# Redemptions

To maintain nUSD’s peg, redemptions allow any user to redeem 1 nUSD for $1 of cBTC.

If the price of nUSD falls below $1, arbitrageurs can buy nUSD on the open market to redeem the underlying collateral from Nectra for a profit.

## Ordering and Risk

When redemptions occur, buckets are redeemed in order of the buckets with the lowest interest rate to the highest interest rate. Collateral is redeemed at 1:1, thereby paying off the associated debt. If a bucket no longer has any remaining collateral, the next lowest bucket will be redeemed against.

Within a bucket, redemptions are applied proportionally across all outstanding positions. The distribution of redemptions ensures that they are distributed fairly among borrowers offering the same interest rate.

## Profitability

The potential profit from redemption arbitrage can be calculated as follows:

$$
\text{Profit} = \text{RedemptionAmount} \times (1 - \text{nUSDPrice} - \text{RedemptionFee})
$$

Where:

* ***Profit*** represents the potential earnings from a single redemption arbitrage transaction.
* ***RedemptionAmount*** is the amount of nUSD being used for the redemption.
* ***1*** is the target peg price of nUSD.
* ***nUSDPrice*** is the current market price of nUSD (which would be below $1 to make arbitrage profitable).
* ***RedemptionFee*** is the fee charged by the Nectra protocol for performing a redemption.

This calculation shows that a profit can be made when the cost of acquiring nUSD (at the market price) plus the redemption fee is less than the value of the cBTC received ($1 per nUSD redeemed).

The larger the trade size and the greater the difference between $1 and the sum of the nUSD price and redemption fee, the higher the potential profit.

## Redemption Fee

To manage the pace of redemptions, a dynamic redemption fee is applied on top of a 0.5% base fee and is paid to the Savings Account module. This fee scales upwards with increasing redemption volume, acting as a temporary deterrent, and then gradually decreases over 6 hours to help the system return to equilibrium.

The redemption fee comprises 3 components: the base fee ($f\_{min}$) of 0.5%, a linearly decaying buffer based on time, and the proportion between the current redemption amount and the nUSD total supply. The linearly decaying factor at time $t\_i$ is calculated as:

$$
\beta\_{(t\_i)} = \beta\_{(t\_{i-1})}\times(1 - \frac{\Delta t}{P})
$$

Where:

* ***B***<sub>***(ti)***</sub> represents the current value for the linearly decaying buffer.
* ***B***<sub>***(ti-1)***</sub> represents the previous value for the linearly decaying buffer.
* ***Delta t*** represents the time since the buffers' last update time, measured in seconds.
* ***P*** represents the configured period that resets the buffer to 0

By combining all 3 components, the redemption fee rate can be calculated for time $t\_i$ using:

$$
f(t\_i) = f\_{min} + \frac{K \times \[(\beta\_{(t\_i)} + T) \times \ln(\frac{T}{T - a}) - a]}{a}
$$

Where:

* ***f***<sub>***min***</sub> is being used to represent the base redemption fee rate of 0.5%.
* ***K*** represents a constant spike scaler, currently set to 1.
* ***a*** is used to represent the amount of nUSD being redeemed.
* ***T*** represents the total supply of nUSD.

After ***B***<sub>***ti***</sub> is used to calculate the redemption fee, the buffer is increased by the redemption amount, and the buffer's last update time is set to the current time.


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