where.
P r o o f: see appendix 3A.
7=0 corresponds to the sum o f investment in asset oo by the three types o f investors. The role o f A resembles the Lagrange multiplier’s role in a problem where there is only unconstrained investors and therefore its name. These two variables play a role in the market variance and the covariance between the market and individual assets in a way that resembles the usual free market equilibrium.
P roposition 6. The total market variance, o h, is equal to;
oh = R
v^{{(5^^+
w^^Y - m l s ) + o l + o l
+ 2 'P ro of: see appendix 3 A.
Although the formula above can be derived directly from applying the three types o f investment to the variance covariance matrix, a more convenient rearrangement o f terms will allow us to relate total market risk to total market return:
^
I o l jcTlI r r +
R V
Also,
('■« + ® . . s ) U = ( T , +CTc
+ (Tr
^
Therefore,
= O"^ - R V J l
P roposition 7. The covariance between one o f the first N assets and the market portfolio, G , is equal to:
^i,M =
P roof: see appendix 3 A
P roposition 8. The return o f one o f the first N assets and the market return are related via:
ai,M
P ro of: B y dividing the formula in proposition 7 by the relationship found before:
and eliminating A w e obtain the result.
The relationship in proposition 8 resembles the usual CAPM risk return trade-off. In particular, the expected return o f any o f the free traded assets is positively related to 142
its diversifiable risk, namely a , ^ . By being able to measure the right indexes:
and ( j h - R V ^ l l , all the usual conclusions about CAPM can be
obtained in this generalisation. Som e additional properties are:
• The proportions o f constrained investors (measured by their holdings o f asset j and ) are irrelevant as determinant o f the risk-retum relationship. Only
the total investment in asset oo, matters.
• The risk tolerance o f each representative investor { ( j ] , and o ^ ) or the distribution o f them is irrelevant, as in the usual CAPM only the total market risk matters.
• The expected return and the variance o f asset oo do not play a direct role in the risk return relationship. Only measurements o f the residual from regressing roo on the returns o f asset 1 i o N are relevant, namely: and r«.,/yv.
• From proposition 6 w e can see that the terms -r».i/v^oo and are
always positive"*^.
• The effect on price o f an increasing amount o f com pulsory holding w ill depend entirely on the realisation o f This can be positive or negative. H owever, the
T hey are equal respectively to:
and
o l + o ] - R V j o l g
Il ^\N^\N^\N
a l
A
and g iven that Q |n is positive definite and o \ — RV^CO^ g is n on -n egative, they are both p ositive.
total market return is positively affected by an increasing loo and this effect is larger than the one depending on roo.;a/- This w ill becom e evident in the next section after using a feasible measurement o f the market return.
• The measurement A w ill overestimate the true market beta (defined
as the market premium) when there is more mandatory investment in asset °o and when the other N assets cannot effectively replicate the return o f asset oo so the residual variance R Voo is large.
• In a w ell developed market, where there is enough diversification with assets 1 to
N, market frictions produced by mandatory holdings will be m ild or even
com pletely ineffective as the two terms i?Foo and roojN w ill go to zero.
• For too large amounts o f mandatory holdings, even a well diversified market w ill be affected. If the asset subject to mandatory holdings cannot be w ell replicated by the free asset 1 to N, the effect over the risk-retum relationship w ill be important.
c.~ A s s e t oo is n o t trad ed in th e m a r k e t
A more realistic assumption in terms o f the relevant market aggregation is that this market index w ill not include any information on asset oo, its variance, its return or the amount o f trade in it. A market index will include stock market trade in equity that is normally not subject to any frictions, like short-sales restriction or mandatory
holdings. Our asset corresponds to claims in assets not traded in the market"''^. Indeed, non-measurable assets like contingent claims on human capital are left out from the market index. Therefore, in the same way as it happens with any feasible test o f the CAPM , the relevant market return is not measurable. The market index w ill therefore consist o f the observable assets; those freely traded in the stock market: asset 1 to N.
This limitation o f the market index w ill introduce problems when trying to estimate or test o f the CAPM. Roll (1977) argues that since the market portfolio must include all claims in the econom y, any test that uses an aggregation o f only marketable assets has a low power. W e identify below an explicit expression for the loss described by Rol^^
Consider and the return and variance o f a market consisting only o f the
investments in assets 1 io N. We can define the covariance between this exclusive market aggregate and one individual asset, /, as cr^*..
P roposition 9. and are related via:
W e identified a few ex cep tio n s to this in section 2 in this chapter. W hen restrictions are im posed on assets that are traded in the estab lish ed market these are norm ally not permanent.
There is som e ev id en ce that this effect m ay not be, in practical term s, as important as R oll indicated. Stambaugh (1 9 8 2 ) studied the properties o f the CA PM using various proxies o f the market portfolio. His identical inferences under different market proxies su ggested that the loss in pow er is not too significant.
P r o o f : see appendix 3A.
P ro po sitio n 10. cr^ and are related via;
and,
Where w e have called (7^ „ and C7^*^ each per unit correlation o f asset with the
respective market index M and A/*.
P ro o f: see appendix 3A.
P rop osition 11. , andcr^, ,. are related via:
P ro of: see appendix 3A.
P roposition 12. The risk return relationship is now:
^ i , M * " ^ ^ / o o - ^ o o X - T -
P ro o f: see appendix 3A.
I f w e call given that this is the best predictor o f and pi the
covariance-variance ratio above, w e have that;
n = A K
From this expression it is clearer what the effect o f the disappearing market frictions like compulsory holdings or short sale restrictions will be, over the risk-retum relationship. Assum ing that the return o f the mandatory holding is positive, positively correlated with the market and asset i and that Pi is not greatly affected by changes in Io., a lower amount o f mandatory investment w ill decrease the return for an equal level o f risk. Consequently, asset prices will rise.