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 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 — Maturity-based borrowing with locked rates

• Rehypothecation — How fixed-rate reserves earn floating yield

• Curation Vaults — Multi-layer capital allocation

• Whitepaper — Full mathematical treatment