Momentum is one of the most popular subjects in the financial media. From a physics perspective it is analogous to a measure of average velocity (the rate of change with respect to time). While measures of instantaneous velocity (the derivative of distance with respect to time) are more accurate they are rarely used in practice. One topic that is rarely discussed is the concept of acceleration- or the rate of change in velocity (instantaneous velocity is the derivative of velocity with respect to time). A simple method to calculate velocity and acceleration is to take the first difference or returns of a time series. The table below shows the difference between velocity and acceleration using this simple method of first differences of price, however returns could also be used:

Acceleration is perhaps a more interesting avenue for quantitative research because it signals changes in momentum. Given two stocks with the same momentum, a parabolic rise tends to be riskier than a short-term deceleration in performance. Furthermore, a stock that rises consistently without much change in velocity tends to be less risky than one that constantly changes directions. Unfortunately, volatility on its own tends to penalize changes around a constant slope which may reflect normal ebbs and flows in a persistent trend. For trend-following, traders want a risk measure that is be able to capture the stability of the trend. Various risk measures have been promoted as superior alternatives such as downside standard deviation or conditional value at risk, but their downside- to excuse the pun- is that they fail to reflect risk that can be created on the upside of the distribution. Furthermore, by focusing on the downside of the distribution you have fewer samples to work with. Ideally an alternative risk metric would be bi-directional and indifferent to up or down returns. One such measure is using the volatility of acceleration- or the volatility of the velocity in returns using the first difference between the return today and the return yesterday as a proxy. The following charts of artificially generated time series data show the difference between the volatility of price differences (rather than returns) and the volatility of acceleration to give readers a sense for how the two measures differ. In this case, both time series start and end at the same price but have different profiles:

In the first chart, the trend is very consistent and both measures are very close- and considering the volatility of acceleration tends to be 50% greater than volatility on average- the volatility of acceleration is showing that risk is actually lower than what volatility is saying. In the second chart, the trend is highly inconsistent with several false moves up and down. In this case both volatility and the volatility of acceleration show an increase in risk versus the first time series. However, the volatility of acceleration shows that risk is significantly higher than volatility- even adjusting for scale- which makes intuitive sense. Essentially, the volatility of acceleration is boosted by the changes in direction in momentum. What is even more interesting is that when we apply this volatility of acceleration measure to real time series data, we see that it tends to lead volatility- in other words it tends to provide an early warning signal for risk. The chart below shows the 10-day volatility on the S&P500 (SPY) versus the 10-day volatility of acceleration during the financial crisis in 2008:

Notice that the volatility of acceleration spikes well in advance of the standard volatility readings. Obviously this is a valuable feature for good risk management. But does this translate to practical use in trading systems? As it turns out, it does, and there are a lot of different ways to apply this concept. One simple method is to look at standard volatility position sizing. In this case we use the same 10-day measure for both and a 1% daily target risk (1.5% for volatility of acceleration to reflect difference in scale):

Using the volatility of acceleration measure results in consistently superior returns over time and also higher risk-adjusted returns (higher sharpe ratio). Creative exploration could yield numerous areas of fruitful research. One additional application can be to create an alternative to the popular sharpe ratio. By calculating the annualized volatility of acceleration and dividing this result by 1.5, one can compute what I like to call the “return-to-uncertainty ratio” which may improve the evaluation of trading strategies.

1. November 28, 2014 3:42 am

David ,
interesting post as always but I am afraid that in this case I find it difficult to come to the same conclusions.
You write “…..results in consistently superior returns over time “.
All I can see by looking at the chart above is that one curve is slightly better in bull markets and the other ( using Acceleration ) slightly better in bear markets. As more time is spend in up-markets overall and due to the starting point the orange curve comes out ahead in your example.
In the past (1990s) the issue of trend stability was addressed – among others – by Kaufman’s Efficiency Ratio (ER) and Chande’s Chande Momentum Oscillator (CMO) .

• November 28, 2014 1:23 pm

Mr. Vollmeier,

I’m not sure if any of those work quickly enough, however. Things like Kaufman’s Efficiency Ratio is known to lag somewhat (for that matter, any trend indicator does). John Ehlers did some work on this, but when I tried it, it wasn’t anything overly mind-blowing.

-Ilya

November 30, 2014 6:01 pm

hi Ilya, I agree with what you are saying– and regarding Ehlers work, i think that it provides a good theoretical background and is very clever but some of the actual applications fail because markets are too noisy to create zero lag type directional indicators without amplifying the bad components that tend to to overwhelm the benefits of using them. of course that is a very general statement.
best
david

November 30, 2014 5:56 pm

Helmuth, i think that the purpose is to highlight alternative ways to look at the same problem. Using derivatives or variations of a concept can yield new areas of research. Towards your point on bull and bear markets– i will be presenting a framework to address how to modify components of the calculation to arrive at a non-linear measure of volatility.
I agree with Ilya completely, and having worked with ER quite a bit, the problem with those measures is additional lag.
best
david

November 28, 2014 3:50 am

hi,
very interesting piece but I’m not very clear what you used to compute the vol of acc?
For vol, the “elementary unit” would be LN(S_i+1 / S_i)^2 but what about the vol of acc?

Great post,
Thanks.

November 30, 2014 6:01 pm

hi jsmith, thanks- i will be doing a follow up post regarding that this week.
best
david

3. November 28, 2014 8:04 am

Suddenly I’m liking Black Friday. It is going to be interesting to drop in VOA instead of volatility into a number of trading systems. Very cool.

November 30, 2014 6:02 pm

hi cm, thanks–nothing like free shopping 🙂 hopefully it leads to new research avenues.
best
david

4. November 28, 2014 1:26 pm

Hm,

One possible application I see with this is with volatility trading shenanigans. Use the ratio between acceleration and volatility to determine XIV or VXX.

-Ilya

November 30, 2014 6:03 pm

Ilya, i agree that there are some interesting applications on that side for sure. good point
best
david

November 28, 2014 1:36 pm

Seems interesting, but like jsmith, i’m not sure how this is calculated. would you calculate the standard deviation of the first differences of the daily log returns?

November 30, 2014 6:03 pm

hi steve, re: jsmith, i will be posting something this week.
best
david

6. November 28, 2014 2:33 pm

Change in price definitely doesn’t make sense to me from a compounding perspective. So I tried this same method on XLF with change in log-price for velocity and then difference of those log-differences to get acceleration.

I don’t see any obvious “spike well ahead” action when plotting one divided by the other. They seem to be exactly on top of one another in timing in most major events and it looks like acceleration even lags coming out in 2009.

Also, hindsight bias may be a large factor of performance differential in your last example. It seems that 1.5x is about the average difference across the entire horizon — but there are periods where it dramatically shifts above or below. For a more accurate test, you’d want to use a look back window to compute the average difference and walk that forward.

November 30, 2014 6:05 pm

hi c, i didn’t use change in price for my example–i used returns. also there was no hindsight bias, if you use a walk forward test (i tested dozens of lookbacks) you will find that using the average of the rolling ratio between the two to derive the hedge ratio and applying that generates nearly identical results. but i agree with you in terms of your comments that this would be important.
best
david

November 28, 2014 6:06 pm

Juicy idea David. I can’t wait to put it through its paces. As always, I hope you are well.

November 30, 2014 6:06 pm

hi John, thanks- i hope you are doing well also. maybe it will yield some interesting avenues of research.
best
david

8. November 29, 2014 2:57 pm

“Notice that the volatility of acceleration spikes well in advance of the standard volatility readings.”

I am not seeing the “well in advance” part.

November 30, 2014 6:07 pm

hi Ed, perhaps that is an overstatement but there appears to be a lead of sorts across lookbacks.
best
david

9. November 30, 2014 7:50 pm

I can’t see the “lead” either. Volatility of Acceleration (VoA) is more volatile than Volatility (Vol) for sure. What if you scale VoA’s volatility to match Vol’s and then plot the ratio?

Have you planned to run the position sizing comparison on multiple asset classes with few similarities (ie relatively uncorrelated) such as gold, oil, REITs, HY bonds, EM equities?
If more than 75% of rolling 12 months periods for each asset class individually shows that VoA position sizing delivers better risk-adjusted performance than Volatility position sizing, then we may be on to something.

Sorry for the extra homework…