Effective Number of Bets and Minimum Torsion

This section reviews the effective number of bets (ENB) and the minimum torsion approach for measuring portfolio diversification, following [Meucci2010] and [Meucci2014].

Variance-Based Risk Decomposition

Let \(X \in \mathbb{R}^n\) be a random vector of asset returns with covariance \(\Sigma\), and let \(w \in \mathbb{R}^n\) with \(w^\top \mathbf{e} = 1\) be the portfolio weights. The portfolio variance is:

\[r_{\operatorname{Var}}(w) = w^\top \Sigma w.\]

The gradient is \(\nabla r_{\operatorname{Var}}(w) = 2 \Sigma w\), so the marginal contribution of asset \(k\) to variance is \(w_k (\Sigma w)_k\). These contributions are generally not independent.

Uncorrelated Factor Decomposition

To obtain independent contributions, we seek an invertible matrix \(T \in \mathbb{R}^{n \times n}\) such that \(T \Sigma T^\top = D\) is diagonal. Let \(Y = TX\) be the transformed returns and \(v = (T^\top)^{-1} w\) the adjusted weights. Then:

\[w^\top X = v^\top Y, \qquad w^\top \Sigma w = v^\top D v = \sum_{k=1}^n d_k v_k^2,\]

where \(d_k\) are the diagonal entries of \(D\). The risk contributions \(d_k v_k^2\) are now independent.

Effective Number of Bets

The normalized risk contributions form a discrete distribution:

(1)\[p_k = \frac{d_k v_k^2}{w^\top \Sigma w}, \quad k = 1, \ldots, n.\]

Since \(p_k \geq 0\) and \(\sum_k p_k = 1\), the ENB is defined as the exponential entropy:

\[N = \exp\!\left(-\sum_{k=1}^n p_k \log p_k\right).\]

The ENB ranges from 1 (risk concentrated in one factor) to \(n\) (equally distributed among all factors). Equivalently, \(-\log N\) is proportional to the KL divergence between \(\{p_k\}\) and the uniform distribution.

Characterizing Diagonalizations

Let \(C = U S U^\top\) be the eigendecomposition of the correlation matrix \(C = \operatorname{diag}(\Sigma)^{-1/2} \Sigma \, \operatorname{diag}(\Sigma)^{-1/2}\), where \(U\) is orthogonal and \(S\) is diagonal.

Proposition. Let \(\Sigma\) be positive definite and \(T\) be invertible with \(T \Sigma T^\top = D\) diagonal. Then there exists an orthogonal matrix \(V\) such that:

(2)\[T = D^{1/2} V S^{-1/2} U^\top \operatorname{diag}(\Sigma)^{-1/2}.\]

Lemma. The ENB \(N\) is independent of the choice of \(D\).

Proof. Let \(u = V S^{-1/2} U^\top \operatorname{diag}(\Sigma)^{1/2} w\) and \(v = D^{-1/2} u\). Then \(p_k = u_k^2 / (w^\top \Sigma w)\), which does not depend on \(D\). \(\square\)

Therefore, it suffices to choose only the orthogonal matrix \(V\). Since \(V\) is a rotation, it can map the vector \(S^{-1/2} U^\top \operatorname{diag}(\Sigma)^{1/2} w\) to any direction. In particular:

• Choosing \(V\) so that all \(v_k\) are equal gives \(N = n\) (maximal diversification).

• Choosing \(V\) to concentrate on one component gives \(N = 1\).

Thus, the diagonalization \(T\) must be chosen carefully.

Minimum Torsion

The minimum torsion approach [Meucci2014] selects \(T\) to minimize the change from the original weights. The rationale is that if \(w\) is close to equally weighted, then \(v = (T^\top)^{-1} w\) should also be close to equally weighted.

The degree of change is measured by the normalized tracking error:

\[\operatorname{NTE}(T) = \sqrt{\frac{1}{n} \operatorname{tr}\!\left(\operatorname{diag}(\Sigma)^{-1/2} (T - I) \Sigma (T - I)^\top \operatorname{diag}(\Sigma)^{-1/2}\right)}.\]

Using representation (2), the minimization problem becomes:

\[\min_{D, V} \; \operatorname{tr}\!\left(D - 2 D^{1/2} V S^{1/2} U^\top\right) \quad \text{s.t.} \quad D \text{ diagonal}, \; V \text{ orthogonal}.\]

If \(D\) is fixed to the identity matrix, the optimal solution is simply \(V^* = U\), giving the constrained minimum torsion transformation:

\[T_{MT} = U S^{-1/2} U^\top \operatorname{diag}(\Sigma)^{-1/2}.\]

For the general case (unconstrained \(D\)), an iterative algorithm that converges rapidly is described in [Meucci2014].

References

[Meucci2010]

Meucci, A. (2010). Managing diversification. Risk Magazine, 22(5), 74-79.

[Meucci2014] (1,2,3)

Meucci, A., Santangelo, A., & Deguest, R. (2014). Measuring portfolio diversification based on optimized uncorrelated factors. SSRN Electronic Journal.