# How much will BNT pay for impermanent loss protection?

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Bancor is not only a pioneer of automatic market makers, but also proposed a solution for impermanent loss protection for the first time, which has made a huge contribution to the development of defi. So how much should it pay for the impermanent loss protection? Whether new BNT will be issued to the market is a question that many people care about for this IL solution. Now we can mathematically deduce how much will BNT pay for impermanent loss protection? First, let’s review the impermanence loss calculation formula of the xyk model:

Here we take BNT-USDT as an example. Assuming that when the price is P0, the liquidity provider deposits y0 USDT to the bancor protocol to provide single sided liquidity, then Bancor will issue an additional x0 BNT for the liquidity provider to deposit into the liquidity pool of BNT-USDT, and when the price moves to Pn, the liquidity provider withdraws USDT equivalent to y0 and LP fees, then the change in the net value of bancor calculated in BNT is: [(xn-x0) + (yn-y0)/pn] /x0;

The net change in USDT is: [(xn-x0)*pn + (yn-y0)]/y0.

Then the loss rate of impermanent loss protection calculated in USDT is 𝐼𝐿𝑃(𝑈𝑆𝐷𝑇 %)=[(xn-x0)*pn + (yn-y0)]/y0/y0–1

According to

We get:

The loss rate of impermanent loss protection calculated in BNT is 𝐼𝐿𝑃(BNT %)=[(xn-x0) + (yn-y0)/pn]/x0/x0–1

Based the same as above we get:

As can be seen from the above calculation, we can know that no matter whether the price of BNT-USDT rises or falls, BNT needs to bear impermanent loss. We can further observe the size of this loss with specific percentage.

If the price falls by 20% and 75%, then

That is to say, when the fluctuation of BNT/USDT exceeds 75%, the amount of BNT additional issuance into the market will double, and for trading pairs with less volatility, the additional issuance of BNT is less obvious. As long as the fluctuation does not exceed 10%, the withdrawal fee of 0.25% can basically cover this part of the loss, but if the price of BNT itself is unstable, its decline will lead to a vicious circle, because the additional issuance of BNT caused by the decline will be infinite, and the upper limit of the additional issuance caused by the increase of BNT is 100%. However, each removal of liquidity will result in an additional issuance of BNT, which can be accumulated in a continuous cycle. It will always increase as same as thermodynamic entropy. For example, if liquidity is removed when BNT/USDT falls by 20%, additional issuance will occur. 1.4%, then added the liquidity again, the liquidity was removed again when and the price rose by 20%, and another additional 1.4% was issued. Although the price of BNT/USDT did not change for the above circle, BNT was issued an additional 2.8%. The cycle repeats, even if the price fluctuates little, the cumulative additional issuance of BNT will also be relatively large, and especially for those tokens with severe fluctuations, the additional BNT will be much more.