Value-at-risk, and moving averages

November 13 update: the Total Stock Market index breached 90%-VaR again today, so for a second day after this article was published.  This validates the point of this article to take care with not appreciating ever-deeper volatility clusters in the tails.

As an overture to a market crash or a market rebound, some market participants follow a signal referred to as the death cross or golden cross, respectively.  The definition of these signals is when the short-term moving average crosses under or over the more stable long-term moving average.  Some market participants (in fact many at the time it happens) suggest that this is a major inflection point in the market’s trend, and it appears we are again close to the start of another market crash now (today itself was the first breach of the 90% value at risk since September).  But how do market participants get misled by the risk measures that are supposed to serve as a protective guide but instead can provide false signals during the risk itself when it is most critical that it work?  Examples that here are shown in the context of months, can also be magnified just as easily to apply to other macroeconomic contexts, over a matter of years or even decades (see this Bloomberg article on CAPE).  Something we will show is that, for example, during a market crash, the market will in fact breach these very low probability risk levels, far more often than one would have expected just at the start of the crash.  We divide this article into three components: a look at the mathematical theory, a look at a simulation, and then a look at an empirical example using the Total Stock Market index in August 2015.

Theory

Say that we have price changes accumulated over 50 days.  We can also generally state that the daily average return (continuously computed) will be called μ.  On the 51^st day we can then look back on the specific 50-day average and call that μ₅₀.  For our exercise here, we’ll detrend the series so that the first 50 days will be averaged to 0%, which for many asset classes today (not in hyper-inflation) is not far off anyway.  We’ll also work with a supposed risk product that has an annual standard deviation (σ) of 16%.  The daily σ for this is equal to 1% (16%/√252).

Let’s also provide for our analytical framework that on this 51^st day, we breach the 90% value-at-risk (VaR).  90% VaR suggests a performance that is in the worst 10% of distribution.  Anyone can compute the theoretical VaR (for a normal distribution) to be -1.3% (a test statistic of -1.3 multiplied by a σ of 1%).  This 1.3% loss is not the expected loss for the day, but rather the smallest loss beyond a specified certain confidence of 90%.  This expected tail loss is also known by other value-at-risk terms such as tail-VaR (TVaR), expected VaR or conditional VaR.  We’ll use TVaR in this article.

TVaR is slightly more complicated to compute as opposed to VaR, regardless of the underlying distribution, and it is more fundamental to study.  For the normal distribution, the formula is shown here:

α% TVaR                                 = μ - σ*ϕ[Φ^-1(α)]/(1- α)

Where α is 90% in this example.

90% TVaR                              = μ - 1%*ϕ[Φ^-1(90%)]/(1-90%)

                                                = 0% - 1%*ϕ[-1.3]/(10%)

                                                = 0% - 1%*16%/(10%)

                                                = -1.6%

We see that TVaR at -1.6% is 0.3% worse than the -1.3% VaR we showed above.  The issue of course for both of these short-term risk measures is that neither is dependable over the course of a trade it is meant to evaluate.  The levels actually get worse either each successive day (e.g., 51, 52, 53) that the VaR is being breached.  

So for example, when we get to the 52^nd day, the 50-day moving average (μ₅₁) can be determined to have deviated from the stationary 0%, as follows:

μ₅₁                                            = 49/50* μ₅₀ + 1/50*(90% TVaR₅₀)

                                                  = 49/50*0 + 1/50*(-1.6%)

                                                  = -0.032%

Of course this is recursive such that μ₁₀₀ would then be equal to the following.

μ₁₀₀                                            = 0/50*μ₅₀ + 50/50*(90% TVaR₁₀₀)

                                                  = 0 + 50/50*(90% TVaR₁₀₀)

                                                  = 90% TVaR₁₀₀

And as we'll show below, this is not equal to the 90% TVaR₅₀ known on the 51^st day

So let's start with our computation of 90% VaR.  During a market crash, the worse 10% of the distribution continues to slide downward, which we can see above as the expected μ_t decreases by 0.032% daily.  So if you stubbornly wait to again reach the original 90% VaR (e.g., VaR₅₁) to buy securities, for example, then you would be mis-estimating the VaR as time passes since the trade on day 51.  

Again on the 51^st day, VaR is -1.3%, but soon enough after that day it will alternatively be between -1.3% and -1.4%.  The exact opposite is false, for when you are thinking in reverse (e.g., during market rebounds).  This is simply because we do not get symmetry with VaR, it only looks at one of the two tails of the distribution.  So one can get a rebound in the index but lack discernible change in the VaR_t shortly thereafter.

To explore this convexity concept further, we can see that there is not a linear transition in VaR in relation to the average μ during market crashes:

90% VaR₁₀₀                                 ≠ 90% VaR₉₉

≠ 0/50 * 90% VaR₅₀ + 50/50*(90% VaR₉₉)

And it then follows backwards in time the following.

90% VaR₅₁                                   ≠ 49/50 * 90% VaR₅₀ + 1/50*(90% VaR₅₀)

Instead what happens is that VaR₅₁ continues to get worse much more quickly than μ (e.g., difference between μ₅₁ and μ₅₀).  This is because in the tail of the original distribution from the first 50 days (μ₅₀=0%, σ₅₀=1%), we are effectively replacing the VaR₅₀ with the next worst performance in the past 50 days (outside of the 90% confidence).  In other words, something like substituting rank 45 with rank 46.  And these data are spread out a lot more, by definition, at the tails.  Therefore, in the initial days of a market crash, the risk measures should quickly worsen.  See the frequency of the worst 50% of the actual stock market changes, over the past 50 days, with attention paid to the area better, and worse than -1.0% (VaR).

0.00% to -0.25%                       8 ********

-0.25% to -0.50%                      8 ********

-0.50% to -0.75%                      0 

-0.75% to -1.00%                      2 **

-1.00% to -1.25%                      1 *           ß VaR

-1.25% to -1.50%                      4 ****     (breached today November 12)

-1.50% to -1.75%                      1 *

-1.75% to -2.00%                      0

-2.00% to -2.25%                      0

-2.25% to -2.50%                      0

-2.50% to -2.75%                      1 *

On the flip side, as we start to approach VaR₁₀₀ we have converged onto the new “crash regime” model centered about the -1.6% change of TVaR₅₀.  And the VaR by that point has already moved well below the -1.3% of VaR₅₀, since we have a new and lower distribution center of price changes.  Since the VaR is much lower, there is an ever slimmer chance that each new day towards the end of the new time window (e.g., day 99, 100, 101) provides new performance data that will rank in the lower 10%, so that the 90% VaR_t towards t=100 is essentially converged.  

With 90% TVaR we have a similar phenomenon to above.  In fact we see from the formula for TVaR above, that this level should always run parallel to the VaR levels (in other words, always be 0.3% worse).  A couple things that are interesting to note here is that a 0 trend in μ does not imply of course that the price level after day 50 is always 0.  The central limit theorem demonstrates that the deviation of the sample average after 50 days would equal σ/√50, or 0.14%.  Consequently, we’d have an error surrounding VaR anyway of about 0.1% based solely on the incoming moving average – or the μ₅₀ at the market crash start – was false by this amount.

The second thing is that for every small daily change in μ, the probability of being in the previous worse 10% increases dramatically.  For example on the 51^st day (μ of 0%) there is by definition a 10% probability of seeing performance <-1.3% (this is VaR).  On the 52^nd day, with μ now of -0.034%, the probability of seeing performance <-1.3% is no longer 10% (recall that VaR worsened and so 1.3% is better than this VaR₅₁).  The probability now is:

Φ[(-1.3% - (-0.034%))/1%]        = Φ[-1.27]

                                                    = 11%

As an interesting piece of probability trivia that is important in this context, the 2^nd derivative of the probability distribution f(x) zeros at about +1 σ.  Therefore we know that the increased probability, increases at a faster clip for nearly a few additional weeks into a market crash!

The main lesson from this note so far is that during market crashes, the 50-day moving average also collapses.  So the initial VaR and TVaR used to manage your risk quickly unravels, during any hedge or trade.  In that sense you would be underestimating the risk of future large drops, unless you readjust your risk along the way.  And also you would have to factor in that further volatile crashes generally occur in clusters, which is vital to know seeing that the market has currently fallen 3 of the past 4 days.  In other words, they are not as independently occurring as some models can suggest.

Simulation

We see on the left, for example, a cloud of 30 simulated price changes over a 50-day period.  Then starting on the 51^st day, we see successive risk hitting beyond the original VaR level from the first 50 days (VaR₅₀).

In the chart below we simply focus on the residual data only.  Concerning μ, VaR, and TVaR, as computed for days 51 through 100.  We see for example that TVaR tracks VaR, though it is more volatile.  

Additionally, we see that while 90% VaR starts on the 51^st day at -1.3%, we see that it slowly matches the -2.9% we noted before that we expect for 90% VaR₁₀₀.  We can get -2.9% on the 100^th day, two different ways:

90% VaR₅₁ + -0.034%*49             ~ 90% TVaR₅₁ + 90% VaR₅₁

-1.3% -0.034%*49                          ~ -1.6% -1.3%

Bear in mind that all of these risk measures are subject to the sensitive +0.1% error we noted in the theory section of this article.  Lastly, we noted previously what was the probability of seeing the original VaR₅₁ (or TVaR₅₁) being breached given the evolving distribution from the 51^st day onwards.

We can simply reverse the question here and ask ourselves what is the reduction in probability of the new VaR (or new TVaR) being breached given a trader still mistakenly assumes the original distribution during the strong market crash.  This is less straightforward as the reverse question, since the spread in the test statistic is not changing at a constant rate.  Instead, we can simply use that as a conservative estimate for the probability reduction.  For example VaR₅₁ could have underestimated as Φ[(-1.3%+(<-0.034%))/1%].  Or about 9% (note it was a larger 10% in the reverse question!)

Empirical

Now for the empirical section, see the world stock market index for 50 days, through August 18, 2015.  Seems fairly stationary (only ~0.8% σ).  Then we see the large market crash that gets further underway on the 51^st day, and continues through the day after Black Monday (August 24).

We are now going to only focus on the price changes (de-trended though the average μ was 0.0% anyway) instead of the shown price levels.  In the chart below we see a similar framework that we showed earlier for the theory or a simulation.  For example, we see the nearly linear slide down in the μ_t, and the accelerated downward drop in both VaR and TVaR.  Of course TVaR -only initially- drops faster than VaR, but this is because the daily drops in the market crash continue to escalate further and further into collapse on a large kurtosis or fat-tail distribution (here, here, here, here).

During the next market crash (which we appear to now be at the start of), when the short-term moving averages start to turn over, it is important to note that larger and more concentrated drops tend to occur.  And that it is exceptionally vital to proactively hedge your portfolio progressively (CFA article).  Assuming ahead that backward looking risk measures will worsen quicker than normal, and that they will typically exacerbate to the TVaR level (not the more attractive VaR level).