Supply & Borrow Interest

Helios keeps the well-tested “index” model of AAVE : deposit yield accrues linearly into a liquidity index, while borrow debt compounds continuously in a variable-borrow index. Both indexes start at 1 RAY (10²⁷) and grow multiplicatively; Whenever anyone supplies, borrows, repays, or withdraws, the contract updates both indexes—so deposit APY compounds discretely on each touch, while borrow APR compounds every second through a gas-saving binomial approximation.

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1 Overview

BTC in, hToken out (for suppliers) Suppliers receive hTokens that rebase upward as the liquidity index grows; borrowers watch their BTC debt rise as the variable-borrow index ticks.

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2 Deposit Interest

2.1 Linear-rate accrual

nextLiquidityIndex = prevIndex * (1e27 + r * Δt);

• r = per-second liquidityRate; Δt = seconds since the last update.

• Because the factor (1 + r Δt) multiplies the previous index, interest already earned also earns interest → true compounding.

Indexes update only on state-changing calls (supply, withdraw, borrow, repay, liquidation). Between touches the index is flat; over many touches the multiplicative chain matches continuous compounding.

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3 Borrow Interest

3.1 Binomial approximation

$$(1 + r)^{Δt} ≈ 1 + rΔt + (Δt(Δt−1) / 2) r² + (Δt(Δt−1)(Δt−2) / 6) r³$$

• Accurate to <0.01 % for multi-year loans, but far cheaper than full exponentiation.

• Variable-rate debt compounds for every borrower interaction; stable-rate debt compounds only when the holder touches their own position.

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4 Putting It Together

Metric	Symbol	Formula	Why it matters
Utilization	U	borrowed BTC ÷ total BTC	Drives both borrow & supply rates.
Borrow APR	rᵦ	interestRateModel(U)	Piece-wise “kink” curve.
LiquidityRate	rₗ	rᵦ × U × (1 – reserveFactor)	Portion paid to suppliers.
LiquidityIndex	Iₗ	Iₗ₍prev₎ × (1 + rₗ Δt)	Updated on every touch.
VariableBorrowIndex	Iᵥ	Iᵥ₍prev₎ × binomial(rᵦ, Δt)	Ticks each second.

Reserve-factor BAC diverts a slice of borrower interest to the Helios treasury; the rest becomes deposit yield.

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5 What Suppliers Should Know

1. hToken balances grow only when the pool is touched—quiet markets mean the bump shows on the next transaction.

2. Borrowers pay every second via the variable-borrow index, even if no one interacts.

3. High utilisation (> 90 %) or congested mempool condition spikes rates. Borrow APR jumps sharply; liquidityRate climbs more gently, attracting fresh BTC to restore balance.

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6 Key Contract Hooks

Function	Role
updateState()	Refreshes both indexes at the start of every mutative call.
calculateLinearInterest()	Computes the supply slice for deposits.
calculateCompoundedInterest()	Applies the binomial borrow interest step.

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Summary

• Deposits: simple-interest slices multiplied into a growing liquidity index → discrete but effectively compound yield.

• Borrows: continuous compounding via binomial math → debt grows each second.

• Indexes: both start at 1 RAY and scale every balance without touching individual accounts, keeping gas low and accounting transparent.