Deriving Mean–Variance Portfolio Weights from a Constraint

Summary

This excerpt discusses deriving portfolio weights in a mean–variance framework, in the context of a paper on market portfolio weights. It sketches a way to solve a linear system involving the covariance matrix, expected returns, a constraint vector, and Lagrange multipliers. Applying the weight constraint yields an expression for one multiplier in terms of the scale parameter and matrix products.

The answer suggests substituting the resulting weights into the objective and optimizing over a single parameter, with the risk-free rate affecting the selected efficient portfolio. It does not complete the derivation of the cited equation or explain where the zero-beta portfolio return comes from, which was the original question. The response is explicitly tentative, so it is best treated as a partial algebraic hint rather than a full account of the paper’s result.

Key ideas

• The proposed weights are expressed using the inverse covariance matrix and a multiplier for the portfolio constraint.
• Applying the constraint that weights sum to one gives an explicit formula for the multiplier.
• The answer suggests reducing the remaining choice to a one-dimensional optimization over a scale parameter.
• The excerpt does not derive the cited equation or explain the zero-beta portfolio return.

Tags

# Mean Variance Investment problem


# Mean Variance Investment problem












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

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/43353

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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