How do you remove expected returns from asset allocation strategies?

The classic mean-variance optimization problem tries to minimize variance of a portfolio for a given expected return:

$$ \underset{w}{\arg \min} \quad w^T \Sigma w \quad \text{s.t} \quad \mu^Tw \geq \bar{\mu} $$

However the expected returns $\mu$ are can be computed in many different ways, but many use very subjective estimations. The problem is that the result of the optimization is very sensitive to these expected returns.

What asset allocation strategies exist where the expected returns are removed from the framework or where the sensitivity of the result (to changes in expected returns) is limited?

For example, the "min-variance" optimization removes the expected returns by return only the portfolio with minimum variance as follows:

$$ \underset{w}{\arg \min} \quad w^T \Sigma w$$

Answer

There is a vast, growing body of literature on risk parity, much of which is predicated on this idea (i.e. of optimizing a portfolio allocation without including expected return). As an example, The Journal of Investing put out an entire issue dedicated to the subject last year: see "Latest Approaches to Risk Parity and Diversification".

From "Risk Parity – Rewards, Risks and Research Opportunities" (2011):

Risk Parity (RP) is a relatively simple idea. Very loosely defined, RP attempts to create a portfolio in which the various asset classes contribute equally to the overall risk of the portfolio.

This is typically viewed against a 60/40 portfolio, and the argument is made that a 60/40 allocation actually allocates 90% of the risk to stocks. There is a some amount of disagreement over whether risk parity is the best approach, as will be clear from reading a selection of these papers. Some argue that there is no theoretical basis for why an RP portfolio should outperform an MVO-portfolio, and that it is inherently inefficient; "Leverage Aversion and Risk Parity" (2011) lays out a positive case based on leverage aversion. There are also many different implementations of this idea (see "Risk Parity for the Masses" for an example).

Some blog posts which have further references:

A related idea is the Most Diversified Portfolio (see, for example, "Properties of the Most Diversified Portfolio", 2011). This optimizes the portfolio based on the diversification ratio, which is simply the sum of the individual asset volatilities over the portfolio volatility.