Generalized Effective Number of Bets

This section extends the ENB framework from Effective Number of Bets and Minimum Torsion to general convex risk measures by diagonalizing the Hessian of the risk function.

Homogeneous Functions and Risk Decomposition

Definition. A function \(r : \mathbb{R}^n \to \mathbb{R}\) is \(\tau\)-homogeneous if \(r(tw) = t^\tau r(w)\) for all \(w \in \mathbb{R}^n\) and \(t > 0\).

Proposition (Tasche [Tasche1999b]). Let \(r\) be a totally differentiable, \(\tau\)-homogeneous function with \(\tau \neq 0\). Then:

  1. \(\partial r / \partial w_k\) is \((\tau - 1)\)-homogeneous.

  2. Euler’s theorem: \(\tau \, r(w) = w^\top \nabla r(w) = \sum_{k=1}^n w_k \, \frac{\partial r}{\partial w_k}(w)\).

This decomposes the risk \(r(w)\) into marginal contributions \(\frac{w_k}{\tau} \frac{\partial r}{\partial w_k}(w)\), analogous to the variance case where \(\tau = 2\).

Local Diagonalization via the Hessian

For a general convex risk function, the marginal contributions \(\frac{\partial r}{\partial w_k}\) are not independent. To extract locally independent contributions, we use the Taylor expansion:

\[r(w + \Delta w) \approx r(w) + \Delta w^\top \nabla r(w) + \frac{1}{2} \Delta w^\top H_r(w) \, \Delta w,\]

where \(H_r(w)\) is the (positive semi-definite) Hessian matrix.

Let \(T(w)\) diagonalize the Hessian: \(T(w) \, H_r(w) \, T(w)^\top = D(w)\), and set \(v = (T(w)^\top)^{-1} w\). Using \((\tau - 1) \nabla r(w) = H_r(w) w\) (from differentiating Euler’s identity), we obtain:

\[(\tau - 1) \, T(w) \nabla r(w) = D(w) \, v.\]

For \(\tau > 1\):

\[r(w) = \frac{1}{\tau} w^\top \nabla r(w) = \frac{1}{\tau(\tau - 1)} v^\top D(w) \, v = \sum_{k=1}^n \frac{d_k(w) \, v_k^2}{\tau(\tau - 1)},\]

where \(d_k(w)\) are the diagonal entries of \(D(w)\). Since \(H_r(w)\) is positive semi-definite, we have \(d_k(w) \geq 0\), which ensures that the risk contributions are non-negative when \(\tau > 1\).

Local Independence

The Taylor expansion in the transformed coordinates is:

\[r(w + \Delta w) &\approx r(w) + \frac{1}{\tau - 1} \Delta v^\top D(w) \, v + \frac{1}{\tau} \Delta v^\top D(w) \, \Delta v,\]

where \(\Delta v = (T(w)^\top)^{-1} \Delta w\). Each component of \(\Delta v\) has an approximately independent contribution to the change \(r(w + \Delta w) - r(w)\), justifying the decomposition.

Generalized ENB

The generalized ENB is defined as:

\[\begin{split}p_k(w) &= \frac{d_k(w) \, v_k^2}{\tau(\tau - 1) \, r(w)}, \quad k = 1, \ldots, n, \\ N(w) &= \exp\!\left(-\sum_{k=1}^n p_k(w) \log p_k(w)\right).\end{split}\]

Unlike the variance case, the Hessian \(H_r(w)\) depends on \(w\), so the transformation \(T(w)\) must be recomputed for each portfolio. However, the structural results from Effective Number of Bets and Minimum Torsion still apply:

Let

\[C(w) = \operatorname{diag}(H_r(w))^{-1/2} \, H_r(w) \, \operatorname{diag}(H_r(w))^{-1/2}\]

be the “correlation” of the Hessian, with eigendecomposition \(C(w) = U(w) \, S(w) \, U(w)^\top\). Then \(T(w)\) has the representation:

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

where \(D\) is diagonal and \(V\) is orthogonal. The ENB is again independent of \(D\), and the constrained minimum torsion transformation is:

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

Application to Coherent Risk Measures

Given a coherent risk measure \(\rho\), define \(r_\rho(w) = \rho(w^\top X)\). By positive homogeneity and subadditivity, \(r_\rho\) is convex and 1-homogeneous (\(\tau = 1\)).

Since the generalized ENB requires \(\tau > 1\), we work with the squared risk \(r_\rho^2(w)\), which is 2-homogeneous. This is analogous to using variance (the square of standard deviation) in the original ENB framework.

The Hessian of \(r_\rho^2(w)\) can be computed from the CVaR gradient and Hessian formulas in CVaR Derivatives for Normal Mixture Distributions:

\[H_{r_\rho^2}(w) = 2 \nabla r_\rho(w) \, \nabla r_\rho(w)^\top + 2 \, r_\rho(w) \, H_{r_\rho}(w).\]

The CVaR-based ENB can reveal tail-risk concentrations that the variance-based ENB misses. For example, in a portfolio of independent assets where one has heavier tails, the variance-based ENB treats all assets equally (since the covariance is diagonal), while the CVaR-based ENB correctly assigns a larger risk contribution to the heavy-tailed asset.

References

[Tasche1999b]

Tasche, D. (1999). Risk contributions and performance measurement. Report of the Lehrstuhl für mathematische Statistik, TU München.