Mathematical Foundations of FCM

This document explains the mathematical models and formulas that power Flow Credit Market. Understanding these fundamentals helps you reason about system behavior and make informed decisions.

Core Variables

Token-Level Variables

Variable	Symbol	Description
Price	$P_t$	Price of token $t$ in MOET terms
Collateral Factor	$CF_t$	Usable percentage of token $t$ value (0 < $CF_t$ ≤ 1)
Borrow Factor	$BF_t$	Multiplier for borrowed token $t$ (typically 1.0)
Amount	$A_t$	Quantity of token $t$

Position-Level Variables

Variable	Symbol	Description
Effective Collateral	$EC$	Total usable collateral value in MOET
Effective Debt	$ED$	Total debt value in MOET
Health Factor	$HF$	Ratio of collateral to debt
Target Health	$HF_target$	Desired health ratio (typically 1.3)
Min Health	$HF_min$	Minimum before rebalancing (typically 1.1)
Max Health	$HF_max$	Maximum before rebalancing (typically 1.5)

Interest Variables

Variable	Symbol	Description
Interest Index	$I_t(n)$	Interest index for token $t$ at time $n$
Scaled Balance	$B_scaled$	Balance divided by interest index
True Balance	$B_true$	Actual balance including accrued interest
Interest Rate	$r$	Annual interest rate

Fundamental Formulas

1. Effective Collateral

The effective collateral is the sum of all collateral assets multiplied by their prices and collateral factors:

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EC = ∑(A_t × P_t × CF_t) for all t in Collateral

Example:

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Collateral assets:

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- 1000 FLOW @ $1 each, CF = 0.8

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- 500 USDC @ $1 each, CF = 0.9

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EC = (1000 × 1 × 0.8) + (500 × 1 × 0.9)

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= 800 + 450

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= $1250 MOET

2. Effective Debt

The effective debt is the sum of all borrowed assets multiplied by their prices and borrow factors:

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ED = \sum_t \in Debt A_t × P_t × BF_t

Example:

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Debt:

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- 800 MOET @ $1 each, BF = 1.0

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ED = 800 × 1 × 1.0

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= $800 MOET

3. Health Factor

The health factor is the ratio of effective collateral to effective debt:

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HF = (EC / ED)

Critical thresholds:

• $HF < 1.0$: Position is liquidatable
• $HF = 1.0$: Exactly at liquidation threshold
• $HF > 1.0$: Position is solvent

Example:

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EC = $1250, ED = $800

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HF = 1250 / 800 = 1.5625

4. Maximum Borrowing Capacity

The maximum amount that can be borrowed to reach target health:

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MaxBorrow = (EC / HF_target)

Derivation:

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We want: HF = EC / ED = HF_target

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Therefore: ED = EC / HF_target

Example:

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EC = $1250

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HF_target = 1.3

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Max Borrow = 1250 / 1.3 = $961.54 MOET

Auto-Borrowing Mathematics

Initial Auto-Borrow Amount

When a user deposits collateral with pushToDrawDownSink=true, the system calculates the initial borrow amount:

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BorrowAmount = (EC / HF_target)

Step-by-step calculation:

1. Calculate effective collateral:

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EC = A_collateral × P_collateral × CF_collateral

2. Calculate target debt:

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ED_target = (EC / HF_target)

3. Borrow to reach target:

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Borrow = ED_target = (EC / HF_target)

Complete example:

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User deposits: 1000 FLOW

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FLOW price: $1.00

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Collateral factor: 0.8

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Target health: 1.3

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Step 1: EC = 1000 × 1.00 × 0.8 = $800

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Step 2: ED_target = 800 / 1.3 = $615.38

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Step 3: Borrow = $615.38 MOET

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Result:

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- Collateral: 1000 FLOW ($800 effective)

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- Debt: 615.38 MOET

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- Health: 800 / 615.38 = 1.30 ✓

Rebalancing Mathematics

Overcollateralized Rebalancing (HF > HF_max)

When health exceeds maximum, calculate additional borrowing capacity:

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AdditionalBorrow = (EC / HF_target) - ED_current

Proof:

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Want: HF_new = HF_target

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HF_new = EC / ED_new = HF_target

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ED_new = EC / HF_target

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Additional borrow = ED_new - ED_current

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= (EC / HF_target) - ED_current

Example:

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Current state:

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- EC = $800

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- ED = $400

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- HF = 800 / 400 = 2.0 (> HF_max of 1.5)

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Calculate additional borrow:

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ED_target = 800 / 1.3 = $615.38

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Additional = 615.38 - 400 = $215.38 MOET

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After borrowing $215.38:

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- EC = $800 (unchanged)

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- ED = $615.38

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- HF = 800 / 615.38 = 1.30 ✓

Undercollateralized Rebalancing (HF < HF_min)

When health falls below minimum, calculate required repayment:

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RequiredRepayment = ED_current - (EC / HF_target)

Proof:

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Want: HF_new = HF_target

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HF_new = EC / ED_new = HF_target

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ED_new = EC / HF_target

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Required repayment = ED_current - ED_new

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= ED_current - (EC / HF_target)

Example:

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Price drops! Collateral value decreases.

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New state:

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- EC = $640 (was $800, FLOW dropped 20%)

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- ED = $615.38 (unchanged)

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- HF = 640 / 615.38 = 1.04 (< HF_min of 1.1)

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Calculate required repayment:

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ED_target = 640 / 1.3 = $492.31

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Repayment = 615.38 - 492.31 = $123.07 MOET

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After repaying $123.07:

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- EC = $640 (unchanged)

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- ED = $492.31

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- HF = 640 / 492.31 = 1.30 ✓

Interest Mathematics

Scaled Balance System

FCM uses scaled balances to efficiently track interest:

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B_scaled = \frac{B_true}{I_t}

Where:

• $B_scaled$: Stored scaled balance
• $B_true$: Actual balance including interest
• $I_t$: Current interest index

Key insight: Scaled balance stays constant while interest index grows.

Interest Index Growth

The interest index grows continuously based on the interest rate:

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I_t(n+1) = I_t(n) × (1 + r × \Delta t)

Where:

• $r$: Annual interest rate (e.g., 0.10 for 10%)
• $\Delta t$: Time elapsed (in years)

For compound interest:

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I_t(n) = I_0 × e^{r × t}

Where $e$ is Euler's number (≈2.718).

True Balance Calculation

To get the current true balance from scaled balance:

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B_true(t) = B_scaled × I_t

Example:

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Initial deposit: 1000 MOET

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Initial index: I_0 = 1.0

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Scaled balance: B_scaled = 1000 / 1.0 = 1000

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After 1 year at 10% APY:

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Interest index: I_1 = 1.0 × e^(0.10 × 1) ≈ 1.105

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True balance: B_true = 1000 × 1.105 = 1105 MOET

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User's debt grew from 1000 to 1105 MOET (10.5% with compound interest)

Why Scaled Balances?

Without scaled balances:

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Every block (every ~2 seconds):

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- Update interest index

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- Iterate through ALL positions

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- Update each position's balance

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- Gas cost: O(n) where n = number of positions

With scaled balances:

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Every block:

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- Update interest index only

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- Gas cost: O(1)

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When position is touched:

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- Calculate true balance: scaled × index

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- Gas cost: O(1) per position

Result: Massive gas savings for the protocol!

Liquidation Mathematics

Liquidation Trigger

A position becomes liquidatable when:

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HF < 1.0

Equivalently:

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EC < ED

Liquidation Target

Liquidations aim to restore health to a target (typically 1.05):

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HF_liquidation = 1.05

Collateral Seized Calculation

Amount of collateral to seize:

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CollateralSeized = \frac{ED_repaid × (1 + bonus)}{P_collateral × CF_collateral}

Where:

• $ED_repaid$: Amount of debt repaid by liquidator
• $bonus$: Liquidation bonus (e.g., 0.05 for 5%)
• $P_collateral$: Price of collateral token
• $CF_collateral$: Collateral factor

Example:

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Liquidatable position:

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- Collateral: 1000 FLOW @ $0.60

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- Debt: 650 MOET @ $1.00

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- HF = (1000 × 0.60 × 0.8) / 650 = 0.738 < 1.0

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Liquidation:

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- Liquidator repays: 150 MOET

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- Liquidation bonus: 5%

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- Collateral seized: (150 × 1.05) / (0.60 × 0.8) = 328.125 FLOW

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After liquidation:

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- Collateral: 671.875 FLOW @ $0.60 = $403.125 effective

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- Debt: 500 MOET

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- HF = 403.125 / 500 = 0.806...

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(May need multiple liquidations or larger liquidation to reach target 1.05)

Required Debt Repayment for Target Health

To restore position to target health factor:

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ED_repay = ED_current - (EC / HF_liquidation)

Example:

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From above, to reach HF = 1.05:

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EC = 1000 × 0.60 × 0.8 = $480

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ED_current = $650

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ED_target = 480 / 1.05 = $457.14

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ED_repay = 650 - 457.14 = $192.86 MOET must be repaid

Price Impact Analysis

Health Factor Sensitivity to Price Changes

Given a percentage change in collateral price:

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HF_new = HF_old × \frac{P_new}{P_old}

Derivation:

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HF_old = EC_old / ED = (A × P_old × CF) / ED

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HF_new = EC_new / ED = (A × P_new × CF) / ED

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HF_new / HF_old = P_new / P_old

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Therefore: HF_new = HF_old × (P_new / P_old)

Example:

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Initial: HF = 1.5, Price = $1.00

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Price drops 20% to $0.80:

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HF_new = 1.5 × (0.80 / 1.00) = 1.5 × 0.80 = 1.20

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Price drops 30% to $0.70:

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HF_new = 1.5 × (0.70 / 1.00) = 1.5 × 0.70 = 1.05 (approaching danger!)

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Price drops 35% to $0.65:

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HF_new = 1.5 × (0.65 / 1.00) = 1.5 × 0.65 = 0.975 < 1.0 (liquidatable!)

Maximum Safe Price Drop

What's the maximum price drop before liquidation?

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MaxDropPercent = 1 - (1.0 / HF_current)

Derivation:

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Want: HF_new = 1.0 (liquidation threshold)

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HF_new = HF_old × (P_new / P_old) = 1.0

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P_new / P_old = 1.0 / HF_old

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P_new = P_old / HF_old

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Drop = P_old - P_new = P_old × (1 - 1/HF_old)

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Drop % = 1 - 1/HF_old

Examples:

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HF = 1.3: Max drop = 1 - 1/1.3 = 23.08%

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HF = 1.5: Max drop = 1 - 1/1.5 = 33.33%

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HF = 2.0: Max drop = 1 - 1/2.0 = 50.00%

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HF = 1.1: Max drop = 1 - 1/1.1 = 9.09% (very risky!)

Multi-Collateral Mathematics

Multiple Collateral Types

With multiple collateral types:

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EC = \sum_i=1^{n} A_i × P_i × CF_i

Where $i$ iterates over all collateral token types.

Effective Collateral with Price Correlation

When collateral types are correlated (e.g., FLOW and stFLOW):

Simplified (no correlation):

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Risk = \sum_i Risk_i

With correlation (advanced):

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Risk = \sqrt{\sum_i\sum_j w_i w_j \sigma_i \sigma_j \rho_ij}

Where:

• $w_i$: Weight of asset $i$
• $\sigma_i$: Volatility of asset $i$
• $\rho_ij$: Correlation between assets $i$ and $j$

Practical impact:

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Scenario 1: Uncorrelated collateral

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- 50% FLOW (volatile)

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- 50% USDC (stable)

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- Effective diversification

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Scenario 2: Correlated collateral

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- 50% FLOW (volatile)

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- 50% stFLOW (volatile, correlated with FLOW)

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- Limited diversification

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- Both can drop together!

Yield Calculations

Simple APY

Annual Percentage Yield without compounding:

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APY_simple = (FinalValue - InitialValue / InitialValue) × (365 / Days)

Compound APY

With continuous compounding:

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APY_compound = e^r - 1

Where $r$ is the continuous annual rate.

Leveraged Yield

When borrowing to increase yield exposure:

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Yield_leveraged = Yield_strategy - Interest_borrowed

Example:

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Deposit: $1000 collateral

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Borrow: $615 at 5% APY

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Deploy $615 to strategy earning 10% APY

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Costs:

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- Interest on borrowed: 615 × 0.05 = $30.75/year

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Returns:

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- Yield from strategy: 615 × 0.10 = $61.50/year

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Net leveraged yield: 61.50 - 30.75 = $30.75/year

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Effective APY on your $1000: 30.75 / 1000 = 3.075% extra

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Total return: Base yield + leveraged yield

Risk Metrics

Liquidation Risk Score

A simplified risk score:

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\text{Risk Score} = (1 / HF - 1.0) × Volatility_collateral

Higher score = higher risk.

Value at Risk (VaR)

Maximum expected loss over time period at confidence level:

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VaR_95% = EC × \sigma × z_0.95

Where:

• $\sigma$: Daily volatility of collateral
• $z_0.95$: Z-score for 95% confidence (≈1.645)

Example:

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Collateral: $1000 FLOW

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Daily volatility: 5%

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Confidence: 95%

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VaR = 1000 × 0.05 × 1.645 = $82.25

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Interpretation: 95% confident that daily loss won't exceed $82.25

Validation & Safety Checks

Health Factor Bounds

All operations must satisfy:

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1.0 ≤ HF_min < HF_target < HF_max

Typical values: $HF_min = 1.1$, $HF_target = 1.3$, $HF_max = 1.5$

Collateral Factor Bounds

For safety:

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0 < CF_t ≤ 1.0

Typically:

• Volatile assets (FLOW): $CF = 0.75 - 0.85$
• Stable assets (USDC): $CF = 0.90 - 0.95$
• Liquid staking (stFLOW): $CF = 0.80 - 0.85$

Maximum Leverage

Maximum theoretical leverage:

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MaxLeverage = (1 / 1 - CF)

Examples:

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CF = 0.8: Max leverage = 1 / (1 - 0.8) = 5x

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CF = 0.75: Max leverage = 1 / (1 - 0.75) = 4x

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CF = 0.9: Max leverage = 1 / (1 - 0.9) = 10x (risky!)

But actual safe leverage is constrained by target health:

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SafeLeverage = (CF / HF_target)

Examples:

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CF = 0.8, HF = 1.3: Safe leverage = 0.8 / 1.3 ≈ 0.615 = ~1.62x

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CF = 0.75, HF = 1.5: Safe leverage = 0.75 / 1.5 = 0.50 = 1.5x

Practical Examples

Complete Position Lifecycle Math

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=== Initial Deposit ===

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Deposit: 1000 FLOW @ $1.00

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CF = 0.8, HF_target = 1.3

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EC = 1000 × 1.00 × 0.8 = $800

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Borrow = 800 / 1.3 = $615.38 MOET

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HF = 800 / 615.38 = 1.30 ✓

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=== Price Drop 20% ===

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New price: $0.80

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EC = 1000 × 0.80 × 0.8 = $640

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ED = $615.38 (unchanged)

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HF = 640 / 615.38 = 1.04 < 1.1 ⚠️

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Rebalance needed:

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ED_target = 640 / 1.3 = $492.31

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Repay = 615.38 - 492.31 = $123.07

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After repayment:

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EC = $640, ED = $492.31

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HF = 640 / 492.31 = 1.30 ✓

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=== Price Recovery to $1.00 ===

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EC = 1000 × 1.00 × 0.8 = $800

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ED = $492.31

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HF = 800 / 492.31 = 1.625 > 1.5 ⚠️

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Rebalance needed:

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ED_target = 800 / 1.3 = $615.38

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Borrow = 615.38 - 492.31 = $123.07

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After borrowing:

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EC = $800, ED = $615.38

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HF = 800 / 615.38 = 1.30 ✓

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Position back to optimal state!

Summary of Key Formulas

Formula	Expression	Use
Effective Collateral	$EC = \sum A_t × P_t × CF_t$	Calculate total collateral value
Health Factor	$HF = EC / ED$	Monitor position safety
Max Borrow	$Max = EC / HF_target$	Auto-borrowing amount
Rebalance Up	$Repay = ED - (EC / HF_target)$	Required debt reduction
Rebalance Down	$Borrow = (EC / HF_target) - ED$	Additional borrowing capacity
Scaled Balance	$B_scaled = B_true / I_t$	Interest-efficient tracking
True Balance	$B_true = B_scaled × I_t$	Current balance with interest
Max Price Drop	$DropPercent = 1 - (1 / HF)$	Liquidation safety margin

Next Steps

• Apply these formulas: ALP Documentation
• Understand architecture: FCM Architecture
• Learn the basics: Understanding FCM Basics

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tip

These mathematical foundations ensure FCM operates predictably and safely. All formulas are implemented on-chain and can be verified by examining the smart contracts.