math.md

Monitoring & Triggering Rebalancing — Core Metrics

1. Time-Weighted Health Factor and Yield

$$ \boxed{ \overline{HF}_t ;=; \frac{\displaystyle\sum_{k=0}^{B-1} w_k ; \dfrac{C_{t-k}}{L_{t-k}}} {\displaystyle\sum_{k=0}^{B-1} w_k} } \qquad \bigl(w_k = \lambda^{k},; \lambda!\in!(0,1],; B = \text{window size}\bigr) $$

$$ \boxed{ \overline{Y}_t ;=; \frac{\displaystyle\sum_{k=0}^{B-1} w_k ; Y_{t-k}} {\displaystyle\sum_{k=0}^{B-1} w_k} } $$

with
$C_{t-k}$ = collateral value,
$L_{t-k}$ = liability value,
$Y_{t-k} = r^{\text{sup}}{t-k} - r^{\text{brw}}{t-k}$ = net APY differential.

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2. Normalisation of Time-Weighted Metrics

$$ \hat{HF}_t ;=; \operatorname{clip} !\Bigl( \frac{\overline{HF}_t - HF_{\min}} {HF_{\max} - HF_{\min}}, ,0,1 \Bigr), \qquad \hat{Y}_t ;=; \operatorname{clip} !\Bigl( \frac{\overline{Y}_t - Y_{\min}} {Y_{\max} - Y_{\min}}, ,0,1 \Bigr) $$

where $\operatorname{clip}(x,0,1)=\max\bigl(0,\min(x,1)\bigr)$.

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3. Composite Rebalancing Score

$$ \boxed{ S_t ;=; \alpha,\hat{HF}_t ;+; (1-\alpha),\hat{Y}_t } \qquad \bigl(\alpha = 0.6,; 1-\alpha = 0.4\bigr) $$

Rebalancing is triggered whenever $S_t < S_{\text{th}}$; the keeper targets $S_t \ge S_{\text{des}}$ after intervention.

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Rebalancing Parameter Calculation

1. Required Debt Reduction

$$ \boxed{ \Delta L = \frac{HF^* \cdot L/\text{LLTV} - C}{HF^*/\text{LLTV} - 1} } $$

where
$HF^*$ = target health factor after rebalancing,
$C$ = current collateral value,
$L$ = current liability value,
$\text{LLTV}$ = liquidation loan-to-value ratio.

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2. Balanced Equilibrium Reserves

Let $V_0$ be the total deposit value expressed in units of the collateral asset (asset0). The balanced equilibrium reserve is:

$$ \boxed{ \text{balEqRsv}_0 = \frac{V_0}{1 - \text{LLTV}} } $$

A symmetric expression holds for asset1: $\text{balEqRsv}_1 = \frac{V_1}{1 - \text{LLTV}}$

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2. Equilibrium Reserve Targets

$$ \boxed{ \begin{aligned} R^{\text{eq}}_{\text{coll}} &= \frac{V_0}{1 - \text{LLTV}} \\ R^{\text{eq}}_{\text{debt}} &= \kappa,\Delta L \end{aligned}} $$

Here $\kappa>1$ is a liquidity-buffer coefficient (empirically $\kappa!\approx!3$) chosen to seed sufficient baseline liquidity on the debt side while avoiding over-leverage.

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3. Reserve Differential

$$ \boxed{ \Delta R = \frac{\Delta L}{P_m} } $$

where $P_m$ is the market mid-price quoted as $\text{asset1}/\text{asset0}$. The debt asset is identified such that $\Delta R$ is measured in its units.

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4. Target Equilibrium Reserves for the New Curve

$$ \boxed{ \begin{aligned} R^{\text{eq}}_{\text{coll}} &= \frac{V_0}{1 - \text{LLTV}} \\ R^{\text{eq}}_{\text{debt}} &= \kappa,\Delta L \end{aligned}} $$

The notation is consistent with §2. These targets are fed into the pool constructor as equilibriumReserveCollateralAsset and equilibriumReserveDebtAsset.

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5. Centre Price Encoding

$$ \boxed{ \begin{aligned} p_x &= 10^{18} \\ p_y &= \frac{10^{18}}{P_m} \end{aligned} } \qquad \text{such that}\quad \frac{p_x}{p_y} = P_m $$

These integers encode the market price in the curve parameters.

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6. Initial Pool State

$$ \boxed{ \begin{aligned} currentReserve_{\text{0}} &= R^{\text{eq}}_{\text{coll}} - \Delta L \\ currentReserve_{\text{1}} &= f(currentReserve_{\text{0}}) \end{aligned} } $$

where $f(\cdot)$ is CurveLib.f()

The system reaches equilibrium after $\Delta R$ debt asset inflow: $$(x_{\text{init}} + \Delta R,, y_{\text{init}} - \Delta R) ;\longrightarrow; (R^{\text{eq}}{\text{coll}},, R^{\text{eq}}{\text{debt}})$$

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