Expected Return Assumptions in Mean–Variance Portfolios
Summary
The document frames expected return estimation as a key input to Markowitz mean–variance portfolio optimization, where allocations balance expected return against variance according to an investor’s risk aversion. It asks what alternatives exist to estimating expected returns with historical average returns, but does not provide a survey or compare forecasting methods.
The response highlights how particular assumptions about expected returns can lead to familiar portfolio constructions: equal expected returns across assets produce a minimum-variance allocation, while expected returns proportional to volatility are associated with risk parity. It also emphasizes that covariance and correlation estimates matter alongside return assumptions. These are conceptual observations rather than a worked derivation or empirical evaluation, and the document does not establish that the assumptions are robust or prescribe a generally preferred estimator.
Key ideas
- Mean–variance optimization requires assumptions about expected returns and the covariance matrix.
- Using equal expected returns across assets leads to a minimum-variance portfolio formulation.
- Assuming expected returns proportional to volatility is associated with risk parity.
- Uncertainty in volatility and correlation assumptions can affect portfolio construction as much as return estimates.
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# Mean estimate in portfolio optimization (Markowitz)
# Mean estimate in portfolio optimization (Markowitz)
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?
## Answer by demully (score 1)
https://quant.stackexchange.com/a/60104
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, DEMShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.