---
tags: general
---

# Element Finance: yield farming

In [Element Finance](https://hackmd.io/@pre-vert/element2), Investors deposit their asset in the protocol, mint $1$ pT and $1$ yT, sell the principal token, which price is $p$ and stake the proceeds. They obtain in expectation:
$$
(1+E(r))p + \text{yT} = (1+E(r))p + E(r)
$$

Let us suppose that the protocol subsidizes the yield by giving away a portion of its token supply. Investors stake the yield token in a special contract which gradually distributes the protocol's native token as a reward. The expeced return of the yT token is now $E(r)+\alpha$ with $\alpha >0$ the return subsidy:
$$
(1+E(r))p + \text{yT} = (1+E(r))p + E(r) + \alpha
$$

At equilibrium (no-arbirage condition), the market is indifferent between staking the yield and obtaining the vanilla return $E(r)$:
$$
(1+E(r))p + E(r) + \alpha = 1+E(r)
$$

The equilibrium price of the principal token is:
$$
p = \dfrac{1-\alpha}{1+E(r)}
$$

The expected return of holding pT at maturity is:
$$
\dfrac{1}{p} -1 = \dfrac{1+E(r)}{1-\alpha}-1 > E(r)
$$

Hence, to profit from the boosted yield, investors mint additional tokens, stake yT and sell pT which price is decreasing as a result. Arbitrage stops when the price of the pT is low enough so that the net return to the mint/split operation is just the market return. The value of the liquidity mining program is transferred to holders of the principal token which has now a greater implicit return.

## Compounding

The yield strategy can be compounded by repeatedly minting and selling the principal token. The $n$-loop strategy is as follows:

1. deposit 1 X, mint 1 PT and 1 YT
2. sell PT against $p$ X
3. deposit $p$ X, mint $p$ PT and $p$ YT
4. sell $p$ PT against $p^2$ X
5. ...

Total expected return with $n$ loops is:
$$
\big( 1+E(r) \big) p^n + (E(r) +\alpha)(1+p+...+p^{n-1})
$$

With an infinite number of loops (provided $p<1$):
$$
\dfrac{E(r) +\alpha}{1-p}
$$

At market equilibrium:
$$
\dfrac{E(r) +\alpha}{1-p} = 1+E(r)
$$

$$
p = \dfrac{1-\alpha}{1+E(r)}
$$

which gives the same equilibrium price.