Multiplicative Contractions, Additive Recoveries: Functional-Form Restrictions on Risk Exposure Dynamics

Abstract

We test a regime-conditional functional-form restriction on aggregate risk-exposure dynamics implied by VaR-constrained intermediary models: exposures contract multiplicatively when capital constraints bind and grow additively (level-independent) when slack. The contraction half follows from binding VaR constraints (Brunnermeier and Pedersen 2009; Adrian and Shin 2010; He and Krishnamurthy 2013). The additive-rebuild prediction is derived under constant-rate capital replenishment; we test the joint restriction on FINRA monthly margin debt (1997-2026). Two findings. First, regime-interacted regression of detrended margin growth on lagged level (T=350 months) yields calm slope -0.040 (p=0.082, additive) and stress slope -0.205 (p<0.001, multiplicative); Wald test on regime x level interaction rejects equal dependence (p=0.0016). Second, the restriction implies drawdown-recovery duration ratio increases with crash depth. On 73 S&P 500 episodes (1950-2026), Cox model gives depth coefficient -13.75 (p<10^{-7}): 75% lower recovery hazard per 10pp deeper drawdown. Continuous-depth regression yields beta=1.22 (p=0.047); beta=1.59 (p<0.001) excluding 1980-82 Volcker. Median duration ratio for crashes >30% is 3.1x; replicates across eight other equity indices. Calibrated Heston, Markov-switching, and block bootstrap nulls match price-level duration asymmetry but lack an exposure state variable, so cannot speak to the regime-conditional flip on direct exposures. We do not claim the exposure test identifies the intermediary mechanism: FINRA margin debt is a noisy proxy. We claim only that the regime-conditional functional form is a sharper target than return-level moments alone, and confirming it on margin debt is consistent with -- not proof of -- the constrained-intermediary mechanism. A companion test on CFTC weekly speculative positioning is left for future work (Sections 5.2 and F).