Drawdown beta and portfolio optimization

Abstract

This paper introduces a new dynamic portfolio performance risk measure called Expected Regret of Drawdown (ERoD) which is an average of the drawdowns exceeding a specified threshold (e.g. 20%). ERoD is similar to Conditional Drawdown-at-Risk (CDaR) which is the average of some percentage of the largest drawdowns. CDaR and ERoD portfolio optimization problems are equivalent and result in the same set of optimal portfolios. Necessary optimality conditions for ERoD portfolio optimization lead to Capital Asset Pricing Model (CAPM) equations. ERoD Beta, similar to the Standard Beta, relates returns of the securities and those of a market. ERoD Beta is equal to [average losses of a security over time intervals when market is in drawdown exceeding the threshold] divided by [average losses of the market in drawdowns exceeding the threshold]. Therefore, a negative ERoD Beta identifies a security which has positive returns when the market has drawdowns exceeding the threshold. ERoD Beta accounts only for time intervals when the market is in drawdown and conceptually differs from Standard Beta which does not distinguish up and down movements of the market. Moreover, ERoD Beta provides quite different results compared to the Downside Beta based on Lower Semi-deviation. ERoD Beta is conceptually close to CDaR Beta which is based on a percentage of worst case market drawdowns. However, ERoD Beta has some advantage compared to CDaR Beta because the magnitude of the drawdowns is known (e.g. exceeding a 20% threshold), while CDaR Beta is based on a percentage of the largest drawdowns with unknown magnitude. We have built a website reporting CDaR and ERoD Betas for stocks and the SP 500 index as an optimal market portfolio. The case study showed that CDaR and ERoD Betas exhibit persistence over time and can be used in risk management and portfolio construction.