# Interest Rate Model

Helios derives interest rates to one on-chain variable—**utilization**.\
A gentle slope applies while utilization is below an *optimal* point; a much steeper slope applies above it, protecting exit liquidity in stress.\
Depositors earn

```
Supply APR = Borrow APR × Utilization × (1 – Reserve-Factor)
```

where the reserve-factor funds the protocol treasury.

***

### 1 Why Utilization Drives Rates

```
Utilization (U) = Borrowed BTC / Total BTC in the reserve
```

* **Low U** → slack liquidity → cheap leverage.
* **High U** → scarce liquidity → rising APR to deter new borrows and attract deposits.

***

### 2 Borrow-Rate Curve

| Parameter           | Symbol | Typical setting\* | Purpose                       |
| ------------------- | ------ | ----------------- | ----------------------------- |
| Base rate           | r₀     | 0 – 1 %           | Floor APR when U ≈ 0          |
| Optimal utilization | U\*    | 70 – 85 %         | “Kink” where slope changes    |
| Slope 1             | m₁     | 2 – 8 %           | Mild gradient below the kink  |
| Slope 2             | m₂     | 30 – 200 %        | Steep gradient above the kink |
| Reserve-factor      | f      | 5 – 20 %          | Protocol share of interest    |

**Piece-wise formula**

```
              { r₀ + m₁ · (U / U*)                      if U ≤ U*
Borrow APR =
              { r₀ + m₁ + m₂ · (U – U*) / (1 – U*)      if U > U*
```

* Below the kink the curve is shallow, encouraging leverage.
* Above the kink the curve is steep, preserving liquidity.

***

### 3 Supply (Liquidity) Rate

```
LiquidityRate (rₗ) = Borrow APR × U × (1 – f)
```

Balances in **hTokens** rebase automatically whenever the pool’s liquidity index updates; users never pay gas to claim yield.

***

### 4 Governance Levers

| Lever   | When to adjust                        | What it does                              |
| ------- | ------------------------------------- | ----------------------------------------- |
| **r₀**  | Macro too tight / loose               | Shifts the whole curve up / down          |
| **U**\* | Volatility changes                    | Moves the kink left / right               |
| **m₁**  | Encourage / dampen moderate borrowing | Tilts slope below kink                    |
| **m₂**  | Strengthen liquidity defense          | Tilts slope above kink                    |
| **f**   | Treasury runway vs. depositor yield   | Widens / narrows the supply–borrow spread |

All levers live in a single **Interest-Rate Strategy** contract that governance can swap out without touching user balances.

***

### 5 Key Takeaways

* **Utilization drives everything**—keep an eye on the pool dashboard.
* **Supply APR = Borrow APR × U × (1 – f).**