**Previously…**

A previous post (https://blog.otastech.com/2015/11/the-dollar-hedge-part-1/) discussed the problem that the average beta of stocks in a typical portfolio is less than 1.

The consequence of this was that if you try to use a dollar hedge, you frequently end up with an overall short position — in other words, the dollar-hedged portfolio should be expected to lose money if the market goes up. The conclusion is that you should typically only hedge about 60%-70% of your dollar position, depending on the exact makeup of your portfolio.

However, you could worry that in times of crisis, the stocks might move together much more — if they’re driven by large-scale macro forces, then perhaps their correlations might go up.

I attempt to answer this question here by plotting average beta vs time. The article is somewhat technical, but the interesting result is in Figure 1, so feel free to scroll to that.

**Methodology**

I picked the 1000 largest Euro-denominated (so that we don’t just measure currency vol) stocks and worked with their beta against the Eurostoxx-50 index, for the last 10 years. Each stock has its beta calculated in a moving (Gaussian) window with width 100 trading days, and the mean is taken.

Beta is hard to estimate based on too little data. This is because it involves dividing two quantities which are both products of returns. Given that stock returns tend to have quite noticeable outliers, the product of two returns (for instance the stock’s return multiplied by the market’s return), can vary wildly from day to day. If you keep outliers in the calculation, then the resulting beta is weighted heavily towards what happened to stocks on just one or two high volatility days, but if you take the outliers out (or clip them to a permissible range), then you don’t answer the question “what happens in high volatility conditions”.

So we need several months’ worth of data realistically, to get a handle on whether beta is currently high or low. There’s also a risk that the numbers we get will be specific to the methodology we choose. For instance, if, during a crash, all the stocks were to crash, but not all on the same day, then the 1-day returns might show low beta, but the 5-day returns might have a much higher beta.

The only way round this is to try lots of methodologies and see if they agree.

**Results**

The average beta has been fairy constant over the last 10 years, and seems not to be particularly correlated to market volatility:

The uncertainty comes from assuming that the beta variation for a 10-day window is completely noise, and scaling the observed noise to the window used here (100 days).

If we try varying the return time, we get a similar shape, but shorter timescales have lower betas. This is completely expected if we take into account the short-term mean reversion: There is a slight tendency for stocks to revert from one day to the next, and although difficult to profit from, the effect is strong enough to increase correlations for longer timescales:

Then, trying median beta rather than mean, and trying a less aggressive outlier reducing process:

So it seems that at least for these 1000 stocks, the beta seems to be fairly unconnected to volatility.

One last test was to try the beta vs a home-made market (the largest 100 stocks in the same universe):

**Conclusion:**

Beta did vary from year to year, and it seems to be significant – but the uncertainty in the beta estimate is difficult to estimate. There seems not to be a strong link between mean beta and volatility, though.

*Underlying data courtesy of Stoxx. The Stoxx indices are the intellectual property (including registered trademarks) of STOXX Limited, Zurich, Switzerland and/or its licensors (“Licensors”), which is used under license. None of the products based on those Indices are sponsored, endorsed, sold or promoted by STOXX and its Licensors and neither of the Licensors shall have any liability with respect thereto.*