# Setting Interest Rates for ETH Instant Unstake Loans

We will need to establish an accurate and fair pricing model for our Instant Unstake Model. Our algorithms must solve for a range of parameters, but arguably the most critical is the borrow interest rate/discount rate at which we will price the Instant Unstake purchases.

### How do We Build the Discount Rate in Our Model?

Within the context of a P2Pool system our interest rate will be determined by available liquidity in the system. Of course that liquidity can take several shapes:

ETH and WETH

aETH, cETH, stETH, and/or other ETH derivatives on Parallel

Within the P2P/P2Pool hybrid model we will likewise add a marketplace for these stakefish staking NFT’s and outright P2P purchases. Below is the schematic of the Parallel Instant Unstake NFT router:

Here we focus on step 1. as this itself will be its own router within the above tokens.

Within Step 1:

Calculate the instantaneous borrow rate $BorrowAPY_{ETH}$ from the Parallel Pool.

Calculate the instantaneous borrow rate $BorrowAPY_{aETH}$, $BorrowAPY_{cETH}$, and Borrow APY’s for stETH/wstETH, cbETH, and rETH. Implicit in this is the assumption that we would be able to redeem/trade via 1inch the ETH derivative at par value via each respective protocol.

Pick the lowest instantaneous $Borrow_{APY}$ for ETH among the various ETH and ETH derivative pools.

NB: Implicit in the above is that instantaneous loans have zero duration risk. But as we will discuss in the next section we will need to adjust borrow rates for the relevant duration risk.

Of course even beyond the stakefish NFT router we want to know how much a specific staking position is worth. And indeed we study pricing models and setting interest rates for these loans below.

### Duration Risk and Overcollateralized DeFi Loans

The well-understood Aave/Compound lending model creates overcollateralized open-ended loans with no specific requirement nor immediate need to assign duration risk to the loan. Or in other words, the borrower pays the same interest rate regardless of how long they maintain their borrow position. Why?

Overcollateralization with quality collateral means the lender can recoup any principal and interest owed upon a successful liquidation.

As long as there is sufficient liquidity available and low utilization, Lenders may at any point withdraw principal + interest with no credit risk. Their discount rate could be infinite but Present Value of cash flow $\frac{Principal + Interest }{(1+\frac{r_{Discount}}{365})^T}$ is always divided by 1 if days $T$ until payment of $Principal + Interest$ is 0.

There nonetheless remains duration risk for both the borrow and the lender, specifically with regard to interest rates:

**Lender**: Given that interest rates vary by Utilization, the Lender may receive a worse return than they anticipated if Utilization within the pool declines. Both of these risks grow as a function of Duration and Utilization:The interest rate may in fact drop below the Lender’s discount rate and thus this may prove unprofitable. This can also be referred to as Convexity risk, or the change in value of an existing lending position on a rise or fall in its interest rate.

Utilization may hit 100% and the Lender will be unable to withdraw their Principal + Interest, extending days $T$ until redemption when liquidity returns.

**Borrower**: The Borrower may ultimately pay a higher interest rate than they expected if Utilization rises.This interest rate may rise above the Borrower’s discount rate and this borrow may prove unprofitable. This risk grows as a function of Duration and Utilization.

### Summarizing Risks Inherent to ETH Unstaking Loans - Duration/Convexity and Slashing

In order to make capital-efficient and risk-controlled loans against ETH Staking, our protocol and liquidity suppliers will need to account for risks specific to Duration and Staking. Our Borrow Interest rate, or $r_{Borrow}$, will thus need to account for three risks which are all a function of or impacted by time to redemption in days $T$:

**Interest rate risk**- the likelihood that interest rates will vary significantly from the time of Instant Unstake to redemption through days $T$. We can likewise refer to this as Convexity Risk—the likelihood that the value of the outstanding loan will change with a change in interest rates.**Liquidity risk**- This is the risk that ETH pool Utilization will surge and leave the ETH supplier without the ability to withdraw their Principal and Interest. Notably Interest Rates are a function of Utilization thus this is closely linked to Interest rate risk.**Slashing risk**- In practice the slashing risk is very low but non-zero. We can safely ignore Slashing risk for shorter-term loans, though it is ultimately a function of validator quality and time $T$.

All three are a function of time $T$ and will be grouped under the broader umbrella of Duration Risk.

In each of these, risk will be greater for lending against lower-liquidity collateral. In practice we will focus on top staking derivative protocols and will not price in Slashing Risk to our valuation algorithms.

### Addressing Interest Rate Risk and Liquidity Risk - the Duration Factor

We reference Interest Rate Risk and Liquidity Risk as two distinct risks and they are, but in practice Interest Rates are a function of Utilization and thus liquidity. Or in short: our pricing of Interest Rate (Convexity) and Liquidity Risk will be one in the same. We will indeed refer to this as Duration Factor, or $DF$ for short.

At time of pricing specific staking derivative Instant Unstake rates we will gather the best borrow rate $r_0$ from our ETH liquidity sources. This will serve as the minimum Discount rate.

We determine that it will take $T$ days for the staking derivative token to be redeemed at the redemption rate in ETH based on our evaluation protocols.

We will thus calculate a Duration Factor $DF$ for duration/convexity risk. $DF$ will be defined as a number $>1$ such that each increment of $T$ will add $\frac{1}{DF}$ interest rate risk to the Instant Unstake request.

Calculate $r_{Discount}$ with the following equation:

The above equation can give us $r_T$ at the origination of the loan, or time $t=0$. If at a future point we would like to calculate the updated $r_T$ we would rewrite the equation as the following, more generalized, equation:

We represent the passage of time $t$ as the days elapsed since the loan was first made and we would re-calculate based on the contemporaneous $r_t$ to represent the best ETH borrow rate available at the time. Necessarily this likewise means the value of the Instant Unstake loan will vary based on ETH pool utilization and time. And indeed there will implicitly be Duration and Convexity risk on the outstanding loan.

### Pricing the Aggregate Borrow and Discount Rate for Instant Token Unstake

With the above information to hand, we can now calculate the aggregate $r_{T}$ and ultimately the $r_{Discount}$ rate we will use to price the Instant Unstake for the staking derivative tokens.

We can then price our instant unstake position of either LSD tokens or a staking position as follows. Since we are working with only the highest quality Staking as a Service providers we will assume $Redemption_{ETH}$ = 1 given a staking risk of 0. We can thus express our Instant Unstake value as follows:

Our on-chain oracle will thus calculate the following figures to a high degree of accuracy to ensure liquidity providers are appropriately compensated for Instant Unstake purchases:

$T$ days until redemption

$Redemption_{ETH}$ if different to 1:1.

Below is a chart of the Present Value/Instant Unstake amount of 1 ETH due to be redeemed in days $T$ at a base $r_0$ of 10%. Notably a $DF$ of 365 implies $r_{Discount}$ would double $r_0$ at $T=365$ days using the equation $r_{Discount_0} = r_{0} + \frac{r_0 \times T }{DF}$.

#### Instant Unstake Swap Rate Projections

Data and chart source: Parallel calculations

By way of example, a user may request an Instant Unstake of a 32 ETH staking NFT with a redemption rate of 1.000 due to be redeemed in 10 days. According to the above equations the Parallel pool would be able to pay them 31.9125 ETH at an $r_{Discount}$ of 10%.

### Establishing DF Value For ETH Staking Derivatives

Given that we are focused on only top-quality Ethereum staking derivatives, we will focus not on the quality of the collateral used but instead liquidity conditions in the ETH money market on Parallel. We expressly build in the current pool Utilization rate $U$ into $r_{Discount_0} = r_{0} + \frac{r_0 \times T }{DF}$ as $r_0$, or the interest rate at $t=0$, is purely a function of pool liquidity as measured by Utilization.

Our setting for $DF$ will aim to make borrowing more expensive if the liquidity will be tied to that specific position for extended times $T$. What is a reasonable rate at which $r_{Discount}$ will double?

It is difficult to estimate the total term of loans in the P2Pool lending model and as such we need to look beyond pure DeFi to estimate average loan duration. According to on-chain data for P2P NFT lending, lenders and borrowers on average agree to a 30-day average lending term for P2P NFT loans. But in this case the quality and liquidity of collateral is ETH staking positions—the more conservative number makes sense here.

We will set $DF$ to a value at which $r_{Discount}$ will be $r_0 \times 2$ at a loan term of 90 days, or $DF = 90$, as our risk-based collateral factor for ETH itself is 3x that of lower-tier Art and Collectible NFT Collections.

Below we see how $r_{Discount}$ and the instant-purchase value of 1 ETH varies by time $T$ until Redemption with an $r_0$ of 10% and $DF$ of 90.

Our model is quite intentionally straightforward but ultimately protocol usage for these fixed-term redemptions will be a function of supply and demand. A $DF$ of 90 thus acts as a starting point and may be adjusted based on community feedback and eventually DAO-governed votes.

### Setting Protocol Insurance Reserve and Supplier Rates

Parallel looks to generate value to the end user and in the process generate funds for future protocol growth. In doing so we will add an insurance reserve to the $r_{Discount}$ rate, currently set at 10% across all ERC20’s.

We can thus calculate the share of each Instant Unstake of NFT discount which goes to the supplier and which share goes to the protocol insurance reserve.

At time of redemption $T$, the supplier pool receives:

We can thus calculate the share of each token sale which goes not to the supplier pool but to the insurance protocol reserve as indicated below. The area under 1.000 indicates the share which goes to the Protocol Reserve:

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