Hazy value-at-risk

November 6 update: we've had 3 90% VaR days for bonds in the past 7 trading days!  Spectacular risk, and less than 0.0005% probability that this is due to chance alone.  We should only see a 90% VaR once every two weeks, by definition.  Let's see what new record surprises next week may bring!

Understanding the capriciousness of financial market statistics can be helpful in guiding portfolio decisions, through what will ultimately be rough waters at some point over one's career.  In the past couple of years we have seen portfolios, invested in both bonds and stocks, trail new record highs.  And more recently we've seen these asset classes tip over, reminding us of how distressing the risks are which lurk within them (especially during these safer periods of market highs).  Is one of these two asset classes easier to grasp than the other, for risk measurement?  In this article we explore the concept of value-at-risk (the largest risk within a certain confidence, or the probability of minimally established amount of risk) in both bonds and stocks.  We'll show, pointedly, that the risks and associated measurement errors are different for bonds and stocks, and what is a logical relationship between them.  Probability theory is in a distinct position to help provide some fascinating characteristics on how uncertain these risk measures, which bankers and news makers trust, actually are.

Let's see the 90% daily value-at-risk (VaR) for both bonds and for stocks.  We'll use the respective Vanguard total market indexes for this analysis, and study them for the past 8 full years (through October 2015).  This time period is just greater than 2000 trading days, and so the outer 10% risk tail is of course 202 trading days.  We use the expression that an asset class "hits" VaR if the risk event exceeds the pre-defined VaR level.

For bonds, the average and median daily changes were both 0.0%, and the 90% VaR was only -0.3%.  For stocks, the average daily change was 0.0% (median was 0.1%), and the 90% VaR was -1.4%.  But as we'll see later, these are simply estimates for VaR and in reality the VaR for stocks could be closer to that for bonds than we initially imagined!

As we've seen many times earlier (here, here, here, here), these risk levels are not uniformly breached (in either asset class) over time.  In the chart below, we see again that most of these risky 202 trading days -over the past 8 years- occurred just beyond the global financial crisis.  And have since come down somewhat, but for 2015 where risk events have since picked up.

We'll see throughout this article's illustrations that U.S. bonds are colored blue, and U.S. stocks are colored red.  Over ¾  of the 202 riskiest trading days in each asset class occurred in the first ½ of the 8-year period shown.  We also have appreciated around the world that stocks have seen an uptick in extreme risk in 2015 (notably in August), while bonds have a much larger uptick of events overall.  Bonds may be of singular significance now given 2 90% VaR days in the past 4 trading days (through November 3).  While for stocks the past 2 90% VaR days was back in mid-September.

However the risk we previously noted in bonds is much more benign versus in stocks by a multiple!  We noted this in a New York Times article a couple years ago when taper tantrum and bonds' rate risk was the only thing on people's minds.  Again the VaR for bonds is expected to be -0.3%, and for stocks it is -1.4%.

It is important now to show the overall distribution of price changes over all trading days, for both asset classes.  We do so on the same scales below.

We notice for example that the 90% VaR for bonds is based off of an empirical sample and hence has inherent uncertainty.  This VaR level is also not comparable with stocks, where -0.3% is about only a 70% VaR.  90% VaR on the other hand for stocks would again be further down at -1.4%.  With the uncertainty which we solve for below however, there is a few percent chance the bond VaR would plainly be much closer to the stock VaR, assuming only that initial estimates only were used!

Let's now get into the calculus to measure this VaR uncertainty.  Probability formulae must be used that begins with an understanding of the probability density function at the expected VaR.  Incidentally, this is the exact same formula build up if we were instead looking at conditional VaR, or expected tail risk, either of which needs to be approximated as no closed form formula is exact there.  We can see from the empirical chart above that the distributions are both leptokurtic.  This implies that our asset class distributions are shown as higher peaked and fatter tailed versus the normal distribution; hence why we are looking at an empirical chart to begin with!  In any event the probability function comes to roughly 0.8 (80%) for bonds, and 0.1 (10%) for stocks.

Now we compute the standard error of the expected deviation of each asset class, for measuring at an individual confidence level:

[c*(1-c)/(n*f(x)^2)]^.5

Where c is the confidence level, n is the sample size, and f(x) is the probability density function.

So for bonds we have 8.6*10^-5; for stocks we have 6.0*10^-4.  Finally, we compute the VaR, with an exposure to plus or minus these standard errors about the expected VaR level.  This 90% confidence comes to:

U.S. bonds 90% VaR of -0.3% + 0.02%, or (-0.33%, -0.29%)

U.S. stocks 90% VaR of -1.4% + 0.08%, or (-1.51%, 1.36%)

So we see that there is great uncertainty in understanding exactly where the 90% VaR is for each asset class.  We start with a point estimate, and often regulators and the media stay stuck on those levels, yet the actual VaR levels often fall outside of that in reality.  We might, for example mispurchase stock market protection at a more benign level, equivalent to the expected -1.4% loss, when it instead should have been at -1.51%.  While this difference seems trivial, it is not for computerized trading and over the course of a week or two, such differences could be greatly amplified.  Alternatively, we might not act quickly enough during any recent temporary bond market drop, thinking the narrow (though proportionally great) -0.29% drop is not significant enough of a risk while waiting for a falsely anchored -0.3% VaR expectation instead (or something more benign relative to stocks!)

These numerous illustrative examples are some of the ways that having too rigid a view of VaR can pose difficulties for otherwise intelligent people who considered a meaningful risk signal.  And also didn't appreciate the substantial, model estimation risk during rapidly-unwinding crashes.

Finally we can note that the 90% VaR shows that just fewer than 10% of the time, the "risk event" in either asset class will just hit VaR.  Jointly pertinent we see from the confidence errors constricted from the estimation errors within these risk measures, that there is a 10% probability (e.g., 1 in 10 times, or for 1 in 10 people perhaps) that the VaR will be completely outside of the interval altogether.  And a small percent of the time such a difference in the VaR estimate implies a large difference in the confidence of VaR itself (e.g., say 80% VaR instead of 90%).