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 Portfolio Construction Based on Management of Tail Risk
5 Summary
Here we summarize the key takeaways from this report.
 Figure 14: GEV-modeled 90-day MDDs pf SPY and the 60/40 portfolio, re-scaled by 90-day expected returns. The separation between the two strategies further diminishes, showing the inability of the 60/40 strategy to manage moderate-to-large MDDs, particularly when adjusted for expected return.
4.2 Dynamic, Not Static
Throughout this paper, we discussed unconditional risk metrics, i.e., metrics that summarize risk over all time, rather than those that are dynamic and may be tied to specific events such as elections, pandemics, or financial crises. In practice, it is important to model risk actively and dynamically in order to most efficiently adjust risk exposure and maximize returns. A variety of different approaches exist, including those based on extreme value theory, GARCH models, autoregressive expectile regression models, and many more. A good overview of a number of these approaches can be found in David Happersberger et al.’s 2020 paper, "Estimating portfolio risk for tail risk protection strategies.”
4.3 Diversification and Tail Dependence
Conventional wisdom tells us that a good portfolio should be diversified, and in fact, Markowitzian optimization approaches tell us the same thing. Volatility can be minimized by selecting uncorrelated or anti-correlated assets. Correlation is a measure of the average extent to which a pair of assets fluctuate in the same or opposite direction about their expected re- turn rates. During crises and extreme events, these average relationships tend to break down, and diversification fails. In these events, a typical strategy that uses diversification to reduce volatility will underestimate the overall risk and experience high drawdowns. Approaches that rely solely on correlation to model dependence structures typically use the underlying machinery of Gaussian copulas, the overuse of which has often been blamed for the 2008 financial crisis.
By shifting away from approaches that manage volatility and towards approaches that manage long-term tail risks, we are forced to confront the tail- dependence structure between assets. Modeling these tail dependencies, for example with non-Gaussian copulas, allows more accurate
estimation of joint risks and enables hedging even in the case of an extreme event.
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The Gaussian probability distribution permeates many aspects of quantitative finance, particularly through the tendency to equate risk with volatility. This is reflected in the use of portfolio metrics such as the Sharpe Ratio, relying on correlation to measure portfolio diversification, and using the Black Scholes model to price options. The use of the Gaussian distribution is due to the mathematical simplicity that it offers rather than the utility that it provides.
The Gaussian distribution is a poor model of actual market returns, with one of its most notable failures being its inability to capture the risk of extreme tail events, leading to overall poor quantification of risk.
Value at Risk (VaR) and Conditional Value at Risk (CVaR) are able to quantify tail risk far more appropriately, but they are restricted to short, fixed, timescales. The Expected Max Drawdown (EMDD), on the other hand, is able to adaptively extend these measures of tail risk to longer timescales.
A major but subtle point is that the most extreme tail events that will ever happen have yet to occur. Naive interpretation of historical data will lead us to underestimate the risk of these “unprecedented" events.
Fortunately, Extreme Value Theory (EVT) is a mathematically rigorous framework that is partic- ularly suited to quantifying tail risk. EVT models, including the Generalized Extreme Value (GEV) are low-dimensional, allowing them to estimate these risks with minimal (and quantifiable) variability.
The holy grail in portfolio optimization is to provide protection against tail risks with minimal impact to the return on investment. The vast majority of portfolio construction approaches fail at this. It is straightforward to construct models that outperform during short-term tail events but underperform at other times. Similarly, it is straightforward to construct models that outperform on average but are fragile and suffer extreme losses during tail events.
Models are required to adapt to and learn from new
information. This is the science and the art behind portfolio construction.
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