Math Reference

All Vanna contracts use WAD fixed-point arithmetic with 10¹⁸ as the representation of 1.0. This page documents every formula used in the protocol.

WAD Arithmetic

Constants

Constant	Value	Meaning
`WAD`	`10¹⁸`	Fixed-point 1.0
`SCALAR_12`	`10¹²`	Blend b_rate divisor
`SECS_PER_YEAR`	`31,556,952 × 10¹⁸`	Seconds in a year in WAD

Core Operations

mul_wad_down(a, b) = floor(a × b / 10¹⁸)
div_wad_down(a, b) = floor(a × 10¹⁸ / b)
rpow_wad(x, n)    = x^n  (via repeated squaring, preserving WAD scale)

All intermediate math uses floor division (mul_wad_down, div_wad_down). This means rounding errors accumulate against the protocol, not in the protocol’s favor.

Decimal Conversion

Every asset has native decimals (XLM: 7, USDC: 6). Balances are converted to WAD at protocol entry and back to native decimals at exit.

native → WAD:   wad_amount = native_amount × 10¹⁸ / 10^decimals
WAD → native:   native_amount = floor(wad_amount × 10^decimals / 10¹⁸)

Example (XLM, 7 decimals):

11 XLM → WAD:   11 × 10⁷ (stroops) × 10¹⁸ / 10⁷ = 11 × 10¹⁸
WAD → stroops:  floor(11 × 10¹⁸ × 10⁷ / 10¹⁸) = 11 × 10⁷ stroops

Interest Rate Model

Utilization Ratio

util = borrows_wad / (liquidity_wad + borrows_wad)

Where liquidity_wad is the pool’s current asset balance in WAD.

Borrow Rate Per Second

Vanna uses a smooth polynomial curve (not a kinked two-slope model):

rate_per_sec = c3 × (util×c1 + util^32×c1 + util^64×c2) / SECS_PER_YEAR

With:

Coefficient	Value	Decimal
`c1`	`10^17`	0.1
`c2`	`3 × 10^17`	0.3
`c3`	`35 × 10^17`	3.5

The util^32 and util^64 terms are computed via WAD exponentiation (rpow_wad). At low utilization the curve is nearly linear. At high utilization the polynomial terms dominate, pushing rates up sharply.

Annual Rate Derivation

To get the annualized rate for display:

borrow_rate_annual = rate_per_sec × SECS_PER_YEAR

Interest Accrual

Rate Factor

Computed at every borrow or repayment (update_state()):

elapsed = now - last_updated_timestamp
rate_factor = elapsed × rate_per_sec    (in WAD)

State Update

interest = floor(borrows_wad × rate_factor / WAD)
borrows_wad += interest
last_updated = now

If now == last_updated the function exits early — no double-accrual within the same ledger.

Live (Preview) Borrows

When computing health factors, the protocol uses live borrows (including interest that has accrued since the last state update but not yet written to storage):

rate_factor_live = (now - last_updated) × rate_per_sec
interest_live    = floor(borrows_wad × rate_factor_live / WAD)
borrows_live     = borrows_wad + interest_live

This ensures borrow shares issued right now correctly reflect the current debt level.

Borrows are tracked via shares, not raw amounts. This lets interest accrue globally without updating each user’s record.

Issuing Shares (on Borrow)

if total_borrow_shares == 0:
    new_shares = borrow_amount           # first borrower, 1:1
else:
    new_shares = floor(borrow_amount × total_borrow_shares / borrows_live)

Burning Shares (on Repay)

shares_to_burn = floor(repay_amount × total_borrow_shares / borrows_live)

Computing User Debt

user_debt = floor(user_borrow_shares × borrows_wad / total_borrow_shares)

Note: user debt is computed from borrows_wad (stored, slightly stale), while share issuance uses borrows_live (fresh). This is intentional — it prevents over-issuance of shares.

Origination Fee

On every borrow, a one-time fee is charged:

origination_fee = floor(borrow_amount × origination_fee_rate / WAD)

The fee is sent from the pool to the protocol treasury. The borrower receives borrow_amount - origination_fee.

vToken Exchange Rate

vTokens represent a proportional share of a pool’s total assets (pool_balance + borrows_wad).

Minting on Deposit

if total_assets == 0 or vtoken_supply == 0:
    vTokens = deposit_amount             # first depositor, 1:1
else:
    vTokens = floor(deposit_amount × vtoken_supply / total_assets)

Redeeming on Withdrawal

underlying = floor(vtoken_amount × total_assets / vtoken_supply)

Exchange Rate

The implicit exchange rate grows as interest accrues:

exchange_rate = total_assets / vtoken_supply

As borrows_wad grows with interest, total_assets grows, and each vToken is worth more underlying.

Health Factor

The health factor is a ratio of total collateral value to total debt value.

HF = (total_collateral_usd × WAD) / total_debt_usd

Healthy: HF > 1.1 × WAD (strictly greater than)
Liquidatable: HF ≤ 1.1 × WAD

Collateral Valuation

Different collateral types are valued differently: Raw tokens (XLM, USDC):

collateral_usd = balance_wad × price_wad / WAD

Blend b-tokens (BLEND_XLM, BLEND_USDC):

underlying    = floor(b_tokens × b_rate / 10¹²)
underlying_wad = underlying × WAD / 10^decimals
collateral_usd = underlying_wad × price_wad / WAD

Where b_rate is fetched from the Blend pool’s reserve data. LP tracking tokens (AQ_XLM_USDC, SS_XLM_USDC):

collateral_usd = 0   # LP positions are tracked but not counted as collateral

Borrow-Time Check (Gross-Asset Model)

When evaluating a new borrow, both the new borrow and existing borrows count as assets:

gross_assets = collateral_usd + existing_debt_usd + new_borrow_usd
new_total_debt = existing_debt_usd + new_borrow_usd

allowed = gross_assets × WAD / new_total_debt > BALANCE_TO_BORROW_THRESHOLD

This means same-asset leverage (borrow XLM, deploy to Blend XLM) correctly shows HF > 1 instead of near zero.

Withdraw-Time Check

new_collateral = collateral_usd - withdraw_value_usd

allowed = withdraw_value_usd ≤ collateral_usd
       AND new_collateral × WAD / existing_debt_usd > BALANCE_TO_BORROW_THRESHOLD

Liquidation Check

Uses pure collateral (not gross-asset model):

liquidatable = collateral_usd × WAD / existing_debt_usd ≤ BALANCE_TO_BORROW_THRESHOLD

Constants

Constant	Value	Description
`BALANCE_TO_BORROW_THRESHOLD`	`1.1 × 10¹⁸`	Minimum health factor (110%)
`dust_debt_threshold`	`10^16`	Debts below this are ignored in health checks

Oracle Price Conversion

Prices from the Reflector oracle come as (price: u128, decimals: u32). The RiskEngine converts to WAD:

price_wad = oracle_price × (10¹⁸ / 10^oracle_decimals)

Example:

Oracle price: 4,000,000 (7 decimals) = $0.40
wad_scale  = 10¹⁸ / 10⁷ = 10¹¹
price_wad  = 4,000,000 × 10¹¹ = 4 × 10¹⁷  (= 0.4 in WAD)

XLM balance: 5 × 10¹⁸ WAD
USD value:   floor(5 × 10¹⁸ × 4 × 10¹⁷ / 10¹⁸) = 2 × 10¹⁸  (= $2.00)

Token Symbol Canonicalization

Before price lookup, the RiskEngine maps wrapped token symbols to their underlying asset:

Input symbol	Canonical symbol	Price feed
`BLUSDC`, `AQUSDC`, `SOUSDC`	`USDC`	USDC/USD
`AQUARIUS_USDC`, `SOROSWAP_USDC`	`USDC`	USDC/USD
`BLEND_XLM`	`XLM`	XLM/USD
`BLEND_USDC`	`USDC`	USDC/USD

Soroswap/Aquarius AMM Formulas

Optimal Liquidity Amount (Add Liquidity)

Given pool reserves (R_a, R_b) and desired inputs (A_desired, B_desired):

B_optimal = floor(A_desired × R_b / R_a)

if B_optimal ≤ B_desired:
    use (A_desired, B_optimal)
else:
    A_optimal = floor(B_desired × R_a / R_b)
    use (A_optimal, B_desired)

Swap Output (Soroswap, 0.3% fee)

amount_out = floor(amount_in × 997 / 1000)   # simplified (before reserves)

Actual output from reserves (constant product, x × y = k):

amount_out = floor((amount_in × 997 × reserve_out) / (reserve_in × 1000 + amount_in × 997))

Storage TTL

All persistent storage keys are extended with a TTL to prevent ledger entry expiry:

Key type	TTL
Contract data (short)	6,307,200 ledgers (~1 year)
Contract data (long)	63,072,000 ledgers (~10 years)

Extensions are applied on every write and on reads for frequently accessed keys.

​WAD Arithmetic

​Constants

​Core Operations

​Decimal Conversion

​Interest Rate Model

​Utilization Ratio

​Borrow Rate Per Second

​Annual Rate Derivation

​Interest Accrual

​Rate Factor

​State Update

​Live (Preview) Borrows

​Borrow Share Accounting

​Issuing Shares (on Borrow)

​Burning Shares (on Repay)

​Computing User Debt

​Origination Fee

​vToken Exchange Rate

​Minting on Deposit

​Redeeming on Withdrawal

​Exchange Rate

​Health Factor

​Collateral Valuation

​Borrow-Time Check (Gross-Asset Model)

​Withdraw-Time Check

​Liquidation Check

​Constants

​Oracle Price Conversion

​Token Symbol Canonicalization

​Soroswap/Aquarius AMM Formulas

​Optimal Liquidity Amount (Add Liquidity)

​Swap Output (Soroswap, 0.3% fee)

​Storage TTL

WAD Arithmetic

Constants

Core Operations

Decimal Conversion

Interest Rate Model

Utilization Ratio

Borrow Rate Per Second

Annual Rate Derivation

Interest Accrual

Rate Factor

State Update

Live (Preview) Borrows

Borrow Share Accounting

Issuing Shares (on Borrow)

Burning Shares (on Repay)

Computing User Debt

Origination Fee

vToken Exchange Rate

Minting on Deposit

Redeeming on Withdrawal

Exchange Rate

Health Factor

Collateral Valuation

Borrow-Time Check (Gross-Asset Model)

Withdraw-Time Check

Liquidation Check

Constants

Oracle Price Conversion

Token Symbol Canonicalization

Soroswap/Aquarius AMM Formulas

Optimal Liquidity Amount (Add Liquidity)

Swap Output (Soroswap, 0.3% fee)

Storage TTL