A displayed Morpho APY is a poor guide once your deposit is large enough to move the pool. Supplied liquidity rises with the ticket while borrowed liquidity stays fixed. Utilization drops, pulling down the borrow rate. The supply rate takes a second hit because it equals utilization multiplied by the borrow rate. Each new dollar earns below the quoted APY and reduces the return on the dollars already deposited. Ranking markets by displayed APY and filling the leader therefore fails by construction when the ticket is large relative to the pool.
Baude, Danos and El Khalloufi formalize this problem as the lender-side companion to their earlier borrower-side paper. A lender divides a budget among n markets that share the same supplied asset, plus an external yield source paying a fixed rate r. The objective maximizes x0 times r plus the sum of xi times si(Bi/(Si+xi)), subject to the budget, non-negative external allocation, and per-market floors and caps.
A multiplier on the budget constraint separates the optimization market by market. In each market, the first-order condition reduces to a depressed cubic in yi = xi + Si, which Cardano solves. When the budget is too small to saturate every market, Brent's method determines the multiplier and nothing remains outside. With enough capital for the residual to flow into the external source, the multiplier equals r. The authors interpret it as the price at which cash becomes preferable. It makes the external source artificially and gradually more profitable until every market reaches saturation.
The kinked, Aave-style curve supplies the paper's sharper result. Borrower-side objectives are concave; lender-side objectives are only piecewise concave. The closed form consequently requires enumeration. For each market, the optimizer chooses a final position below or above target utilization. A pre-kink regime survives only when the budget can reach the kink and doing so beats the external rate. The best feasible configuration then wins.
An allocation landing exactly on the kink is discarded. The authors prove it can never be optimal for a lender whenever the post-kink slope over (1 minus u) exceeds the pre-kink slope over u. Borrowers can optimally stop there. After its time dynamics are removed, Morpho's AdaptiveCurveIRM is a reparametrization of the same kinked curve, with u* = 0.9 and kd = 4 hardcoded. One condition in the proposition follows automatically, while the other holds whenever utilization exceeds 30%.
The source of the 5.5%
The backtest covers Morpho markets on Ethereum from January 1, 2026 to April 1, 2026. It uses data from Morpho's GraphQL API, rebalances daily, and feeds one-day moving averages of market state into the allocation. The authors test Two USDC universes: the three markets in the Gauntlet USDC Prime vault and ten markets the Gauntlet USDC Core vault had already funded. Pendle PT-collateralized markets are excluded because their collateral has a fixed maturity and would require rolling. An appendix runs the same exercise on the four largest WETH markets.
With a $100k budget, the core-inspired strategy earns 5.5% APY. The prime-inspired version earns 3.3%, equal to passively placing the full amount in USDC/cbBTC. At $100k, then, the three-market prime universe merely matches the single most liquid market. All 220bp comes from the seven additional markets in the core set. Supply rates across the ten core-inspired markets reach about 25% during the sample (Figure 3b), compared with about 17% across the three prime-inspired markets (Figure 2b).
Size changes the result quickly. At $100m, the core set produces 2.8%, prime produces 2.4%, and cbBTC produces 2.2%.
The 220bp becomes 60bp.
This contraction is the authors' headline finding. Section 4.2 assigns it to dilution, and the conclusion says the backtests emphasize the effect. Their answer follows immediately: dilution is the lender-side equivalent of the borrower-side size effect in the companion paper, a modelled feature of the framework rather than a defect. Fair. The economic consequence remains.
The numerical results section contains the mechanics. High-rate markets have little liquidity, so larger allocations steadily consume their capacity and push their rates toward those of the main markets. Beyond a certain investment size, every rate has converged to a common level. APY then behaves as though the capital sat in a single market with aggregated liquidity. At $100m, the prime set earns 20bp over the benchmark because it has greater aggregate liquidity. The WETH exercise traces the same curve, with returns falling from 2.2% to about 1.6% between zero and 10k WETH.
Small, high-rate pools supply the capacity this strategy harvests. Its size ceiling is already plain in the reported figures: 220bp at $100k and 60bp at $100m, with the curve flattening across that range. The mechanism is the contribution. The 220bp edge does not scale.
Why the optimizer concentrates
At $100k in the core set, the allocation is bang-bang. Almost all capital enters one market, then switches discontinuously into another. Prime remains diversified because its market rates move together. The authors call the core-set spreads "significant and persistent" and interpret them as market pricing of heterogeneous collateral risk.
The objective leaves collateral risk unpriced. Per-market floors and caps are the proposed control, yet the authors classify them as lender-specific inputs beyond the paper's scope. The paper does not report which limits, if any, were imposed in the backtest.
At $100m, positions instead form allocation bands that track pool depth. Liquidity has become the binding input, matching the capacity explanation.
Closed form wins only on synthetic markets
The clean solver advantage appears on five synthetic kinked markets. Their parameters are u* = 90%, no base rate, a 5% pre-kink slope, post-kink slopes from 16.6% to 20.0%, and 920 to 943 borrowed against 1000 supplied. The closed form earns 4.6% in 0.04s. Multi-start SLSQP reaches 4.3% in the same 0.04s. Single-start SLSQP begins with the full budget in the external source, earns 3.0% in 0.01s, and leaves 161.7 idle there.
A generic solver paired with a lazy initial point fares badly on this piecewise-concave objective. The closed-form allocation is also concentrated at (83.6, 23.6, 81.9, 84.2, 26.6). Multi-start ends near uniform at (56.6, 63.8, 54.9, 57.1, 67.6).
The authors are candid about the real data. Across the USDC backtests, all three optimizers produce nearly identical results, with differences largely lost in market noise. The synthetic advantage, 160bp over single-start SLSQP and 30bp over multi-start, disappears from the real USDC backtests. Practical value lies in the size-aware objective and the kink result.
The kink result supports less than the abstract's suggestion that lender-borrower asymmetry "may create tension" and "exacerbate rate volatility". The word may carries the qualification. This is a single-agent own-impact model, and the authors identify a multi-agent Nash problem as the important extension. We found no equilibrium construction and no test of rate volatility around the kink. The volatility claim therefore remains a conjecture drawn from comparative statics.
The evidence is narrow. Realized APY over one three-month window, on one protocol and one chain, is the only return metric reported anywhere. There is no volatility, drawdown, or dispersion across sub-periods. The setup acknowledges costs by assuming frictionless reallocation "apart from gas fees", though we did not find gas deducted from any reported APY. A gross 220bp edge on $100k equals $2,200 a year, against daily reallocation on Ethereum mainnet with all-in switching.
We also did not find the value assigned to the external rate r, a free input that determines whether markets saturate at all. The universes consist of markets that two Gauntlet vaults had already funded, conditioning selection on the study period. Position figures are captioned January 1, 2025 to April 1, 2025, while the text and every other figure give 2026.
We could not test any of this ourselves. Reproduction requires historical Morpho state for each market, including total supplied, total borrowed and the IRM parameters. Our crypto coverage consists of token price series, and a supply position in USDC/stcUSD is not a tradeable ticker.
The dilution arithmetic disciplines any DeFi lending yield quoted without size: 5.5% at $100k versus 2.8% at $100m across the same ten markets. The closed form also justifies itself against a lazy solver, earning 4.6% against 3.0% for single-start SLSQP on the synthetic five. A longer window on the same universe, with gas charged and caps reflecting collateral risk, would change my view of the edge itself. On three months of gross APY, the paper demonstrates the mechanism.