CAPM and the risk free rate

In a recent post, I was musing about Sharpe and Lintner's decision to treat the risk free rate as an external fact about the world, and not endogenous to their model.  I noted that if you do this, then the curvy efficient frontier flattens down and becomes the capital market line.  Instead we have a set of risky assets and an additional tangent mechanism whereby the line running from the risk free (a.k.a. zero volatility) rate to the tangent point on the efficient frontier is introduced into the CAPM world.

I found out subsequently, re-reading the excellent Fischer Black biography that he considered it endogenous.   In his 1969 initial extension of CAPM, Black sees no role for the monetary authority, and models the risk free rate as the equilibrating rate which satisfies those timid investors who prefer to have larger fractions of their wealth in safe assets and who therefore want to lend all their money (via depositing it with the bank) to leveraged and more aggressive investors, who are happy to borrow that money and climb up past the tangency point on the CML.  

In this model, having a monetary authority messing with the risk free rate in order to stabilise the price level or even worse, to manipulate the economy, implied the system was in a non-equilibrium state, and hence had opportunities for profit.

There's something temptingly beautifully simple in this model of people having two choices for their wealth, and their changing distribution of animal spirits driving not only the price of money but also the price of risk.  It certainly doesn't much correspond to reality but then again nor did option prices much coalesce around the famous Black-Scholes formula pre-1973.  On the other hand, it says nothing about the origins of those animal spirits which stir up price discovery for the prices of money and risk.  Perhaps the only institution the libertarian radical markets theorist needs is human psychology.

Sharpe Hacks the Efficient Frontier Diagram

Markowitz didn't add the capital market line to his famous efficient frontier diagram.  The idea of a capital market line dates back at least as far as Irving Fisher, and it seems that Tobin, in his 1958 "Liquidity Preference as Behavior toward Risk", in which he references the earlier work of Markowitz, and asks the question: why would a rational investor ever own his government's 0 yield obligations (cash, to you and I) versus that same government's non-0 yielding bonds (or bills).

He could have considered, and perhaps he did think this way, interest-bearing government obligations as just one more asset to drop into the portfolio.  But, according to the following informative blog post on the subject of Tobin's separation theorem, there's a better way of doing this.  Before going into it in detail, note that the theory is becoming more institutionalised here by treating risk free lending as a separate element to the portfolio selection problem.  In effect, we've added a rather arbitrary and uniquely characterised asset.  Not only that, I've always thought the capital market line is a weird graft for Lintner (1965) and Sharpe (1964) to add on to the efficient portfolio diagram.

The efficient portfolio diagram is a $(\sigma, r)$ space, it is true, but in Markowitz's formulation, each vector in this space is also a collection or zero or more different portfolio combinations.  Whereas, when you add the CML, whilst it is true that the proportion $p$ of cash and $1-p$ of the market portfolio is at each point on the line is at that higher level a unique portfolio in its own right, we are coming from a semantic interpretation of the efficient frontier where each distinct point contains a different portfolio of risky assets.  On the CML, every point has precisely the same market portfolio, more or less watered down by a mix of the risky market portfolio (in general, the tangency portfolio, before we get to the CAPM step).  

As a side note Markowitz has been noticeably critical of the CAPM assumption on limitless lending and borrowing at the risk free rate, and unbounded short assumptions.  In other words, he's likely to approve of the CML from $R_f$ up to the point it hits the market portfolio, at which point, like a ghost train shunted on to the more realistic track, he would probably proceed along the rest of the efficient frontier.

Clearly, the tangency portfolio has the highest sharpe ratio, in any line emanating from the risk free rate at the ordinate.  Sharpe and Lintner were to argue that point happens to be the market portfolio (on the assumption that all investors had the same $E[r]$ and $E[\sigma^2]$ expectations and all cared only for these two moments in making their investment decisions.

The addition in this way of the tangency line always felt as if it were a geometric hack around the then costly step of having to run a brand new portfolio optimisation with treasuries added as the $N+1$th asset.  Remember, at this point the CAPM hasn't been postulated and the tangency portfolio was not necessarily the market portfolio, so the tangency portfolio was still going to have to be optimised.  However, the CML from $R_f$ to $M$, as it approaches $M$ is actually above (and hence better than) the original efficient frontier.  And again, if one was happy with the assumption that one can borrow limitlessly, then all points to the right of $M$ would be objectively higher and hence more preferred than those on the original efficient frontier.

However, how would they look when you plot the CML plus original efficient frontier together with the new efficient frontier with treasuries as the $N+1$th asset?  Clearly that new frontier would be closer to the line, and flatter.    And Markowitz would then ask the investor to chose where they want to be on the new $N+1$ (nonlinear) efficient frontier.  Also, linear regression of stocks via a sensitivity of $\beta_i$ to the market would not be such a done deal.

A further problem I have with the CML is that treasuries, even bills, most surely do have variance, albeit very small, and only if the one-period analysis matches precisely the maturity of the bill will there be no variance.  Perhaps on an $N+1$th efficient frontier, the CML isn't the one with the highest $\frac{R_p - R_f}{\sigma}$.  I can well imagine that, leaving the original CML on the graph, as you chart the new Markowitz $N+1$ frontier, then there'd be points along that new frontier which have better risk-return profiles that those of the CML associated with an $N$ portfolio.

As a matter of academic fact, Sharpe actually attached the CML to the efficient frontier first in his 1963 paper "A Simplified Model for Portfolio Analysis", where he very much sees the regression step which ultimately later leads to his concept of beta and which makes attachment to economic equilibrium theory as an optimisation step to reduce the number of estimable parameters.  In the same vein, he sees the CML (the idea for which he doesn't credit Tobin/Fisher, whereas a year later in his classic CAPM paper, he does credit Tobin - who himself doesn't credit Fisher) as a speedup only.  He says:

There is some interest rate $r_i$ at which money can be lent with virtual assurance that both principal and interest will be returned; at the least, money can be buried in the ground ($r_i=0$).  Such an alternative could be included as one possible security ($A_i = 1+r_i, B_i=0, Q_i=0$) but this would necessitate some needless computation.  In order to minimise computing time, lending at some pure interest rate is taken into account explicitly in the diagnonal code.

Wow.  What a poor reason, in retrospect, for doing it this way.  By 1964 he found his economic justification, namely that it theoretically recapitulated a classical Fisherian capital market line. But in 1963 it was just a hack.  Even his choice of variable name $A_i$ for $E[r_i]$ and $Q_i$ for $\sigma_i$ showed where his head was at - namely he was an operations research guy at this point, working with Markowitz in an operations research private firm.

At the very least, it seems to me, there's no theoretically good reason why we can't just add a risk free asset into the mix and do away with the CML.  That way, we'd get a touch of variance in the asset, and a degree of purity back, Markowitzian framework purity.  CAPM certainly is needed to produce beta, the major factor, but that after all is a function of the security market line, a different line.

Markowitz the micro-economist of the investor

In 1990 Markowitz was awarded the Nobel prize, so I had a read of his short acceptance speech, which quite clearly sets the scene for his work.  He describes microeconomics as populated by three types of actor - the firm, the consumer and the investor (that last one being the actor he focuses on).  He then also interestingly creates binary divisions on work in on each of these three actors.  First, the individual and then the generalised aspect of their ideal behaviour.   How ought a firm best act?  A consumer?  An investor.  After having answered these questions, the generalisation is, how would the economy look if every firm, every consumer and every investor acted in the same way.

It is worth pausing on just this point about generalisation alone.  Clearly the question of uncertainty must raise its head to our modern ear.  Can one model all firms as following he same basic template, a so-called rational template?  If we can, then we may identify an economic equilibrium state.  Likewise, with consumers, how does an economy look if everybody is consuming according to the same basic utility function.  In both of these cases, whilst uncertainty is present, and known about by economic modellers, it is given a back seat.  Markowitz accepts this, but shows how it is literally impossible to background when it comes to the actions of the rational investor, since doing so leads to a model where every investor picks the single security with the largest expected return.  This does not happen, so any model which treats risk/uncertainty poorly is insufficient.

I think it is probably widely agreed that today, models of the firm's behaviour and of consumers' behaviour is best done with uncertainty built into the model.  The old linear optimisation models accepted that variability in firms, or consumers could be averaged away.  That is, that it was a valid approach to assume minimal uncertainty and see how, under those simplifying model assumptions, equilibrium models of the economy might be produced.

But fundamentally, portfolio investing in the absence of risk makes no sense at all.  In this case, in the limit, we find the portfolio with the best expected return, and put all our money in this.  However, not many people actually do that.  So, in the sense that the micro-economic models of the investor make claims to model actual behaviour, then uncertainty must play a more prominent role.

Markowitz also hands off on 'the equilibrium model of the investor' to Sharpe and Lintner's CAPM. He is happy to see basic portfolio theory as the element which attempts to model how people actually act (hence, a normative model) and leaves positive elements to Sharpe's theory, which I think he does so with only partial success.  But  certainly I see how he's keen to do so, especially since his mean variance functions are not in themselves utility functions, and in that sense don't touch base with economic theory as well as Arrow-Pratt.

Rather, looking back on his achievement, he makes a contrast between Arrow-Pratt and his own, perhaps more lowly contribution and praises his approach as computationally simpler.  This may be true, but it isn't a theoretically powerful defence.  However, I like Markowitz, I like his lineage, Hume, Jimmy Savage and the Bayesian statistical approach.  I'm happy to go along with his approach.

I notice how Markowitz gently chides John Burr Williams for describing the value of an equity as the present value of its future dividends, instead of describing it as the present value of its expected future dividends, that is to say, Markowitz draws out that these dividends ought to be modelled as a probability distribution, with a mean and with a variance.

Markowitz also highlights early on in his career that he reckons that downside semi-variance would be a better model of risk in the win-lose sense, but he notes that he's never seen any research which shows semi-variance captures a better model than variance.  This is a rather passive backing off of his original insight into semi-variance.  Did he not consider doing any real work on this?  Is it enough for him to note that he hasn't seen any papers on this?  However, it is certainly true that there isn't a huge numerical difference in equity index returns, usually, so I could well believe this doesn't matter as much as it sounds, though it would be good to know if someone has confirmed it isn't an important enough distinction.

What Markowitz in effect did was replace expected utility maximisation with an approximation function, which is a function of portfolio mean and portfolio variance, and then he, and others later, try to reverse this back in to particular shapes of utility function.  This is where the computer science algorithm of simplex, together with the ad hoc objective function involving maximising returns and minimising variance attempt to meet top quality economic theory, as expressed in Morgenstern and Von Neumann. 

Markowitz then spends the rest of his lecture showing how strongly correlated mean-variance optimisation is with believable utility functions.

He wraps up, as I'm sure many good Nobel laureates do, by talking about new lines of research.  Here, he lists three: applying mean variance analysis to data other than just returns.  He refers to these as state variables.  They too could have a mean-variance analysis applied to them.  Semi-variance, as mentioned already, is another possible new line of development, and finally he mulls over the seemingly arbitrary connection between certain utility functions and his beloved mean-variance approach.    The slightly point here is that all three of these potential lines of investigation were already candidates back in 1959, yet clearly here is Markowitz in 1990 repeating them as issues still.