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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

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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

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    $\begingroup$ All portfolios on the fromtier can be written as a linear combination of two portfolios on the frontier. Here the two base portfolios are:(1)The market portfolio, (2)The Zero Beta portfolio, which is by definition the portfolio orthogonal to the market portfolio. That is why $\bar{r}_Z$ enters into the equation. It plays a role similar to the risk free rate in the CAPM. You may want to look up Orthogonal Portfolios and zero beta portfolio . Or the 4 references given. $\endgroup$ – noob2 Jan 5 at 20:43
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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.

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