Borrow Rate

as of 02/01/2025

Differently to other assets, USH won’t follow the standard interest rate model based on supply and demand dynamics, as the supply part does not exist. For this reason, the borrowing rate is given as a smart contract parameter, which will be fully controllable by the Governance smart contract once admin rights are transferred to it. Additionally, there is a time lock as well as a maximum change allowed for increasing the rate.

As we have already mentioned, not all borrowers share the same borrowing rate: some of them might have a discount applied to the base borrowing rate. In the following section we will explain how to:

• Simulate an account borrowing without a discount: Simulate the behavior of an account borrowing over time with a variable borrow rate.

• Simulate an account borrowing with a discount: Simulate the behavior of an account borrowing over time with a variable borrow rate, including a variable discount applied on top of it.

Discount Rate Model

We need to prepare the framework of meaningful equations before going deep into the Discount Rate Model implementation. In that sense, let’s describe the money market dynamics first:

The continuously compounded money market account $\beta \left( t \right)$ satisfies the following differential equation:

$$d\beta(t) = r(t) \cdot \beta(t) \cdot dt,$$

in which:

• $\beta( t_0 ) = 1$ is the initial condition, and

• $r(t)$ is the borrow rate, deterministic in our case.

This equation describes an account that has an initial deposited amount of $\beta( t_0 )$ which increases over time at a rate $r(t)$. The exact solution for this differential equation is nothing more than:

$$\beta(t) = \exp \left( \int_{t_0}^t r(u) \cdot du \right).$$

However, we can discretize the previous differential equation in a tenor structure $0 = T_0 < T_1 < \dots < T_N$ using the Euler explicit method, i.e. the most simple approximation, such that:

$$I(T_{i+1}) - I(T_{i}) = r(T_{i}) \cdot I(T_{i}) \cdot \left( T_{i+1} - T_{i} \right),$$

in which $i$ goes from $0, 1, 2, \dots, N$. $I(t)$ is known as a rolling certificate of deposit and can be interpreted as a discrete-time equivalent of $\beta (t)$. More precisely, $I(t)$ will approach $\beta (t)$ as the time spacing of the tenor structure is made increasingly fine. We can reorder the previous equation and get:

$$I_{i+1} = I_i \cdot \left( 1 + r_i \cdot \tau_i \right),$$

in which we have abused notation by dropping $T$ and with $\tau_i = T_{i + 1} - T_i$. We can even go a step further and obtain:

$$I_{i + 1} = \prod_{n = 0}^i \left( 1 + r_n \cdot \tau_n \right).$$

Now, suppose there is a security that pays $1 at maturity . This security is known as the discount bond or zero coupon bond, and its value at any given time $t \leqslant T$ in a deterministic environment is just:

$$P(t,T) = \frac{\beta(t)}{\beta(T)} = \exp\left( \int_{t_0}^t r(u) \cdot du - \int_{t_0}^T r(u) \cdot du \right),$$

which yields:

$$P(t,T) = \exp\left( - \int_t^T r(u) \cdot du \right).$$

Based on what we have learned, its discrete value $D\left(T_{i+1}, T_{j+1} \right)$ known as the discount factor is:

$$D(T_{i+1}, T_{j+1}) = \frac{I(T_{i+1})}{I(T_{j+1})} = \prod_{n = i+1}^j \left( 1 + r_n \cdot \tau_n \right)^{-1},$$

with $i \leqslant j$. At this point, it is important to realize that $I(t)$ is what is usually called the Borrow Index in smart contract jargon.

Simulation of an Account’s Borrow Without a Discount Rate

An account borrowed at any time $t$ can be translated to a time $T \geqslant t$ using the discount factor, such that:

$$D(t, T) \cdot B(T) = B(t),$$

so that the account borrowed amount at time $T$ is:

$$B(T) = B(t) \cdot D(t, T)^{-1} = B(t) \cdot \frac{I(T)}{I(t)}.$$

In other words, computing the dynamics of $I(t)$ at all interaction times allows the translation of any borrow amount across time.

Simulation of an Account’s Borrow With a Discount Rate

In the previous scenario, the Discount Factor or Borrow Index can be used to translate any borrow amount because all accounts are exposed to the same borrow rate. However, we now need to compute an account’s borrow even if each account has a different effective borrowing rate. This is because we apply a discount rate to the base rate and this discount rate is per user. The account borrows with discount, named $\hat{B}(t)$, is affected by a discount amount proportional to the accrued interest between two infinitesimally close time intervals, such that:

$$d\hat{B}(t) = r(t) \cdot \hat{B}(t) \cdot dt - d\delta(t) \quad \text{with} \quad d\delta(t) = d(t) \cdot r(t) \cdot \hat{B}(t) \cdot dt,$$

with a discount rate $0 \leqslant d(t) \leqslant 1$. After rearrangement, we get:

$$d\hat{B}(t) = \left ( 1 - d(t) \right) \cdot r(t) \cdot \hat{B}(t) \cdot dt.$$

We can implement the same discretization that we did for the money market account and obtain the following expression:

$$\hat{B}_{i+1} = \hat{B}_i + \left( 1 - d_i \right) \cdot r_i \cdot \hat{B}_i \cdot \tau_i = \hat{B}_i \cdot \left[ 1 + \left( 1 - d_i \right) \cdot r_i \cdot \tau_i \right].$$

With a bit of algebra and using the proven fact that $I_{i+1} = I_i \cdot \left( 1 + r_i \cdot \tau_i \right)$ , we get:

$$\hat{B}_{i+1} = \hat{B}_i \cdot \left[ \frac{I_{i+1}}{I_i} \cdot \left( 1 - d_i \right) + d_i \right].$$

This last equation allows using the conventional Borrow Index $I(t)$, obtained through the money market dynamics without any kind of discount, to determine an account’s borrow with a discount rate $d(t)$.

Discount Rate

The discount rate $d(t)$ plays a crucial role in the protocol because it allows any user to borrow USH with a discount applied to the borrow rate. In this context, the discount rate is defined as a function of the collateral distribution deposited by the user in the protocol, such that the discount rate $d$ of an account $a$ with deposited collaterals $C$ of asset types $j$ behaves as:

$$d_a = d_a(t, C_{a,j}(t)) \quad \text{for } j=1, \dots, N.$$

Each market $j$ has a conversion rate $h_j$ (a.k.a the coverage) which determines the amount of USH that can be borrowed at a discount based on the amount of collateral $C_j$:

$$b^\text{cov}_j = h_j \cdot C_j.$$

Then, an account with a borrowed amount $B_a$ has a borrowed amount at a discount $B_{a,d}$ given by:

$$B_{a,d} = \min \left( B_a, \sum_{j=1}^{N} h_j \cdot C_{a,j} \right).$$

It follows that the borrow amount without a discount is just:

$$B_{a,\sout{d}} = B_a - B_{a,d}.$$

If $B_{a,d} = 0$, then the account’s discount rate is zero:

$$d_a = 0.$$

If $B_{a,d} \leqslant B_a$ , we have:

$$d_a = \frac{1}{B_a} \cdot \sum_{j=1}^{N} d_j \cdot h_j \cdot C_{a,j},$$

in which each $d_j$ is the discount rate per asset defined as a market parameter. In order to demonstrate this last equation, one can reason about the discounts being applied to each part of the account’s borrow, such that the following relationship must be true:

$$\small r \cdot \left( 1 - d_a \right) \cdot B_a = r \cdot B_{a,\sout{d}} + r \cdot \left( 1 - d_1 \right) \cdot h_1 \cdot C_{a,1} + \dots + r \cdot \left(1 - d_{N} \right) \cdot h_{N} \cdot C_{a,N}.$$

Finally, if there is too much coverage such that $\sum_{j=1}^{N_j} h_j \cdot c_{a,j} > B_a$ and $B_{a,\sout{d}} = 0$, the protocol applies the coverage in descending order, i.e. that largest discounts are taken into consideration first.