine the unknown parameters. Since the objective function is not differentiable, we use a differential evolutionary genetic algorithm for the numerical optimization. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the Dynamic Quantile test. Applications to simulated and real data provide empirical support to this methodology and illustrate the ability of these algorithms to adapt to new risk environments.