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.