Wednesday, 16 November 2016

Two interpretations of the Security Market Line

The security market line (SML) is a straight line drawn to represent at a high level the conclusions of the CAPM.  I have two distinct ways of reading it.  First the standard one.

The chart documents some relationships about some particular stock or portfolio.  It is interesting that for the purposes of the SML it doesn't matter of you're considering a single stock or a portfolio of some socks with a bunch of weightings.

The Y axis shows you what the CAPM model output at any given moment thinks a set of portfolios is likely to return.  The X axis shows the beta of that portfolio.  So each <x,y> co-ordinate represents a set of portfolios where each member shares the same expected return and the same beta as as its cohabitees.  This set of portfolios behind a single point can be considered infinite.  And of course there are an infinite number of points on he Cartesian plane. 

Only the set of portfolios which give the best return over risk profile exist on a single upwardly sloping line referred to as the SML.  These on-line portfolios are expected to return the market return (plus the risk free rate).

The slope of the line is referred to as the Treynor ratio and equals the excess return to be expected from the market now in excess of a (fairly) risk free rate.

Passive index trackers have as their job the task of residing at the point <1, E[R_m ] +R_f> in as cheap a way as possible.  That is to say, there are many portfolios which have a beta of (about) 1.0 and which have an expected return of the return of the market.  Passive fund managers try to implement being on this point in as cost effective way as possible.    Passive fund managers of leveraged offerings try to do the same thing but at betas of 2.0, 3.0, 0.5 etc.

CAPM tells you there's no point being off-the-line as you're taking diversifiable risk and hence shouldn't be getting paid for it.  You only get paid for diversifiable risk, that is, risk which is correlated with the market.

Active portfolio management believes that some portfolios exist above and below the SML and can be exploited to make returns greater than the market.

Deciding which value of x you'd like is not something the model can help you with.  That represents an exogenous 'risk appetite choice.  Once you've made that choice the SML tells you, assuming it is based on a well functioning and calibrated CAPM, how much you can expect to make. 

Let's imagine you have a normal risk appetite and set x=1.  There are many ways of constructing a portfolio which delivers that return but the one where you're fully invested in the market portfolio is a natural choice.  You could be fully invested in a number of other portfolios which do the same.  Or you could be under-using your notional capital investment and using market weights for all other stocks; or you could be borrowing money and over-gearing your unit of capital to raise the beta greater than 1.

That is, by using financing gearing, you can travel with a fixed market portfolio up and down the x-axis, in theory to any value of x.  Of course you can't get infinite leverage but still, the theory assumes you can. 

You can achieve the same by using leveraged products - equity options or equity futures, for example.  These narrow your time horizon (theta) but in theory you don't need to worry about that.

If you try to be, e.g., fully invested and then try to tilt the beta by owning long more high beta stocks than the market, you will indeed see your Y value increase but this will also be taking risk which is diversifiable.  So you'll be taking risk you are not getting paid for.  Achieving this same level of return can be achived in a more efficient way with the market portfolio and some form of leverage, and this approach is theoretically to be preferred on this basis.

In practice there are costs associated with gaining any leverage to achieve a desired return.  Perhaps a better model is a CAPM with funding costs burned in.

Also you won't see a SML with negative x values.  There's no reason why not.  Sometimes you my be seeking a portfolio which returns less than the risk free rate (and perhaps even negative returns) in certain circumstances.  In this case you'd see the return go negative as your beta goes negative.

A question arises in my head.  Long term, which is the best value of x to sit on together with a market portfolio, if your goal is to mximise expected excess returns across all time periods and business cycles.  I think this is a permutation of asking whether there's a way of forecasting the Treynor ratio (the equity risk premium).  If you could, then you could move to a x>1 portfolio construction and move likewise to a x<1 construction when your model calls a decreasing equity risk premium.

What if my equity risk premium forecaster was a random process which swept randomly through the 0.9-1.1 range?  Long term, would this not be equivalent to a steady 1.0?  Could the algoithm have a degree of mean reversion at the back.  That is to say, if a long term random peppering of the 0.9-1.0 space delivers 1.0 results, then if your active algorithm has placed you at 0.9 for a while, might it then increase the hit rate at the >1.0 space?

So the SML is an SML for today, and the slope of that curve may steepen or become shallow though time.  Probably within a very tight range.

Calculating and predicting the equity risk premium seems to be perhaps an even more valuable thing to do than trying to do active equity factor portfolio modelling.