Element Deep Dive

How Element Works

The Element Protocol wraps yield bearing assets and splits the resulting shares into a principal and a yield token.

Basic Features:

• PTs can be purchased at a discount to lock in a fixed yield rate
• Minting and Staking for additional rewards

Pricing Principal Tokens

Constant Power Sum Invariant

$$x^{1-t}+y^{1-t}=k$$

where

• $x$ is the reserves of the base asset
• $y$ is the reserves of the PT
• $t$ is the time to maturity
• $k$ is a constant

The Behavior of the Constant Power Sum Invariant Over Time

To calculate the result of a swap, solve for

$output$:

$$\begin{array}{rl}& (x-output{)}^{1-t}+(y+input{)}^{1-t}=k\\ & output=x-{(k-{(y+input)}^{(1-t)})}^{\frac{1}{1-t}}\end{array}$$

Fees are calculated as a percent of the yield:

$$\begin{array}{r}fee=(input-output)\times \varphi \end{array}$$

Currently,

$\varphi$ is set to .1

See Appendix A in the Element Construction Paper for more details

Configuring the Curve

Time Stretch

Intuitively, this parameter can be thought of like this:

$$\begin{array}{rl}t& =\frac{termlength}{365\times timestretch}\end{array}$$

Goal 1: Optimize the staking requirements for a particular APY

Time Stretch vs Reserve Ratio

Goal 2: Price Discovery

Time Stretch vs Price Discovery

Selecting the Time Stretch

The following plot shows what time stretches result in the ratio of the reserves (base/bond) approximately equal to the spot price of the associated APY:

the green dashed line is the following equation:

$$\begin{array}{r}{t}_{stretch}=\frac{3.09396}{0.02789\cdot AP{Y}_{PT}}(21)\end{array}$$

and approximates the suggested

$t_{stretch}$ for a given

$AP{Y}_{PT}$.

See Appendix B in the Element Construction Paper for more details

Market Simulations

The following represents a ycrvSTETH position, at 20% APY, with PTs trading at 10%:

The following table shows the various scenarios at different trading volumes. The final column shows the resulting output APR.

target_liquidity	trade_volume_sum	mean_daily_volume	apr
10,000,000	140,244,378	1,558,271	12.38
10,000,000	282,196,292	3,135,514	21.51

See Section 4.3 in the Element Construction Paper for more details

Leverage: Yield Token Compounding

Calculate the number of PTs and YTs after N Compounds

$$\begin{array}{rl}NumberofPTs& =P\cdot {(1-R_{fixed})}^N\\ NumberofYTs& =\frac{P\cdot (1-{(1-R_{fixed})}^{N+1})}{R_{fixed}}\\ AP{Y}_{boosted}& =\frac{P\cdot (1-{(1-R_{fixed})}^{N+1})}{R_{fixed}}\cdot R_{variable}+P\cdot {(1-R_{fixed})}^N\end{array}$$

$where$

• $P$ is the initial principal amount deposited
• $R_{fixed}$ is the fixed yield rate
• $R_{variable}$ is the variable yield rate

APY vs Number of Compounds Given different Rates

See Appendix C & Appendix D in the Element Construction Paper for more details

Element State Machine

Symbology

is a decision point

is an interaction with the Element Protocol

What Can Be Built On Element?

• Automated Leveraging
• Self paying loans
• Yield Ladders
• etc

We Are Hiring

https://element.fi/careers

Shoutouts

• Balancer Team
• Yearn Team
• YieldSpace authors:
  Allan Niemerg
  Dan Robinson
  Lev Livnev

The End