Nonparametric Estimation and Backtesting of Financial Risk Measures

NIHLANI, KIRAN. Nonparametric Estimation and Backtesting of Financial Risk Measures. (Under the direction of Peter Bloomfield.) Measuring risk is an important question in finance, and the recent financial crisis has led to an overwhelming academic and regulatory response, so that we now have a theoretical corpus on risk measurement techniques. An important component in the risk measurement exercise is the evaluation of these measurements against the outcomes. This procedure is known as backtesting. Two measures of risk recommended by the Basel Committee on Banking Supervision (BIS) are Value at Risk (VaR) and Expected Shortfall (ES). VaR has been the preferred risk measure for its computational ease and simplicity. However, the lack of coherence and the inability to capture “tail risk” has led the regulators to shift to ES. The major challenge with the proposed change is the unavailability of efficient yet easy to implement tools to backtest ES. The early VaR models were mostly parametric and imposed strong theoretical assumptions and rules on the underlying properties of the data. One such assumption is that the distribution of the data conforms to a known theoretical distribution. While these assumptions provided computational simplicity, the empirical evidence and VaR failures during the Mexico (1996), Asian (1997), and Russian (1998) market crises spurred the search for better VaR models. A number of non-parametric and semi-parametric VaR models were subsequently developed. The most commonly used of these include historical simulation, CAViaR, and extreme value analysis.There is a growing number of VaR measures based on quantile regression and its adaptations. Since VaR is a quantile of loss, it can be implicitly defined as a conditional quantile function making the quantile regression approach a natural environment to study and investigate VaRs. Merging the quantile regression approach with RiskMetrics, we get an exponentially weighted quantile regression framework that can be viewed in a kernel based framework. We proposed a modified way of implementing the exponentially weighted double kernel framework that improves the quality of the VaR forecasts and simplifies it for computational ease. The quantile regression framework along with being a preferred base for VaR measures is a significant addition to the VaR and ES backtesting literature. A large portion of the VaR backtesting techniques including the recommended traffic-light based system by the Basel Committee on Banking Regulations are based on binary variables, such as whether or not there was an exception. This approach sacrifices too much information especially in finite samples. The quantile regression framework provides for a random coefficient model that can be used to perform a Wald type inference for the null hypothesis that the VaR model is correctly specified. Furthermore, since ES is the aggregated loss exceeding the VaR level, it can be approximated using VaR at a finite number of quantile levels. This helps propose a backtesting framework for ES using a multi-quantile regression approach. We explore these backtests for VaR and ES through a comparative analysis with other backtests already established in the literature. The purpose here is to shed light on common issues underlying the implementation of these backtests and suggest modifications that provide significant improvements in computational ease and accuracy.