A note on the CAPM: 
The market as a whole cannot have negative weights, but submarkets might

Introduction

We arrive at the paradox that individuals are advised to choose their own portfolio using short selling, with a part going to the certain rate and the other part going to the market portfolio; while the market portfolio would be without short selling ! Indeed, part of the paradox is that, since the market portfolio has to be on the efficiency frontier, this frontier should also be calculated with nonnegative weights; and this is an idea that many will reject who are able to sell short.

This paradox is not mentioned nor resolved in standard textbooks. Luenberger (1998:174) writes: "Hence the weight of an asset in the market portfolio is equal to the proportion of that asset's total capital value to the total market capital value. These weights are termed capitalization weights." These are the weights that many index funds rely on. Bodie & Merton (1998, preliminary edition) also regard this indexing as one of the main conclusions of CAPM: "Whether or not the CAPM is strictly true, it provides a rationale for a very simple passive portfolio strategy: (*) Diversify your holdings of risky assets according to the proportions of the market portfolio, (*) Mix this portfolio with the risk-free asset to achieve a desired risk-reward combination." (p309). For someone who can sell short, the latter advice is clearly inoptimal.

The paradox can be resolved by noting that there will be submarkets for those who lack the credibility to sell short, and submarkets for those who can (each up to a specific level of credibility). Each submarket will have its submarket portfolio. The One Fund theorem must be qualified to the submarkets.

Each submarket can be seen as consisting of a representative individual times the number of those - and then there can be negative weights since those individuals can have these. The notion of a representative individual is not restrictive here, since the representation only concerns the credit position. Also, since there is trade across submarkets, a negative weight in one submarket is balanced by another, and the total market will again have nonnegative weights.

Hence, in CAPM theory we should replace "market portfolio" by "submarket portfolio". And investment advisors should take account of the short selling capacities of their clients, rather than advise to take an index fund based on capitalisation weights.

It will be useful to clarify the issues with some algebra, that will also allow us to derive some conclusions on the weights and the beta's.

For an individual

• Own worth = W.
• Borrow x. Revenue on sale now is X.  Available for investments is W + X
• Invested Y = W + X,   or  W = Y - X  (so division by W gives 1, with negative weights)
• End of year result after returning the borrowed x is Y[1] - X[1]
• Rate of return = (Y[1] - X[1] - W) / W        (so we indeed still can divide by W)

Going short means that more people are involved

Let us look at the accounting. We take cash instead of treasury bonds since it has an easy rate of return. With cash positions Ci (first row), assets X and Y (second row), and three agents (columns), the start positions are:
 

CAPM would argue that everybody invests in X and Y with the same weights w = X / (X+Y) and (1-w) = Y / (X+Y), multiplied by their investment capital reserved for such uncertain assets. This would mean that no individual would have negative weights, and thus that nobody would take short positions. But some people do.

Let us assume that agent 2 borrows x from 1, sells to 3; and buys Y from 3. So C2 + X == Y in period 0.
 

-	C1		-			-
-				Y		-	X

Gross market values in period 1 are:
 

-	C1		-			-
-						-

Officially, individual 2 will have to buy back the original stock X. With this limited number of individuals, this implies that individual 3 buys back Y at price Y[1]. If there is a loss, then we presume a fourth credit institution.
 

The agents take their original positions as their base, not those from after the short position. The rates of return per individual are:

•      (X[1] + C1) / (X + C1) - 1        (Y[1] - X[1]) / C2 - 1                       (X[1] + C2 + C3) / (Y + C3) - 1
• or  (X[1] - X) / (X + C1)               (Y[1] - X - (X[1] - X)) / C2 - 1         (X[1] - X + Y + C3) / (Y + C3) - 1
• or  (X[1] - X) / (X + C1)               (Y[1] - Y -  (X[1] - X)) / C2             (X[1] - X) / (Y + C3)

We find, as proper, that individual 1 and 2 take the risk for x, while individual 2 takes the risk for "y - x". The risk for x has been multiplied, with a balancing act by individual 2. But the amount of shares "x" remains x.

This multiplication of risks is the same as when a million people bet that a toss of a die will give 6: This does not mean that there are a million tosses. There is just one toss (and there must be someone else who takes the bet).

The relevant conclusion here is that the total investment in the market is W* = C1 + C2 + C3 + X + Y. Clearly the weights of C, X and Y are nonnegative in W*. Or similarly if we concentrate on X + Y only. Contrary to what seems to be a common thought, going short is no argument to allow for negative weights in the market.

Note: One question on the side was: Did the short position increase the input ? One might argue, along the lines of an individual investor, that the real investment in the market was W* + X. However, since the agents only take their original positions, the argument is in favour of W* only. The short position concerns a redistribution of the profits, not a real new investment.

Note: There are only negative weights for the total for an individual in W* if the market regulators allow people to go short who have no capital or to go short beyond their capital. For individual 2 this would mean C2 < 0. Portfolio weights for the totals of individuals are, with Y - X = C2, and here C2  > 0:

            (C1+ Y) / W*                        (Y - X) / W*                                    (C3 + Y) / W*

Note: To compare with the former paragraph: For individual 2 we have individual portfolio weights {Y / C2, -X / C2}, so there are negative weights for this individual.

Taking account of submarkets

One particular important group consists of those who cannot go short but who will also accept a greater spread than their submarket portfolio or than their capital market line allows: They will select a point on the efficiency frontier to the right of their submarket portfolio. Another group can go short for a little, and their capital market line goes a little bit beyond their submarket portfolio - but not ad finitum. Another group may be fully free.

Let us limit the complexity, and assume here only two submarkets: (1) Those who can go short at will, (2) Those who cannot and who will prefer a section on their capital market line (i.e. to the left of their submarket portfolio; note that the and is important). Then there are the following steps:

• An individual i will invest an amount  in the certain asset and  in the uncertain assets.
• For each individual there is a submarket kind k = k(i), and k is in {S, N} (for Short and Nonnegative).
• Let  be the weight of asset j in the submarket portfolio for market k (i.e. under efficiency).
• The submarket capitalisation weights  =  are equal to the submarket portfolio weights  if the submarkets are efficient.  (Thus, it is no use for an individual to copy the submarket portfolio if it were inefficient; but if it is and everybody knows this, then everybody copies it, and thus they have the same weights. But if a market is inefficient, then the individual still can compute the w and use them.)
• Under efficiency  =   is the total investment of individual i in asset j.
• Let  =   be the total amount invested in submarket k.
• We have M =  +  for the total amount invested in uncertain assets; similarly for F.
• There is also  =  for the total amount invested in asset j in submarket k.
• We find that the total amount invested in uncertain asset j is .
• The total amount invested in uncertain assets also is X = M.
• The capitalisation weights for the whole market are weighted averages  =  / X, and these differ from the .
• The S market can have negative weights as long as the N market compensates. The  translates as  =  for s =  the relative size of the short market.
• With the latter, the story of the capital market line of the short market becomes a bit different. Conventionally the story is that one's utility function could select any point. Now however there is this balance constraint. It would seem that the amount of investment capital starts to become relevant, while this is not so for conventional CAPM.
• There is not one beta for an asset: these also depend upon the submarket.

The existence of submarkets also has consequences for the beta's. With the law of one price, for each asset the rate of return, the current and the future price must be the same for all submarkets. From this solves a relationship for the beta's. Using Cool (1999):

The price condition for the short submarket is:

Conclusion

That paradox is resolved by allowing for submarkets.

It appears that we cannot simply uphold the CAPM analysis for the submarkets. Some modification is required.

The optimal weights of one market enter the solution of the other market - namely via the weight constraints in the optimalisation of the efficiency frontier and the slope of the capital market line. The relative size s of the short market enters both solutions. If the s-adjusted constraints are binding, then there may also be some "Nash" situation, and perhaps we need additional assumptions to solve the case. One possible assumption is that s is a function of the profitability of the market, for example if banks allow more people to go short if the expected returns are getting more interesting. This will make the model far complexer, of course.

Next, there is a restriction on the betas. Price equality over submarkets is a real condition and not just some consequence, and thus the restriction on the betas enters the optimalisation procedure.

It may be, though, that in practice only few can go short, so that the N market is large enough to absorb this, and that the CAPM analysis can be applied independently. However, with the current size of the market for derivatives, it is doubtful that this is the case.

It is my impression that this analysis is a basic aspect in any introduction into the CAPM. An individual who might go short would need to know it, even when he or she would not affect the market. A researcher who studies the real economy would benefit from it too.

Literature