Math Primer

0. Notation & Symbols

Symbol	Meaning
$p$	BTC spot price in USD
$p_o$	Oracle BTC/USD price used by the Rebalancing‑AMM
$V_c$	Value of collateral (Curve LP tokens)
$V_*$	Value of the leveraged position (after borrowing)
$d$	Outstanding crvUSD debt
$L$	Compounding leverage factor (fixed to 2 in YieldBasis)
$y$	LP token amount held by the Rebalancing‑AMM
$x_0(p_o)$	Anchor term in the Rebalancing‑AMM pricing formula
$I(p_o)$	Invariant constant of the Rebalancing‑AMM for a fixed oracle price
$\phi$	Flash‑loan size (crvUSD) solved via a quadratic
$f_a$	Admin‑fee fraction routed to veYB
$f_{\min}$	Minimum admin fee when nobody stakes
$s, T$	Staked vs total supply of ybBTC
$r_{\text{pool}}$	Fee APR earned by the underlying Curve pool (unlevered)
$r_{\text{borrow}}$	Borrow APR on the crvUSD CDP line
$r_{\text{rebalancing}}$	Effective APR “cost” of maintaining leverage (rebalancing losses/subsidy)

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1. The Constant‑Product Problem (Why $\sqrt{p}$ Shows Up)

Classic AMMs (e.g. Uniswap v2) enforce

$$x \cdot y = k \tag{1.1}$$

where $x$ is the USD side and $y$ is the BTC side. Let the external BTC price be $p$ (USD per BTC).

1.1 Normalising the start

Assume you deposit equal USD value at $p_0 = 1$:

• $x_0 = 1$ USD
• $y_0 = 1$ BTC (worth $1 at $p_0$)

So

$$k = x_0 \cdot y_0 = 1. \tag{1.2}$$

1.2 After a price move

Arbitrage pushes the AMM price to $p$. In a constant‑product AMM,

$$p = \frac{x}{y}. \tag{1.3}$$

Combine (1.1) and (1.3):

$$x = \sqrt{k p}, \qquad y = \sqrt{\frac{k}{p}}. \tag{1.4}$$

With $k = 1$:

$$x = \sqrt{p}, \qquad y = \frac{1}{\sqrt{p}}. \tag{1.5}$$

1.3 LP value scales as $\sqrt{p}$

Ignoring fees, LP USD value is

$$V_{\text{LP}}(p) = x + p\,y = \sqrt{p} + p \cdot \frac{1}{\sqrt{p}} = 2\sqrt{p}. \tag{1.6}$$

We started with $2 total, so per‑initial‑dollar scaling is

$$\boxed{V_{\text{LP,norm}}(p) = \sqrt{p}} \tag{1.7}$$

1.4 Impermanent loss (IL)

If you just held instead:

$$V_{\text{hold}}(p) = p + 1. \tag{1.8}$$

Impermanent loss (relative underperformance) is

$$\text{IL}(p) = \frac{V_{\text{LP}}(p)}{V_{\text{hold}}(p)} - 1 = \frac{2\sqrt{p}}{1+p} - 1. \tag{1.9}$$

Since $2\sqrt{p} < 1+p$ for any $p \neq 1$, $\text{IL}(p) < 0$: the LP underperforms HODLing whenever price drifts.

Takeaway. Constant‑product LP value $\propto \sqrt{p}$. Other, more complicated, AMM formulae are not immune to the IL problem either, and the source thereof is the sublinearity of the value function, for which reason the only way of eliminating IL is to design an AMM with an at least a linear value function. YieldBasis provides such a design by engineering exposure to scale linearly with $p$, not as $\sqrt{p}$.

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2. Compounding Leverage — Turning $\sqrt{p}$ into $p$

YieldBasis borrows against the LP collateral to keep a constant compounding leverage of $L$.

2.1 Defining “compounding leverage”

By construction, the protocol keeps debt at a fixed fraction of collateral:

$$d = V_c \left(1 - \frac{1}{L}\right). \tag{2.1}$$

So the net position value is

$$V_* = V_c - d = \frac{V_c}{L}. \tag{2.2}$$

Consider a small change in collateral value $dV_c$. After re‑leveraging to restore the ratio, the relative change of $V_*$ is amplified by $L$:

$$\frac{\delta V_*}{V_*} = L \, \frac{\delta V_c}{V_c}. \tag{2.3}$$

Integrate both sides:

$$\int \frac{\delta V_*}{V_*} = L \int \frac{\delta V_c}{V_c} \;\Longrightarrow\; \ln V_* = L \ln V_c + \text{const}. \tag{2.4}$$

Exponentiate:

$$\boxed{V_* \propto V_c^{\,L}} \tag{2.5}$$

Why is the relative change amplified?

Growth of the price of collateral

If the collateral grows in price, the value $V_*$ changes as ($d$ is kept constant at this step): $V_* + \delta V_* = V_c - d + \delta V_c - \delta d = (V_c - d) + \delta V_c = V_* + \delta V_c = V_* + y\delta p$ $\frac{\delta V_*}{V_*} = \frac{y \delta p}{\frac{yp}{L}} = L\frac{\delta p}{p} = L\frac{\delta V_c}{V_c}$

Adjustment of the debt value and of the collateral amount

By definition: $d = V_c(1-\frac{1}{L})$ Then: $V_*+\delta V_* = V_c - d +\delta V_c - \delta d = (V_c + \delta V_c)-d-(1-\frac{1}{L})\delta V_c = V_*+\frac{1}{L}\delta V_c = V_*+\frac{1}{L}p\delta y$ Whence it follows that $\frac{\delta V_*}{V_*} = \frac{\frac{p\delta y}{L}}{\frac{py}{L}} = \frac{\delta y}{y} = \frac{\delta V_c}{V_c}$

As in this step the value of the leveraged position tracks the value of the collateral linearly, the total scaling factor is exactly equal to the scaling factor of the previous step. For more details, refer to the General Idea section of the Whitepaper.

2.2 Pick $L = 2$ to linearise BTC exposure

Curve Cryptoswap LP value scales approximately as $V_c \propto \sqrt{p}$. Plug that in:

$$V_* \propto (\sqrt{p})^{\,L} = p^{\,L/2}. \tag{2.6}$$

Set $L = 2$:

$$\boxed{V_* \propto p} \tag{2.7}$$

Result: With $L = 2$, your position tracks BTC 1:1 — the $\sqrt{p}$ curvature (and thus impermanent loss) is eliminated while you still earn trading fees.

2.3 What if $L \neq 2$?

• $L > 2$: $V_* \propto p^{L/2}$ — outperform on the way up, underperform on the way down (re‑introduces divergence vs. “just hold”).
• $L < 2$: You under‑hedge the $\sqrt{p}$ curve and still suffer residual IL.

Hence, locking $L = 2$ is the sweet spot: linear BTC exposure, zero IL, and manageable debt that can always be unwound with the LP’s USD side.

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3. Rebalancing‑AMM & Virtual Pool

To keep leverage fixed at $L = 2$ (i.e. debt = 50% of value), YieldBasis uses a second AMM (“Rebalancing‑AMM”). It holds Curve LP tokens on one side and implicit crvUSD debt on the other, and is anchored to an oracle price $p_o$ (BTC/USD).

We first state the target (“ideal”) state at $p = p_o$, then derive the invariant and solve for the anchor variable $x_0(p_o)$.

3.1 Ideal debt at the oracle price

If the live market price equals the oracle price $p_o$, “perfect” leverage is:

$$\tilde{d} \;=\; \frac{L-1}{L}\,p_o\,\tilde{y} \qquad\text{(ideal debt when } p = p_o\text{)}, \tag{3.1}$$

where $\tilde{y}$ is the LP amount in the AMM at that oracle touch point (tildes = “ideal” values).

For $L = 2$: $\tilde{d} = \tfrac{1}{2} p_o \tilde{y}$.

3.2 Rebalancing‑AMM Invariant & Pricing

Define $x \equiv x_0(p_o) - d$. The Rebalancing‑AMM uses a constant‑product‑like invariant:

$$\big(x_0(p_o) - d\big)\; y \;=\; I(p_o), \tag{3.2}$$

with $I(p_o)$ constant as long as $p_o$ stays fixed.

At $p = p_o$, the AMM’s instantaneous price should equal the oracle:

$$\frac{x_0(p_o) - \tilde{d}}{\tilde{y}} = p_o. \tag{3.3}$$

Substitute $\tilde{d}$ from (3.1):

$$x_0(p_o) = \frac{2L - 1}{L}\,p_o\,\tilde{y}. \tag{3.4}$$

3.3 Solving for $x_0$ with arbitrary $y, d$

Away from the ideal point, we don’t know $\tilde{y}$ directly. From (3.4):

$$\tilde{y} \;=\; \frac{L}{2L - 1}\,\frac{x_0}{p_o}. \tag{3.5}$$

Evaluate the invariant at the ideal point:

$$I(p_o) \;=\; \big(x_0 - \tilde{d}\big)\,\tilde{y} \;=\; \left(x_0 - \frac{L-1}{L} p_o \tilde{y}\right)\tilde{y}. \tag{3.6}$$

Set that equal to the current‑state invariant $(x_0 - d)\,y$, substitute (3.5), and rearrange. You obtain a quadratic in $x_0$:

$$x_0^{2}\left(\frac{L}{2L-1}\right)^{2} - p_o y\,x_0 + p_o y\,d = 0. \tag{3.7}$$

Choose the larger root (matches the ideal‑state intuition):

$$\boxed{ x_0(p_o) = \frac{p_o y + \sqrt{(p_o y)^2 - 4\,p_o\,y\,d\,\left(\tfrac{L}{2L-1}\right)^{2}}} {2\left(\tfrac{L}{2L-1}\right)^{2}} } \tag{3.8}$$

For $L = 2$, you can simplify numerically in code, but keep the general form for clarity.

3.4 Useful derived quantities

Pool value at oracle price. It’s cleaner to value the AMM at $p = p_o$ to avoid sandwich games:

$$V \;=\; \tilde{y}\,p_o - \tilde{d} \;=\; \frac{1}{L}\,\tilde{y}\,p_o \;=\; \frac{x_0}{2L - 1}. \tag{3.9}$$

Invariant in closed form.

$$I(p_o) \;=\; \big(x_0 - \tilde{d}\big)\,\tilde{y} \;=\; \frac{x_0^{2}}{p_o}\left(\frac{L}{2L-1}\right)^{2}. \tag{3.10}$$

An alternative expression for $V$ follows from (3.10):

$$V \;=\; 2\,\sqrt{I(p_o)\,p_o} - x_0. \tag{3.11}$$

3.5 Flash‑Loan Sizing ($\phi$)

When the arbitrageur comes with $x$ crvUSD (or needs extra), the Virtual Pool computes how much crvUSD to flash‑borrow so that:

• Curve liquidity is added/removed symmetrically (no spot price move),
• The flash loan is repaid inside the same tx.

Let:

• $f$ — Rebalancing‑AMM fee
• $d$ — current debt
• $y$ — LP amount in the Rebalancing‑AMM
• $x_0$ — anchor term from Eq. (3.8)
• $x$ — user‑provided crvUSD for this swap
• $x_c, y_c, S_c$ — Curve pool’s stablecoin reserves, crypto reserves, total LP supply

Define a convenience constant

$$r \;=\; \frac{x_c}{S_c}\,(1 - f). \tag{3.12}$$

The required flash amount $\phi$ solves the quadratic (whitepaper Eq. 39):

$$\phi^{2} \;+\; \phi\Big(x_0 - d + x - r\,y\Big) \;-\; y\,x\,r \;=\; 0. \tag{3.13}$$

Discriminant:

$$D \;=\; \Big(x_0 - d + x - r\,y\Big)^2 + 4\,y\,x\,r. \tag{3.14}$$

Choose the positive root:

$$\boxed{ \phi = \frac{\sqrt{D} - (x_0 - d + x - r\,y)}{2} } \tag{3.15}$$

This is evaluated inside the Virtual Pool before executing the swap, ensuring the flash loan is exactly covered by the LP redemption and AMM trade proceeds.

3.6 Directional intuition (no new math)

• Debt too low (price up): $d/V < 0.5$. Then $x_0 - d$ is large → LP is “expensive” in crvUSD. Arb brings in crvUSD, mints LP, increases $d$.
• Debt too high (price down): $d/V > 0.5$. Then $x_0 - d$ is small → LP is “cheap”. Arb buys LP, unwraps, repays debt.

All legs (flash‑loan, LP mint/burn, CDP mint/repay) run atomically in the Virtual Pool wrapper.

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4. Net APR & Cost Terms

We want an explicit expression for the BTC‑denominated return of an unstaked ybBTC position.

4.1 Components

Let

• $r_{\text{pool}}$ — APR the Curve BTC/crvUSD pool would earn without leverage (pure trading fees, in BTC terms).
• $L$ — compounding leverage (YieldBasis: $L = 2$).
• $r_{\text{borrow}}$ — effective borrow APR on the protocol’s crvUSD line (set by governance).
• $r_{\text{rebalancing}}$ — drag from maintaining constant leverage (AMM fees/price edge paid to arbs, flash‑loan interest, slippage).

Because we supply twice as much notional liquidity (BTC + borrowed crvUSD) to Curve, fee flow scales ~linearly with $L$:

$$\text{Gross fee APR} \;=\; L \cdot r_{\text{pool}}. \tag{4.1}$$

For $L = 2$:

$$\text{Gross fee APR} \;=\; 2\,r_{\text{pool}}. \tag{4.2}$$

We then subtract the two costs:

1. Borrow cost — interest paid on the crvUSD debt.
2. Rebalancing cost — continuous “leak” to arbitrageurs that keep the ratio at 50%.

Putting it together:

$$\boxed{ \text{APR}_{\text{net}} = L\,r_{\text{pool}} - \bigl(r_{\text{borrow}} + r_{\text{rebalancing}}\bigr) } \tag{4.3}$$

For YieldBasis’ fixed $L = 2$:

$$\boxed{ \text{APR}_{\text{net}} = 2\,r_{\text{pool}} - \bigl(r_{\text{borrow}} + r_{\text{rebalancing}}\bigr) } \tag{4.4}$$

4.2 Heuristic for $r_{\text{rebalancing}}$

Backtests (Jan 2019 – Oct 2024) show:

• Optimised constant‑product pools (no leverage) earn $\sim 3\%$ APR over that window.
• Empirically, $r_{\text{rebalancing}} \approx 2\times$ that value — the “cost” of keeping leverage perfect is roughly twice what a plain Uniswap‑style pool can earn under the same conditions.

This is why the method fails on a pure $x\cdot y = k$ AMM (fees not big enough), but works on Curve Cryptoswap where $r_{\text{pool}}$ is materially higher.

We do not hard‑code this heuristic on‑chain; it’s just a modelling shortcut:

$$r_{\text{rebalancing}} \approx 2 \cdot r_{x y = k}^{\text{opt}} \quad \text{(empirical)}. \tag{4.5}$$

4.3 Continuous vs. discrete compounding

All rates above are quoted as simple APRs for intuition. If you want continuous compounding:

$$\text{APY}_{\text{net}} = \exp\!\big(\text{APR}_{\text{net}}\big) - 1. \tag{4.6}$$

Given the relatively small magnitudes (single‑digit percentages for costs, double‑digit for fees), APR ≈ APY within a few bps, so we stick to APR in docs.

Takeaway. YieldBasis only “wins” if the doubled fee stream outpaces borrow + rebalance drag. Governance controls those drags (via borrow rate and AMM params), keeping net APR attractive in most regimes.

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5. Token Incentives & Admin‑Fee Function

YieldBasis splits real fee flow between unstaked LPs (they want BTC fees) and veYB stakers (they want protocol value capture). The split is governed by an admin‑fee curve that depends on how much of the liquidity is staked.

5.1 Admin‑fee formula

Let

• $T$ = total ybBTC supply (all LP shares),
• $s$ = amount of ybBTC staked (earning YB emissions),
• $f_{\min}$ = minimum admin fee when nobody stakes,
• $f_a$ = actual admin‑fee fraction skimmed from fees and sent to veYB.

The curve:

$$\boxed{ f_a \;=\; 1 - (1 - f_{\min}) \, \sqrt{1 - \frac{s}{T}} } \tag{5.1}$$

Limits:

• $s = 0 \Rightarrow f_a = f_{\min}$ (LPs keep $1-f_{\min}$ of fees)
• $s = T \Rightarrow f_a \to 1$ (all fees to veYB; stakers opted out of fee yield anyway)

The square‑root makes the increase gentle at first (so early stakers don’t massively penalize LPs), then accelerates near full staking to avoid “everyone stakes, no one gets fees” equilibria.

5.2 Splitting fee gains / sharing losses

After each accounting tick (or harvest), let

• $v_{-1}$ = total value of the leveraged position (in BTC terms) before fees this period,
• $v_1$ = total value after market moves + fees, before admin fee is taken,
• $\Delta v_{\text{use}} = (v_1 - v_{-1})(1 - f_a)$ = value left for LPs after admin fee,
• $\Delta a = (v_1 - v_{-1}) f_a$ = admin fee sent to veYB.

We split $\Delta v_{\text{use}}$ between staked and unstaked portions:

• If $\Delta v_{\text{use}} < 0$ (loss): share it pro‑rata by stake.

  $$\Delta v_s = \Delta v_{\text{use}} \cdot \frac{s}{T}, \qquad \Delta v_{us} = \Delta v_{\text{use}} - \Delta v_s \tag{5.2}$$

• If $\Delta v_{\text{use}} > 0$ (profit): first bring staked value $v_s$ back up to its “ideal” target $v_s^*$ (if previous losses hit stakers), then split the remainder pro‑rata.

  $$\Delta v_s = \min\!\left(\Delta v_{\text{use}} \cdot \frac{s}{T}, \; \max(v_s^* - v_s, 0)\right) \tag{5.3}$$

Unstaked side gets whatever is left:

$$\Delta v_{us} = \Delta v_{\text{use}} - \Delta v_s \tag{5.4}$$

5.3 Negative rebase for staked supply

Staked ybBTC rebases down when value allocated to stakers (after fees) is less than proportional to total value. Let

• $v_1'$ = new total value after admin fee (i.e. $v_{-1} + \Delta v_{\text{use}}$),
• $v_s'$ = new staked value after sharing profit/loss,
• $s$ = staked token supply before rebase,
• $T$ = total token supply before rebase,
• $\delta s$ = how many staked tokens are burned (negative rebase).

We want the value share to match the token share:

$$\frac{s - \delta s}{T - \delta s} \;=\; \frac{v_s'}{v_1'}. \tag{5.5}$$

Solve for $\delta s$:

$$\boxed{ \delta s = \frac{s\,v_1' - T\,v_s'}{\,v_1' - v_s'\,} } \tag{5.6}$$

So the post‑rebase supplies become:

$$s_{\text{new}} = s - \delta s, \qquad T_{\text{new}} = T - \delta s. \tag{5.7}$$

Unstaked tokens never rebase; they simply accumulate (or lose) value via the $(1-f_a)$ fee stream.

Intuition. Stakers opt into emissions instead of cash fees. If their side under‑earns (because they took earlier losses), their token count is cut (negative rebase) so each remaining token still represents its fair slice of $v_s'$.

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