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 Portfolio Construction Based on Management of Tail Risk
 ables do not exhibit pathological dependence structures or infinite variance. This makes the Gaussian distribution the choice to model average behaviors, for example the average monthly return of an asset. Incredibly (and luckily for us!), a similar theorem holds for the maximum of many random variables. The Fisher- Tippet-Gnedenko theorem from Extreme Value Theory (EVT) states that maxima of such a collection of random variables tends to the Generalized Extreme Value (GEV) distribution, making the GEV distribution the choice to model these kinds of extremes.
While the GEV distribution was originally formulated for maxima of independent random variables, it can be extended to stationary time-series data, giving theoret- ical justification for its use in modeling quantities such as the largest one-day loss of an asset in a month or its MDD over a quarter. Despite the potential nonstationarity of real-world financial returns, we find that, in practice and over long time horizons, the GEV fits to this data very well. For example, we show the close fit of a GEV model to unconditional quarterly MDDs of SPY since 1993 in a histogram in Figure 3 and a Q-Q plot in Figure 4.
Figure 3: Histogram of SPY quarterly MDDs since 1993. The density of a fit GEVmodel is also shown, and the two distributions display similar structure.
The GEV is described by only three parameters, one governing location (related to the mean), one governing scale (related to the standard deviation), and one governing shape (related to tail thickness). Due to their very nature, the frequency and magnitude of rare and extreme events are difficult to estimate, but the simplicity of the GEV model and its suitability to the task makes it possible to infer these quantities with minimal statistical bias and relatively low variance. The GEV distribution takes on three special cases, depending on whether the underlying distribution before taking the maximum was fat-tailed, thin-tailed, or bounded from above, corresponding to the shape parameter being positive, zero, or negative, respectively. Because financial data is so typically fat- tailed, we tend to use the fat-tailed special case, known as the Fréchet distribution, in practice.
3.2 Case Study: Covid Unemployment
In order to examine the ability of GEV models to estimate the probabilities of events more extreme than any previously
Figure 4: Quantiles of empirical SPY quarterly MDDs since 1993 as a function of fit GEV model quantiles. Systematic deviations from the diagonal line mark discrepancies between the model and the observed data. We can see agreement over the support of the distribution (with some variability in the upper tail), indicating a good model fit.
recorded, we take a look at the United States unemployment rate since 1948, reported at a monthly resolution by the U.S. Bureau of Labor Statistics, which is shown Figure 5. The unemployment rate reached 14.7% in April 2020 due to the Covid-19 pandemic, 36% higher than its previous maximum of 10.8% in 1982. This large increase is a good example of a fat- tailed phenomenon, where, for the same reason that the CVaR is much larger than the VaR, we can expect each maximum to surpass the previously observed maximum by a considerable amount.
To assess the GEV model’s ability to evaluate the ex-ante probability of a spike in unemployment of the magnitude experienced in April 2020, we separate the data into a training-set, which includes unemployment data up to December 31, 2019, and a testing set, including only data from January 1, 2020 onward. By taking the maximum unemployment level of each calendar year, we are able to coax this data into a form that can be accurately modeled by the GEV distribution. To confirm this accuracy, we can examine a QQ-plot of empirical training-data quantiles as a function of fit GEV quantiles, shown in Figure 6. These scattered points lie close to the diagonal line, indicating a good model fit.
Figure 5: United States unemployment rate since 1948. The spike in unemployment associated with the 2020 Covid-19 pandemic reached a maximum of 14.7%, surpassing the previous all-time high of 10.8% reached in 1982, marking a 36% in- crease. Source: U.S. Bureau of Labor Statistics, Unemployment Rate [UNRATE], retrieved from FRED, Federal
Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/UNRATE
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