Does "standard Markowitz approach" include only mean-variance approach or does it also include other approach such as minimum-variance approach?


2 Answers 2


The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years:

The standard model (Markowitz, 1952, 1959) originally only considered:

  • Constrained model where short sales are forbidden
  • Only risky assets considered for investment (no risk-free asset)

Scenarios that the mean-variance model can be extended to include one, or a combination of, the following:

  • Unconstrained model where short sales are allowed (Black, 1972)
  • Inclusion of a risk-free asset (Tobin, 1958, 1965)
  • Dynamic model that looks at multi-period rebalancing of portfolios
  • Inclusion of transaction costs

Classical characteristic portfolios that can be constructed by optimizing different objective functions for many combinations of the scenarios above, including obviously the standard model, are:

  • Global Minimum Variance portfolio (GMV): has minimum portfolio volatility
  • Tangency portfolio: has maximum reward-to-risk Sharpe ratio
  • all other frontier portfolios that lie on the efficient frontier

Portfolio risk optimization and risk parity, two sub-fields of modern portfolio theory that further explore risk contributions, correlations, diversification and concentration, introduced the following:

  • Maximum Diversification portfolio
  • Maximum Decorrelation portfolio
  • Risk Parity portfolio
  • Volatility Targeting portfolio
  • Hierarchical Risk Parity (HRP) portfolio

Common active strategies that modify the optimization problem to take into account investor views are:

  • the Treynor-Black model (1973)
  • the Black-Litterman model (1992)

To deal with estimation error in the classical mean-variance model, especially due to an ill-conditioned covariance matrix, the following techniques are popular:

  • Nested Clustering Optimization (NCO) (de Prado 2020)
  • Covariance shrinkage (Ledoit-Wolf, 2003, 2004 and Jagannathan-Ma, 2003)
  • Denoised or detoned covariance by random matrix theory (de Prado 2020)
  • Robust portfolio optimization (Goldfarb and Iyengar, 2003)
  • Portfolio regularization (de Miguel et al, 2009 and Brodie et al, 2009)
  • Mean-VaR and mean-CVaR models that replace variance with Value-at-risk
  • Resampled efficient frontier (Michaud and Michaud, 1998)

Although all of the above tend to focus on the optimization approach, there are alternative approaches to obtain equivalent solutions to the optimization approach:

  • the optimization approach
  • analytical closed-form solutions (Merton, 1972)
  • the regression approach (Britten-Jones, 1999 and Kempf and Memmel, 2006)
  • system of linear equations

Finally, heuristic portfolios, shown below, are commonly used as benchmarks and don't require optimization:

  • the equally-weighted portfolio
  • market capitalization weighted portfolio
  • the inverse-volatility weighted portfolio
  • $\begingroup$ Thank you for your answer. In a academic paper, would it be appropriate to refer to "minimum variance approach" as being included in "mean-variance approach"? $\endgroup$
    – Aqqqq
    Nov 24, 2019 at 15:07
  • 1
    $\begingroup$ An efficient portfolio is one that has the lowest portfolio risk for a given level of expected return. In the mean-variance model, all portfolios, including the GMV, tangency and other portfolios along the efficient frontier have the lowest variance for their respective levels of expected return. Therefore, mean-variance speaks to the overall trade-off the model tries to affect, by defining portfolios by their mean and variance, while minimum variance is an optimal criterion of this trade-off. no reason to separate these concepts into approaches because they are both inherent $\endgroup$
    – develarist
    Nov 24, 2019 at 15:27
  • $\begingroup$ @develarist This is very helpful, thank you! $\endgroup$
    – Guifan Li
    Nov 20, 2020 at 17:17

In its strict form, "mean-variance" is a sub-component, albeit the default normative one, of the "standard Markowitz approach". Just like eg there are lots of regression techniques out there; but OLS is by far the default one.

Markowitz simply put on the table a method of working out expected portfolio risk from expected asset volatility and expected correlations. Combining these with expected returns produces a set of portfolios that maximise expected return and minimise expected vol for any given level of the other. IE your "efficient frontier". By definition, minimum vol/variance and max-return will always be at the bounds of this set. But for instance, "risk parity" may well be "inefficient".

Strictly speaking all of these portfolios, efficient or not, will be "mean-variance". The returns are a simple weighted average; the variance is a weighted function of the covariance matrix. Efficient or not, it's been measured thus.

But when people talk about "mean-variance", they are usually defaulting to Bill Sharpe's contributions to the Markowitz framework. That is, the hunt for the "tangency portfolio" point within the efficient frontier. That is, the maximum Sharpe Ratio point. Blending this portfolio with cash or levering it up, will produce a better outcome than riding up or down the curve to a different point on the frontier.

This kind of "mean-variance" is essentially trying to find a particular sweet spot in the "standard-Markowitz" set of portfolios, that are also "mean-variance" in how they are measured. The inevitable confusion arises because people often don't specify, or even feel they need to specify (because the shorthand is so readily used by so many other people).


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