Squaring the Square Root: A New Category of DeFi Yield

Every liquidity provider in DeFi has felt it. You deposit into a pool, fees trickle in, and then the market moves. When you check your position, it's worth less than if you'd just held the tokens in your wallet. Impermanent loss — the invisible tax on providing liquidity — has been the defining problem of automated market makers since their inception.

But what if the very mechanic that creates impermanent loss also contains the key to eliminating it?

A new category of DeFi yield is emerging that does exactly this. We call it Squaring the Square Root, and it turns the math of AMMs on its head — transforming a structural loss into a structural yield engine.

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The Square Root Problem

The math behind impermanent loss is well understood. In a constant product AMM like Uniswap V2, the bonding curve follows the formula x · y = k. When the price of one asset changes relative to the other, the pool automatically rebalances — selling the appreciating asset and accumulating the depreciating one.

The value of a liquidity position after a price change follows the square root of the price movement. If BTC doubles in price, your LP position doesn't double. It grows by √2, or roughly 41%. The remaining 9% relative to simply holding the assets? That's an impermanent loss.

This square root function is the source of the problem. But it's also the source of the solution.

Squaring the Square Root

If the price of a liquidity position is proportional to √p (where p is the price of the volatile asset), then applying 2x leverage to that position gives you (√p)² = p. The square root gets squared. Impermanent loss vanishes.

Here's why this works mechanically. A 50/50 BTC/USD liquidity position at 2x leverage means borrowing dollars equal to the value of your deposit and adding them to the pool. Your collateral is the LP position itself, and at 2x leverage (50% loan-to-value), the borrowed USD portion is roughly equal to the USD half of the liquidity. The borrowed asset's exposure cancels out that same asset's presence within the pool.

What remains is pure exposure to the non-borrowed asset — in this case, BTC — plus trading fees earned on twice the capital.

The concept is elegant. The execution is where it gets interesting.

The Revenue and Expense Model

Squaring the Square Root isn't just about canceling impermanent loss. To be a compelling yield strategy, it needs to operate profitably. The economics break down into a simple framework:

Revenue is the income earned by the liquidity pool — trading fees, arbitrage capture, and any yield from the underlying assets — doubled by the 2x leverage.

Expenses are twofold: the cost of borrowing to create the leverage, and the friction of maintaining 50% LTV as prices move. Every time the volatile asset's price shifts, the leverage ratio drifts from its target and must be corrected through rebalancing trades.

For the strategy to produce yield, revenue must exceed expenses. And this is where the choice of underlying AMM becomes critical.

Why Uniswap V2 Doesn't Work

A naive implementation using a standard constant product pool like Uniswap V2 is unlikely to generate positive returns after expenses.

The problem is capital efficiency. In a V2-style pool, liquidity is spread uniformly across the entire price curve from zero to infinity. This means most of the capital sits idle at price ranges far from the current market. The result is enormous slippage for traders, which sounds good for LPs — except that it also means very low fee revenue relative to the capital deployed.

Worse, the pool is a sitting target for arbitrageurs. Every time the external market price moves, the pool's price is stale. Arbitrageurs trade against this stale price, extracting value from LPs in a process known as Loss-Versus-Rebalancing, or LVR. Research has shown that LVR is responsible for more value extraction than all other forms of MEV combined, costing liquidity providers an estimated 5–7% of their capital annually. When you account for LVR, many of the largest liquidity pools on Uniswap are not profitable for LPs at all.

Doubling the exposure to an unprofitable pool via leverage simply doubles the losses. The square root gets squared, but there's no yield left to capture.

Yield Basis: Optimizing the Pool

Yield Basis, designed by Curve Finance founder Michael Egorov, takes a fundamentally different approach to the underlying liquidity pool. Rather than accepting the capital inefficiency of x · y = k, Yield Basis builds on Curve's cryptoswap AMM and optimizes every parameter for profitability.

Concentrated liquidity with dynamic fees. Yield Basis pools concentrate liquidity around the current price, dramatically increasing capital efficiency and fee revenue per dollar deployed. But unlike other concentrated liquidity designs where LPs must actively manage their ranges, the Curve cryptoswap mechanism automatically repositions liquidity toward the current price. Fees are set dynamically — high enough that the pool captures a meaningful share of each arbitrage trade, but low enough that arbitrageurs still have incentive to keep the pool balanced.

This is a delicate optimization. Simulations across six years of BTC/USD price data show that with the right parameters, the leveraged pool can generate an average APR of approximately 20% on BTC — meaning you hold BTC exposure and earn yield on top of it.

The releverage AMM. Maintaining constant 2x leverage as prices move is one of the largest potential expense items. Doing it manually — triggering rebalances at fixed price thresholds — is inefficient and creates large swings in returns. Yield Basis solves this with a purpose-built releverage AMM: a secondary automated market maker that continuously adjusts the leverage position through arbitrage.

The re-leverage AMM holds LP tokens as collateral and borrows stablecoins as debt. It uses an oracle-informed variant of the constant product formula where one side of the equation is defined by the current debt rather than a token reserve. When the market price diverges from the oracle price, the AMM creates an arbitrage opportunity. Traders who correct this arbitrage are simultaneously adjusting the protocol's leverage ratio back toward target. In effect, the market itself maintains the leverage, and it does so continuously and efficiently — far more so than discrete manual rebalances.

The beauty of this design is that the releverage fee parameter has a remarkably flat impact on returns, meaning the system is robust to a wide range of settings. Simulations confirm that re-leverage losses via this AMM are substantially lower than through manual approaches.

Structural cost advantages. Yield Basis also benefits from its close relationship with Curve Finance in ways that directly reduce the expense side of the equation. Yield Basis pools are exempt from the standard fees paid to the Curve DAO that other pools incur. And rather than borrowing from external lenders, Yield Basis borrows crvUSD to pair with user BTC deposits for the leveraged position. The interest on this loan stays within the Yield Basis ecosystem, subsidizing pool rebalancing rather than flowing out to third-party lenders. Curve is compensated through YB governance tokens and the indirect benefits of increased crvUSD supply and deeper liquidity across Curve pools that do generate DAO fees.

The tradeoff: Temporary Redemption Discount. The optimization comes with a constraint. Because liquidity is highly concentrated and fees are set to capture maximum value from arbitrage, significant price moves can cause the pool to become temporarily imbalanced. During these periods, a liquidity provider withdrawing their position may face a haircut known as a Temporary Redemption Discount (TRD). Once market movements and accumulated fees rebalance the pool over time, withdrawals return to full value. This is the cost of concentration — you earn more, but you accept periods where immediate exit isn't at par.

Bitcoin Dollar: Squaring the Square Root Without a Pool

Bitcoin Dollar introduces a different implementation of the same fundamental principle. Where Yield Basis optimizes a liquidity pool, Bitcoin Dollar builds a managed portfolio — and turns the rebalancing activity itself into a yield engine.

At its core, Bitcoin Dollar is simply a portfolio that always holds 50% Bitcoin and 50% US dollars. Its token design mirrors Ethena's: BTCD is the base token, always redeemable through the protocol for USDC at the current 50/50 portfolio value. sBTCD is the yield-bearing staking token that earns any excess the portfolio generates above the peg — just as sUSDe earns whatever Ethena generates above the $1 peg.

Three Sources of Yield

The Bitcoin Dollar portfolio generates yield from three distinct sources:

The first is yield on USD assets. The dollar half of the portfolio doesn't sit idle in USDC. It's deployed into productive dollar strategies — including assets like sUSDS and sUSDe — that earn yield while maintaining dollar exposure.

The second is yield on BTC assets. Similarly, the Bitcoin half of the portfolio holds yield-bearing Bitcoin assets, including Yield Basis's own ybBTC, earning returns while maintaining Bitcoin exposure.

The third — and perhaps most important — is profitable rebalancing. This is where Bitcoin Dollar diverges most sharply from a traditional AMM.

In a standard liquidity pool, the pool itself is passive. It sits and waits. When the market price moves, the pool's price becomes stale. Arbitrageurs arrive, trade against the stale price, pocket the difference, and leave the pool rebalanced but poorer. The pool earns fees but loses to LVR. The arbitrageur captures the spread.

Bitcoin Dollar isn't a liquidity pool waiting to be arbitraged. It has its own trading engine with rebalancing bands optimized to generate a positive return. When BTC price rises, the portfolio's BTC holdings grow to 51% of value. This triggers a trade: sell a small amount of Bitcoin for USD, restoring balance. When BTC price falls, the portfolio becomes 51% USD. This triggers the opposite: buy a small amount of Bitcoin, restoring balance.

Over time, this constant cycle of buying low and selling high — executed on the portfolio's own terms rather than at the mercy of external arbitrageurs — produces a return in excess of the passive 50/50 peg. Combined with the yield from productive underlying assets, this return flows to sBTCD holders.

The result: sBTCD always maintains the same 50/50 BTC/USD exposure as BTCD, but it appreciates in value over time relative to BTCD as yield accrues.

The USD Vault: Bringing It Together

With a yield-bearing 50/50 BTC/USD token in hand, Bitcoin Dollar applies the Squaring the Square Root mechanic through its USD Vault.

The vault accepts USDC deposits, acquires sBTCD, and then borrows wBTC to create 2x exposure to its sBTCD collateral at a 50% LTV. Because sBTCD is 50% BTC and 50% USD, the wBTC debt cancels out the Bitcoin exposure within the collateral. What remains is pure, yield-bearing dollar exposure — amplified by leverage.

Revenue: The vault earns 2x sBTCD yield. If sBTCD generates 10% APY from its combined sources — productive assets and profitable rebalancing — the gross yield on the leveraged position is 20%.

Borrowing expense: The vault borrows wrapped BTC, and this is where the Bitcoin Dollar has a notable structural advantage. In DeFi lending markets, BTC is one of the cheapest assets to borrow. Everyone wants to lend it as collateral, but few have a reason to borrow it. This dynamic keeps wBTC borrow rates low — often less than 1% — making the cost of leverage minimal.

Rebalancing expense: This is where the architecture gets clever.

Every time BTC price increases, the USD Vault's wBTC debt grows faster than its sBTCD collateral. The vault needs to deleverage — sell some sBTCD, acquire BTC, and repay a portion of its debt. Every time BTC price falls, the opposite occurs: the vault needs to borrow more wBTC and acquire more sBTCD to bring leverage back to target.

Executing these trades on the open market would incur slippage and MEV extraction. But the USD Vault and the underlying BTCD Portfolio share a Coincidence of Wants.

When BTC price rises, the BTCD Portfolio needs to sell a small amount of BTC for USD to maintain its 50/50 balance. At the exact same moment, the USD Vault needs to sell sBTCD to acquire BTC so it can repay wBTC debt. These two entities sit on opposite sides of the same trade.

Through an internal atomic swap mechanism, the portfolio broadcasts a trade intent with a price quote based on current market rates. The vault evaluates whether accepting the trade moves its LTV back toward the 50% target at a price equal to or better than what external markets would offer. If it does, the swap executes internally — no slippage, no external fees, no MEV.

Because the portfolio maintains tight rebalancing bands (approximately ±1%) while the vaults operate with slightly wider tolerance bands, the vast majority of portfolio rebalancing volume can be internalized. External markets serve as a backstop for residual imbalances, not the primary execution venue.

This internalization transforms what would otherwise be a major expense line — the cost of maintaining leverage — into a source of efficiency. Every dollar saved on rebalancing is a dollar that flows to vault depositors as yield.

A New Category

Squaring the Square Root isn't a single protocol trick. It's a design pattern — a category of yield strategies built on a shared mathematical insight: that 2x leverage on a 50/50 portfolio eliminates the exposure to one of its component assets while doubling the yield.

Yield Basis implements it through an optimized AMM, concentrating liquidity and using a purpose-built re-leverage mechanism to maintain leverage efficiently. It delivers BTC-denominated yield to holders who want to earn on their Bitcoin without giving up price exposure.

Bitcoin Dollar implements it through a managed portfolio with an internal trading engine and a vault system that leverages sBTCD. Its USD Vault delivers dollar-denominated yield to holders who want high returns on stablecoins without directional crypto exposure — and it does so while internalizing the majority of its rebalancing costs.

Both protocols share the same foundational math. Both face the same economic equation of revenue versus expenses. And both have found different, complementary ways to tip that equation decisively toward yield.

The square root created the problem. Squaring it is the solution.