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Essays on Quantitative Risk Management
The costly lessons from global crisis in the past decade reinforce the importance as well as challenges of risk management. This thesis explores several core concepts of quantitative risk management and provides further insight.
We start with rating migration risk and propose a Mixture of Markov Chains (MMC) model to account for stochastic business cycle effects in credit rating migration risk. The model shows superior in-sample estimation and out-of-sample predication than its rivals. Compared with the naive approach the economic application suggests banks with MMC estimator will increase capital requirement in economic expansion and free up capital during recession hence it is aligned with Basel III macroprudential imitative by reducing the recession-vs-expansion gap in capital buffers.
Subsequently we move to the key concept of dependence by investigating the importance of dynamic linkages between credit and equity markets. We propose a flexible regime-switching copula model to explore the dynamics of dependence and possible structure breaks with special consideration on tail dependence. The study reveals a high-dependence regime that coincides with the recent financial crisis. The backtesting results acknowledge the new model's superiority on out-of-sample VaR forecasting over purely dynamic or static copula. It can serve to emphasise the relevance for risk management of appropriately modeling complex dependence structures.
Finally we discuss the risk measures and how they affect the portfolio optimisation. We contend that more successful portfolio management can be achieved by combining extreme value analysis to describe downside tail risk and dynamic copulas to model nonlinear dependence structures. Conditional Value-at-Risk is adopted as pertinent measure of downside tail risk for portfolio optimisation. Using both realised portfolio returns and a set of out-of-sample Monte Carlo experiments, our novel portfolio strategy is confronted with the de facto mean-variance approach. The results suggest that the MV approach produces suboptimal portfolios or a less desirable risk-return tradeoff
Topics in kernal hypothesis testing
This thesis investigates some unaddressed problems in kernel nonparametric hypothesis testing. The contributions are grouped around three main themes: Wild Bootstrap for Degenerate Kernel Tests. A wild bootstrap method for nonparametric hypothesis tests based on kernel distribution embeddings is proposed. This bootstrap method is used to construct provably consistent tests that apply to random processes. It applies to a large group of kernel tests based on V-statistics, which are degenerate under the null hypothesis, and non-degenerate elsewhere. In experiments, the wild bootstrap gives strong performance on synthetic examples, on audio data, and in performance benchmarking for the Gibbs sampler. A Kernel Test of Goodness of Fit. A nonparametric statistical test for goodness-of-fit is proposed: given a set of samples, the test determines how likely it is that these were generated from a target density function. The measure of goodness-of-fit is a divergence constructed via Stein's method using functions from a Reproducing Kernel Hilbert Space. Construction of the test is based on the wild bootstrap method. We apply our test to quantifying convergence of approximate Markov Chain Monte Carlo methods, statistical model criticism, and evaluating quality of fit vs model complexity in nonparametric density estimation. Fast Analytic Functions Based Two Sample Test. A class of nonparametric two-sample tests with a cost linear in the sample size is proposed. Two tests are given, both based on an ensemble of distances between analytic functions representing each of the distributions. Experiments on artificial benchmarks and on challenging real-world testing problems demonstrate good power/time tradeoff retained even in high dimensional problems. The main contributions to science are the following. We prove that the kernel tests based on the wild bootstrap method tightly control the type one error on the desired level and are consistent i.e. type two error drops to zero with increasing number of samples. We construct a kernel goodness of fit test that requires only knowledge of the density up to an normalizing constant. We use this test to construct first consistent test for convergence of Markov Chains and use it to quantify properties of approximate MCMC algorithms. Finally, we construct a linear time two-sample test that uses new, finite dimensional feature representation of probability measures