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Aggregated APY (AAPY)

Overview

"Aggregated APY" (Annual Percentage Yield) represents a consolidated APY calculation across multiple Cauldron liquidity pools, referred to as "micro-pools." Each micro-pool follows the Constant Product Market Maker (CPMM) model, contributing liquidity individually. Aggregating APY provides a comprehensive view of yield across all pools in a period, while weighting pools proportionally to their liquidity to ensure that larger pools have a more substantial impact on the final APY.

Key Concepts

  1. Constant Product Market Maker (CPMM):

Each micro-pool is governed by the CPMM model, where the product of the reserves of two assets remains constant:

\[ k = x \times y \]
  • \( x \) represents the reserve of one asset (e.g., BCH).
  • \( y \) represents the reserve of the paired token.

  • \( k \) represents the constant product, which increases over time as liquidity and yield accumulate.

  • Annual Percentage Yield (APY):

  • APY is calculated for each pool based on the yield generated over time.

  • Weighted Aggregation:

  • Larger pools that have existed for a longer time contribute more to the aggregated APY. To achieve this, the calculation uses the square root of each pool’s \( k \) value at the end of the period as a proxy for the pool’s size. This weighting ensures that larger pools, which have more liquidity and more stable and less gameable yield, have a proportionally larger impact on the aggregated APY.

Calculation Methodology

1. Calculate Pool-Specific APY

See Individual Pool APY

2. Weight Each Pool’s APY by Size and Active Duration

To accurately aggregate APY across all pools, we use the square root of the final \( k \) value to weight each pool. This approach considers the pool’s liquidity and yield accumulated up to the end of the period:

\[ \text{Weighted APY}_{\text{pool}} = \text{APY}_{\text{pool}} \times \sqrt{k_{\text{final}}} \times \text{Days Active} \]

Where:

$$ \sqrt{k_{\text{final}}} $$ represents the square root of the pool’s final $ k $ value, providing a size-based weight.

3. Calculate Aggregated APY

Finally, the aggregated APY is computed by summing each pool’s weighted APY and dividing by the total weight across all pools:

\[ \text{Aggregated APY} = \frac{\sum \left( \text{Weighted APY}_{\text{pool}} \right)}{\sum \left( \sqrt{k_{\text{final}}} \times \text{Days Active} \right)} \]

This formula ensures that the aggregated APY reflects contributions from all pools, with larger, longer existing pools having a proportionally greater impact on the result.