How does LP token accrue yield?

The yield tokenization and yield accrual mechanism for LP tokens are similar to that of Compound's c-tokens.

In a typical X * Y = K AMM like Sushiswap / Uniswap, k represents the size of the pool. Upon liquidity addition, the liquidity provider receives LP tokens to represent his share of the pool. The share of the pool is quantified by sqrt(K), a unit called "Liquidity".

$Liquidity = sqrt(qtyTokenX*qtyTokenY)$

LP tokens represent the amount of Liquidity that an LP is entitled to. Learn more.

Swap fee is incurred on the transaction sender whenever a swap occurs. The swap fee added to the pool increases the size of the pool, increasing the value of Liquidity. Hence, when swaps occur, the Liquidity owned by each LP increases.

In other words, the amount of Liquidity per unit LP token increases as swaps occur. (1 LP tokens will be worth more unit of Liquidity)

LP Token Yield Tokenization

1 YT-LP represents the yield from 1 unit of Liquidity.

Yield for LP position is equivalent to the swap fees generated in a fixed time period. As swap fee incurred is added to the liquidity pool, yield accrued will be in the form of LP tokens.

1 OT represents the ownership of 1 unit of Liquidity (redeemable for 1 unit of Liquidity at expiry, NOT number of LP tokens).

To simply illustrate this, when LP token is deposited into Pendle, the amount of Liquidity it represents at that point in time is represented by OT. As swap happens, YT holders receive the extra Liquidity that the LP token accrues.

Example

The following example is to illustrate the concept explained above.

1st Jan 2022:

Let's say that at the very start, the PENDLE/ETH pool on Sushiswap has 10 ETH and 1000 PENDLE. We'll subsequently refer to this pool as PE.

• Amount of liquidity of the pool = sqrt(10*1000) = 100 liquidity

• Inititially, the total amount of LPs is also exactly 100 LPs

• 1 LP = 1 liquidity = 0.1 ETH + 10 PENDLE

1st July 2022

After some swaps, the PE pool could have 11 ETH and 1100 PENDLE

• Amount of liquidity of the pool now = sqrt(11*1100) = 110 liquidity

• The total amount of LPs is still 100 LPs (we assume no-one has entered or left the pool, for the sake of simplicity)

• 1 LP = 1.1 liquidity = 0.11 ETH + 11 PENDLE

Let's say user A tokenizes 10 PE SLP (=11 liquidity) in Pendle now, A will receive 11 OT-PE-Dec2022 and 11 YT-PE-Dec2022. A transfers the 11 YT to B

• By holding 11 OT-PE-Dec2022, A is entitled to receive 11 liquidity after the expiry (which will be paid in terms of LP tokens)

• Currently, 11 liquidity is equivalent to 1.1ETH + 110 PENDLE (10% of the Sushiswap pool)

• By holding 11 YT-PE-Dec2022, B is entitled to receive swap fees (in terms of LP tokens) from the moment B started holding the YT, until the expiry. If B transfers away the YT at some point, B will only receive the swap fees up to that point

1st Jan 2023:

After more swapping on Sushiswap happened, the PE pool could have 6 ETH and 2400 PENDLE

• Amount of liquidity of the pool = sqrt(6 * 2400) = 120

• The total amount of LPs is still 100 LPs

• 1 LP = 1.2 liquidity = 0.06 ETH + 24 PENDLE

A can now redeem his 11 OT-PE-Dec2022:

• 11 liquidity is quivalent to 11/120 = 9.17% of the Sushiswap pool = 0.55 ETH + 220 PENDLE

• 11 liquidity is equivalent to 9.17 LP now

• As such, A will get back 9.17 LP token

B has held 11 YT-PE-Dec2022 from 1st July 2022 to 1st Jan 2023, and will get the swap fees on 11 liquidity for this whole period:

• 1st July 2022: 11 liquidity was equivalent to 10 LPs

• 1st Jan 2023: 10 LPs now will be worth 10*1.2 = 12 liquidity. Therefore, the swap fees gained was 12-11 = 1 liquidity. Exchanged into LPs, this is equivalent to 1/12 = 0.83 LPs

We can see that 9.17 + 0.83 = 10. Essentially, the ownership of 10 LPs on 1st Jul 2022 was split into the ownership of the underlying liquidity for OT, and the swap fees for YT.