Monday, 28 November 2016

A feeling for equity factors


At this point I wanted to spend a little time thinking about how realistic equity factor modelling is.  To the degree that it isn't realistic, what can be said about the limitations.

The first point is an obvious one about data quality.  Stocks have corporate actions.  They split, they have regular and irregular dividends.  They become involved in m&a activity.  They dual list.  They enter and exit indices.    Each of these, and many more, real world effects can be considered as data quality challenges.  Some of these can also be considered to be the normal everyday lived experience of the average stock, and on that basis ought to be dealt with squarely by the model.

This double approach - the degree to which you pre-filter your universe, clearly can have results ramifications.  Statistically, what is it exactly we do when we remove outliers, and how justified are we to do that.  People tend to be informed by the economic and theoretic reality of stocks and the CAPM in deciding how to treat data issues.  

But in the end we are trying to do the following: find a single, more or less stable, relationship - a linear one - which captures this primary 'like Jagger/not like Jagger' distinction in stocks.  In other big data enterprises, sparse data is a problem, but with equity factors, for the life of each stock, you will often have continuous (end of day) prices over the examination period.  Clearly some stocks are going to be more liquid than others, but they're all likely to be liquid enough to provide an end of day price.  And thanks to the very idea of beta, we can be assured that we're always finding end of day correlations with the whole market, which means that the correlation data embedded in the stock's current beta number is also not going to run into the sparse data problem.

The whole idea of CAPM and equity factors is underpinned  by the idea that it is meaningful to talk about the average properties of stocks - that there is, in a sense, an average stock.

If you imagine the primary regression chart underlying CAPM, perhaps imagine whether some shape other than linear might be applied to the regression.    The security market line (SML) shows you what expected return you ought to expect from the equity market you just analysed knowing only what that stock's beta is.  Or alternatively how leveraged you chose to be in any given stock.  But imagine this line isn't linear.

How would it deviate from linearity?  With high beta stocks (and when operating with stocks, the assumption is that they are usually going to be positive beta, with a notional holding of a single unit of them long) what that do to the expected payoff.  

Expected payoffs, if they partition a stock space, can be considered to be additive.  In other words, pretend you divide all stocks into 2 groups - those whose corporate name begins with the letter M or lower, and that the other group represents all other stocks.  When you calculate their SMLs separately, you'd first of all expect them to look identical.  But in any case, you could combine them to reach an average SML for both halves.  If i performed the same analysis for all companies whose name started with M,  versus all the rest, you'd want to weight these two SMLs to account for the fact that the market is clearly mostly like the 'non M' category. So perhaps you can weight by market capitalisation proportion.

Now start imagining some interesting partitions.  If you found a really significant partition based on some economically relevant measure, M, you could always immediately know what the 'non M' SML must look like, since you know that when combined with M, the two together combine to give you the SML associated with the market.

To repeat, there is often considered to be one SML, 'the' SML, which allows you to work on leveraged passive ETFs, for example.  Namely that if you were oblivious to the real effects of leverage, you'd be indifferent to where on the straight line you chose to be.  But think of non-random partitions of the stock universe.  To the degree that these partitions are information rich, the resulting partitions could be considered to be different SMLs.  CAPM's spin on all this is that you're a fool to want any line other than the SLM line, since you're not going to get paid for concentration risk.

You can think of all these component SMLs as linear combinations of their respective partition component SML.  Or as non-linear.

The idea that you can leverage the market to achieve whichever level or return you like and it is identical to selecting high beta stocks only to reach the same expected return is clearly a poor assumption.  It is kind of like the idea that portfolio insurance and puts are the same thing.  In theory only.

The degree to which high beta stocks are better is the degree to which you might imagine that the SML will droop on the upside.  I.e. that you would be happy to take less for the high beta stocks implementation when compared to the leverage way of getting to that return.

Put another way.  Leverage is costly.  So the high beta end of the SML is likely to droop as it tries and fails to live up to the theory  of CAPM.  Perhaps there's an argument for saying that the low beta end of the SML must be perky in contrast to the high beta leverage droop, to make the final theoretical SLM be the particular gradient of line it is.  Another thought.  If high beta stocks are the preferred way to achieve leverage at reasonable cost, then perhaps these stocks are more prized for this very reason, and bid up?  In being too expensive, perhaps their returns are poorer as a result?