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 Portfolio Construction Based on Management of Tail Risk
 far into the tail, but the confidence band is wide in this region, just as we previously observed. Note that this widening occurs
for the same reason that VaR and CVaR cannot easily be extended to periods as large as quarters or years: it takes tens of periods worth of data to reliably estimate the 10% level and hundreds of periods to reliably estimate the 1% level. The relative accuracy and confidence-band coverage of GEV models even into this region show that they can be used to reasonably assess ex-ante probabilities of extreme risks, and perhaps just as critically, to accurately quantify our model uncertainty, captured by the widening of the block-bootstrap confidence band. To drive this point home, we take a look at the role that correct model specification, in particular fat- tailedness, played in this analysis.
Figure 9: Estimated probability of SPY exceeding quarterly (90-day) MDD thresholds. The model is a GEV distribution fit to data in the 1993-2008 training period. The model is able to accurately estimate the probabilities of exceeding the out of sample (test) data, including the single largest quarterly MDD of approximately 35%, which occurred in 2008.
We restrict our GEV model to the special case where the underlying random variables have thin tails, giving us the Gumbel distribution. The Gumbel model probabilities are presented in Figure 10. We can see that the Gumbel model is able to accurately model the small MDDs (less than about 10% and happening in roughly 80% of cases), but it fails to model the tails of even the training dataset. Even more egregiously, not only is the Gumbel model wrong, it is confidently wrong: in contrast to the GEV model, the Gumbel confidence band does not widen in the tail, completely failing to cover either train or test data in this region. This failure underscores the importance of using a simple yet correctly specified probabilistic model. If VaR, CVaR, EMDD, or MDD distributions are estimated using thin- tailed models, they provide no more utility than the standard deviation alone.
Figure 10: In contrast to the GEV model, a Gumbel distribution (i.e., a thin-tailed GEV) fails to model SPY MDDs. Not only is the model inaccurate, it is confidently inaccurate, corresponding to a thin confidence band that fails to cover the tails of either in-sample or out-of-sample data.
4 Risk In Context
4.1 Risk and Reward
Risk and reward are meaningless in isolation. Markowitz’s expected reward at a given level of risk makes it clear that portfolio evaluation metrics and selection techniques must
consider the two simultaneously. This idea was further developed into the Sharpe ratio in William Sharpe’s 1966 paper, “Mutual Fund Performance," which measures the ratio of expected return to volatility, allowing the direct comparison of portfolios across different return and volatility levels. Many similar ratios have since been developed, replacing the role of volatility with different risk measures. These ratios are given the general name of Return on Risk-Adjusted Capital (RORAC) ratios. These include the (confusingly named) Risk-Adjusted Return on Capital (RAROC) ratio, which uses VaR as its risk measure; the Stable Tail-Adjusted Return Ratio (STARR), which uses CVaR; and the Sterling-Calmar ratio, which uses EMDD; amongst many others.
There is no such thing as a free lunch. It is notoriously difficult to build portfolios that achieve high returns with small levels of risk. Chi Keong Lee’s 2016 paper, "Expected Drawdown Management: An Ex-Ante, Long-Term Approach to Portfolio Construction," argues that traditional portfolio optimization techniques that maximize returns while controlling volatility necessarily expose themselves to tail risks, making them exceedingly fragile. However, risk- management approaches that instead focus on mitigating long-term tail risks are able to achieve overall high returns by strategically taking on some amounts of short-term volatility risks.
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