Quadratic Solvency Feedback and Death-Spiral Risk in a Reserve System

Summary

The document defines a stochastic reserve system in which a fund backs liabilities and a solvency ratio scales an issuance adjustment. The adjustment grows with the square of the ratio up to a cap, while reserve inflows, outflows, and shocks drive changes in reserves and liabilities. A decay or breakage parameter reduces the liability impact of issuance. The author asks whether this feedback can keep solvency positive under high, Brownian-like volatility and whether issuance and decay can create a threshold associated with a death spiral.

No answer or analysis is included, so the document does not establish stability, boundedness, a ruin probability, or a critical parameter relationship. It supplies a model structure and identifies questions for stochastic stability and risk-of-ruin analysis. Any conclusions would depend on assumptions about the distributions and dependence of inflows and shocks, the time scale, parameter constraints, and behavior near zero solvency. The stated update rules alone do not provide evidence that quadratic feedback prevents failure.

Key ideas

• The solvency ratio compares reserves with liabilities scaled by a safety margin.
• Issuance is regulated by a capped quadratic function of the solvency ratio.
• Random inflows and outflows affect reserves, while issuance and decay affect liabilities.
• The author asks whether feedback prevents solvency from approaching zero under volatile shocks.
• No stability result or threshold condition is derived in the document.

Tags

# Stability and risk of ruin in a stochastic reserve system with quadratic feedback control


# Stability and risk of ruin in a stochastic reserve system with quadratic feedback control












I am modeling a closed-loop liquidity system where a reserve fund ($R$) covers a total liability ($L$). I am seeking to verify the asymptotic stability and the risk of ruin under high volatility.

The Model

Let $S_t$ be the Solvency Ratio at time $t$, defined as:

$$S_t = \frac{R_t}{L_t \cdot \gamma}$$

where $\gamma$ is a safety margin constant (e.g., $1.2$). The system issues new units based on a base rate $\alpha$, regulated by a Quadratic Feedback Loop: $$E_{adj} = \alpha \cdot \min(1, S_t^2)$$

State Updates

The state updates are stochastic, where $V_t$ (inflow) and $W_t$ (outflow/shocks) are independent random variables:

$R_{t+1} = R_t + V_t + E_{adj} - W_t$$L_{t+1} = L_t + E_{adj}(1 - \beta) - W_t$

where $\beta$ is a constant decay or breakage rate.

The Challenge

Does the quadratic damping factor $S^2$ effectively ensure that $S_t$ remains bounded away from zero ($S_t > 0$) for $t \to \infty$ under Brownian motion-like volatility? Is there a critical threshold for the ratio between emission ($\alpha$) and decay ($\beta$) that could lead to a "death spiral" despite the feedback loop?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.