Momentum strategies

Momentum is an age-old feature of financial markets. It is perhaps the simplest and also the most puzzling of the “anomalies” discovered.  It is simply the tendency for assets (for example shares of some company) that did well (or poorly) in the past to continue to do so for a time in the future. It has been extensively examined in academia and has been found to be present in virtually all markets and going as far back as we have data. It has even persisted some decades after being extensively investigated for the first time. And still, it seems, we do not understand it very well. In today’s post I just want to highlight some different momentum strategies and their uses.

A property and a strategy

Momentum is a property of asset prices in markets and momentum strategies try to benefit from this property. One way of understanding momentum is to consider different momentum strategies and the profits they make, which gives an indirect means of understanding how asset prices work. For investors, of course, this is perhaps the most convenient way to study momentum as they are inherently interested in the strategies. They only care about momentum as a property if they can exploit it. The distinction between momentum as a property and as a strategy is not always clear because academics have not yet, I think, deemed it important to make the distinction explicit and thus both are simply called momentum. 

How to construct a momentum strategy

Momentum strategies come in all shapes and forms. Basically all of technical analysis is some kind of momentum strategy. A very general way of thinking about constructing a momentum strategy is depicted in the picture below. One starts by identifying some kind of trend (or signal) for each of the assets you are considering. This gives the direction of the momentum for the asset (for instance up or down). One can then assign a strength (or score) to this signal, which can be related to the magnitude of the momentum or the confidence you place in it. Then based on the signal and strength one makes an allocation decision – you decide how to bet in order (hopefully) to profit.

Time-series and cross-sectional momentum

Momentum strategies come in two main forms (though they are related). The first is to consider momentum for individual assets – the tendency for an asset’s price to go up if it went up in the past.  Here the signal and strength are evaluated for assets in isolation. This is time-series momentum. This form of momentum can be contrasted with cross-sectional momentum, which considers the momentum of assets relative to each other, e.g. the tendency of one asset to perform better than other assets if it also did so in the past, for instance. Here the signal and strength depends on how assets compare to each other.

Time-series momentum (strategy) tends to do well if an asset’s return is related positively related to its own past (property), for instance in what is called an AR(1) process:

Thus a higher return in the past predicts a higher return in the future.

Cross-sectional momentum (strategy) tends to do well if one asset’s return is negatively related to the past return of another asset (property), for instance if (numbering the assets 1 and 2)

This means that a high return on the one asset predicts a lower return for the other asset in the future.

Some simple strategies

Here are some simple strategies, based on a simple taxonomy:

Signed time-series momentum: buy any asset that went up in the past; sell any asset that went down.

Signed cross-sectional momentum: this is analogous to the above, but now invest in deviations from the average return or the market return. For instance the deviation of asset i’s return from the average is

If the asset did better than the average, buy the asset and sell the market and do the opposite if it did worse. This is a bet that assets that had above average performance in the future will continue to do so in the future.

Linear time-series strategy: again buy any asset that went up and sell any asset that went down, but invest more in assets with larger returns (invest proportionally to the asset’s past return)

Linear cross-sectional strategy: the same as above, but for deviations from the average (or market) return.

Quantile cross-sectional strategy: buy, for instance, the top third of assets and sell the bottom third.

In practice only the signed time-series and quantile cross-sectional strategies are used. The other strategies are, however, useful in formulating theory. For instance the linear strategies are easier to cope with mathematically, but amplify volatility too much to make them useful in practice.

Some references

My thesis:

• Du Plessis, J. (2013). Demystifying momentum: time-series and cross-sectional momentum, volatility and dispersion. University of Amsterdam. Retrieved from http://www.johandp.com/downloads/johandp_momentum_final.pdf?attredirects=0

Some empirical articles looking at momentum strategies:

• Jegadeesh, N., & Titman, S. (1993). Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. The Journal of Finance, 48(1), 65–91. doi:10.1111/j.1540-6261.1993.tb04702.x
• Lewellen, J. (2002). Momentum and Autocorrelation in Stock Returns. (C. H. Schiller, Ed.)Review of Financial Studies, 15(2), 533–564. doi:10.1093/rfs/15.2.533
• Moskowitz, T. J., Ooi, Y. H., & Pedersen, L. H. (2012). Time series momentum. Journal of Financial Economics, 104(2), 228–250. doi:10.1016/j.jfineco.2011.11.003

"Just a little longer": the curse of finite capital

Warning: this post contains some mathematics. However, non-technical readers may ignore the equations and focus on the concepts, which are far more important in any case.

One thing that hit me about LTCM’s failure is this: they failed not because their models were wrong, but because they ran out of capital. I came across a simple mathematical example of a strategy which illustrates a strategy that runs out of capital and will share it with you.

The double or nothing strategy

We consider a discrete time market which is both fair and efficient. There is, however, a way of making a guaranteed profit. What is the catch? You need an infinite amount of capital.

Consider the shares of Fairness Company. The company has an equal (and independent) chance of doing well (in which case the share price goes up) or poorly (in which case it falls) every period. We take out a derivative which pays 2 if the shares go up and 0 if they go down in the next time period. Naturally, the cost of one unit of this derivative is 1, the expected gain. So if we buy one unit of this derivative we will make a profit of 1 with probability ½ and a loss of 1 with probability ½. The expected profit is zero.

Let us call the profit on the derivative at time t (which we buy at time t-1) d_t. So


Suppose we hold Y_t units at time t, which we buy at time t-1 and we start off by buying one derivative. So with probability ½ we gain or lose Y_t units at each period. Our accumulated loss/gain at time n is then


If the share goes up after the first period, we stop and get a profit of 1. If it goes down we bet more, so that when the share does go up we recover our loss and make a profit. That is we invest 1 (our original holding), plus what we lost, also 1. So we invest 2.

So we set


It is important that Y_n depends only on n-1, which is something we already know when we buy our next set of derivatives.

Note that if d_n = 1 then

and Y_n = 0. So we naturally stop when we make a profit.

If we do not make a profit, but a loss, we get


So for our next bet we need to invest


So after each period we double our bet, as long as we are losing. Let us call the time when we stop T.


T is finite with probability 1. Which can be seen because


Adding up for all n gives


So eventually the strategy will stop and you will get your guaranteed reward of 1. An example of how this happens is shown in the following graph. Note how large your losses become before you eventually make your (by that time miniscule) profit.

But how much money should you have, exactly?. Well your last bet has value

and the expected amount you need, just for this last bet, is then


The moral

This example is, of course, very simple and somewhat contrived. It does illustrate one salient point: A strategy that must “eventually” work can fail if you cannot stay solvent long enough to see it through.

In this case the problem is, of course, that the market is fair. If markets actually are fair you cannot beat them. Another reason your strategy may fail is because of market irrationality. The market may be completely irrational in the short term. LTCM was waiting, quite reasonably, for the prices of Royal Dutch and Shell to converge (they WERE shares in the same company after all). However, they just did not have the ability to wait long enough.

Doubling your bet at each period may seem insane, but there are many investors out there who believe that if a stock they hold goes down, this is a sign to buy even more of it as it is now even more undervalued. Of course, if markets are mean-reverting and the stock truly is undervalued then this may be legitimate.

The problem comes when you borrow in order to buy the stock, then you may be expected to repay your debt before your profits have come. And no one has unlimited borrowing capacity.

Never devise a strategy without considering how much capital you might need. Never assume that you will not need to make a (dis)graceful exit. I leave you with the words of John Maynard Keynes, whose wisdom mocks us even today:

"Markets can remain irrational a lot longer than you and I can remain solvent."

Some references

• Spreij, P. J. C. (2011). Measure Theoretic Probability. Retrieved from http://sites.google.com/site/mtp1112/mtp.pdf?attredirects=0
• Wikipedia. (2011). Long-Term Capital Management. Wikipedia. Retrieved from http://en.wikipedia.org/wiki/LTCM