Regularizing Portfolio Risk Analysis: A Bayesian Approach

It is important for portfolio manager to estimate and analyze the recent portfolio volatility to keep portfolio's risk within limit. Though number of financial instruments in the portfolio are very large, some times more than thousands, however daily returns considered for analysis is only for a month or even less. In this case rank of portfolio covariance matrix is less than full, hence solution is not unique. It is typically known as "large $p$ - small $n$" or "ill-posed" problem. In this paper we discuss a Bayesian approach to regularize the problem. One of the additional advantages of this approach is to analyze the source of risk by estimating the probability of positive `conditional contribution to total risk' (CCTR). Each source's CCTR sum upto total volatility risk. Existing method only estimates CCTR of a source, but it does not estimate the probability of CCTR to be significantly greater (or less) than zero. This paper presents Bayesian methodology to do so. We use parallalizable and easy to use Monte Carlo (MC) approach to achieve our objective.