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.

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

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	$TargetHF$	Desired health ratio (typically 1.3)
Min Health	$MinHF$	Minimum before rebalancing (typically 1.1)
Max Health	$MaxHF$	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	$ScaledBalance$	Balance divided by interest index
True Balance	$TrueBalance$	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), t ∈ 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

Borrow Factor (BF) is a risk adjustment multiplier applied to debt. While typically set to 1.0 for standard assets like MOET, the borrow factor can be increased above 1.0 for riskier or more volatile borrowed assets. This makes the effective debt higher than the nominal debt, requiring more collateral to maintain the same health factor. For example, a BF of 1.2 means borrowing $100 of that asset counts as $120 of effective debt, providing an extra safety margin for the protocol.

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 / TargetHF

Derivation:

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We want: HF = EC / ED = TargetHF

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Therefore: ED = EC / TargetHF

Example:

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EC = $1250

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TargetHF = 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 / TargetHF)

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 / TargetHF)

3. Borrow to reach target:

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Borrow = ED_target = (EC / TargetHF)

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 > MaxHF)

When health exceeds maximum, calculate additional borrowing capacity:

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AdditionalBorrow = (EC / TargetHF) - CurrentED

Proof:

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Want: HF_new = TargetHF

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HF_new = EC / ED_new = TargetHF

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ED_new = EC / TargetHF

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Additional borrow = ED_new - CurrentED

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= (EC / TargetHF) - CurrentED

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 (> MaxHF 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 < MinHF)

When health falls below minimum, calculate required repayment:

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RequiredRepayment = CurrentED - (EC / TargetHF)

Proof:

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Want: HF_new = TargetHF

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HF_new = EC / ED_new = TargetHF

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ED_new = EC / TargetHF

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Required repayment = CurrentED - ED_new

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= CurrentED - (EC / TargetHF)

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 (< MinHF 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

Instead of updating every position's balance when interest accrues, FCM stores a "scaled" version of each balance that remains constant over time. This scaled balance is the actual balance divided by a global interest index, allowing the protocol to track interest for thousands of positions with minimal gas costs.

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ScaledBalance = \frac{TrueBalance}{I_t}

Where:

• $ScaledBalance$: Stored scaled balance
• $TrueBalance$: 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 is a global multiplier that starts at 1.0 and grows over time based on the current interest rate. Every time the protocol updates, the index increases slightly, and this single update effectively compounds interest for all positions simultaneously.

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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

When you need to know the actual current balance of a position (including all accrued interest), you multiply the stored scaled balance by the current interest index. This calculation happens on-demand only when the position is accessed, not on every block.

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TrueBalance(t) = ScaledBalance × 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: ScaledBalance = 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: TrueBalance = 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?

The scaled balance system is a gas optimization that makes FCM economically viable even with thousands of active positions. By storing balances in a scaled form and only updating a single global index, the protocol avoids the prohibitive cost of updating every position on every block.

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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Target HF = 1.05

Collateral Seized Calculation

The amount of collateral to seize depends on the implementation approach used by the protocol.

Simplified Formula (used in basic examples and documentation):

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CollateralSeized = (DebtRepaid × (1 + bonus)) / PriceCollateral

Where:

• $DebtRepaid$: Amount of debt repaid by liquidator (in MOET or debt token)
• $bonus$: Liquidation bonus (e.g., 0.05 for 5%)
• $PriceCollateral$: Market price of collateral token in MOET terms

Complete Formula (actual FCM implementation):

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CollateralSeized = [(DebtRepaid × PriceDebt) / BorrowFactor] × (1 + bonus) / (PriceCollateral × CollateralFactor)

Where:

• $DebtRepaid$: Amount of debt repaid by liquidator
• $PriceDebt$: Oracle price of debt token (in MOET terms, typically 1.0 for MOET)
• $BorrowFactor$: Risk parameter for debt (typically 1.0)
• $bonus$: Liquidation bonus (e.g., 0.05 for 5%)
• $PriceCollateral$: Oracle price of collateral token (in MOET terms)
• $CollateralFactor$: Risk parameter for collateral (e.g., 0.8 for FLOW)

Why the difference? The simplified formula assumes debt price = 1.0, borrow factor = 1.0, and ignores the collateral factor in seizure (treating it as only affecting borrowing capacity). The complete formula properly accounts for all risk parameters and token prices as implemented in the actual protocol.

For MOET debt with typical parameters:

• $PriceDebt$ = 1.0 (MOET)
• $BorrowFactor$ = 1.0
• This simplifies the numerator to: $(DebtRepaid × 1.0)/ 1.0 = DebtRepaid$

The collateral factor appears in the denominator because it affects how much collateral value must be seized to repay the effective debt. Since effective collateral is calculated as $CollateralAmount × PriceCollateral × CollateralFactor$, seizing collateral to cover debt requires dividing by the CollateralFactor to get the actual token amount.

Example using simplified formula:

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Liquidatable position:

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- Collateral: 1000 FLOW @ $0.60 = $600 total value

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- Effective collateral: $600 × 0.8 CF = $480

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- Debt: 650 MOET @ $1.00

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- HF = 480 / 650 = 0.738 < 1.0

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Partial liquidation using simplified formula:

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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 = 262.5 FLOW

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- Value of seized collateral: 262.5 × $0.60 = $157.50

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After partial liquidation:

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- Remaining collateral: 1000 - 262.5 = 737.5 FLOW @ $0.60 = $442.50

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- Effective collateral: $442.50 × 0.8 = $354.00

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- Remaining debt: 650 - 150 = 500 MOET

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- New HF = 354.00 / 500 = 0.708 (still liquidatable)

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Liquidator's profit:

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- Paid: $150 (debt repayment)

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- Received: $157.50 worth of FLOW

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- Profit: $7.50 (5% bonus on $150)

Example using complete formula:

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Same liquidatable position as above.

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Partial liquidation using complete formula:

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- DebtRepaid: 150 MOET

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- PriceDebt: 1.0 (MOET)

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- BorrowFactor: 1.0

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- Liquidation bonus: 5% (0.05)

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- PriceCollateral: 0.60 (FLOW in MOET terms)

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- CollateralFactor: 0.8

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CollateralSeized = (150 × 1.0 / 1.0) × 1.05 / (0.60 × 0.8)

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= 157.50 / 0.48

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= 328.125 FLOW

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Value of seized collateral: 328.125 × $0.60 = $196.875

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After partial liquidation:

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- Remaining collateral: 1000 - 328.125 = 671.875 FLOW @ $0.60 = $403.125

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- Effective collateral: $403.125 × 0.8 = $322.50

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- Remaining debt: 650 - 150 = 500 MOET

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- New HF = 322.50 / 500 = 0.645 (still liquidatable, but lower than simplified)

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Liquidator's profit:

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- Paid: $150 (debt repayment)

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- Received: $196.875 worth of FLOW

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- Profit: $46.875 (31.25% effective bonus!)

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Note: The complete formula gives the liquidator significantly more collateral because

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it divides by the CollateralFactor. This compensates for the risk discount applied

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to the collateral. In practice, the actual FlowCreditMarket implementation uses the

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quoteLiquidation() function which calculates the exact amounts needed to reach the

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target health factor of 1.05.

Required Debt Repayment for Target Health

To restore a position to the target health factor (typically 1.05), we need to find how much debt to repay. This is complex because seizing collateral also reduces the effective collateral simultaneously.

Goal: Achieve a specific target health factor after liquidation:

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HF_target = EC_after / ED_after

The challenge: Both EC and ED change during liquidation:

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EC_after = EC_current - (Collateral Seized × Price × CF)

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ED_after = ED_current - Debt Repaid

Using the simplified seizure formula:

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Collateral Seized = (Debt Repaid × (1 + bonus)) / Price

The effective collateral value seized is:

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EC_seized = Collateral Seized × Price × CF

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= [(Debt Repaid × (1 + bonus)) / Price] × Price × CF

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= Debt Repaid × (1 + bonus) × CF

Substituting into the target health equation:

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HF_target = [EC_current - Debt Repaid × (1 + bonus) × CF] / [ED_current - Debt Repaid]

Solving for Debt Repaid:

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HF_target × (ED_current - Debt Repaid) = EC_current - Debt Repaid × (1 + bonus) × CF

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HF_target × ED_current - HF_target × Debt Repaid = EC_current - Debt Repaid × (1 + bonus) × CF

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HF_target × ED_current - EC_current = HF_target × Debt Repaid - Debt Repaid × (1 + bonus) × CF

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HF_target × ED_current - EC_current = Debt Repaid × [HF_target - (1 + bonus) × CF]

Final formula:

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Debt Repaid = (HF_target × ED_current - EC_current) / [HF_target - (1 + bonus) × CF]

Working example:

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Initial position (severely undercollateralized):

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- Collateral: 1000 FLOW @ $0.50 (price dropped significantly!)

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- EC = 1000 × 0.50 × 0.8 = $400

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- ED = 615.38 MOET

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- Current HF = 400 / 615.38 = 0.65 < 1.0 (liquidatable!)

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- Target HF = 1.05

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- Liquidation bonus = 5% (0.05)

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- Collateral Factor (CF) = 0.8

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Step 1: Calculate required debt repayment

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Debt Repaid = (1.05 × 615.38 - 400) / [1.05 - (1.05 × 0.8)]

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= (646.15 - 400) / [1.05 - 0.84]

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= 246.15 / 0.21

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= 1,172.14 MOET

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This is more than the total debt! This means the position cannot be restored to HF = 1.05

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because there isn't enough collateral. This would be a full liquidation case.

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Step 2: Calculate maximum achievable HF

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If all debt is repaid (615.38 MOET):

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Collateral seized = (615.38 × 1.05) / 0.50 = 1,292.30 FLOW

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But we only have 1000 FLOW, so this is a full liquidation.

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In full liquidation:

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- All 1000 FLOW seized → value = $500

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- Effective value for liquidator = $500

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- Debt repaid = 500 / 1.05 = $476.19 MOET (limited by collateral available)

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- Remaining debt = 615.38 - 476.19 = $139.19 (bad debt for protocol)

Better example with partial liquidation: k

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Initial position (moderately undercollateralized):

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- Collateral: 1000 FLOW @ $0.78

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- EC = 1000 × 0.78 × 0.8 = $624

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- ED = 650 MOET

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- Current HF = 624 / 650 = 0.96 < 1.0 (liquidatable)

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- Target HF = 1.05

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- Bonus = 5% (0.05)

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- CF = 0.8

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Step 1: Calculate debt repayment

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Debt Repaid = (1.05 × 650 - 624) / [1.05 - (1.05 × 0.8)]

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= (682.5 - 624) / [1.05 - 0.84]

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= 58.5 / 0.21

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= 278.57 MOET

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Step 2: Verify the calculation

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Collateral seized = (278.57 × 1.05) / 0.78 = 375.33 FLOW

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EC seized = 375.33 × 0.78 × 0.8 = $234.21

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EC after = 624 - 234.21 = $389.79

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ED after = 650 - 278.57 = $371.43

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HF after = 389.79 / 371.43 = 1.049 ≈ 1.05 ✓

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Step 3: Liquidator's outcome

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Collateral received: 375.33 FLOW @ $0.78 = $292.76

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Debt repaid: $278.57

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Profit: $292.76 - $278.57 = $14.19 (5.09% return)

Key insights:

1. The formula works when there's sufficient collateral to reach target HF
2. When Debt Repaid > ED_current, it indicates a full liquidation scenario
3. The denominator [HF_target - (1 + bonus) × CF] is typically small (0.21 in this example), meaning small changes in EC/ED require large debt repayments
4. The liquidation becomes more efficient (smaller debt repayment needed) when the current HF is closer to the target HF

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 = ∑(A_i × P_i × CF_i)

Where the sum is taken over all n 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) × \text{Volatility Collateral}

Higher score = higher risk.

Value at Risk (VaR)

Maximum expected loss over time period at confidence level 95%:

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VaR(95) = EC × σ × z(0.95)

Where:

• σ: 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 ≤ MinHF < TargetHF < MaxHF

Typical values: $MinHF = 1.1$, $TargetHF = 1.3$, $MaxHF = 1.5$

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 = \frac{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 borrowing is constrained by target health:

Safe Debt Ratio

Maximum debt-to-collateral ratio while maintaining target health:

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SafeDebtRatio = CF / TargetHF

Examples:

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CF = 0.8, TargetHF = 1.3: Safe debt ratio = 0.8 / 1.3 ≈ 0.615

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CF = 0.75, TargetHF = 1.5: Safe debt ratio = 0.75 / 1.5 = 0.50

This means with CF = 0.8 and TargetHF = 1.3, you can safely borrow up to $0.615 for every $1 of collateral value.

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, TargetHF = 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!

Next Steps

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