Bayesian Non-Parametric Risk Metric
By Kiwan Hyun
Abstract
This thesis constructs completely non-parametric Risk Metric models through Dirichlet process in order to account for both the parametric uncertainty and model uncertainty that a Risk Metric may bring.
Value at Risk (VaR), along with its integrated form Continuous Value at Risk (CVaR) / Expected Shortfall (ES), is one of the most frequently used risk metrics in finance. VaR is a quantile value of forecasted return of a portfolio—linear and non-linear. [Siu, et. al., 2006] According to the Basel 95% and 99% VaR are recommended to be posted by the financial institutions for portfolios and assets; 97.5% CVaR/ES value needs to be set aside when making an investment for “capital buffer”. [Obrenovic & Akhunjonov, 2016] Therefore, an accurate estimation of risk is critical for VaR models and CVaR/ES models.
The traditional approach of a normal approximation to VaR and CVaR/ES has been discredited—especially for daily returns—and even blamed by some for causing the 2008 Financial Crisis [Nocera, 2009] Many advancements have been made to the VaR model including Bayesian inference to the normal model [Siu, et. al., 2006], Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH) VaR model [Bollerslev, 1986], and Conditional Autoregressive Value at Risk (CAViaR) model [Engle & Manganelli, 2004]. When tested against 6 years (Jan, 2001 – Jan, 2005) of daily returns data of 10 different market indexes, the Bayesian CAViaR model has shown to be the most accurate in predicting daily 95% and 99% VaR. [Gerlach, et. al., 2011]
However, there were certain years for certain indexes where the 99% Bayesian CAViaR VaR did not perform well, especially for years that had multiple > 5% daily drops. Moreover, the Bayesian CAViaR models—though are almost non-parametric—follow a Skewed-Laplace distribution. To even account for the uncertainty of the likelihood model, this thesis constructed daily 97.5% VaRs for 7 different country indexes for 7 years (Jan, 2012 – Dec, 2019) using the completely non-parametric Dirichlet Process.
The Dirichlet Process 97.5% VaR outperformed all Bayesian Normal, Bayesian GARCH, and Bayesian CAViaR models of years when CAViaR models underperformed. The model may be inefficient for normal years since it is overly conservative. Nevertheless, the non-parametric model still seems to be significantly more accurate during fluctuant years.
Professor Kyle Jurado, Ph.D., Faculty Advisor,
Assistant Professor Simon Mak, Ph.D., Faculty Advisor
Dealing with Data: An Empirical Analysis of Bayesian Black-Litterman Model Extensions
By Daniel Roeder
Portfolio Optimization is a common financial econometric application that draws on various types of statistical methods. The goal of portfolio optimization is to determine the ideal allocation of assets to a given set of possible investments. Many optimization models use classical statistical methods, which do not fully account for estimation risk in historical returns or the stochastic nature of future returns. By using a fully Bayesian analysis, however, this analysis is able to account for these aspects and also incorporate a complete information set as a basis for the investment decision. The information set is made up of the market equilibrium, an investor/expert’s personal views, and the historical data on the assets in question. All of these inputs are quantified and Bayesian methods are used to combine them into a succinct portfolio optimization model. For the empirical analysis, the model is tested using monthly return data on stock indices from Australia, Canada, France, Germany, Japan, the U.K.
and the U.S.
Advisor: Andrew Patton | JEL Codes: C1, C11, C58, G11 | Tagged: Bayesian Analysis Global Markets Mean-Variance Portfolio Optimization
