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bch_qp3hvzy09lzcm46qm4cd 66e0acc0be snapshot: docs.riftenlabs.com/cauldron/ (28 pages) - author: Riften Labs 
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Includes: whitepaper, liquidity/swap/yield/wallets, developer resources,
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contract-subscribe, ido, moria_v11, oracle, root, tokenbch, tokentoken).
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<article class="md-content__inner md-typeset">
<h1 id="aggregated-apy-aapy">Aggregated APY (AAPY)</h1>
<h2 id="overview">Overview</h2>
<p>"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.</p>
<h2 id="key-concepts">Key Concepts</h2>
<ol>
<li><strong>Constant Product Market Maker (CPMM)</strong>:</li>
</ol>
<p>Each micro-pool is governed by the CPMM model, where the product of the reserves of two assets remains constant:</p>
<div class="arithmatex">\[
k = x \times y
\]</div>
<ul>
<li><span class="arithmatex">\( x \)</span> represents the reserve of one asset (e.g., BCH).</li>
<li><span class="arithmatex">\( y \)</span> represents the reserve of the paired token.</li>
<li>
<p><span class="arithmatex">\( k \)</span> represents the constant product, which increases over time as fee income accumulates.</p>
</li>
<li>
<p><strong>Annual Percentage Yield (APY)</strong>:</p>
</li>
<li>
<p>APY is calculated based on the fee yield generated across all pools over the period.</p>
</li>
<li>
<p><strong>Weighted Aggregation</strong>:</p>
</li>
<li>
<p>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 <span class="arithmatex">\( k \)</span> 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.</p>
</li>
<li>
<p><strong>Capital Injections</strong>:</p>
</li>
<li>A pool owner can add capital to both sides of their pool simultaneously. This causes an immediate jump in <span class="arithmatex">\( k \)</span> that is not the result of fee income. Including this jump in the yield calculation would inflate the reported APY.</li>
<li>To handle this, each pool period is split at injection boundaries into sub-periods. Each sub-period spans between two real pool states with no capital injection crossing its boundary, ensuring only fee growth contributes to the reported yield.</li>
</ul>
<h2 id="calculation-methodology">Calculation Methodology</h2>
<h3 id="1-calculate-pool-specific-yield">1. Calculate Pool-Specific Yield</h3>
<p>For each pool (or sub-period), the yield is calculated from the growth in <span class="arithmatex">\( \sqrt{k} \)</span> over the period. See <a href="../yield/">Individual Pool APY</a>.</p>
<h3 id="2-compute-weighted-average-yield-and-duration">2. Compute Weighted Average Yield and Duration</h3>
<p>Rather than annualizing each pool individually and averaging the resulting APYs, the yield and active duration are weighted and averaged first. This prevents short-duration pools with extreme annualized returns from distorting the aggregate.</p>
<p>The weight for each pool is:</p>
<div class="arithmatex">\[
\text{weight} = \sqrt{k_{\text{final}}} \times \text{Days Active}
\]</div>
<p>The weighted average yield and duration across all pools are then:</p>
<div class="arithmatex">\[
\text{avg yield} = \frac{\sum \left( \text{yield}_{\text{pool}} \times \text{weight} \right)}{\sum \text{weight}}
\]</div>
<div class="arithmatex">\[
\text{avg days} = \frac{\sum \left( \text{Days Active} \times \text{weight} \right)}{\sum \text{weight}}
\]</div>
<h3 id="3-annualize-once-to-produce-aapy">3. Annualize Once to Produce AAPY</h3>
<p>The final AAPY is computed by annualizing the average yield over the average duration:</p>
<div class="arithmatex">\[
\text{AAPY} = \left(1 + \frac{\text{avg yield}}{100}\right)^{365.25 \;/\; \text{avg days}} - 1
\]</div>
<p>This ensures that the aggregated APY reflects contributions from all pools, with larger, longer-active pools having a proportionally greater impact on the result, and with no single short-duration outlier able to dominate the calculation.</p>
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