> For the complete documentation index, see [llms.txt](https://vaultbook.gauntlet.xyz/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://vaultbook.gauntlet.xyz/vaults/morpho-vaults/morpho-adaptive-curve-irm.md).

# Morpho Adaptive Curve IRM

The AdaptiveCurveIRM is Morpho Blue's native interest rate model. It targets a **90% utilization ratio** by continuously adjusting an internal rate anchor. This page explains how the curve works, how the anchor adapts over time, and how quickly rates can move at different utilization levels.

### The Rate Curve

At any point in time, the borrow rate is a function of two inputs:

* **Current utilization** `U` - the share of supplied assets currently borrowed
* **`rateAtTarget`** - an internal state variable representing the borrow rate at exactly 90% utilization

The curve is piecewise linear with a kink at the 90% target. Below the kink it is shallow; above it is steep.

```
// err = (U - 0.90) / normFactor,  bounded to [-1, +1]
// normFactor = (1 - 0.90) above target, 0.90 below target

if err < 0:
  r = ((1 - 1/C) × err + 1) × rateAtTarget

if err >= 0:
  r = ((C - 1) × err + 1) × rateAtTarget

// C = 4 (curve steepness)
```

This produces a fixed multiplier structure regardless of where `rateAtTarget` sits:

| Utilization | Multiplier | Example (rateAtTarget = 4%) |
| ----------- | ---------- | --------------------------- |
| 0%          | 0.25×      | 1.00%                       |
| 90% (kink)  | 1.00×      | 4.00%                       |
| 100%        | 4.00×      | 16.00%                      |

The **shape of the curve never changes** — only the anchor moves. If `rateAtTarget` doubles, every point on the curve doubles with it.

#### Key Parameters

| Parameter             | Value  | Description                                        |
| --------------------- | ------ | -------------------------------------------------- |
| `TARGET_UTILIZATION`  | 90%    | Kink point; borrow rate equals `rateAtTarget` here |
| `CURVE_STEEPNESS` (C) | 4      | Max rate = 4× anchor; min rate = 0.25× anchor      |
| `ADJUSTMENT_SPEED`    | 50/yr  | Controls how fast `rateAtTarget` adapts            |
| `MIN_RATE_AT_TARGET`  | 0.1%   | Floor on `rateAtTarget`                            |
| `MAX_RATE_AT_TARGET`  | 1,200% | Ceiling on `rateAtTarget`                          |

***

### The Adaptive Mechanism

Between market interactions, `rateAtTarget` drifts continuously based on the signed error between current utilization and the 90% target.

```
speed            = ADJUSTMENT_SPEED × err       // 50/yr × [-1, +1]
elapsed          = block.timestamp - lastUpdate

endRateAtTarget  = startRateAtTarget × exp(speed × elapsed)
```

The further utilization deviates from target, the faster the anchor moves:

* **U > 90%** - error is positive, `rateAtTarget` rises. Higher rates attract new supply and discourage borrowing until utilization falls back to target.
* **U < 90% -** error is negative, `rateAtTarget` falls. Lower rates stimulate borrowing and may cause suppliers to exit until utilization recovers.
* **U = 90%** - error is zero, `rateAtTarget` is unchanged. This is the only stable fixed point.

### Rate Velocity

#### Borrow Rate at Each Utilization Level

The table below shows the instantaneous borrow rate and the rate after 5 days of sustained utilization, starting from `rateAtTarget = 4%`.

| Util | err  | rAT t=0 | rAT t=5d | Borrow t=0 | Borrow t=5d | Change | Max rate t=0 | Max rate t=5d |
| ---- | ---- | ------- | -------- | ---------- | ----------- | ------ | ------------ | ------------- |
| 90%  | 0.00 | 4.00%   | 4.00%    | 4.00%      | 4.00%       | +0%    | 16.00%       | 16.00%        |
| 91%  | 0.10 | 4.00%   | 4.28%    | 5.20%      | 5.57%       | +7%    | 16.00%       | 17.13%        |
| 92%  | 0.20 | 4.00%   | 4.59%    | 6.40%      | 7.34%       | +15%   | 16.00%       | 18.35%        |
| 93%  | 0.30 | 4.00%   | 4.91%    | 7.60%      | 9.33%       | +23%   | 16.00%       | 19.65%        |
| 94%  | 0.40 | 4.00%   | 5.26%    | 8.80%      | 11.57%      | +32%   | 16.00%       | 21.04%        |
| 95%  | 0.50 | 4.00%   | 5.63%    | 10.00%     | 14.08%      | +41%   | 16.00%       | 22.53%        |
| 96%  | 0.60 | 4.00%   | 6.03%    | 11.20%     | 16.89%      | +51%   | 16.00%       | 24.13%        |
| 97%  | 0.70 | 4.00%   | 6.46%    | 12.40%     | 20.03%      | +62%   | 16.00%       | 25.84%        |
| 98%  | 0.80 | 4.00%   | 6.92%    | 13.60%     | 23.52%      | +73%   | 16.00%       | 27.68%        |
| 99%  | 0.90 | 4.00%   | 7.41%    | 14.80%     | 27.41%      | +85%   | 16.00%       | 29.64%        |
| 100% | 1.00 | 4.00%   | 7.93%    | 16.00%     | 31.74%      | +98%   | 16.00%       | 31.74%        |

*rAT = rateAtTarget. Max rate = borrow rate at 100% utilization given the adapted rateAtTarget. Protocol fee not applied.*

#### Time for `rateAtTarget` to Reach a Level

Starting from `rateAtTarget = 4%`, days required to reach each target rate at constant utilization.

```
t = ln(targetRate / startRate) / (ADJUSTMENT_SPEED × err) × 365
```

| Constant UR | To 7%     | To 8%     | To 9%     | To 10%    |
| ----------- | --------- | --------- | --------- | --------- |
| 91%         | 40.4 days | 50.0 days | 58.5 days | 66.1 days |
| 92%         | 20.2 days | 25.0 days | 29.2 days | 33.0 days |
| 93%         | 13.5 days | 16.7 days | 19.5 days | 22.0 days |
| 95%         | 8.1 days  | 10.0 days | 11.7 days | 13.2 days |
| 97%         | 5.8 days  | 7.1 days  | 8.3 days  | 9.4 days  |
| 100%        | 4.1 days  | 5.1 days  | 5.9 days  | 6.7 days  |

*Assumes constant utilization throughout. In practice, rising rates will attract new supply and reduce utilization before these levels are reached.*

#### Doubling Time by Utilization

How many days of sustained utilization before `rateAtTarget` and every point on the curve doubles.

| Utilization | err  | Speed   | Doubling Time |
| ----------- | ---- | ------- | ------------- |
| 91%         | 0.10 | 5.0/yr  | 50.6 days     |
| 92%         | 0.20 | 10.0/yr | 25.3 days     |
| 93%         | 0.30 | 15.0/yr | 16.9 days     |
| 94%         | 0.40 | 20.0/yr | 12.6 days     |
| 95%         | 0.50 | 25.0/yr | 10.1 days     |
| 96%         | 0.60 | 30.0/yr | 8.4 days      |
| 97%         | 0.70 | 35.0/yr | 7.2 days      |
| 98%         | 0.80 | 40.0/yr | 6.3 days      |
| 99%         | 0.90 | 45.0/yr | 5.6 days      |
| 100%        | 1.00 | 50.0/yr | 5.1 days      |

***

### Supply APY vs Borrow APY

`rateAtTarget` is not the supply APY. The relationship between the two is:

```
supplyAPY = borrowAPY × utilization × (1 - protocolFee)
```

At 90% utilization, a 4% borrow APY with a 15% protocol fee yields a supply APY of approximately 3.0&#x36;**%**. The gap between borrow and supply APY narrows at higher utilization but never fully closes due to the protocol fee.

***

### Risk Considerations

**Slow-moving markets.** A market pinned at 91% adapts slowly - `rateAtTarget` takes \~50 days to double. This is insufficient pressure for rapid rebalancing. Curators should treat sustained utilization above 91% as a signal to add supply capacity proactively.

**Fast-moving markets.** Above 95% utilization, the anchor can double in under 10 days. Combined with the 4× curve multiplier, the max borrow rate can reach multiples of its starting value quickly, which may trigger liquidations if borrowers are not monitoring positions.

**The 4× leverage on volatility.** Because the max rate is always `C × rateAtTarget`, any drift in the anchor is amplified fourfold at the top of the curve. A `rateAtTarget` move from 4% to 8% moves the max rate from 16% to 32%.

**`rateAtTarget` as a market health signal.** Rising `rateAtTarget` indicates sustained over-utilization. Falling `rateAtTarget` indicates sustained under-utilization!


---

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