How Is the Invariant Formula for a Multi-Asset Pool, like Balancer’s Value Function, Different from the Constant Product Formula?

Balancer's Value Function is a generalization of the constant product formula (x y=k). It is a weighted geometric mean, expressed as V = product(B_i ^ W_i), where B is the balance of each token and W is its weight.

Unlike the 50/50 split in the constant product model, this allows for pools with custom weights, such as 80% of one asset and 20% of another. This flexibility enables the creation of pools that can act as index funds and alters the impermanent loss profile based on the asset weightings.