Tightening the LTV Leash: Direct Effects on Liquidity and Liquidation Prices

25 Oct, 2023

In this post, we outline two key findings for combined trading and lending markets: a direct relationship of the loan-to-value ratio with the loan-to-reserves ratio and the post-liquidation price.

Current Lending Markets Are Unfavorable for Lenders

One main challenge in lending markets is handling loans that might not have enough collateral to back the borrowed amount. These are often termed "bad loans". Most of lending market smart contracts on blockchains currently refuse to deal with such problems themselves, and instead offload this task to lenders. This means lenders must assess factors like liquidity and decide if they're comfortable taking on the associated risks.

In contrast, Hoyu prioritizes lender safety. Specifically, Hoyu's smart contracts auto-liquidate loans right before they become problematic. It is possible to implement this mechanism in a set of smart contracts due to some surprisingly simple relationships governing loan liquidations. We derive those next.

Starting From A Loan

Think about two tokens: CUR, which stands for currency, and ALT for an altcoin. Imagine there's a trading market based on the usual formula $k = x \times y$ of UniswapV2 fame, along with a related lending market. A trading market of the $k = x \times y$ style is also known a constant product Automated Market Maker (AMM) or more colloquially as a decentralized exchange (DEX).

A user puts $L_{A L T}$ ALT tokens as collateral to borrow $L_{C U R}$ CUR tokens. You can think of $L_{C U R}$ as the main amount borrowed (the "principle") since we're not taking interest into account in this derivation. Consider the marginal situation, where this loan is just barely collateralized, that is, if one were to sell these ALT tokens, the CUR one'd get would be equal to the CUR borrowed.

Note that this single loan can be interpreted as an aggregate of all the outstanding loans for the CUR-ALT market, for example, when it comes to system-wide requirements or worst-case liquidation scenarios. We will see that issuing this loan at a given loan-to-value (LTV) ratio and a certain CUR/ALT spot price as computed from the reserves of the trading market gives a minimum CUR reserve requirement and a maximum bound on the price impact of the liquidation.

Two Important Results

Initial Setup and Price Definition

The trading market starts off with CUR and ALT reserves, labeled as $R_{C U R}$ and $R_{A L T}$ . Once again, we're setting aside interest and fees to keep things simple. At the time of borrowing, the spot price as set by the reserves of the trading market is defined as

p \equiv R_{C U R} / R_{A L T} .

The Importance of Loan-to-Value (LTV)

As a reminder, the loan consists of the principle $L_{C U R}$ and collateral $L_{A L T}$ . The LTV ratio is simply

LTV \equiv \frac{L_{C U R}}{L_{A L T} \times p} .

Liquidating the collateral, however, does not happen at the reserve price $p$ due to finite reserves. Instead, the amount of currency that one can obtain by selling the whole collateral is

Δ_{C U R} = \frac{R_{C U R} \times L_{A L T}}{R_{A L T} + L_{A L T}},

which can be derived by requiring that the product of the reserves $R_{C U R} \times R_{A L T}$ remains the same before and after the swap. Requiring that liquidating the collateral covers the principle of the loan

Δ_{C U R} \geq L_{C U R}

gives the minimum reserve ratio requirement for CUR:

\frac{L_{C U R}}{R_{C U R}} \leq 1 - L T V .

This is an important result, as it shows that a single system parameter, the LTV requirement, fully determines the loan size restriction in the protocol. It is also encouraging that the relationship is simple and straightforward to interpret. In particular, the more stringent (lower) the LTV requirement, the more CUR can be taken out as loans. This is because a liquidation causes a negative price impact, and a stringent LTV requirement establishes resilience against this price drop. In other words, there are two components to a successful liquidation: the magnitude of the price drop and the resilience against it. This resilience is determined by the LTV requirement.

Understanding Liquidation and Price Drop

Every liquidation leads to a price drop since collateral is being sold off. We can compute this price drop by examining the reserve price after the liquidation:

\tilde{p} \equiv \frac{R_{C U R} - Δ_{C U R}}{R_{A L T} + L_{A L T}} .

By comparing this with the starting price $p$ , we get

\frac{\tilde{p}}{p} \geq {L T V}^{2} .

This equation, our second big takeaway, tells us that the LTV alone sets the possible price drop size. We see from this inequality that a stringent (low) LTV requirement implies a large price drop. As discussed above, a large price drop during a liquidation is acceptable as long as the LTV requirement is stringent enough, since this allows to sell a large number of collateral relative to the initial collateral price.

Visualizing Key Insights

Let's paint a clearer picture by showing these insights visually.

It is useful to have in mind that a small swap (as compared to the trading market reserves) barely moves the price in the trading market. Therefore, to the first approximation, the execution price is equal to the reserves price for such a swap. With that in mind, let's look at these visuals.

Loan-To-Reserves Ratio Goes Down Linearly With LTV

See the figure below

Loan-to-reserves

for the minimum reserve ratio requirement $L_{C U R} / R_{C U R} \leq 1 - L T V$ . This figure shows that for large LTV ratios, only a small amount of CUR can be taken as loan. So, when the LTV ratio is nearing 1, you're restricted to tiny loans. Why? Because liquidating anything bigger would noticeably shift the price. This difference between the reserve price and the liquidation price requires a more conservative LTV ratio. On the contrary, for very low LTV ratios, large loans (again, as compared to trading market reserves) are allowed, since the conservative LTV absorbs the price impact. The shaded area in the figure shows the whole space of allowed loan sizes and LTV ratios.

Post-Liquidation Price Ratio Depends On LTV Quadratically

A similar line of thinking works for the second key result, too. See the figure below

Post-liquidation price

for the post-liquidation price ratio $\tilde{p} / p \geq {L T V}^{2}$ . This plot highlights that a larger LTV ratio leads to a smaller price shift during liquidation. Remember our first visual? It's because the loan size has to be really, really tiny. On the other hand, if loans with conservative LTV ratios are allowed, the post-liquidation price may well go to zero as the CUR reserves are exhausted due to large amounts of collateral being liquidated. The space that the lending protocol can exist in is an area since the LTV ratio of a loan can be equal to or smaller than the maximum LTV required by the protocol.

Conclusion and Implications for Hoyu

In conclusion, by merging trading and lending markets, and then analyzing them in a single framework, we can better understand how liquidity reserves interact with loan liquidations. Two simple results come out from that interplay: an increasingly stringent LTV requirement (1) linearly restricts lending liquidity and (2) quadratically decreases the price impact of the liquidation. These results are important for the Hoyu protocol, as they allow in-depth understanding and automation of the process of liquidations, as well as ensuring that no bad loans arise in the protocol.