I attach a part of a paper explaining how the weights of a market portfolio are derived. I do not understand how equation 5 has been derived and, in particular, where the zero beta portfolio's return comes from. Many thanks in advance
For the curious, this is an excerpt from Capital Asset Pricing Compatible with Observed Market Value Weights by Michael J. Best and Robert R. Graber, The Journal of Finance, Vol. 40, No. 1 (Mar., 1985), pp. 85-103
you can observe that if:
$$ \Sigma x + \lambda t = s \mu $$
then:
$$ x = \Sigma^{-1}su - \Sigma^{-1}\lambda t $$
satisfies this. This is of a similar form to (5) without explicit constants.
Then,
$$ t'x=1 $$
gives
$$ t'\Sigma^{-1}su - t '\Sigma^{-1}\lambda t = 1 $$
So $ \lambda = \frac{s t' \Sigma^{-1} u - 1}{t' \Sigma^{-1} t} $
Now that you have $x$ in terms of $u$ and $s$ I suppose it is substituted back into (2) and solved as a 1-dim maximisation problem for parameter s. The above formula essentially finds the efficient frontier and the precise optimal efficient portfoli0 depends on the risk free rate $r_f$.
Anyway, this was a quick 5 minute job, sorry I didn't manage to get the full answer. hope it helps.
