Comparison with ESD, Basis, and Frax

Expansion
ESD/Basis
If token price > $1.0, ESD/Basis use the following equation to issue new tokens and distribute them to holders as expansion reward:
​expansion_reward=new_supply×(price−1) expansion\_reward = new\_supply \times (price - 1)expansion_reward=new_supply×(price−1)
​
or
​new_total_supply=total_supply×price new\_total\_supply = total\_supply \times pricenew_total_supply=total_supply×price
​
One concern such model is that it may over-issue tokens that the current market may not be able to support.  Further, if the LPs removes liquidity from the Uniswap pools and thus liquidity is low, selling the newly issued token can worsen the price and makes it harder to recover.
Example A: Suppose the algorithmic stablecoin is XSD (XSD = ESD/BAC/VSD),  Uniswap pool XSD/USDC has 100M XSD / 103M USDC with XSD price = $1.03, and total supply of XSD is 120M XSD.  Upon expansion, ESD/Basis will create 120M * 0.03 = 3.6M XSD as expansion reward.  If all of tokens are sold to the pool, then the pool will become 103.6M XSD / 99.42 USDC with XSD price = $0.9596. By contrast, VSD will issue and sell 1.488M VSD to the pool so that the pool becomes 101.488M XSD / 101.488 M USDC.  The received 1.51108 M USDC will be deposited into the collateral pool, and suppose collateral ratio (cr) = 90%, 1.51108M / 0.9 (cr) - 1.488M = 0.19 M VSD will become expansion reward. In this way, VSD always ensures that all newly issued VSD are backed by partial collateral at target collateral ratio.
Frax
In Frax, then the price of FRAX > $1.0, FRAX allows users to perform an arbitrage trade by mining $1.0 FRAX with
cr USDC
(1 - cr) worth of FXS (Frax share token)
and by selling new minted FRAX to the pool to bring price
This creates a buying power of FXS, which serves as algorithmic reserve of the Frax system, and the arbitrageurs will get significant part of the profit of seigniorage.
Example B: Suppose the algorithmic stablecoin is XSD (XSD = ESD/BAC/VSD),  Uniswap pool XSD/USDC has 100M XSD / 103M USDC with XSD price = $1.03, and collateral ratio (cr) is 90%.  In Frax, an arbitrageur is able to mint 1.488M FRAX, sell to the pool,  get 1.51108M USDC, and enjoy the profit 1.51108M - 1.488M = 0.023M.  To mint FRAX, the arbitrageur needs to buy (1 - cr) * 1.488M = $0.1488M worth of FXS. By contract, after performing auto-sell and partial collateralization,  VSD will have $0.19M VSD as expansion reward, which can be used to buy and burn the share token (VSS) from market or distribute the reward to LPs.  This means that in this scenario, VSD can create $0.19M buying power of the share token compared to $0.1488M of Frax.
Contraction
If price < $1.0, to encourage users to buy and burn excess amount of token, ESD/Basis/Frax offer the following ways
ESD/Basis issues debt and offers a coupon/bond market to encourage users to buy the debt using XSD at discounted price.  When the system enters into expansion stage, the coupon/bond holders will be prioritized to redeem coupon/bond for expansion reward.
Frax allows users to redeem one FRAX as an exchange for cr USDC and (1 - cr) worth of FXS.  The FXS is directly issued and thus the total supply of FXS can be increased.
VSD implements a hybrid contraction mechanism, where when price < $1.0, VSD will issue debt and a user
can buy coupon using VSD at discounted price; and/or
redeem the collateral at no extra cost.
(Note that if there is no debt, redeeming VSD will incur redemption fee).

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Expansion

ESD/Basis

Frax

Contraction