# 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. 

***

### 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.

***

### 2 Deposit Interest

#### 2.1 Linear-rate accrual

```solidity
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. 

***

### 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.&#x20;
* Variable-rate debt compounds for **every** borrower interaction; stable-rate debt compounds only when the holder touches their own position.&#x20;

***

### 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.&#x20;

***

### 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.&#x20;

***

### 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.                  |

***

#### 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.


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