Impermanent Loss

Executive Summary

Impermanent Loss (IL) is a fundamental risk that liquidity providers face in Automated Market Maker (AMM) pools. However, our analysis demonstrates that FIVA's yield tokenization pools exhibit unique IL characteristics that make them significantly more predictable and bounded compared to traditional AMM pools. Based on extensive research and empirical data, we've found that:

1. Maximum IL in FIVA pools typically ranges between 1-3%, substantially lower than conventional crypto trading pairs

2. IL follows predictable patterns related to the pools' lifecycle and naturally decreases as maturity approaches

3. Trading fees often outweigh potential IL, especially for positions held for longer periods

This section provides a detailed explanation of IL in FIVA's yield tokenization pools, including methodology, empirical evidence, and strategies for liquidity providers.

The Special Nature of IL in Yield Markets

In standard AMM pools (like those for trading pairs such as TON/USDT), impermanent loss occurs when the price ratio between the two assets changes from the initial deposit ratio. However, in yield tokenization pools, IL has unique properties due to the convergence behavior of Principal Tokens (PTs).

Unlike regular trading pairs where assets can move independently indefinitely, PTs and their underlying assets have a mathematically determined convergence point: at maturity, the PT price will equal the face value. This creates a "pull to par" effect that fundamentally changes the IL dynamic.

In FIVA pools, IL can still occur but is generally more bounded and predictable than in standard pools due to this convergence property. Based on our analysis of yield tokenization protocols, the maximum IL in these pools typically ranges from 1-3%, which is significantly lower than what might be experienced in standard crypto trading pairs.

Understanding Our Custom AMM Formula

FIVA utilizes a specialized AMM formula designed for yield tokenization markets:

x^3 × price^3 × y + y^3 × x × price = k

Where:

• x: quantity of SY tokens (underlying yield-bearing asset)

• y: quantity of PT tokens (principal tokens)

• price: the price ratio between the tokens

• k: the invariant constant maintained by the AMM

This formula was specifically chosen to optimize token rebalancing behavior for yield markets, providing several advantages over traditional constant product (x*y=k) formulas:

1. Better handling of the convergence property of PTs

2. More capital-efficient price curves near the expected equilibrium

3. Reduced IL during typical market movements

4. Improved stability during early pool price discovery

IL Calculation Methodology

To provide a comprehensive understanding of impermanent loss in FIVA pools, we employ two complementary calculation approaches:

Theoretical IL Calculation

Theoretical IL represents the mathematically expected impermanent loss based purely on our custom AMM formula. This calculation:

1. Starts with the initial token quantities and price ratio

2. For each possible new price ratio:

   • Solves the AMM invariant equation to determine new token quantities

   • Calculates what these new quantities would be worth (AMM value)

   • Calculates what the original tokens would be worth at the new price (HODL value)

   • Computes IL as: (AMM value / HODL value) - 1

The formula used in our calculation is:

x^3 × price^3 × y + y^3 × x × price = k

Where:

• x = SY token quantity

• y = PT token quantity

• price = price ratio

• k = invariant constant

This mathematical approach allows us to construct a theoretical IL curve across all possible price ratios, identifying the theoretical maximum IL that could occur in a pool.

Empirical IL Calculation

Empirical IL represents the actual impermanent loss observed in live pools based on real market data. This calculation:

1. Uses historical data of LP token prices, PT prices, and SY prices from our pools

2. For each time point:

   • Calculates the current value of the LP position based on LP token price

   • Calculates what the value would be if the original tokens were held separately

   • Computes IL as: (LP position value / HODL position value) - 1

This real-world approach captures all market factors, including trader behavior, time effects, and actual pool compositions. It verifies that our theoretical models align with actual LP experiences.

Empirical Analysis of IL in FIVA Pools

We've conducted extensive analysis of IL across multiple FIVA pools. Here's a summary of our findings:

Pool	Theoretical Worst IL	Theoretical Best IL	Empirical Worst IL	Empirical Best IL
tsTON	-0.9160%	1.1589%	-0.2062%	0.3756%
EVAA	-0.0111%	6.9098%	-1.3294%	1.7749%
USDT SLP	0.0000%	0.3474%	-0.0392%	0.0704%
TON SLP	-0.1679%	0.1852%	-0.0222%	0.0080%
NOT SLP	-0.9577%	0.3756%	-0.0073%	3.7433%

In this table:

• Theoretical Worst/Best IL: The minimum and maximum IL values mathematically possible based on our AMM formula across all observed price ratios

• Empirical Worst/Best IL: The minimum and maximum IL values actually observed in the live pools based on real market data

As this data demonstrates, both theoretical and empirical IL in FIVA pools typically remains well below 3%, with many pools experiencing maximum IL below 1%. The empirical results closely align with theoretical expectations, confirming the effectiveness of our custom AMM formula for yield tokenization markets.

How IL Develops in Yield Tokenization Pools

Impermanent loss in FIVA pools can develop through several patterns:

Early Pool Volatility

When a new pool launches, there's often a period of price discovery as the market determines the appropriate discount rate for the PTs. This period may see higher volatility and consequently higher IL for early liquidity providers. Our data suggests this initial IL can reach up to 2% during the first few weeks.

Interest Rate Shifts

When market interest rates change significantly, the relative value of fixed-rate PTs versus variable-rate underlying assets also changes. This creates divergence that translates to IL for liquidity providers.

Market Sentiment Shifts

Changes in sentiment about future yield prospects can cause temporary price movements in both PTs and YTs. These sentiment-driven movements can create temporary IL, though they typically mean-revert over time.

Yield Token Hype Cycles

When there's significant hype around the YT component (perhaps due to speculation about future protocol incentives), this can indirectly affect PT pricing and create temporary IL for liquidity providers in PT/underlying pools.

Quantifying IL Through Time

An important characteristic of IL in yield tokenization pools is its time-dependent nature. The IL experience depends significantly on when you enter and exit the pool:

Early Entry IL Pattern

Liquidity providers who enter pools very early (first few days after launch) may experience higher IL as the pool undergoes initial price discovery. Our analysis shows this IL typically peaks around 2% between 3-4 weeks after pool launch, but often recovers within 1-2 months as prices stabilize.

Mid-Lifecycle IL Pattern

LPs who enter after the initial volatility period generally experience much lower IL, typically below 1% throughout their holding period.

Maturity Convergence Effect

As the maturity date approaches, IL naturally decreases and approaches zero. This occurs because the PT price converges to its face value, eliminating the price ratio volatility that creates IL.

Numerical Example

Let's consider a simplified example to illustrate IL in a yield tokenization pool:

Imagine you provide liquidity to a PT/underlying pool with a 50/50 value split, with the PT initially trading at 95% of face value (reflecting the time value of money until maturity).

Now assume a significant market interest rate change causes the PT price to drop to 92% of face value. The AMM automatically rebalances your position:

• You now hold relatively more PTs and less of the underlying asset compared to when you entered

• If you withdrew at this point, the value of your position would be about 1-1.5% less than if you had simply held the original 50/50 position without providing liquidity

However, if you continue providing liquidity until maturity, this IL disappears as the PT price converges to 100% of face value. Additionally, throughout this period, you would be earning trading fees on all pool activity, potentially offsetting or exceeding the temporary IL.

Risk Management Strategies for Liquidity Providers

Given these risk factors, liquidity providers can employ several strategies to optimize their experience:

For Managing Impermanent Loss:

• Consider the pool's lifecycle stage when entering (early pools may have higher initial IL)

• Align your expected holding period with the maturity date when possible

• Monitor changes in market interest rates that might affect PT pricing

• Be prepared for temporary IL during periods of high market volatility

• Remember that IL is generally bounded in yield tokenization pools compared to standard AMMs

Conclusion

FIVA's custom AMM formula and the unique convergence properties of yield tokenization pools create a fundamentally different IL profile compared to traditional AMM pools. Our empirical analysis confirms that IL is typically bounded between 1-3% in most unfavorable conditions, and approaches zero as maturity nears.

This makes FIVA's pools an attractive option for liquidity providers seeking to minimize IL risk while still capturing trading fee revenue. By understanding these dynamics and employing appropriate risk management strategies, LPs can make more informed decisions about how yield tokenization pools fit within their broader investment approach.