Diversification Ratio

Most diversified portfolio and CAPM-style relations

Let’s review the 2008 paper¹ where Y. Choueifaty and Y. Coignard introduce the notion of diversification ratio of a portfolio, defined by

\((1)\ \ \ \ \ \ D(p) = \frac{p^T\sigma}{\sqrt{p^T \Omega p}},\ \ \ \ \sum p_k = 1,\)

where p=(p_1,…,p_n) is the vector of portfolio p’s weights on assets x = (x_1,…,x_n), σ = (σ_1,…,σ_n) their vector of volatilities, and Ω their covariance matrix. D represents the ratio of the average of asset volatilities to the volatility of the average (i.e. of the portfolio).

This ratio can also be written in terms of the weights of the equivalent portfolio of normalized assets y=(y_1,…,y_n) (where y_k = x_k / σ_k)

\((2)\ \ \ \ \ \ D(\bar{p}) = \frac{\bar{p}^T\mathbb{1}}{\sqrt{\bar{p}^T\rho\bar{p}}} = \frac{1}{\sqrt{\bar{p}^T\rho\bar{p}}},\)

where ρ is the correlation matrix of x (and y). It is clear that, for long-only portfolios (p_k >= 0, k=1,…,n), we have D(p) >= 1, with equality if and only if p is concentrated on a single asset. One natural question is: what is the portfolio of maximal diversification? Clearly by (2), it is the min-variance portfolio, also given by

\((3)\ \ \ \ \ \ p^* = \lambda \Sigma^{-1}\rho^{-1}\mathbb{1},\ \ \lambda>0,\)

with λ chosen so that weights sum up to 1, and Σ is the diagonal matrix formed from vector σ.

Let’s now compute the correlation of any portfolio p with the maximum diversification portfolio:

\((4)\ \ \ \ \ \ \begin{equation*} \begin{split} \rho_{p,p^*} &= \frac{p^T\Sigma\rho\Sigma p^*}{\sigma_p \sigma_{p^*}} = \frac{p^T\Sigma\rho\Sigma\lambda \Sigma^{-1}\rho^{-1}\mathbb{1}}{\sigma_p \sigma_{p^*}}\\ \\ &= \frac{p^T\sigma}{\sigma_p}\cdot \frac{\lambda}{\sigma_{p^*}} = D(p) \frac{\lambda}{\sigma_{p^*}}\\ \\ &= \boxed{\frac{D(p)}{D(p^*)}} \end{split} \end{equation*}\)

where σ_p and σ_p* are the respective volatilities of portfolios p and p*; the last equality being obtained by setting p = p* (and thus ρ_p,p* =1).

With (4) in mind, we can now regress the returns r_p of any portfolio p on r_p* to get the CAPM-style formula

\((5)\ \ \ \ \ \ \boxed{r_p = \alpha_p + \frac{\sigma_p}{\sigma_{p^*}}\frac{D(p)}{D(p^*)} r_{p^*} + \varepsilon_p},\)

which gives us a new insight on the interpretation of the coefficients.

Quantitatively Yours,

1. “Toward Maximum Diversification”, Y. Choueifaty, Y. Coignard, J. Portfolio Management (Fall 2008)