# fCash Interest Rate Model

Each active fCash Market has its own liquidity pool. Each liquidity pool is made up of two assets:

• Prime Cash (ex: Prime DAI)

• fCash (ex: fDAI 1 Dec 2023)

Each pool acts as an AMM to facilitate the exchange of fCash. The exchange rate between fCash and Prime Cash is based on the following formula:

$fCashExchangeRate = e^{interestRate*timeToMaturityInYears}$
// 100 fCash value at one year to maturity with 5% interest rate
fCash Exchange Rate = e^(.05 * 1) = 0.9512
A user would need to pay 0.9512 USDC to purchase 1 fUSDC dated one year out

// 100 fCash value at six months to maturity with 5% interest rate
fCash Exchange Rate = e^(.05 * .5) = 0.9753
A user would need to pay 0.9753 USDC to purchase 1 fUSDC dated 6 months out

Time to maturity is defined as the difference between the market's maturity date and the current time. The annualized interest rate is based on the pool's interest rate model. Notional V3 uses a two-kink interest rate model to determine fCash interest rates.

## fCash Interest Rate Model

Each fCash market's interest rate can be configured by governance using the following parameters:

• Kink1 Interest Rate

• Kink2 Interest Rate

• Max Interest Rate

• Kink1 Utilization Rate

• Kink2 Utilization Rate

fCash interest rate is a function of the pool's current utilization rate. A pool's utilization rate is defined as:

$utilization Rate= totalfCash / (totalfCash + primeCashInUnderlying)$

Notional's governance can also set the following parameters to set how much lenders and borrowers have to pay in fees to the protocol's reserves when they trade fCash:

• Fee Rate Percent (percentage of the current fCash interest rate)

• Min Fee Rate

• Max Fee Rate

The borrower fee is based on the current fCash interest rate and the Fee Rate Percent. For example, if the current fCash interest rate is 3% and the Fee Rate Percent is 8% then the fCash fee will be 0.24% (3% * 8%) for lenders and borrowers on fCash trades.

The Min Fee Rate acts as a lower bound on the fCash fee rate that has to be paid by fCash borrowers and lenders. Similarly, the Max Fee Rate acts as a higher bound on the fCash Interest Rate Fee that has to be paid by borrowers and lenders.

The reserve fee share parameter then dictates the proportion of trading fees that go to the reserve versus the proportion that goes to nToken holders. For example, if the reserve fee share is set to 80, then 80% of fees will go to the protocol's reserve while 20% will go to nToken holders.

## fCash Interest Rate Modelisation

This spreadsheet model can be used to model the fCash Interest Rate curve under different parameter assumptions.

The main objectives when setting fCash Interest Rate Model parameters are to:

• Optimize capital efficiency

• Minimize liquidity risk

#### Capital Efficiency

The two kink interest rate curve can be configured to minimize slippage for large fCash trades between kink1 and kink2. To do so, the interest rate has to increase rapidly until utilization reaches kink1 and then increase at a slower pace until utilization reaches kink2. By setting a relatively flat slope between kink1 and kink2 Notional's governance can allow traders to trade large amounts of fCash with relatively low slippage. Above kink2 the interest rate increases rapidly to incentivize lenders to lend and borrowers to close out their positions (See picture above).

Alternatively, governance can also set the interest rate model to increase slowly until kink1 utilization, then increase at a more rapid pace until kink2 utilization and then increase very rapidly above kink2 (See picture below). Setting the fCash interest rate model in such a fashion can mitigate liquidity risks more gradually for assets with changing interest rate regimes.

#### Liquidity risk

For fCash lenders to be able to close out their position fCash markets need to maintain a sufficient amount of liquidity.

To mitigate this risk, Notional's governance can set the Max Interest Rate at a rate that is attractive to potential new fCash lenders and penalizing to fCash borrowers. The kink2 utilization rate parameter can also mitigate liquidity risk concerns by determining the percentage of total liquidity available between kink2 utilization and 100% utilization.

For fCash borrowers to be able to close out their position, they must be able to lend fCash. To do so, there must be fCash liquidity in the market (utilization needs to be above 0%). Setting the kink1 interest relatively low makes it attractive for borrowers to borrow at low fixed rates and discourages lenders to lend, thereby mitigating this liquidity risk.

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