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:

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


3 Answers 3


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.


Do you know the Black-Litterman Model?

In principle Modern Portfolio Theory (the mean-variance approach of Markowitz) offers a solution to this problem once the expected returns and covariances of the assets are known. While Modern Portfolio Theory is an important theoretical advance, its application has universally encountered a problem: although the covariances of a few assets can be adequately estimated, it is difficult to come up with reasonable estimates of expected returns. In other words, composing a portfolio based only upon statistical measures of risk and returns yields simplistic results; these are known as unconstrained optimizations. Black–Litterman overcame this problem by not requiring the user to input estimates of expected return; instead it assumes that the initial expected returns are whatever is required so that the equilibrium asset allocation is equal to what we observe in the markets. The user is only required to state how his assumptions about expected returns differ from the market's and to state his degree of confidence in the alternative assumptions. From this, the Black–Litterman method computes the desired (mean-variance efficient) asset allocation.

More references on this Black-Litterman model can be found easily on Internet


Enclosed is a longer comment on risk parity:

A risk parity portfolio is one with the objective to equalize each asset's marginal contribution to the portfolio's total risk. At first glance, this objective function does not require returns estimates to construct portfolio weights, only estimates of returns variance, which is a feasible task.

However, it is impossible to entirely divorce portfolio construction from making key assumptions about the returns of assets in your portfolio.

For instance, one has to choose which assets to include in a risk parity portfolio. Amongst the numerous choices of assets, one would intuitively hope to end up with a mixture with the same properties desirable for any portfolio: (1) positive expected returns and (2) imperfect correlation amongst one another. The higher the returns, and the lower the correlation, the better the outcome.

So be mindful of this when thinking of risk parity. If you believe you can select a robust mix of assets with both (1) and (2) without relying on statistical estimates, then risk parity may be the best way forward.


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