# Cyclical Notes: Borrower Premium Framework

## **Objective**

Develop a framework to calculate fixed rates for a Cyclical Note product. The challenge is to translate volatile variable rates into a predictable fixed yield while incorporating utilization and borrower premiums.

## **Methodology**

### **Fixed Rate Formula**

The Cyclical Note defines a fixed rate as:

$$
\text{FixedRate} = \text{VariableRate} + (\text{UtilizationPremium} + \text{BorrowerPremium})
$$

This ensures fixed rates are anchored in prevailing variable rates, with adjustments for utilization dynamics and borrower demand.

### **Vasicek Model**

To provide a standardized approach for determining fixed rates, we recommend the Vasicek stochastic interest rate model. The model treats interest rates as a mean-reverting process: rates move randomly but tend to drift back toward a long-run average b. The speed of this reversion is controlled by a, and short-term fluctuations are captured by $\sigma$. This balance between randomness and stability makes Vasicek a natural choice for translating floating rates into fixed rates.

The fair fixed rate over a horizon T is given by:

$$
R_{\text{fixed}}(T) = b + (r_t - b)\,\frac{1 - e^{-aT}}{aT}
$$

Where:

- $r_t$: today’s observed variable rate
- b: long-run mean rate
- a: mean-reversion speed
- T: note duration (e.g., 90/365 years)

In practice, this means the fixed rate reflects the time-averaged expectation of future floating rates. If today’s rate is above the long-run mean, the fixed rate comes in lower; if today’s rate is below, the fixed rate comes in higher.

### **Estimation of Parameters**

We estimate the Vasicek parameters directly from historical rate data using a simple AR(1) regression:

$$
r_{t+1} = c + \phi r_t + \varepsilon_t
$$

Note: We have used Compound USDC Ethereum Comet Data for the past 90 days to fit on the above model

The regression outputs $\hat c, \hat \phi, \widehat{\sigma}_\varepsilon^2$, which map to Vasicek parameters as follows:

- $a = -\ln(\hat\phi)/\Delta$
- $b = \hat c / (1 - \hat\phi)$
- $\sigma = \sqrt{\tfrac{2a\hat\sigma_\varepsilon^2}{1-\hat\phi^2}}$

This approach provides a data-driven way to estimate the fair fixed rate.

### **Application to Market Data**

We applied this framework to USDC rates from Compound (with plans to extend to Aave + Compound). The resulting fixed rate series was much smoother and more stable than the raw variable rates, as expected.


The chart above also validates the Vasicek intuition: variable rates oscillate around a steadier long-run mean, and the fixed rate sits in between, providing a predictable anchor.

### **Borrower Premium Analysis**

While the Vasicek model gives us a way to estimate fair fixed rates, our main objective is to determine the **Borrower Premium**. We define it as the spread between fixed and variable rates:

$$
\text{BorrowPremium} = \max(\text{FixedRate} - \text{VariableRate}, 0)
$$


The graph shows how this premium behaved over the last 90 days. Our recommendation is to build an array of these daily premiums and map them to utilization levels using percentiles:

- **25% utilization →** 25th percentile of the borrower premium array
- **50% utilization →** 50th percentile
- **75% utilization →** 75th percentile

This method scales easily across different vaults. We initially tested a clustering-based approach, but found it less robust and harder to generalize. The percentile mapping approach is simpler, more transparent, and consistent in practice.

## **Key Takeaways**

- The **Vasicek model** offers a rigorous approach to translating volatile variable rates into smooth, predictable fixed rates.
- Fixed rates naturally sit between today’s spot rate and the long-run mean, capturing the time-averaged expectation of future yields.
- **Borrower Premiums** are best measured as the spread between fixed and variable rates, mapped to utilization levels using percentiles.
- The percentile-based method is scalable, simple, and consistent across vaults, making it preferable to clustering-based approaches.