Formulas and Calculations

The Throttle Module relies on various formulas and calculations to achieve its functionality

Throttle function

$$t(r, s, m, \left(\frac{L}{D}, \frac{\Delta V_b}{\Delta V_s}\right)) \rightarrow (\Delta B, H) < (x, y)$$

Rule for comparing pairs is as following:

$\forall a, b, c, d \in \mathbb{Q},\ (a, b) < (c, d) \iff (a < b \text{ and } c < d)$

Where:

• $r$: Borrowing rate

• $s$: Throttle surcharge

• $m$: Multiplier on borrowing amount

• $\frac{L}{D}$: Potential liquidation amount at risk over market depth

• $\frac{ΔV_b}{ΔV_s}$: Rate of increase in borrowing velocity versus supply velocity

• $(ΔB,H)$: Output tuple representing delta in borrowing/supply and market health factor

• $(x,y)$: Threshold values for output control

• $\mathbb{Q}$ is set of all rational numbers

This means that:

1. Throttle Function is calculated

2. Output values are compared with Threshold Values

3. Further actions are based on the result of comparison

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Borrowing and supply velocities

$$\text{Borrowing velocity, } V_b = \frac{\text{Net increase in borrowing (USD)}}{\text{Wallets transacted per epoch}}$$

$$\text{Supply velocity, } V_s = \frac{\text{Net increase in supply (USD)}}{\text{Wallets transacted per epoch}}$$

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Rate of change of these velocities

$$\Delta B = \text{Rate of change of } V_b, \quad \Delta S = \text{Rate of change of } V_s$$

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Borrow Rate Calculations

$$\text{If } \frac{\Delta B}{\Delta S} \geq \text{Critical level} \Rightarrow r_{\text{new}} = m \cdot r + s$$

Where $r_{new}$ is a new borrow rate after applying multiplier and surcharge

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Total pool at risk ratio

If sudden 50% downside risk, then:

$$if\ \frac{P_{\text{at risk}}}{P_{\text{total}}} \lt x \Rightarrow \ no \ action \ needed \newline if \ x\leq \frac{P_{\text{at risk}}}{P_{\text{total}}} \lt y \Rightarrow adjust\ borrow/supply\ rates\ gradually+emit\ warnings \newline if\ \frac{P_{\text{at risk}}}{P_{\text{total}}} \geq y \Rightarrow adjust\ rates\ immediately\ and\ liquidate$$

Where:

$x$ - primary threshold

$y$ - secondary threshold

$P_{at-risk}$ - pool value at risk

$P_{total}$ - total value of pool

$x < y$

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Rate Adjustment Formula

$$Adjusted\ borrow\ rate = Base\ Borrow\ Rate×(1+Multiplier×(\frac{ΔS}{ΔB}−CR))\\ + \ Surcharge$$