Most people seem to know that the developed-world economies appear sluggish, yet the “tail risks” to growth or of relevant financial markets are still less than what they were during the peak effects of the financial crisis. In the U.S., we have seen some evidence of this through continuous positive monthly job growth, and through the ongoing "reduced downside risks" comments from the Federal Reserve (themselves may be part responsible for the intermediate evolution of market risks). We have also seen automobile manufacturing and housing construction, both at recovery highs. National deficits are now stable, and recently we have seen corporate earnings and the stock market push through new record highs. So why is there apparent unease in accepting these current lower tail risks? Part of it lies with a sense that perhaps we’ve come too far along this dimension and have overshot what is the typical equilibrium level.
This note focuses on the topic of tail risks, through the prism of the U.S. stock market (S&P 500). And we look at the recent history to provide some context, while showing some advanced mathematical models that can quantify the amount of tail risk reduction we have experienced, and the stability of these risk results and implications to investors.
The parameters established for this research are the past 40 full months: from May 2010, through August 2013. The 840 days sample was divided into 40 blocks of 21 days each, in order to provide equal sensitivity to approximate month durations. And then the sample was divided into two separate periods, with the first 20 months labeled “~2010-2011” and the last 20 months labeled “2012-~2013”. The former period provides a slightly greater than typical volatility period, given the random oscillation in same, over the broader history of the U.S. markets, and keep in mind that it also ignores the peak tail risks from exactly five years ago. And it ignores any possible low, tail risk distortion, from the most recent period. Other volatility studies in this blog suggests that this is a reasonable assumption.
The median (50th percentile) performance return over the 840 days was 0.06% (0.05% in ~2010-2011, 0.06% in 2012-~2013), and the 5th percentile return was -1.74%. So there were 42 days (~5%*840) where the market pulled back 1.74%, or greater. With 42 of these “worst-5%” days spread across 40 months (with one risk day a month equaling roughly 5% of the month), we have a good, statistically significant sample for looking at tail risk.
The distribution of these 42 worst-5% trading days: 36 were in ~2010-2011, while only 6 were in 2012-~2013. Now let’s take a look at the distribution of these worst-5% days in the 20 months of each period. While we clearly see a heavy distribution of tail risks concentrated during the ~2010-2011 period, we have simply experienced too few months with tail risk recently during the 2012-~2013 period.
Due to this convolution of risk only in the quarter or so time when it generally occurs, next we will look at a generalized extreme value (GEV) distribution to understand the nature of these worst-5% tail risks. This differs from more basic probability analysis, which relies heavily on the exponential tail of the normal distribution. We should note that in recent testimony before Congress concerning the failures of the London Whale trade, J.P. Morgan’s chief stated he didn’t even like to consume the risk alerts based on these more basic probability distributions, even as it was his firm that had invented these "groundbreaking" tools decades earlier.
But to still lay some groundwork in this area, let’s see how the tail risks have empirically differed from what we’d expect if we used a normal (or Gaussian) distribution. The 840 daily returns have an average geometric return of 0.04%, and a standard deviation of 1.12%. So if we want to estimate, for example, the number of days we’d expect to see a drop of 2.5% or more, we’d follow the following steps. The t-score would be (-2.5-0.04)/1.12=-2.3. So the distribution associated with this critical level is 0.012, implying 10 (~0.012*840) of these extreme risk days. But in reality we had nearly twice as many extreme risk days. The S&P 500 had a daily loss of 2.5% or more, a total 21 times in the 840 day period.
So to properly look at these extreme risks, we will sample each of the 40 month blocks, and instead eye the most extreme risk experienced in each of their 21 days. These worst days by month average a 2.5% drop during ~2010-2011, but average only a 1.4% drop during the 2012-~2013 period. But this is just the average, so now we’ll look at expressing the range of the GEV, a class of models that include: (A) the Pareto distribution following the power law, (B) the tails of the normal distribution, and (C) one variation of the Weibull distribution.
If γ ≠0, then F(x) = e^(1+γx’)^[-γ^(-1)].
If γ =0, then F(x) = e^[-e^(-x’)].
Both where:
x’ = (x-α)/β.
This is a subsuming family of expressions discussed in other popular applications of extreme probabilities, such as in the books Outliers, and The Black Swan. And includes specific variations of extreme modeling argued by mathematician Mandelbrot, who is also considered the father of modern fractal geometry. Given the complexity of the equations above, which sometimes lack higher-order moments in which to parameter match (depending on the γ selection), a maximum likelihood estimate was used to fit the parameters to the empirical data. For both periods, we get values of roughly α~0.4% (a location indication suggesting fat tails), and β~0.2% (a scale indication). Now the shape parameter γ was ~1/3 for the earlier ~2010-2011, but an even lower ~1/7 for the 2012-~2013 period. The higher values of γ>0 indicate a fatter tailed, distribution shape. Still we will also contrast these models with similar shapeless (parameter γ=0). For most parity along the optimization frontier, in the shapeless version we will keep the α the same, while only doubling β (serves as “sigma” in the illustration below) to ~0.4% for the earlier ~2010-2011 period.
We see two things occurring in the illustration above. The first is that while the shapeless distributions (e.g., similar to the normal distribution) both do an ok job in other parts of the distribution, they are problematic since they fail to properly model tail risks in either of the two periods, by consistently under-representing them. More importantly we see that the characteristics of the GEV distributions fitted for the most recent time period of 2012-~2013 (in orange) is significant less than the fit of the earlier time period of ~2010-2011 (in blue). This is why we have seen only 6 of the 42 tail risk events occur in this most recent period, and why it is important to know that this recent 2012-~2013 period is of an unusually low volatility that can only later revert to a more typical regime. More importantly, the lower likelihood fit of the GEV models in the most recent period suggests a psychological disparity in normal modeling distribution of risk (and we’ll see this same result later in the research when looking at a new model).
In addition to the GEV, for the fat tailed distributions we are looking at, we can explore the shapes of a generalized Pareto distribution (not to be confused with the specific Pareto approximations noted above), which can better hone in on the tail risk distribution we have illustrated above. The generalized Pareto distribution, conditional on a critical level of risk, is a limiting distribution with the following linear approximations:
If γ’ ≠0 and x>α, then F(x) = 1-(1+x’/γ’)^(-γ').
If γ’ =0 and x>α, then F(x) = 1-e^(-x’).
Both where:
x’ = (x-α)/β.
Now the parameters will also be different from the GEV distributions above (e.g., we relabel a new parameter γ’ to avoid confusion), though we’ll still see that they do have the same directional impact on this generalized Pareto distribution. We can consider α the hurdle rate, while β is an important scaling parameter.
Again in order to maintain parity with the GEV distribution models above, we allowed β to remain the same but the γ’ shape parameter was adjusted to 2 for the ~2010-2011 period, and to 6 for the 2012-~2013 period. The inverse of the γ’ values here are 1/2 and 1/6, which are both the approximate magnitude of the γ values from the GEV models. This is not only in the correct range for our empirical financial data set, but since the sign on γ’ remains consistent even when inverted, γ’ and γ both represent fat tail shaped distributions in this case.
The challenge here is to discover the range of hurdle risk rates to understand the relationship with the likelihood function for both periods in the illustration below. Those periods, again, are 2012-~2013 in orange, and ~2010-2011 is blue. The objective of this illustration below is not to match to the maximum likelihood α, which anyway decays inconsistently as less months eventually contain a maximum risk sampled that is greater then α. But instead the objective here is to show that up to a certain level of extreme values there is a nice, approximate linear distribution in the tails. We see this level of stability in an α range of 0.6%, to 0.8%, for both time periods as shown below. Contrast this with an α of ~0.4% in the GEV models above.
The generalized Pareto model is slightly weaker of a statistical fit versus the GEV models above, but the value here is that the linear approximation -of the expected risk when it happens- works well for higher order modeling. Particularly for the slightly greater than typical results from ~2010-2011. This higher order modeling from a risk stand-point means a cohesive array of conditional risk expectations.
In conclusion, we have suspected that the tail risks, which are low right now, are not at a stable equilibrium level. They will now rise in the future just to be in a more typical regime. Ideas that “this time it’s different”, are almost never true in the realm of the best statistics models. In the 20th century, essayist T.S. Eliot captured this point of reliving the range of history and our collective human psyche:
That the past experience revived in the meaning
Is not the experience of one life only
But of many generations - not forgetting
Something that is probably quite ineffable
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