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 Portfolio Construction Based on Management of Tail Risk
 Figure 6: Empirical quantiles of 1-year maxima of U.S. un- employment rate over the training set, plotted as a function of GEV model quantiles. We see agreement over the support of the distribution, indicating a good model fit.
We visualize this model in Figure 7 by plotting the probability of exceeding any threshold 1-year maxi- mum as a function of the threshold. The dashed line corresponds to a GEV model fit to the training data, and the shaded region represents a 90% pointwise confidence band, estimated with a 1-year rolling block bootstrap over the training data. Solid dots represent the training data, while the empty dot corresponds to the testing data (2020 only). The empirical probabilities of these dots are given by Weibull plotting positions computed over the full dataset.
We can clearly see that the model underestimates the empirical probability of the spike in unemployment rate associated with the Covid-19 pandemic. However, we note that this point is well within the limits of model tail- variability, captured by the widening confidence band.
Figure 7: Value on y-axis gives the estimated probability of the maximum unemployment rate over a 1-year period exceeding thresholds given on the x-axis. The model (dashed line) fit to the training data (solid dots) underestimates the empirical probability of the Covid-19 spike in the testing data (empty dot), but this point lies well inside the 90% confidence band (shaded region).
Although we are unable to accurately “forecast" the probability of such an extreme event, this analysis indicates that its occurrence could not have been ruled to be extremely improbable before the fact, with the upper limit of the confidence band indicating a probability of approximately 10%, corresponding to an exceedance occurring on average once every 10 years. In contrast, an
empirical examination of the historical data would have ruled this event as highly unlikely, as there were no similarly extreme events over the 72 years spanned by the data.
It is informative to examine the impact that this extreme event has on the model. Figure 8 compares the GEV model probabilities and the associated confidence band before and after the addition of the Covid-19 spike to the training dataset. As we would hope, observing an extreme event with higher empirical probability than predicted by the model influences the model to expect such extreme events more frequently in the future. Furthermore, it is able to do so without significantly compromising the model’s accuracy on previ- ously observed data. Although the addition of a single data- point only contributes so much information, this far-tail observation is accompanied by a slight reduction in model uncertainty in this region, which manifests as a slight shrinking of the confidence band. These intuitive observations reflect that this model is able to appropriately incorporate new information as it becomes available.
Figure 8: GEV model with 90% confidence band using only pre-Covid data (blue) and using both pre- and post-Covid data (orange). The addition of the Covid observation both increases the model’s predicted probability of large unemployment rates and slightly decreases model uncertainty, marked by tighter confidence bands.
3.3 Case Study: 2008 Financial Crisis
We follow a similar procedure to examine the properties of a GEV model fit to SPY quarterly MDDs. Because the largest single-quarter MDDs of roughly 35% occurred during the 2008 financial crisis, we constructed a training dataset from the start of 1993 to the end of 2007 and a testing dataset from the start of 2008 to October 2020. Figure 9 shows a plot of the estimated probability of exceeding given quarterly MDD levels, where the model and block-bootstrapped confidence interval are again constructed over the training data only.
Just as we saw in the example of the unemployment rate, the empirical probabilities of exceeding our out-of-sample extreme values – including an extreme much larger than any previously observed – are well covered by our confidence band. In this case, the model itself is also very accurate even
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