Allocation at Kinks, Atoms, and Non-Smooth Risk Measures

Why Allocation at a Kink Needs Special Care

Marginal capital contributions are often introduced as derivatives: increase one business unit’s exposure slightly and measure the resulting change in aggregate capital. That interpretation is straightforward when the capital function is differentiable. It needs more care when the portfolio lies at a kink, or when the loss distribution has an atom at the VaR threshold. In those cases, the change can depend on which direction the portfolio moves, and a single gradient may not exist.

This matters in practice because an allocation may be used for pricing, performance assessment, or risk limits. A precise calculation should identify the risk measure, the direction of change, and any convention used to resolve ties. In particular, VaR is a quantile, not generally a maximum of scenario losses; the active-scenario subgradient rule for a maximum of affine functions cannot automatically be applied to VaR.

VaR with Discrete Scenario Losses

Suppose there are finitely many scenarios \(s\), with fixed probabilities \(p_s\), and scenario loss \(L_s(w)\). The loss distribution at weights \(w\) is discrete. At confidence level \(\alpha\), define \(\operatorname{VaR}_{\alpha}(w)=\inf\{x:P(L(w)\le x)\ge\alpha\}\). This is the lower quantile convention. At an atom, several scenarios can have the same loss as VaR, and their combined probability may straddle the quantile level.

For a useful directional calculation, assume scenario losses are affine in the weights near the portfolio: \(L_s(w+th)=L_s(w)+t a_s\), where \(a_s\) is the scenario’s loss change per unit \(t\) in direction \(h\). Let \(q\) be the VaR, \(p_<\) the probability of losses strictly below \(q\), and \(p_=\) the probability of losses equal to \(q\). If \(p_< < \alpha \le p_<+p_=\), the quantile lies within the mass at \(q\). For small positive \(t\), scenarios initially at \(q\) separate according to their slopes \(a_s\). The right directional derivative is the weighted quantile of those slopes at conditional rank \(r=(\alpha-p_<)/p_=\). This result assumes fixed scenario probabilities and that scenarios not tied at \(q\) remain separated from \(q\) for sufficiently small \(t\). It does not say that VaR is globally convex or that its derivative is a subgradient.

The scenario selected at the quantile can change with the direction. Consequently, “the VaR scenario” need not specify a unique marginal contribution for every business unit. If a scenario-based allocation is required, one transparent convention is to choose a quantile scenario and allocate its loss across units using that scenario’s component losses. At a tie, the choice or weighting of scenarios must be stated. In this example, scenario B’s losses at the base weights are 60 and 40, which sum to its VaR loss of 100; scenario C’s losses are 20 and 80, which also sum to 100. Both reconcile to VaR, but they give different unit allocations. Neither allocation is implied uniquely by the VaR value alone.

Subgradients for Convex Kinks

A different situation arises when the aggregate capital function itself is a convex, piecewise-linear function on the full weight space. For example, a maximum of finitely many affine functions is convex. At a point where several affine functions attain the maximum, there may be no unique gradient, but the convex subdifferential describes the supporting slopes.

The full-space condition matters. If weights are restricted to a feasible convex domain \(C\), the subgradient inequality is required only for \(y\in C\). At a boundary point, this restricted subdifferential can be larger than the convex hull of active gradients. Under the usual convex-analysis convention it includes the domain’s normal cone, defined here as \(N_C(w)=\{n:n\mathbin{\cdot}(y-w)\le 0\text{ for all }y\in C\}\). For instance, on \(C=[0,\infty)\), \(K(x)=x\), and \(w=0\), the inequality \(y\ge gy\) for every \(y\ge0\) holds for all \(g\le1\); thus the restricted subdifferential is \((-\infty,1]\), not just \(\{1\}\).

The active-gradient result follows from the supporting-slope condition. Each active affine function \(l_j\) satisfies \(l_j(w)=K(w)\), and \(K(y)\ge l_j(y)=K(w)+\nabla l_j\mathbin{\cdot}(y-w)\). Thus each active gradient is a subgradient, as is any convex combination of them. Conversely, for a finite maximum of affine functions on the full space, a vector outside their active-gradient convex hull cannot support the function at \(w\): a separating direction would make its proposed linear support exceed the maximum’s directional slope. This gives precisely the convex hull. The conclusion depends on the full-space setup; a restricted feasible domain changes the supporting inequalities as described above.

For a positively homogeneous convex capital function on the full space, a subgradient \(g\) at \(w\) satisfies \(g\mathbin{\cdot}w=K(w)\). Indeed, applying the subgradient inequality at \(y=0\) gives \(g\mathbin{\cdot}w\ge K(w)\), while applying it at \(y=2w\) gives \(g\mathbin{\cdot}w\le K(w)\). This explains why a subgradient can produce a full allocation in this setting. It does not establish that a subgradient exists for a nonconvex measure such as VaR, nor that an arbitrary finite-difference estimate must fully allocate capital.

Common Mistakes and Practical Interpretation

• Using one tied scenario for every direction. As the first example shows, increasing different weights can select different tied scenarios and produce different VaR derivatives.
• Calling a directional derivative a subgradient. A directional derivative is a slope along a specified direction. A convex subgradient must satisfy a supporting inequality for all points in its domain. VaR need not be convex.
• Applying the active-gradient formula on a restricted domain without adjustment. The convex-hull formula stated here is for the full ambient space; boundary constraints can add normal-cone directions.
• Assuming every marginal contribution is unique or that the components automatically sum to aggregate capital. At a kink, one may select a subgradient or a scenario convention, but the choice and its reconciliation should be explicit.

A sound allocation analysis therefore reports the capital measure and confidence level, identifies any atom or active pieces, specifies the direction or allocation convention, and checks the resulting contributions against aggregate capital when the chosen method supports full allocation. These distinctions are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.