# Yield Curve

### Understanding the Yield Curve

The yield curve plays a crucial role in determining borrowing rates and managing liquidity supply within a lending pool. An efficient yield curve aims to maintain market equilibrium by adjusting rates based on the utilization rate of the liquidity pool. The shape of the curve significantly influences the behavior of market participants, such as borrowers and Loan Providers, by incentivizing or disincentivizing certain actions. A well-designed yield curve helps ensure that traders can access liquidity when needed while maintaining a balance between supply and demand, fostering a healthy and sustainable lending ecosystem

### **Modeling the Yield Curve**

Modeling the yield curve, which represents the relationship between interest rates and supply, can be approached through:

* Linear Function: The yield is modeled as a simple linear function of the utilization rate of the lending pool. While straightforward, this approach may fail to capture the nuanced shape of the curve.
* Nonlinear Function: More flexible nonlinear functions, such as exponential or polynomial forms, can be employed to better represent the curvature and complexities of the yield curve. 

The choice between linear and nonlinear models involves a trade-off between simplicity and accuracy. Nonlinear functions offer greater flexibility in capturing the intricacies of the yield curve but may introduce additional complexity and computational overhead

### **The Role of the Yield Curve in Market Equilibrium**

A yield curve adjusts yields more efficiently in response to changes in demand. It benefits borrowers by slowing the pace of rate hikes, while benefiting lenders by slowing the pace of rate decreases. This dynamic allows for a more gradual and measured impact on borrowing costs and lending returns, mitigating sudden shocks to either party.

* When utilization is low, borrowing is cheap, and supplying is underperforming, leading to an increase in utilization ; 
* When utilization is high, borrowing is expensive, and supplying is overperforming, leading to a decrease in utilization.

The shape of the curve is adjusted to incentivize lenders to supply into the market and to encourage borrowers to take debt. 

### **Interpolated Yield Curves**

In practice, a great approximation of the complexity of the yield curve is to create an interpolated yield curve. This is done by combining multiple linear yield curves, where the directrix coefficient (slope) increases with utilization. The more linear yield functions included in the interpolated curve, the better is the approximation. 

<figure><img src="/files/75BwkzhthmyoYKIZPL35" alt=""><figcaption></figcaption></figure>

$$
\begin{cases}
\boldsymbol{R(U)=R\_{slope0}+R\_{slope1}\frac{U}{U\_{T1}}} & \text{if } U \le U\_{T1} \\
\boldsymbol{R(U)=R\_{slope0}+R\_{slope1}+R\_{slope2}\frac{U-U\_{T1}}{1-U\_{T1}}} & \text{if } U\_{T1} < U \le U\_{T2} \\
\boldsymbol{R(U)=R\_{slope0}+R\_{slope1}+R\_{slope2}\frac{U\_{T2}-U\_{T1}}{1-U\_{T1}}+R\_{slope3}\frac{U-U\_{T2}}{1-U\_{T2}}} & \text{if } U\_{T2} < U \le U\_{T3} \\
\boldsymbol{R(U)=R\_{slope0}+R\_{slope1}+R\_{slope2}\frac{U\_{T2}-U\_{T1}}{1-U\_{T1}}+R\_{slope3}\frac{U\_{T3}-U\_{T2}}{1-U\_{T2}}+R\_{slope4}\frac{U-U\_{T3}}{1-U\_{T3}}} & \text{if } U\_{T3} < U
\end{cases}
$$

| Parameter | Initial Value |
| --------- | ------------- |
| Rslope0   | 4%            |
| Rslope1   | 2%            |
| Rslope2   | 4%            |
| Rslope3   | 4,5%          |
| Rslope4   | 6%            |
| UT1       | 40%           |
| UT2       | 60%           |
| UT3       | 90%           |

The exact shape of the function will set by the protocol’s decentralized governance and adjusted over time.


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