# Payoff, APY & Performance

Smilee Vault replicates the payoff of a DEX Concentrated Liquidity position on the pair with a base token (typically USDC) and the reference token.

The range is automatically selected using a multiple of the reference token volatility and it is auto re-balanced at the end of each epoch (i.e., at the maturity of Impermanent Gain positions).

Mathematically, the vault range is set as:

where:

sigma is the volatility at the beginning of the epoch

K is the price of the side token expressed in base token at the beginning of the epoch

tau is the duration of the epoch expressed in years

n is the volatility multiplier

This range is log-symmetrical with respect of K.

At the moment Smilee does not allow users to set specific ranges to minimize liquidity fragmentation and simplify the UX as much as possible. Nonetheless, future release will allow for more flexibility.

## Vault Payoff

Graphically, given a certain range ka and kb, the following chart represents the Equal Weight portfolio payoff and a DEX LP payoff:

As in Uniswap, DEX LP Payoff does NOT include APY. APY is considered an additional component earned on top of the LP Payoff as mathematically defined.

Smilee vault payoff will be the same as:

Equal Weight portfolio, if Smilee vault utilization is 0%

DEX LP, if Smilee vault utilization is 100%

In between the two, in proportion to the utilization level

This is possible thanks to Smilee Delta Hedging engine, which ensures that if utilization is lower than 100% (or when utilization changes due to traders buying & selling Impermanent Gain positions) vault gets back an "Impermanent Loss rebate" proportional to the part of the vault not utilized.

More details on delta hedging in Delta Hedging.

You can find an in-depth description of the delta hedging mechanism here.

## APY

Smilee benchmark for comparison is the DEX LP. Therefore, we defined the Vault APY as the performance of the vault on top of the LP payoff as mathematically defined (same as what Uniswap itself does).

In formulas, the Smilee APY is given by:

Where:

Smilee Earn is the performance of the Smilee vault

DEX LP is the performance of DEX LP with the same range of the vault

TVL0 is the vault TVL a the beginning of the epoch

As a result of such definition it is possible to easily **compare Smilee and Uniswap **by comparing **Smilee Vault APY and Uniswap APY**, using for Uniswap the same pair of tokens, the same time horizon and the same range.

Financially, Smilee APY is the result of multiple components:

Premium paid by Impermanent Gain buyers

Bid Ask spread in the Synthetic AMM

Delta Hedge PnL and Impermanent Loss Rebate

## Performance

APY focuses on the overperformance on top of the mathematically defined DEX LP.

Vault Performance instead captures all the vault performance components, which are: APY, Impermanent Loss and Positive Delta exposure.

To better understand, looking at the payoff chart at the top:

The Equal Weight portfolio shows clearly the positive delta exposure: if the market goes up, the portfolio grows in value, and viceversa

The DEX LP shows, in addition to the delta exposure, the impact of Impermanent Loss: if the market goes up, the LP makes less money than the Equal Weight portfolio and viceversa

As said in the Payoff chapter, Smilee sits in between depending on the utilization rate.

As a result a vault may have:

Positive APY and even more positive performance, when market goes up

Positive APY but negative performance, when market goes down causing losses bigger than the APY earned

Any combination of the above

It is important to understand that Smilee users can easily hedge vault delta and, thus, remain exposed only to Impermanent Loss in exchange for APY.

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