Balancer Math

Balancer 2-asset weighted pools use a generalized constant-product invariant with configurable token weights:

\[B_x^{w_x} \cdot B_y^{w_y} = k\]

where \(B_x\) and \(B_y\) are the pool’s token balances and \(w_x\), \(w_y\) are the normalized weights (summing to 1). When \(w_x = w_y = 0.5\), this reduces to the standard Uniswap V2 constant-product invariant \(x \cdot y = k\).

Spot price

\[P = \frac{B_y / w_y}{B_x / w_x}\]

Impermanent loss with weights

For a price change \(\alpha = P_{\text{new}} / P_{\text{entry}}\), the impermanent loss for a weighted pool is:

\[IL(\alpha, w_x) = \frac{\alpha^{w_x}}{w_x \cdot \alpha + (1 - w_x)} - 1\]

When \(w_x = 0.5\), this reduces to the standard \(2\sqrt{\alpha}/(1+\alpha) - 1\) formula used in Uniswap V2. Asymmetric weights compress or amplify IL exposure depending on the direction of price movement.

DeFiPy implementation

DeFiPy’s Balancer math lives in BalancerPy. The weighted-pool invariant, spot-price calculation, and swap math are implemented in balancerpy.cpt.exchg.BalancerExchange. The impermanent loss formula is implemented in balancerpy.analytics.risk.BalancerImpLoss.

The agentic primitives that consume Balancer math:

• AnalyzeBalancerPosition — full position decomposition using the weighted IL formula

• SimulateBalancerPriceMove — project position value at a hypothetical price change

See agentic_primitives_by_category for executable examples.

Note

Full derivation notebook (matching the depth of the Uniswap V2 and V3 math pages) is planned for v2.1.