# FeeAMM vs Invariant AMM Under Depegging: An LP-Centric Liquidity and Tail Risk Model

## 1. Objective and Scope

This note develops a **comparative tail risk model** of FeeAMMs (as introduced in [Tempo](https://tempo.xyz/)) and standard invariant AMMs under stablecoin depegging.

The analysis is deliberately **LP-centric**:

> *What is the economic exposure of LPs under depegging?

The core claim is that **identical external price shocks induce fundamentally different loss profiles** across the two designs:

- **Invariant AMMs**: losses are continuous, bounded by concave value function, with automatic rebalancing.
- **FeeAMMs**: losses become linear in price after rapid concentration, with no automatic recovery.

We provide:
1. Mechanistic analysis of why concentration occurs,
2. Quantitative comparison of loss functions,
3. Break-even analysis,
4. Game-theoretic analysis of LP run incentives.

---

## 2. Mechanism Comparison

### 2.1 Invariant AMM (CPMM)

A constant-product AMM with reserves $(x, y)$ satisfies:

$$
xy = K
$$

Arbitrage enforces the internal price to match external price $p$:

$$
p_{AMM} = \frac{x}{y} = p
$$

Solving yields reserve dynamics:

$$
x(p) = \sqrt{Kp}, \qquad y(p) = \sqrt{\frac{K}{p}}
$$

**LP value** (in numeraire terms):

$$
V_{AMM}(p) = x(p) + p \cdot y(p) = 2\sqrt{Kp}
$$

Key property: $V_{AMM}(p)$ is **concave** in $p$.

---

### 2.2 FeeAMM

A FeeAMM holds user token $U$ and validator token $V$ with **fixed conversion rules**:

| Flow | Direction | Rate |
|------|-----------|------|
| Fee payment | $U \rightarrow V$ | $\kappa$ (e.g., 0.997) |
| Arbitrage | $V \rightarrow U$ | $\kappa'$ (e.g., 0.9985) |

Both directions operate at **fixed exchange rates** independent of external price $p$.

There is:
- no invariant linking $(U, V)$,
- no endogenous price discovery,
- no automatic rebalancing mechanism.

---

### 2.3 Core Structural Difference

| Property | CPMM | FeeAMM |
|----------|------|--------|
| Price adjustment | Endogenous | None (fixed rates) |
| Arbitrage effect | Rebalances pool | Concentrates pool |
| Loss profile | Concave in $p$ | Linear in $p$ (post-concentration) |
| Recovery | Automatic via arbitrage | Only through slow fee flow |

---

## 3. Concentration Dynamics Under Depeg

### 3.1 Why V Accumulates in the Pool

When $V$ depegs ($p < \kappa'$), arbitrage becomes profitable:

1. Buy $V$ externally at price $p$
2. Deposit $V$ into FeeAMM, receive $\kappa' \cdot U$
3. Profit: $\kappa' - p > 0$

This arbitrage:
- extracts $U$ from the pool,
- deposits $V$ into the pool,
- continues until $U$ is depleted or $p$ recovers.

**Result**: Pool inventory converges toward 100% $V$.

---

### 3.2 Fast-Slow Asymmetry

Define:
- **Fast process**: Arbitrage (clears in minutes/hours)
- **Slow process**: Fee flow (clears over days/weeks)

| Process | Direction | Speed | Trigger |
|---------|-----------|-------|---------|
| Arbitrage | $V$ into pool | Fast | $p < \kappa'$ |
| Fee payment | $V$ out of pool | Slow | User transactions |

Under depeg:

$$
\text{Rate}_{arbitrage} \gg \text{Rate}_{fee\ flow}
$$

**Implication**: Concentration happens almost instantly; recovery (if any) takes orders of magnitude longer.

---

### 3.3 Numerical Example

Assume:
- Initial pool: 50% $U$, 50% $V$
- Depeg: $p$ drops from 1.0 to 0.9
- Arbitrage rate: $\kappa' = 0.9985$

**Arbitrage profit per unit $V$**: $0.9985 - 0.9 = 0.0985$ (9.85%)

**Outcome**: Rational arbitrageurs extract all $U$ within hours.

Post-arbitrage pool composition:
- $U$: ~0%
- $V$: ~100%

LP exposure to depegged asset: **100%**

---

## 4. Loss Function Comparison

### 4.1 CPMM Loss Function

For CPMM, LP value as a function of price:

$$
V_{AMM}(p) = 2\sqrt{Kp}
$$

Normalizing to $V_{AMM}(1) = 1$:

$$
V_{AMM}(p) = \sqrt{p}
$$

**Properties**:
- Concave: $\frac{d^2 V}{dp^2} < 0$
- At $p = 0.5$: loss = $1 - \sqrt{0.5} = 29.3\%$
- At $p = 0.1$: loss = $1 - \sqrt{0.1} = 68.4\%$
- At $p = 0$: loss = $100\%$

---

### 4.2 FeeAMM Loss Function (Post-Concentration)

After arbitrage concentrates the pool to 100% $V$:

$$
V_{FeeAMM}(p) = p
$$

**Properties**:
- Linear: $\frac{d^2 V}{dp^2} = 0$
- At $p = 0.5$: loss = $50\%$
- At $p = 0.1$: loss = $90\%$
- At $p = 0$: loss = $100\%$

---

### 4.3 Tail Exposure Comparison

| Price $p$ | CPMM Loss | FeeAMM Loss | Ratio |
|-----------|-----------|-------------|-------|
| 0.90 | 5.1% | 10.0% | 1.9× |
| 0.80 | 10.6% | 20.0% | 1.9× |
| 0.50 | 29.3% | 50.0% | 1.7× |
| 0.25 | 50.0% | 75.0% | 1.5× |
| 0.10 | 68.4% | 90.0% | 1.3× |

**Key insight**: FeeAMM losses are consistently higher, with the gap largest in moderate depeg scenarios—which are more frequent than total collapse.

---

## 5. Fee Sufficiency: Does the Fee Compensate for Tail Risk?

Even if FeeAMM losses are structurally larger, could sufficiently high fees compensate LPs?

### 5.1 LP Break-Even Condition

LP expected profit over a period can be decomposed as:

$$
\mathbb{E}[\Pi] = \underbrace{f \cdot C}_{\text{fee income}} - \underbrace{\mathbb{E}[L_{depeg}]}_{\text{tail loss}}
$$

Where:
- $f$: effective fee rate per transaction (e.g., 0.3%)
- $C$: cumulative fee notional processed (as multiple of LP capital)
- $L_{depeg}$: loss from depeg events

For LP participation to be rational:

$$
f \cdot C \geq \mathbb{E}[L_{depeg}]
$$

---

### 5.2 Expected Depeg Loss

Let:
- $\lambda$: annual depeg frequency (events/year)
- $\mathbb{E}[1 - p \mid depeg]$: expected loss severity given depeg

Then:

$$
\mathbb{E}[L_{depeg}] = \lambda \cdot \mathbb{E}[1 - p \mid depeg]
$$

Substituting into break-even:

$$
f \cdot C \geq \lambda \cdot \mathbb{E}[1 - p \mid depeg]
$$

---

### 5.3 Implied Depeg Frequency

Rearranging for $\lambda$:

$$
\lambda_{implied} = \frac{f \cdot C}{\mathbb{E}[1 - p \mid depeg]}
$$

This is the **maximum depeg frequency** that the fee structure can sustain.

**Interpretation**: Given observed fees, FeeAMM LPs are implicitly pricing depeg events as occurring no more frequently than $\lambda_{implied}$.

Further analysis is needed to calculate rational $\lambda_{implied}$ for LPs in FeeAMM, which is quite difficult given that FeeAMM is not on mainnet yet and historical depeg statistics are not occasional and too survivor-biased.

## 6. LP Exit Dynamics: The Closing Window

### 6.1 Two Phases (state variable: remaining U)
Arbitrage under depeg p<\kappa' monotonically depletes U. Define the pre-arbitrage window by the fraction of U remaining:
$$
\frac{U(t)}{U_0}>\varepsilon
\quad(\text{pre-arb}),
\qquad
\frac{U(t)}{U_0}\le \varepsilon
\quad(\text{post-concentration})
$$

---

### 6.2 Arbitrage speed (exponential depletion)
Model fast arbitrage as proportional depletion of the remaining U:
$$
\frac{dU}{dt}=-\rho\,U(t)
\qquad\Longrightarrow\qquad
U(t)=U_0 e^{-\rho t}
$$
Here $\rho$ is the arbitrage depletion rate (per unit time).

---

### 6.3 Inventory mapping (fixed-rate arbitrage $V\to \kappa' U$)

Each unit of U extracted corresponds to ($1/\kappa'$) units of V deposited into the pool:
$$
V(t)=V_0+\frac{U_0-U(t)}{\kappa'}=
V_0+\frac{U_0}{\kappa'}\bigl(1-e^{-\rho t}\bigr)
$$

---

### 6.4 LP exit value during the window

Pool value in U-numeraire (LP exit value up to pro-rata share) is:

$$
X(t)=U(t)+p\,V(t)
$$

Substitute U(t),V(t):
$$
X(t)=
pV_0+\frac{p}{\kappa'}U_0
+\Bigl(1-\frac{p}{\kappa'}\Bigr)U_0 e^{-\rho t}
$$

Since $p<\kappa'$, we have $1-\frac{p}{\kappa'}>0$, hence
$$
\frac{dX(t)}{dt}=
-\rho\Bigl(1-\frac{p}{\kappa'}\Bigr)U_0 e^{-\rho t}<0,
$$
so exit value declines monotonically as concentration proceeds.

---

### 6.5 The closing window length

Define the “closing time” $T_\varepsilon$ as the time until only an \varepsilon-fraction of U remains:

$$
T_\varepsilon=
\inf\Bigl\{t:\frac{U(t)}{U_0}\le\varepsilon\Bigr\}=
\frac{1}{\rho}\ln\frac{1}{\varepsilon}
$$

---

### 6.6 Post-concentration: timing no longer matters (only p matters)
As $t\to\infty$ (i.e., $U(t)\to 0$), exit value converges to

$$
X(\infty)=pV_0+\frac{p}{\kappa'}U_0
$$

and the residual timing premium from exiting at time t instead of after full concentration is
$$
X(t)-X(\infty)=
\Bigl(1-\frac{p}{\kappa'}\Bigr)U_0 e^{-\rho t},
$$

which decays exponentially at rate $\rho$.

---

### 6.7 LP reaction time vs. window capture probability

Let $T_{LP}$ be the LP’s reaction time (random) with CDF $F_{LP}$. The probability an LP exits before the window closes is:

$$
\Pr(T_{LP}\le T_\varepsilon)=F_{LP}(T_\varepsilon)
$$

and expected exit value is
$$
\mathbb{E}[X(T_{LP})]=
pV_0+\frac{p}{\kappa'}U_0
+\Bigl(1-\frac{p}{\kappa'}\Bigr)U_0\,\mathbb{E}\!\left[e^{-\rho T_{LP}}\right].
$$

---

## 7. Why This Is Structural, Not Parametric

This section explains why fee adjustment **cannot solve** the fundamental problem.

### 7.1 Fee Income Is Linear, Loss Is Unbounded

Fee income over time $T$:

$$
\text{Fee Income}(T) = f \cdot C \cdot T
$$

Expected loss over time $T$:

$$
\mathbb{E}[L(T)] = \lambda \cdot \mathbb{E}[1-p] \cdot T
$$

Both scale linearly with $T$.

However, there exists **no finite fee** that makes the following true:

$$
f \cdot C > \lambda \cdot \mathbb{E}[1-p] \quad \forall \lambda
$$

If depeg probability is uncertain, there is no fee that guarantees positive expected value.

---

### 7.2 The Infinite Maturity Problem

FeeAMM LP positions have **no expiry**.

For any finite fee $f$, there exists a sufficiently long horizon $T$ where:

$$
P(\text{at least one depeg in } [0,T]) \to 1
$$

Since loss given depeg can be arbitrarily severe (up to 100%), and LP exposure is **unbounded in time**:

$$
\lim_{T \to \infty} \mathbb{E}[\Pi(T)] = -\infty \quad \text{for any finite } f
$$

---

### 7.3 Increasing Fees Does Not Restore Convexity

The core problem is that LP loss at FeeAMM is linear.

| Property | CPMM | FeeAMM |
|----------|------|--------|
| Value function | $V(p) = \sqrt{p}$ (concave) | $V(p) = p$ (linear) |
| Marginal loss at $p=0.5$ | 0.71 | 1.00 |
| Tail protection | Built-in via convexity | None |

CPMM's concavity provides **implicit downside protection**: as price falls, the rate of loss decreases.

FeeAMM's linearity provides **no such protection**: loss rate is constant regardless of price level. Increasing fees shifts the profit curve **up**, but does not change its **shape**. The tail exposure remains unhedged.

## 8. Design Implications

The preceding analysis shows that FeeAMM fragility is **contractual**, not behavioral or parametric.

LPs implicitly provide:
- A perpetual conversion service
- With unbounded downside exposure
- At a fixed price that does not reflect tail risk

This section outlines design modifications that address these structural issues.

---

### 8.1 Bounding the Tail

**Notional-Based Expiry**:
- FeeAMM liquidity is provided in discrete epochs.
- Each epoch is valid only up to a fixed cumulative processed-fee notional $C_{\max}$.
- Once cumulative notional $C(t)$ reaches $C_{\max}$:
- the epoch closes,
- LP positions are settled,
- any continuation requires fresh opt-in liquidity.

**Effect**:
- Maximum LP loss in an epoch is mechanically bounded by $\kappa C_{\max}$.
- Exposure is capped in volume, not in time.
- Persistent depegging no longer implies unbounded tail accumulation.

---

### 8.2 Restoring Convexity

**State-Dependent Conversion Rate**:
- Let $\kappa = \kappa(V)$ with $\kappa'(V) > 0$
- As $V$ inventory falls, conversion becomes less favorable
- Creates soft resistance to concentration

**Effect**:
- Fast arbitrage is no longer costless when inventory is scarce.
- Concentration pressure weakens endogenously as $V$ declines.
- The run-to-zero dynamic is replaced by a soft barrier.

## 9. Conclusion

FeeAMMs expose LPs to a structural form of tail risk that is fundamentally different from invariant AMMs. This risk does not arise from parameter choices, temporary market inefficiencies, or insufficient fees, but from the contractual shape of the LP payoff itself.

Under depegging, fixed-rate conversion combined with fast arbitrage forces FeeAMMs into near-complete inventory concentration. Once this occurs, LP value becomes linear in the depegged price, eliminating the concavity that protects LPs in invariant AMMs. Fees can shift expected returns upward, but they cannot change the loss geometry or hedge the downside. As a result, FeeAMM LP positions resemble perpetual short tail-risk exposures with no natural maturity, making eventual large losses statistically inevitable over long horizons.

This creates a predictable dynamic: when confidence weakens, rational LPs face a shrinking exit window and strong incentives to run early. The resulting fragility is therefore endogenous and unavoidable under the current design.

Robust FeeAMM designs must explicitly complete the LP contract by bounding exposure or restoring convexity. Mechanisms such as notional-based expiry, state-dependent conversion rates, or other endogenous feedbacks can transform unbounded, implicit risk into finite, explicit, opt-in exposure. Without such structural safeguards, FeeAMMs will systematically overburden LPs with tail risk and remain unstable in the presence of even modest depegging events.