Tall Tails: An extreme value theory approach to modelling and predicting extreme risk in long-run factor returns

Abstract

Throughout history, equity markets have famously seen catastrophic declines in times of distress: understanding, predicting, and managing extreme losses in these times remains one of the great challenges facing financial institutions today. Motivated by modern factor theory and the idea that the market risk of any given equity depends only on exposure to fundamental, underlying factor risks, I look at the tail risk in the six tradeable factors identified by \cite{fama2015five} and \cite{Carhart1997}. This paper's implications for predicting risk in these factors should therefore have broader implications for predicting risk in US equities more generally. The industry standard risk measure, Value-at-Risk (VaR), requires the accurate modelling of return distributions well into their extreme tails; therefore, this seems like a natural place to apply methods grounded in extreme value theory. Knowing that the fat-tailed behaviour of these distributions makes risk estimation challenging, I show that when the standard normal approximation is applied to estimate VaR in factors, this method systematically underestimates extreme tail risk. Normal-VaR estimates are rejected by every VaR backtest, and underestimate risk at the 99.9\% level by factors of 7 to 12. In contrast, extreme value based VaR estimates, when used in conjunction with conditional volatility models, continue to accurately predict extreme factor losses, even as we move into the highest loss quantiles. This suggests that extreme value based models can very accurately capture extreme loss behaviour across a broad range of equities, and holds interesting implications for financial regulators and investors alike.