Critical Dependence Stress for Portfolio Diversification

I. Introduction

How much of a portfolio's diversification benefit survives uncertainty about dependence? A risk desk may have agreed marginal models for its individual positions without an equally well-determined model for their joint losses with the background book [31]. We ask how a proposed reallocation performs when these marginal models are retained but the coupling changes. Specifically, we seek the smallest dependence perturbation that reverses its Expected Shortfall (ES) advantage when both allocations face the same stressed law.

Let $X_1,X_2$ be the unit losses of two positions of equal initial cost. A transfer from holdings $(1+\varepsilon,1-\varepsilon)$ to $(1,1)$, with an unchanged background loss $\beta X_3$, changes the portfolio loss from $A_\varepsilon=(1+\varepsilon)X_1+(1-\varepsilon)X_2+\beta X_3=B+\varepsilon H$ to $B$, where

At upper-tail probability $p\in(0,1)$, we compare them through $\Phi_\varepsilon(P)=ES_p^P(B)-ES_p^P(A_\varepsilon)$, optimizing the two ES thresholds separately under $P$ [32], so that $\Phi_\varepsilon(P)<0$ favors the transfer under that law. Although $\beta X_3$ cancels from the payoff difference $A_\varepsilon-B$, its dependence with the traded positions determines which states enter each portfolio's upper tail. Put differently, a trade assessed in isolation can rank differently once embedded in the whole book [8].

Writing $\Pi$ for the Fréchet class of joint laws with these marginals, we measure distance from the reference law $P_0$ by weighted $W_1$ with ground cost $c_\gamma(x,y)=|y-x|_\gamma=\sum_i\gamma_i|y_i-x_i|$, where each $\gamma_i>0$ specifies the budget penalty per unit coordinate loss displacement as losses are re-paired across joint states. At budget $\rho$, the stress problem is

Under an exchangeable reference law, the traded positions have equal marginal ES contributions, so the nominal gain is of second order: $m_\varepsilon=(J/p)\varepsilon^2+o(\varepsilon^2)$, with $J>0$. For example, when $\gamma_1\le\min\{\gamma_2,\gamma_3/\beta\}$, the local exchange in Proposition 2.2 attains an initial stress derivative $K_\varepsilon$ of order one, suggesting the reversal budget $r_{0,\varepsilon}=m_\varepsilon/K_\varepsilon=O(\varepsilon^2)$. The derivative at zero budget does not determine whether this efficiency persists until the nominal gain is exhausted. Since the predicted reversal already occurs at scale $\rho=s\varepsilon^2$, we need the risk profile over a nonvanishing interval of the rescaled budget $s$, rather than its tangent at zero alone.

Three effects contribute at the critical scale. Moving the $O(\varepsilon)$ mass in an $O(\varepsilon)$-layer around the reference tail boundary by distances of order $\varepsilon$ changes both transport cost and tail expectation by $O(\varepsilon^2)$, matching the nominal gain. This local movement need not preserve the marginals. For a fixed regular local plan, the marginal CDF discrepancies are $O(\varepsilon^2)$; by Vallender's formula [39], their $L^1$ norms give the one-dimensional restoration costs, even when compensation occurs far from the tail boundary. An $O(\varepsilon)$ shift of either ES threshold also contributes at order $\varepsilon^2$ through the curvature of its objective. We must therefore optimize the local displacement, marginal restoration, and both thresholds together.

When the reference law has a positive continuous density on a compact box $I^2\times I_3$ and is exchangeable in its first two coordinates, Theorem 3.1 gives a finite, continuous profile

with convergence locally uniform in $s\ge0$. The transport energies have a $\Gamma$-limit [11] that includes both the local displacement cost and the cost of restoring its marginal projections. In the recovery argument, coordinatewise quantile maps attain the latter cost for each fixed regular local plan. Although this correction moves losses throughout the box, its leading effects on the two ES values cancel outside the threshold layer. Approximation then extends the paired-risk convergence to kernels with finite displacement moment.

Let $\rho^-_\varepsilon$ and $\rho^+_\varepsilon$ be the infima of budgets permitting a tie and a strict reversal. The strict-growth theorem identifies their common leading coefficient as the unique zero $s^*$ of $F$:

Initial efficiency and its limit. The effective unit costs are $\gamma_1,\gamma_2$, and $c_3=\gamma_3/\beta$, with $c_{\min}=\min\{\gamma_1,\gamma_2,c_3\}$. If the first traded coordinate is cheapest, Proposition 3.2 constructs a positive interval $[0,s_{\rm lin}]$ on which $F(s)=-J/p+s/(p\gamma_1)$. To reach zero at this rate, however, efficient displacement must remain possible throughout the required interval. Theorem 4.2 characterizes attainment of the universal efficiency bound:

We call the nonnegative gap $s^*-Jc_{\min}$ the marginal-restoration premium (normalized by $Jc_{\min}$ when reported as a percentage). When background displacement is cheapest, exchangeability permits opposite shifts to balance on every fixed-background slice. When only traded directions are cheapest, a corner region requires a displacement moment of order $\delta^2$, but its coordinate projections admit only $O(\delta^4)$ of compensating moment near the diagonal (Figure 2). Under the attained-slope condition of Proposition 2.2, with $c_{\min}=\gamma_1<c_3$, this gives $\rho^\pm_\varepsilon/r_{0,\varepsilon}\to s^*/(\gamma_1J)>1$.

An analytical benchmark. Consider independent uniform losses on $[0,2]^3$, with $\beta=1$, $p=1/2$, and $\gamma=(1,1,2)$. Here $g=3/8$ and $J=5/32$, so Proposition 2.2 gives $m_\varepsilon=(5/16)\varepsilon^2+o(\varepsilon^2)$. By Corollary 5.2, the profile is exactly $F(s)=2s-5/16$ for $0\le s\le1/96$, yet its zero satisfies $13/64\le s^*\le29/96$. Even in this uniform model, the proved lower bound on reversal is already $30\%$ above the local-efficiency prediction $5/32$, despite an attainable initial slope.

Sensitivity at equal effective costs. Along the cost family $\gamma_1=\gamma_2=\gamma$, $c_3=\gamma(1+\eta)$ for $\eta\ge0$, the baseline $Jc_{\min}=\gamma J$ is constant, so $s^*$ and the marginal-restoration premium have the same right derivative $\gamma Ja_+$ at $\eta=0$. Here $a_+\in(0,1)$ is the least background share among efficient stresses, characterized in Theorem 4.3 by continuous maximum-flow/minimum-cut duality; Section 5 shows how this share and the budget sensitivity $\gamma Ja_+$ can move in opposite directions as the reference dependence changes.

Our question lies between extremal risk aggregation and local distributional sensitivity. Subadditivity bounds aggregate risk [1, 25], while rearrangements within a Fréchet class describe how diversification depends on the coupling [33, 15, 5, 16, 41]. Partial information leads to convex-order and mixability constraints [22], but can still permit comonotonic upper tails [12], and fixed-marginal MES bounds quantify uncertainty in individual tail contributions [9]. For allocation comparisons, the symmetry imposed on the candidate laws is decisive. Chen et al. [10] and Liu and Liu [23] use exchangeability and majorization to identify diversification protection. Liu and Liu also analyze when complete dependence ambiguity favors concentration and how Wasserstein uncertainty modifies that conclusion. In our model, stressed couplings need not remain exchangeable. Symmetry at the reference gives stationarity, which does not by itself protect the reallocation once symmetry is broken under stress. We use this setting to measure the distance to a specified decision reversal, in the spirit of reverse sensitivity analysis [28, 29].

Transport duality provides a natural way to preserve the marginal information [40, 34]. In Wasserstein robust optimization [7, 19], adding marginal constraints confines the perturbation to dependence changes [18, 13]. Such constraints encode calibrated risk-neutral laws in multiasset robust pricing [14]; Wasserstein and copula restrictions also arise in benchmark portfolio optimization [30]. Here the constraint acts on a common stressed loss law for two allocations. After rescaling, its full coordinate projections remain as a transport penalty, and the efficient allocation of displacement reduces to a capacity-constrained flow problem [20].

Local Wasserstein expansions relate the sensitivity of optimized decisions and AVaR criteria to changes in the reference law [2], with adapted transport incorporating the information available in multiperiod decisions [4]. When a first-order term vanishes, higher-order expansions become relevant. Fiechtner and Blanchet [17] derive a second-order regret expansion under a unique empirical-risk minimizer in this situation, and Sauldubois [35] develops higher-order Wasserstein and adapted-Wasserstein expansions, including martingale constraints. Dependence restrictions also affect local sensitivity, as in Lam [21], while Namkoong [26] evaluates the first-order response over restricted transport directions. In our problem the trade size and stress radius vanish together, so locating a decision reversal at scale $\rho=s\varepsilon^2$ requires the full rescaled profile, with global marginal compensation and both optimized ES thresholds retained over the budget interval needed to reach zero.

Section 2 establishes the nominal gain and the initial stress slope. Section 3 derives the variational limit at scale $\rho=s\varepsilon^2$ and identifies the unique zero of the limiting profile. Section 4 characterizes zero premium and represents the boundary derivative by continuous projected flow–cut duality. Section 5 develops analytical examples and their implications for reverse stress testing. Appendix A contains the analytic proofs, and Appendix B gives the potential estimates and section calculations for the examples.

Notation and conventions We write $T_\#\mu$ for a pushforward and measure displacements by $|d|_\gamma=\sum_i\gamma_i|d_i|$. The constant in $O_\alpha(\varepsilon^k)$ may depend on the fixed kernel and model, but not on $\varepsilon$. Scalar section integrands are extended by zero outside their domains, and projection identities are equalities of finite signed Borel measures. We use $\inf\varnothing=+\infty$; allocation ratios are assigned the value zero where both numerator and denominator vanish.

II. Model formulation and the local benchmark

We first formulate the comparison for arbitrary integrable marginal laws, which suffice for existence of a worst stressed model. The critical-scale analysis will use a positive continuous density and exchangeability of the two traded positions, introduced in Section 2.2.

A. Fixed marginals and a common stress law

Let $n\ge2$, let $\mu_1,\ldots,\mu_n$ have finite first moments, and fix $P_0$ in the Fréchet class $\Pi(\mu_1,\ldots,\mu_n)=\{P\in\mathcal P(\mathbb R^n):P_i=\mu_i,\ 1\le i\le n\}$.

For exposure vectors $b,h\in\mathbb R^n$, the losses before and after the proposed reallocation are $A_\varepsilon=B+\varepsilon H$ and $B$, with $B=b\cdot X$ and $H=h\cdot X$. We use the upper-tail mass $p\in(0,1)$ and the Expected Shortfall representation of Rockafellar and Uryasev [32],

We write $\mathscr E_L(P;t)=t+p^{-1}E_P(L-t)_+$ for the threshold objective. For an integrable loss, its minimum is attained on the nonempty compact interval characterized by

Without a continuity assumption on the stressed loss law, support gaps can leave a nondegenerate interval of minimizing thresholds; we retain the set-valued characterization (2.2) so that subsequent estimates apply to every minimizer, including for laws with atoms.

Under a common candidate law $P\in\Pi$, we compare $\Phi_\varepsilon(P)=ES_p^P(B)-ES_p^P(A_\varepsilon)$, minimizing each loss over its own threshold. Maximizing this paired difference tests whether a single dependence scenario can erase the nominal gain; separate maximizations of the two ES values may instead select different joint laws. We measure changes in dependence with positive weights $\gamma_i$ and ground cost

Within $\Pi=\Pi(\mu_1,\ldots,\mu_n)$, transport changes how component losses are paired across joint states. Holding the weights and all marginal laws fixed, we define the worst comparison at budget $\rho$ and the nominal gain by

Proposition 2.1 guarantees attainment in (2.3), allowing atoms, singular dependence, and unbounded marginal supports. Since a nondecreasing profile can vanish on an interval, we distinguish the infimal budgets for a tie and a strict reversal,

which have the same leading asymptotics in the exchangeable box model of Section 2.2 by Theorem 3.3.

Proposition 2.1 (Existence under integrable marginal laws). Let $n\ge2$, let $\mu_1,\ldots,\mu_n$ have finite first moments, and fix $P_0\in\Pi(\mu_1,\ldots,\mu_n)$, positive weights $\gamma_i$, exposure vectors $b,h\in\mathbb R^n$, and $p\in(0,1)$. Then $\Pi$ and every $\mathcal A_\rho$ are compact for $W_\gamma$, and the maximum in (2.3) is attained. For each fixed $\varepsilon$, the profile $D_\varepsilon$ is nondecreasing and Lipschitz on $[0,\infty)$.

Proof. The fixed-marginal class is tight and weakly closed. Moreover, along any weakly convergent sequence $P_k\to P$ in $\Pi$, the first moments satisfy $\int|x|_\gamma\,dP_k=\sum_i\gamma_i\int|x_i|\,d\mu_i=\int|x|_\gamma\,dP$, so the Wasserstein convergence criterion [40] implies $W_\gamma$-compactness of $\Pi$ and its closed subsets $\mathcal A_\rho$. For a linear loss $L=a\cdot X$, coupling the laws and minimizing the uniformly Lipschitz hinges over their thresholds gives

Applying this bound to $B=b\cdot X$ and $A_\varepsilon=(b+\varepsilon h)\cdot X$, we find that their ES difference $\Phi_\varepsilon$ is Lipschitz with constant $C_\varepsilon=p^{-1}(\|b\|_{\gamma,*}+\|b+\varepsilon h\|_{\gamma,*})$. Compactness of $\mathcal A_\rho$ then ensures attainment. To compare two budgets $\rho_2>\rho_1$, take a maximizer $P$ at $\rho_2$. The mixture $P_a=(1-a)P_0+aP$, $a=\rho_1/\rho_2$, satisfies $W_\gamma(P_a,P_0)\le a\rho_2$ and $W_\gamma(P_a,P)\le(1-a)\rho_2$, by mixing an optimal plan with the corresponding identity plans. Comparing $\Phi_\varepsilon(P_a)$ and $\Phi_\varepsilon(P)$ yields $0\le D_\varepsilon(\rho_2)-D_\varepsilon(\rho_1)\le C_\varepsilon(\rho_2-\rho_1)$. Q.E.D.

For a transfer $H=X_i-X_j$ between positions with the same marginal law, replacing $X_j$ by $X_i$ preserves every marginal and gives $A_\varepsilon=B$ at cost $\gamma_jE_{P_0}|X_i-X_j|$; reversing their roles yields

This guarantees a finite tie budget but does not identify its small-trade scale: under the exchangeable box assumptions introduced next, Section 3 shows that both critical budgets are of order $\varepsilon^2$.

B. The exchangeable box model

To isolate how dependence with the rest of a portfolio affects a small reallocation, we keep the two traded losses separate and collect all unchanged positions into their aggregate loss $\beta X_3$. We take the marginal law of $X_3$ as retained information and specify the transport metric directly on $(X_1,X_2,X_3)$. This aggregate model fixes the information used in the stress test; its marginal and displacement cost are not equivalent to separately fixing the marginals and transport costs of every underlying background position.

Standing assumptions (exchangeable box model) Let

We also write $I_1=I_2=I$ when treating coordinate projections together. The reference law has a continuous density $f>0$ on the closed box $S$, normalized to integrate to one and satisfying $f(x,y,z)=f(y,x,z)$. We take

with arbitrary $p\in(0,1)$ and $\gamma_i>0$, so that $b=(1,1,\beta)$ and $h=(1,-1,0)$ in the preceding notation. All model parameters remain fixed as $\varepsilon\downarrow0$. The first two unit positions are expressed at equal initial cost, allowing the trade to change their holdings from $(1+\varepsilon,1-\varepsilon)$ to $(1,1)$ without changing the initial outlay. The position $\beta X_3$ represents the background book, whose holding remains unchanged during the trade.

The stressed law may be any coupling in $\Pi((P_0)_1,(P_0)_2,(P_0)_3)$; exchangeability and absolute continuity are assumptions on the reference law only. Reference symmetry makes the equal allocation stationary under $P_0$, while equality of the first two marginals gives $E_P[A_\varepsilon]-E_P[B]=\varepsilon(E_P[X_1]-E_P[X_2])=0$ under every candidate. Before implementation costs, the ES comparison therefore distinguishes tail risks between portfolios with the same expected loss.

The compact box and positive continuous density let us study this dependence mechanism with uniform control of marginal restoration. Positivity on the closed box gives both the interior mass needed for local exchanges and lower bounds on the marginal densities, hence Lipschitz bounds for their inverse CDFs. These bounds enter the quantile recovery in Section 3.2 and Appendix A.5; Section 5.3 considers the additional difficulties posed by unbounded marginals.

Loss units and effective stress costs A shift $\Delta X_3$ changes the portfolio loss by $\beta\Delta X_3$ at cost $\gamma_3|\Delta X_3|=(\gamma_3/\beta)|\beta\Delta X_3|$. Accordingly, $c_1=\gamma_1$, $c_2=\gamma_2$, and $c_3=\gamma_3/\beta$ are the costs per unit change of $B$ along the three coordinate axes. The ratio $c_3/\gamma_1$ compares a background displacement with a first-position displacement having the same effect on portfolio loss.

Canonical box normalization Writing the original coordinates as $X=T_{\rm box}(u)=(\ell+\kappa u_1,\ell+\kappa u_2,\ell_3+\kappa_3u_3)$, with $\kappa=\Delta_I/2$ and $\kappa_3=\Delta_3/2$, gives a normalized model on $[0,2]^3$ with $\widehat\beta=\beta\kappa_3/\kappa$ and $\widehat\gamma=(\kappa\gamma_1,\kappa\gamma_2,\kappa_3\gamma_3)$. By cash invariance and positive homogeneity of ES, the original paired risk is $\kappa$ times its normalized counterpart, while corresponding transport plans have equal cost; hence the critical budgets agree and $p$ is unchanged. A common rescaling of all transport weights rescales those budgets.

C. Nominal curvature and the exact initial slope

Under the reference model of Section 2.2, let $v$ be the upper-tail $p$ threshold of $B$. The marginal expected shortfall (MES) of a unit position $X_i$, namely $E_{P_0}[X_i\mid B>v]$, is the ES derivative with respect to its holding, and multiplying by that holding gives its Euler contribution [36, 38]. Since the first two holdings in $B$ both equal one, the directional derivative along $H=X_1-X_2$ is their MES difference: with $T_H=E_{P_0}[H1_{\{B>v\}}]$, it equals $p^{-1}T_H=E_{P_0}[X_1-X_2\mid B>v]=0$ by exchangeability. To determine the gain from a nonzero trade, we must then differentiate the moving tail boundary, which leads to a conditional variance on the tail section.