🔬Asset Liability Management

The concept of Asset Liability Management is fundamental to understanding the Möbius Exchange algorithm. Let's dive in.

Asset Liability Management (ALM), a concept from finance and accounting, uses assets and liabilities to assess risk. Platypus Finance has introduced the use of ALM within an Automated Market Maker (AMM).

In a pool with multiple assets (e.g., USDT, USDC, DAI), each asset is held by a token account. Within each token account:

• Liability is the amount of tokens deposited by Liquidity Providers (LPs).

• Asset is the amount of tokens in the pool.

Initially, the liability and asset amounts are equal. However, as traders swap assets, some token accounts may end up with assets exceeding liabilities, while others have assets that are less than their liabilities.

The Coverage Ratio for each token account, denoted “r” is calculated as Asset ÷ Liability.

Definition - 2.1

$$\text{coverage ratio}_i = r_i = \frac{\text{asset}_i}{\text{lliability}_i} = \frac{a_i}{l_i}$$

The tables below provides an example.

	Asset	Liability	Coverage ratio
USDT	100,000	100,000	100% (equilibrium)
USDC	100,000	100,000	100% (equilibrium)
DAI	40,000	40,000	100% (equilibrium)

Table 2.1. The initial assets of the Pool are provided by LPs deposits. Each token account starts with a 100% coverage ratio. Note that each token account has a different size.

	Asset	Liability	Coverage ratio
USDT	103,000	100,000	103% (over covered)
USDC	100,000	100,000	100% (equilibrium)
DAI	37,001	40,000	92.5% (under covered)

Table 2.2. This example illustrates the impact of a trade on coverage ratios. A trader exchanges 3000 USDT for 2999 DAI, the USDT coverage ratio increases from 100% to 103%, while the DAI coverage ratio decreases from 100% to 92.5%.

Coverage Ratio and Equilibrium

When a token account's coverage ratio is greater than or equal to 100%, it possesses sufficient assets to fully reimburse all LPs upon their withdrawal. Hence the equilibrium state is defined as when all token accounts have the same coverage ratio. Note that equilibrium doesn't equate to all token accounts having a 100% coverage ratio. This is because trading fees accumulate within the pool, the assets will gradually increase over time even if liability remains the same, i.e. no further deposit or withdrawal.

At equilibrium, all token accounts have the same coverage ratios.

$$r_1 = r_2 = ... = r_n$$

The advantage of using this definition of equilibrium is that each token can have varying liquidity sizes. As shown in Table 2.1, even if DAI has lower liquidity, the pool won't perceive it as an imbalance and attempt to absorb more DAI, as long as the coverage ratios are balanced.

As we will see in the next sections, the pool would provide an attractive rate to encourage trades that align its state with equilibrium, while offering a less favorable rate to deter trades that deviate from it.