# Floating-Rate Markets

Floating-rate markets provide variable-rate borrowing and lending without maturity constraints. Rates adjust continuously based on utilization. These markets serve two purposes in Fira: standalone variable-rate lending, and the foundation for [rehypothecation](/protocol/rehypothecation.md) of fixed-rate market reserves.

## How It Works

* Lenders deposit assets into a pool and earn variable yield
* Borrowers post collateral and borrow from the pool, accruing interest continuously
* Both sides can enter and exit at any time — no maturity date
* Rates move with utilization: more borrowing means higher rates

### Lend Rate

$$
Lend\_{APR} = Borrow\_{rate}\times Utilization
$$

Because both rates are volatile, floating-rate markets suit short- to medium-term positions.

## Interest Rate Model (IRM)

The borrow rate is determined by a utilization-based Interest Rate Model. The core formula:

$$
r(t)=r\_T(t)\times curve(u(t))
$$

Where:

* $$r(t)$$ is the instantaneous spot borrowing rate
* $$r\_T(t)$$ is the rate target — the rate when utilization equals the target
* $$curve(u(t))$$ shapes how rates respond to utilization deviations

### Utilization

Represented as the fraction of supplied liquidity currently borrowed.

$$
u(t)=\frac{Borrowed(t)}{Supplied(t)}
$$

Further, using utilization, we compute an abstract value from our utility function $$e(u)$$.

$$
e(u) = \begin{cases} \dfrac{u - u\_{\text{target}}}{u\_{\text{target}}} & \text{if } u \leq u\_{\text{target}} \\\[8pt] \dfrac{u - u\_{\text{target}}}{1 - u\_{\text{target}}} & \text{if } u > u\_{\text{target}} \end{cases}
$$

### Rate Target

The rate target starts at a configured value and evolves dynamically:

$$
r\_T(t) = r\_T(t\*) × exp(k\_p × e(u) × (t-t\*))
$$

When utilization exceeds the target, the rate target increases — making borrowing more expensive and encouraging repayment. When utilization is below target, the rate target decreases — making borrowing cheaper and encouraging demand.

The utility function $$e(u)$$ symmetrically scales deviations above and below the target utilization $$u\_{target}$$.

### Utilization Curve

The curve function adds a second layer of rate responsiveness. Below target utilization, rates decrease gradually. Above target, rates increase sharply — creating natural pressure to keep utilization near the target.

$$
\text{curve}(u) = \begin{cases} \left(1 - \dfrac{1}{k\_d}\right) \cdot e(u) + 1 & \text{if } u \leq u\_{\text{target}} \\\[8pt] (k\_d - 1) \cdot e(u) + 1 & \text{if } u > u\_{\text{target}} \end{cases}
$$

The slope is controlled by $$k\_d$$, the curve steepness parameter.

### Rate Bounds

Rates are bounded between $$r\_{min}$$ and $$r\_{max}$$ to prevent extreme values:

$$
r(t) = \max!\bigl(\min!\bigl(r\_T(t) \cdot \text{curve}(u(t)),; r\_{\max}\bigr),; r\_{\min}\bigr)
$$

## Parameters

All IRM parameters are global across floating-rate markets and governed by the DAO:

| Parameter                   | Description                          |
| --------------------------- | ------------------------------------ |
| $$u\_{target}$$             | Target utilization ratio             |
| $$k\_p$$                    | Speed of rate target adjustment      |
| $$k\_d$$                    | Steepness of the utilization curve   |
| $$r\_T(0)$$                 | Initial rate target at market launch |
| $$r\_{min}$$ , $$r\_{max}$$ | Rate floor and ceiling               |

## Role in Fira's Architecture

Floating-rate markets integrate with the rest of Fira through:

* **Rehypothecation** — Idle reserves from fixed-rate markets (FW tokens) are deployed into floating-rate vaults to earn yield, which flows to CT holders
* **Curation** — Curated vaults can allocate capital across both fixed-rate and floating-rate markets
* **Composability** — The same collateral types and oracle infrastructure serve both market types

## Related

* [Fixed-Rate Markets](/protocol/fixed-rate-markets.md) — Maturity-based borrowing with locked rates
* [Rehypothecation](/protocol/rehypothecation.md) — How fixed-rate reserves earn floating yield
* [Curation Vaults](/protocol/curation-vaults.md) — Multi-layer capital allocation
* [Whitepaper](/resources/whitepaper.md) — Full mathematical treatment


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