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  1. Equally-weighted portfolio: weights each asset the same $w_i = 1/N$
  2. Maximum diversification portfolio: maximizes the ratio, $\frac{w' \sigma}{\sqrt{w' \Sigma w}}$
  3. Maximum decorrelation portfolio: minimizes portfolio correlation, $w' C\hspace{1mm} w$, where $C$ is the correlation matrix

All of the above supposedly diversify optimally. But their solutions, $w$, are never equal. You would think that the maximum diversification portfolio must be the most diversified by name, but it actually has a lower portfolio correlation than portfolio #3. and we know that the more negative-leaning portfolio correlation is, the stronger is "diversification".

So which of the three rules is the strongest claim to "diversification"? how to explain the type of diversification they comparatively enact to someone who has been trained to believe that there can only be one definition of diversification. i.e. How can we unify/relate the above clashing rules in terms of one overarching concept of diversification?

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First of all, I am not sure what you mean by the ratio in your second point. However, I will try to give you a partial answer at least.

There is a very comprehensive overview of these by EDHEC, page 4. What is particularly interesting is that they give you conditions under which these diversification portfolios are optimal in a classical/sharpe ratio sense.

That can be useful because they can serve as a common ground to unify all of these approaches. Please bear in mind that all of these only hold in a completely unconstrained case but that should be enough to get a good intuition.

For your convenience, I will reproduce some of them here as they relate to the ones you posted above:

  • Maximum Deconcentration: Your equally-weighted portfolio. $w_i = 1/N$. Is optimal if all assets have the same expected return, the same risk and pairwise correlations are the same. In addition to that, in the fixed income case, if all your bonds have the same default risk and you expect to recover the same amount from each, then this is the most diversified portfolio in terms of default risk!

  • Maximum Decorrelation: $w = \frac{C^{-1}1}{1^{'}C^{-1}1}$. Its optimal if your assets have same expected return and volatilities, but you have to estimate correlations somehow.

  • Efficient Minimum Volatility: We get the minimum vol portfolio by calculating $w = \frac{\Sigma^{-1}1}{1^{'}\Sigma^{-1}1}$ For this to be optimal, we need to assume the expected returns to be equal, but we have to estimate volatility and correlations.

  • Efficient Maximum Sharpe Ratio: Optimum Portfolio is the sharpe ratio portfolio - the only difference is the expected return $\mu$: $w=\frac{\Sigma^{-1}\mu}{1^{'}\Sigma^{-1}\mu}$. We have to estimate everything. Expected returns, volatilities, correlations.

So how can we uniform all these approaches?

Instead of always focusing on one singular risk measure as your diversification method and then optimize (by definition, this ptf will be the best/optimal in that regard) and then comparing these risk measures by taste/preference in different situations, we could treat them all as variants of the same problem - the problem to find an optimal portfolio. On our way to this portfolio, we will have to estimate some parameters. Some might be hard to estimate, potentially giving rise to other problems in the process. So here's how we could make these approaches comparable:

  • If I am confident I can estimate all parameters (expected return, volatilities and correlations) correctly, lets go for the maximum sharpe ratio portfolio. It diversifies in terms of risk diversification but looks at the risk return tradeoff as well.
  • If I fear I could get my expected return wrong or am agnostic to expected return, I can just implicitly assume them to be equal. In this case, lets get the risk down as low as possible. That would be the Efficient Minimum Volatility case.
  • In addition to that, if I fear I cannot estimate volatility correctly, lets assume them to be equal. Then the only way to reduce risk is to maximize the decorrelation.
  • If, in addition to that, I am also not comfortable with making correlation estimates, we can resort to the Maximum Deconcentration, aka naive diversification or equal weighted portfolio.

Honorable mentions go to the Diversified Risk Parity case not mentioned here (where we only know the assets' volatilities and assume returns to be equal and correlations to be constant to be an optimal portfolio. (see also the reference I posted))

All in all, we have moved from the comparison of correlation measures to a successively simplified portfolio optimization tasks. It is possible that these diversification portfolios emerged because of the popular notion that sometimes its better for investors to accept the fact you know nothing than to forcefully try to estimate something.

There are other diversification methods as risk parity, maximum entropy or diversification across different skewness/payoff profiles and some of them (especially the latter) will not fit into the concept outlined above in a straightforward way. However, I think that this change of viewpoint can be a start for the comparison of these notions of diversification.

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  • $\begingroup$ i agree about the ratio of the maximum diversification portfolio. Not sure what maximizing weighted asset volatilities in its numerator accomplishes in the way of diversification, but it is a well published strategy $\endgroup$ – develarist Sep 28 '20 at 14:33
  • $\begingroup$ Sorry, my bad. Knowing that the $\sigma$ is the vector of volatilities, it makes sense now. Weighted assed volatilities is the case of all correlations being equal to one. The denominator is then the volatility of the portfolio. Maximizing this ratio is maximizing the diversification benefit. $\endgroup$ – vanguard2k Sep 28 '20 at 14:40
  • $\begingroup$ by max entropy portfolio, do you mean Bera and Park 2008? $\endgroup$ – develarist Sep 28 '20 at 15:05
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    $\begingroup$ No, as in here, p. 10: papers.ssrn.com/sol3/papers.cfm?abstract_id=1358533 In all honesty, I dont know the Bera and Park paper, it might be the same. $\endgroup$ – vanguard2k Sep 28 '20 at 15:07
  • $\begingroup$ What would you say to someone who says that the equally-weighted portfolio is the most diversified you can get? $\endgroup$ – develarist Sep 28 '20 at 16:48

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