There are several design differences between Compound III and Aave V3 such as

• many-to-many vs many-to-one lending

  • both protocols allow for multiple collateral assets but Compound III only has one base asset available to borrow per comet

• rehypothecation

  • on Aave V3, borrowers get additional yield on collateral assets at the cost of increased risk exposure from more asset pairs along with liquidity crunches and liquidation failures at high utilization rates

• interest rate curves

  • Aave V3 has a single interest rate curve for both borrowing and lending with the rate spread determined by the reserve factor whereas Compound III has separate borrow and supply rate curves, which allows for more parameterization but introduces the possibility of negative reserve growth due to inverted rates

• risk segmentation

  • Aave V3 has efficiency mode to allow for more borrowing power for certain pairs e.g. LRT vs base asset and isolation mode which places borrowing restrictions to contain the risk of new or volatile collaterals

While these are all worth noting, it’s Compound III’s liquidation design that is it’s main limiting factor when raising CFs, putting it at a disadvantage when competing with Aave V3 on capital efficiency

For loans with a single collateral and borrow asset on Aave V3, the maximum borrowable amount is determined by the loan to value ratio (LTV), while liquidations are triggered when the debt-to-collateral ratio exceeds the liquidation threshold (LT).

During a liquidation, a portion of the borrower's debt is repaid, and they forfeit the corresponding amount of collateral, scaled by the liquidation bonus (LB). Of this bonus, a liquidation protocol fee (LPF) is collected by the protocol, while the remainder is paid to the liquidator. Additionally, the amount of debt that a liquidator is allowed to repay in a single transaction is constrained by the close factor ($c$).

We can derive the following relationship

$\displaystyle LTV \leq LT \leq \frac{1}{1+LB}$

where the left hand side of the inequality is set to prevent opening loans that are immediately liquidatable and the right hand side is set to ensure liquidations improve the collateralization of the loan.

Comparing this from a previous post covering absorptions on Compound III we note some parallels

1. $LTV$ on Aave V3 is analogous to $CF$ on Compound III

2. $LT$ corresponds to $LF$

3. $LB$ serves a similar purpose to $LP$

along with several key differences

1. Compound III takes over the full position when absorb is called and pays out the difference between the debt and collateral sans liquidation penalty as base asset to the borrower out of the protocol reserves. This leads to warehousing a lot more immediate price risk compared to Aave V3 where the close factor ensures that liquidations occur more gradually

2. The insolvency buffer on Compound III is independent of $LF$ and entirely determined by $LP$ and $SFPF$. On Aave V3 the insolvency buffer is dependent on $LT$, $LB$, $LPF$ and $c$. As such, Aave V3 gets shielded from insolvency from the close factor and the gap between $LT$ and $LT_{max}$ whereas Compound III does not

For Aave V3 we define the insolvency buffer (IB) as the maximum % the collateral price can drop relative to the debt between the loan being liquidatable and insolvent.

Assuming that a liquidator liquidates up to the full close factor to maximize their profit, the protocol breaks even on the loan if the post liquidation collateral plus the protocol proceeds from the liquidation equals the remaining debt.

As a proportion of collateral, the value of the collateral in a price drop is given by $1 - IB$ and the size of the liquidation is $LT \cdot c (1+LB)$ since it's the debt value scaled by the liquidation bonus and the close factor. The protocol liquidation gains are $LT \cdot c \cdot LB \cdot LPF$ and the remaining debt is $LT(1-c)$. Putting this altogether gives the relation

$1 - IB - LT \cdot c (1 + LB) + LT \cdot c \cdot LB \cdot LPF = LT (1 - c)$

which when rearranged gives

$IB = 1 - LT (1+c \cdot LB (1- LPF))$

There is an edge case to consider when the liquidation is bottlenecked by the loan's collateral value rather than the close factor. In this case, all the collateral is repurchased by the liquidator, and the protocol breaks even if the protocol liquidation proceeds equals the remaining debt.

Here the liquidation size in debt terms is

$\displaystyle\frac{1-IB}{1+LB}$ so the protocol liquidation proceeds are $\displaystyle\frac{LB \cdot LPF(1-IB)}{1+LB}$ and the remaining debt is given by $LT - \displaystyle\frac{1-IB}{1+LB}$ so equating them gives the relation

$\displaystyle\frac{LB \cdot LPF(1-IB)}{1+LB} = LT - \frac{1-IB}{1+LB}$

which when rearranged gives

$IB = 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF}$

This edge case occurs when $1 - IB \geq LT \cdot c (1+LB)$ and this edge bound is valid if $c \geq \displaystyle\frac{1}{1 + LB \cdot LPF}$ which is also the intersection point of the two bounds. Putting this altogether gives

$IB = \begin{cases} 1 - LT (1+c \cdot LB (1- LPF)) & \text{if } c < \displaystyle\frac{1}{1 + LB \cdot LPF} \ 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF} & \text{if } c \geq \displaystyle\frac{1}{1 + LB \cdot LPF} \ \end{cases}$

Now, with these formulations in place, let's gain deeper insight into how into how raising $LF$/$LT$ impacts the insolvency buffer for the two protocols. To help reduce dimensionality, let's define the aggressiveness (a) of each protocol as$\displaystyle a_{\text{Aave V3}} = \frac{LT}{LT_{\max}} = LT(1+LB)$ and

$\displaystyle a_{\text{Compound III}} = \frac{LF}{LF_{\max}} = \frac{LF}{1-LP}$

This ratio normalizes the LT and LF relative to their respective maximum values, providing a standardized measure of how aggressively the protocol is setting its risk parameters.

Currently, the highest aggressiveness on Aave V3 is just under 96.5% for wstETH, with a 95.5% LT and a 1% LB in the ETH correlated emode on their mainnet Prime market. For Compound III, the highest aggressiveness is slightly below 97% for weETH, with a 93% LF and a 4% LP in the mainnet WETH comet. Using these empirical values to set LB/LP based on LT/LF along with SFPF at 0.7 as per the mainnet WETH comet, LPF at 0.1 as set for ezETH and a close factor of 0.5 as set in the Aave V3 contracts produces the following plot.

As an aside, the x-axis is cut off at 70% LF since with these aggression values LFs below 70% correspond to unrealistically high LPs (30%+).

As shown, with current parameters the insolvency buffer declines with higher liquidation factors as expected. More notably, the IB is substantially higher on Aave V3 than Compound III pretty much irrespective of LF.

This begs the question as to whether Compound III can make parameter changes to reduce or even flip this difference. Since both protocols are already near max aggression, the only parameter Compound III can really tweak is the store front price factor.

The difference in insolvency buffer between the two protocols can be expressed as

$\delta = \begin{cases} 1 - LT (1+c \cdot LB (1- LPF)) - \bigg(1 - \displaystyle\frac{1-LP}{1-LP \cdot SFPF} \bigg) & \text{if } c < \displaystyle\frac{1}{1 + LB \cdot LPF} \ 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF} - \bigg(1 - \frac{1-LP}{1-LP \cdot SFPF} \bigg) & \text{if } c \geq \displaystyle\frac{1}{1 + LB \cdot LPF} \ \end{cases}$

Equating analogous parameters so $LT = LF$, $LB = LP$ and $LPF = 1 - SFPF$ and simplifying this becomes

$\delta = \begin{cases} \displaystyle\frac{1-LP}{1-LP \cdot SFPF} - LF (1+c \cdot LP \cdot SFPF) & \text{if } c < \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \displaystyle\frac{1-LP}{1-LP \cdot SFPF} - \displaystyle\frac{LF(1+LP)}{1+LP(1-SFPF)} & \text{if } c \geq \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \end{cases}$

Taking the partial derivative with respect to $SFPF$ gives

$\frac{\partial\delta}{\partial SFPF} = \begin{cases} \displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} - c \cdot LF \cdot LP & \text{if } c < \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} - \frac{LF \cdot LP (1+LP)}{(1 + LP (1-SFPF))^2} & \text{if } c \geq \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \end{cases}$

In the first case

$\displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} \geq (1-LP)LP \geq LF \cdot LP \geq c \cdot LP \cdot LF$

where all the inequalities come from $LP, SFPF, LF$ $\in [0, 1]$ and $LF \leq 1 - LP$

In the second case

$\displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} \geq \frac{LF \cdot LP}{(1-LP \cdot SFPF)^2}$

so showing

$\displaystyle\frac{1}{(1- LP \cdot SFPF)^2} \geq \frac{1+LP}{(1 + LP (1-SFPF))^2}$

would show that

$\displaystyle\frac{\partial\delta}{\partial SFPF} \geq 0$

over the parameter region

Equality holds when $LP = 0$ and otherwise cross multiplying and simplifying gives the inequality

$1 + LP \geq LP^2 \cdot SFPF^2$

which is strict. Thus

$\displaystyle\frac{\partial\delta}{\partial SFPF} \geq 0$ so $\delta$ is minimized when $SFPF = 0$

$\delta$ is also univariately minimized when $c=1$. This is because $IB$ for Aave V3 decreases as the close factor increases. Ultimately, when the close factor reaches 1, it becomes equivalent to not having a close factor at all (which does occur on Aave when loan collateralization drops below 95% of LT).

Using these extreme parameters gives the following plot

Unfortunately, even in this extremely favorable setting, the insolvency buffer remains greater for Aave V3 across the board. However, it is worth noting that the gap is much smaller, highlighting the value of having a close factor to facilitate partial liquidations.

In practice, this scenario would almost never occur since it would essentially require liquidators to work for free while also disabling a key safety feature in Aave V3. This is why the plot is labeled as a near best-case comparison, rather than a realistic setting.

The protocol can also be examined from the lens of the cost to the user getting liquidated. By inverting the insolvency buffer equations above to solve for the liquidation penalty and converting costs to collateral terms, we obtain the following plot using the existing parameters.

For a fixed insolvency buffer, Compound III does eventually charge the user less than Aave V3, but only when the collateral cost exceeds ~65%. At all cost levels that a user would realistically find acceptable, Aave V3 charges less for the same insolvency buffer as Compound III.

Applying the same favorable extrema as before gives

The cost gap shrinks, but Compound III only overtakes Aave V3 when the collateral cost exceeds 35%, which is still too high in practice to be actionable. Even in this case, Aave V3 still outperforms Compound III in terms of user costs.

As a concrete example of the results shown above, let's examine what happens when we try to provide the same amount of leverage for ezETH on Compound III as is currently allowed on Aave V3.

Currently ezETH has an LTV of 0.93, LT of 0.95 and LB of 0.01 in LRT wstETH emode on Aave V3's mainnet Prime market. This corresponds to an IB of ~5%.

Now, suppose Compound III raised CF and LF to 0.93 and 0.95, matching Aave V3. The maximum possible LP in this case is 0.05, which results in an IB of ~1.6% with the existing SFPF—less than 1/3 of Aave’s IB while incurring over 5× the user cost. To achieve IB parity with Aave by modifying the SFPF, it would need to be reduced from 0.7 to ~0.01. Additionally, SFPF is set per comet, rather than per asset so such a change would drastically affect all other collateral assets in the comet.

If Compound III were to match Aave V3 on user cost with an LP of 0.01, it would yield an insolvency buffer of ~30 bps, which is more than 16× smaller than Aave’s IB.

Alternatively, if Compound III attempted to match Aave V3’s IB by raising the LP, it would require an LP of nearly 0.15. Not only would this be >15× costlier to users than Aave, but it would also constrain the maximum CF to 0.85, reducing available leverage.

Because of Compound III's liquidation mechanism, it is essentially impossible in practice to match the leverage offered by Aave V3 without either costlier liquidations for borrowers or a smaller insolvency buffer for the protocol.

There are several design differences between Compound III and Aave V3 such as

• many-to-many vs many-to-one lending

  • both protocols allow for multiple collateral assets but Compound III only has one base asset available to borrow per comet

• rehypothecation

  • on Aave V3, borrowers get additional yield on collateral assets at the cost of increased risk exposure from more asset pairs along with liquidity crunches and liquidation failures at high utilization rates

• interest rate curves

  • Aave V3 has a single interest rate curve for both borrowing and lending with the rate spread determined by the reserve factor whereas Compound III has separate borrow and supply rate curves, which allows for more parameterization but introduces the possibility of negative reserve growth due to inverted rates

• risk segmentation

  • Aave V3 has efficiency mode to allow for more borrowing power for certain pairs e.g. LRT vs base asset and isolation mode which places borrowing restrictions to contain the risk of new or volatile collaterals

While these are all worth noting, it’s Compound III’s liquidation design that is it’s main limiting factor when raising CFs, putting it at a disadvantage when competing with Aave V3 on capital efficiency

For loans with a single collateral and borrow asset on Aave V3, the maximum borrowable amount is determined by the loan to value ratio (LTV), while liquidations are triggered when the debt-to-collateral ratio exceeds the liquidation threshold (LT).

During a liquidation, a portion of the borrower's debt is repaid, and they forfeit the corresponding amount of collateral, scaled by the liquidation bonus (LB). Of this bonus, a liquidation protocol fee (LPF) is collected by the protocol, while the remainder is paid to the liquidator. Additionally, the amount of debt that a liquidator is allowed to repay in a single transaction is constrained by the close factor ($c$).

We can derive the following relationship

$\displaystyle LTV \leq LT \leq \frac{1}{1+LB}$

where the left hand side of the inequality is set to prevent opening loans that are immediately liquidatable and the right hand side is set to ensure liquidations improve the collateralization of the loan.

Comparing this from a previous post covering absorptions on Compound III we note some parallels

1. $LTV$ on Aave V3 is analogous to $CF$ on Compound III

2. $LT$ corresponds to $LF$

3. $LB$ serves a similar purpose to $LP$

along with several key differences

1. Compound III takes over the full position when absorb is called and pays out the difference between the debt and collateral sans liquidation penalty as base asset to the borrower out of the protocol reserves. This leads to warehousing a lot more immediate price risk compared to Aave V3 where the close factor ensures that liquidations occur more gradually

2. The insolvency buffer on Compound III is independent of $LF$ and entirely determined by $LP$ and $SFPF$. On Aave V3 the insolvency buffer is dependent on $LT$, $LB$, $LPF$ and $c$. As such, Aave V3 gets shielded from insolvency from the close factor and the gap between $LT$ and $LT_{max}$ whereas Compound III does not

For Aave V3 we define the insolvency buffer (IB) as the maximum % the collateral price can drop relative to the debt between the loan being liquidatable and insolvent.

Assuming that a liquidator liquidates up to the full close factor to maximize their profit, the protocol breaks even on the loan if the post liquidation collateral plus the protocol proceeds from the liquidation equals the remaining debt.

As a proportion of collateral, the value of the collateral in a price drop is given by $1 - IB$ and the size of the liquidation is $LT \cdot c (1+LB)$ since it's the debt value scaled by the liquidation bonus and the close factor. The protocol liquidation gains are $LT \cdot c \cdot LB \cdot LPF$ and the remaining debt is $LT(1-c)$. Putting this altogether gives the relation

$1 - IB - LT \cdot c (1 + LB) + LT \cdot c \cdot LB \cdot LPF = LT (1 - c)$

which when rearranged gives

$IB = 1 - LT (1+c \cdot LB (1- LPF))$

There is an edge case to consider when the liquidation is bottlenecked by the loan's collateral value rather than the close factor. In this case, all the collateral is repurchased by the liquidator, and the protocol breaks even if the protocol liquidation proceeds equals the remaining debt.

Here the liquidation size in debt terms is

$\displaystyle\frac{1-IB}{1+LB}$ so the protocol liquidation proceeds are $\displaystyle\frac{LB \cdot LPF(1-IB)}{1+LB}$ and the remaining debt is given by $LT - \displaystyle\frac{1-IB}{1+LB}$ so equating them gives the relation

$\displaystyle\frac{LB \cdot LPF(1-IB)}{1+LB} = LT - \frac{1-IB}{1+LB}$

which when rearranged gives

$IB = 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF}$

This edge case occurs when $1 - IB \geq LT \cdot c (1+LB)$ and this edge bound is valid if $c \geq \displaystyle\frac{1}{1 + LB \cdot LPF}$ which is also the intersection point of the two bounds. Putting this altogether gives

$IB = \begin{cases} 1 - LT (1+c \cdot LB (1- LPF)) & \text{if } c < \displaystyle\frac{1}{1 + LB \cdot LPF} \ 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF} & \text{if } c \geq \displaystyle\frac{1}{1 + LB \cdot LPF} \ \end{cases}$

Now, with these formulations in place, let's gain deeper insight into how into how raising $LF$/$LT$ impacts the insolvency buffer for the two protocols. To help reduce dimensionality, let's define the aggressiveness (a) of each protocol as$\displaystyle a_{\text{Aave V3}} = \frac{LT}{LT_{\max}} = LT(1+LB)$ and

$\displaystyle a_{\text{Compound III}} = \frac{LF}{LF_{\max}} = \frac{LF}{1-LP}$

This ratio normalizes the LT and LF relative to their respective maximum values, providing a standardized measure of how aggressively the protocol is setting its risk parameters.

Currently, the highest aggressiveness on Aave V3 is just under 96.5% for wstETH, with a 95.5% LT and a 1% LB in the ETH correlated emode on their mainnet Prime market. For Compound III, the highest aggressiveness is slightly below 97% for weETH, with a 93% LF and a 4% LP in the mainnet WETH comet. Using these empirical values to set LB/LP based on LT/LF along with SFPF at 0.7 as per the mainnet WETH comet, LPF at 0.1 as set for ezETH and a close factor of 0.5 as set in the Aave V3 contracts produces the following plot.

As an aside, the x-axis is cut off at 70% LF since with these aggression values LFs below 70% correspond to unrealistically high LPs (30%+).

As shown, with current parameters the insolvency buffer declines with higher liquidation factors as expected. More notably, the IB is substantially higher on Aave V3 than Compound III pretty much irrespective of LF.

This begs the question as to whether Compound III can make parameter changes to reduce or even flip this difference. Since both protocols are already near max aggression, the only parameter Compound III can really tweak is the store front price factor.

The difference in insolvency buffer between the two protocols can be expressed as

$\delta = \begin{cases} 1 - LT (1+c \cdot LB (1- LPF)) - \bigg(1 - \displaystyle\frac{1-LP}{1-LP \cdot SFPF} \bigg) & \text{if } c < \displaystyle\frac{1}{1 + LB \cdot LPF} \ 1 - \displaystyle\frac{LT(1+LB)}{1+LB\cdot LPF} - \bigg(1 - \frac{1-LP}{1-LP \cdot SFPF} \bigg) & \text{if } c \geq \displaystyle\frac{1}{1 + LB \cdot LPF} \ \end{cases}$

Equating analogous parameters so $LT = LF$, $LB = LP$ and $LPF = 1 - SFPF$ and simplifying this becomes

$\delta = \begin{cases} \displaystyle\frac{1-LP}{1-LP \cdot SFPF} - LF (1+c \cdot LP \cdot SFPF) & \text{if } c < \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \displaystyle\frac{1-LP}{1-LP \cdot SFPF} - \displaystyle\frac{LF(1+LP)}{1+LP(1-SFPF)} & \text{if } c \geq \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \end{cases}$

Taking the partial derivative with respect to $SFPF$ gives

$\frac{\partial\delta}{\partial SFPF} = \begin{cases} \displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} - c \cdot LF \cdot LP & \text{if } c < \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} - \frac{LF \cdot LP (1+LP)}{(1 + LP (1-SFPF))^2} & \text{if } c \geq \displaystyle\frac{1}{1 + LP (1-SFPF)} \ \end{cases}$

In the first case

$\displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} \geq (1-LP)LP \geq LF \cdot LP \geq c \cdot LP \cdot LF$

where all the inequalities come from $LP, SFPF, LF$ $\in [0, 1]$ and $LF \leq 1 - LP$

In the second case

$\displaystyle\frac{(1-LP)LP}{(1-LP \cdot SFPF)^2} \geq \frac{LF \cdot LP}{(1-LP \cdot SFPF)^2}$

so showing

$\displaystyle\frac{1}{(1- LP \cdot SFPF)^2} \geq \frac{1+LP}{(1 + LP (1-SFPF))^2}$

would show that

$\displaystyle\frac{\partial\delta}{\partial SFPF} \geq 0$

over the parameter region

Equality holds when $LP = 0$ and otherwise cross multiplying and simplifying gives the inequality

$1 + LP \geq LP^2 \cdot SFPF^2$

which is strict. Thus

$\displaystyle\frac{\partial\delta}{\partial SFPF} \geq 0$ so $\delta$ is minimized when $SFPF = 0$

$\delta$ is also univariately minimized when $c=1$. This is because $IB$ for Aave V3 decreases as the close factor increases. Ultimately, when the close factor reaches 1, it becomes equivalent to not having a close factor at all (which does occur on Aave when loan collateralization drops below 95% of LT).

Using these extreme parameters gives the following plot

Unfortunately, even in this extremely favorable setting, the insolvency buffer remains greater for Aave V3 across the board. However, it is worth noting that the gap is much smaller, highlighting the value of having a close factor to facilitate partial liquidations.

In practice, this scenario would almost never occur since it would essentially require liquidators to work for free while also disabling a key safety feature in Aave V3. This is why the plot is labeled as a near best-case comparison, rather than a realistic setting.

The protocol can also be examined from the lens of the cost to the user getting liquidated. By inverting the insolvency buffer equations above to solve for the liquidation penalty and converting costs to collateral terms, we obtain the following plot using the existing parameters.

For a fixed insolvency buffer, Compound III does eventually charge the user less than Aave V3, but only when the collateral cost exceeds ~65%. At all cost levels that a user would realistically find acceptable, Aave V3 charges less for the same insolvency buffer as Compound III.

Applying the same favorable extrema as before gives

The cost gap shrinks, but Compound III only overtakes Aave V3 when the collateral cost exceeds 35%, which is still too high in practice to be actionable. Even in this case, Aave V3 still outperforms Compound III in terms of user costs.

As a concrete example of the results shown above, let's examine what happens when we try to provide the same amount of leverage for ezETH on Compound III as is currently allowed on Aave V3.

Currently ezETH has an LTV of 0.93, LT of 0.95 and LB of 0.01 in LRT wstETH emode on Aave V3's mainnet Prime market. This corresponds to an IB of ~5%.

Now, suppose Compound III raised CF and LF to 0.93 and 0.95, matching Aave V3. The maximum possible LP in this case is 0.05, which results in an IB of ~1.6% with the existing SFPF—less than 1/3 of Aave’s IB while incurring over 5× the user cost. To achieve IB parity with Aave by modifying the SFPF, it would need to be reduced from 0.7 to ~0.01. Additionally, SFPF is set per comet, rather than per asset so such a change would drastically affect all other collateral assets in the comet.

If Compound III were to match Aave V3 on user cost with an LP of 0.01, it would yield an insolvency buffer of ~30 bps, which is more than 16× smaller than Aave’s IB.

Alternatively, if Compound III attempted to match Aave V3’s IB by raising the LP, it would require an LP of nearly 0.15. Not only would this be >15× costlier to users than Aave, but it would also constrain the maximum CF to 0.85, reducing available leverage.

Because of Compound III's liquidation mechanism, it is essentially impossible in practice to match the leverage offered by Aave V3 without either costlier liquidations for borrowers or a smaller insolvency buffer for the protocol.