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The Markowitz mean-variance portfolio optimization problem is to find the optimal allocation, $w_{optimal}$ by solving:

\begin{equation} w = \mathrm{argmax} \ \mu_{t}^Tw - \frac{\gamma}{2}w^{T}\Sigma_{t}w \end{equation}

where $\mu_t$ and $\Sigma_t$ are the conditional expected value and conditional variance respectively at time t. $\gamma$ is the risk aversion parameter.

So, doing this in practice, one has to get an estimate of $\mu$. The most naive solution is to use the mean of previous returns. Can anyone provide what other standard methods that are available?

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Hi and welcome MathStat2718,

The "other standard methods" you ask about are EXACTLY the cute financial products of the last decade.

Assume mu is constant across constituents, this gives you the "minimum variance portfolio". Assume mu is proportional to sigma, this gives you the "risk parity portfolio". Tbere are lots of MV and RP funds out there ;-)

You can take this thought process further and argue that the sigma and correlation assumptions are just as critical to portfolio construction as your return assumptions... why should these historic values/assumptions be any more robust. Honest to God, there are portfolios you can buy (albeit with a good wealth manager) that are volatility and/or correlation-indifferent as well as historic-return indifferent ;-)

Couldn't make this shit up, DEM

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