Liquidity Provision Fundamentals

Constant Product vs Concentrated Liquidity

Constant Product Liquidity (Uniswap V2 Model)

Liquidity is distributed uniformly across the entire infinite price range $(0, \infty)$, which means price cannot go zero as liquidity of each asset cannot be completely depleted. But it also means capital is allocated to all possible prices even though most trading occurs within a narrow range.

The constant product model follows the formula $x \times y = k$, where:

• $x$ and $y$ are the reserves of two tokens
• $k$ is the liquidity expressed as a constant product of token amounts, e.g. $k = \text{Token}_{X,0} \times \text{Token}_{Y,0}$

After price change from $P_0$ to $P_1$ (where $P_i = \frac{\text{Token}_{Y,i}}{\text{Token}_{X,i}}$):

• New amount for $\text{Token}_{X,1} = \sqrt{\frac{k}{P_1}}$
• New amount for $\text{Token}_{Y,1} = \sqrt{k \times P_1}$
• Liquidity stays constant with new amounts: $k = \text{Token}_{X,1} \times \text{Token}_{Y,1}$

Concentrated Liquidity (Uniswap V3 Model)

The concentrated liquidity pool is comprised of multiple price ranges, each having finite reserves. The entire price range from 0 to infinite is split into shorter price ranges, with each of them having its own amount of liquidity. Instead of spreading liquidity across all prices, LPs can concentrate their capital where they expect trading to occur. Note, that full price interval is also possible.

Similarly to the constant product model, here the liquidity is the combination of token reserves. The measure of liquidity is expressed as the geometric mean of reserves $x$ and $y$: $L = \sqrt{xy}$. More useful is to express it as $L = \Delta y / \Delta \sqrt{P}$, which illustrates that the change in square root of price $P$ is tied to the change in token $y$ amount. Since token prices in a pool are reciprocals of each other, we use only one of those in calculations (by convention Uniswap V3 uses $P = y / x$).

$L$ remains constant as price moves within the interval $[a, b]$, only the token amounts change. For a position with liquidity $L_{a,b}$ in a range $[P_a, P_b]$, when market price P is within range:

$x = L \times \lparen \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_b}} \rparen$

$y = L \times \lparen \sqrt{P} - \sqrt{P_a} \rparen$

Therefore, at range boundaries the position consists only of single token reserves:

• At $P = P_a$ (lower bound), hold 100% $\text{Token}_X$, 0% $\text{Token}_Y$
• At $P = P_b$ (upper bound), hold 0% $\text{Token}_X$, 100% $\text{Token}_Y$

warning

This is not a comprehensive overview of how liquidity works. This is beyond the scope of this guide. If you want to dive deep into math, this resource is a good place to start.

Capital Efficiency Comparison

Constant Product:

• Capital efficiency: low (~5-20% typically utilized)
• If you provide $10,000 of liquidity, only $500-2,000 might be actively used for swaps
• Simple to manage, no rebalancing needed

Concentrated Liquidity:

• Capital efficiency: high (up to 4000x more efficient)
• If you concentrate liquidity in a ±20% range around current price (typical for altcoins), nearly 100% of capital is utilized
• Requires active management and rebalancing

Real-World Example: for a $0.01 altcoin with $1M daily volume:

• Constant Product: Might need $500k+ TVL spread across entire range
• Concentrated (±20%): Only ~$33k TVL needed for 10% max slippage
• This 15x difference explains why concentrated liquidity dominates for volatile pairs

Risks with Concentrated Liquidity

1. Out-of-Range Risk: If price moves outside your range, liquidity becomes inactive (earning no fees). Particularly acute for volatile altcoins that can move ±50% in a day.
2. Increased Impermanent Loss: Concentrated positions experience amplified IL within the range. A ±20% range experiences ~5x the IL of full-range positions.
3. Rebalancing Costs: As prices move, positions might need rebalancing, consuming gas fees. On Ethereum mainnet, gas costs can exceed profits for small positions.
4. Complexity: Requires sophisticated strategy and monitoring.
5. Range Wars: Other LPs can provide tighter ranges to capture more fees.

Liquidity Required to Support $1 Million Volume

Formula for Concentrated Liquidity

The relationship between liquidity $(L)$, volume $(V)$, and price impact $(\delta)$ in concentrated liquidity is:

$L = \frac{V \times \sqrt{P}}{2 \times \delta \times \text{fee}}$

Where:

• $L$ = Liquidity required
• $V$ = Trading volume
• $P$ = Current price
• $\delta$ = Acceptable price impact (as decimal)
• fee = Trading fee tier (as decimal)

Sample Calculation

Scenario: SYNT/USDC pool

• Current price: $0.01 per SYNT
• Trading volume: $1,000,000
• Acceptable price impact: 10% (=0.1)
• Fee tier: 0.3% (=0.003)
• Liquidity concentrated in ±20% range (wider range for volatile altcoin)

Step 1: Calculate base liquidity requirement (if spread across entire range):

$L = \frac{1{,}000{,}000 \times \sqrt{0.01}}{2 \times 0.1 \times 0.003} = \frac{1{,}000{,}000 \times 0.1}{0.0006} = \$166{,}667$

Step 2: Adjust for concentration Concentrating liquidity in a ±20% range instead of the full range $(0, \infty)$ provides approximately 5x capital efficiency:

$\text{Required TVL} = \frac{\$166{,}667}{5} = \$33{,}333$

Result: Only ~$33,000 in concentrated liquidity (±20% range) needed to support $1M daily volume with 10% max price impact. This demonstrates how small-cap altcoins can have high volume relative to liquidity.

Note: For comparison, with tighter requirements:

• 5% price impact: ~$67,000 TVL needed
• 2% price impact: ~$167,000 TVL needed
• 1% price impact: ~$333,000 TVL needed

The concentration efficiency factor depends on the range width:

• ±50% range: ~2x efficiency vs full range
• ±20% range: ~5x efficiency
• ±10% range: ~10x efficiency
• ±5% range: ~20x efficiency
• ±1% range: ~100x efficiency

Revenue from Liquidity Provision

Theoretical Earnings Calculation

Using our example of $33,333 TVL supporting $1M daily volume with 0.3% fees:

• Total daily fee generation

  $\text{Daily Fees} = \text{Volume} \times \text{Fee Rate} = \$1{,}000{,}000 \times 0.003 = \$3{,}000$

• Individual liquidity provider earnings

  For different position sizes:

  • Full pool ($33,333): $3,000/day → $1,095,000/year → 3,285% APR
  • 10% of pool ($3,333): $300/day → $109,500/year → 3,285% APR
  • 1% of pool ($333): $30/day → $10,950/year → 3,285% APR

Real-World Considerations

These astronomical returns (3,285% APR) are theoretically possible but face significant reductions.

Factors Reducing Actual Returns

1. Impermanent Loss:

   • Volatile altcoins can experience 20-50% IL over months
   • Each 2x price move = -5.72% IL (see examples in the section below)
   • Multiple volatile swings compound losses

2. Out-of-Range Positions:

   • ±20% range might be inactive 30-50% of the time
   • No fees earned when price exits your range
   • Requires gas costs to rebalance

3. Volume Decay:

   • $1M daily volume rarely sustains for a full year
   • Bear markets can reduce volume by 80-95%
   • Project lifecycle affects trading interest

4. Competition & Dilution:

   • High APRs attract more LPs
   • Your pool share decreases as TVL grows
   • Mercenary capital floods profitable pools

5. Protocol Risks:

   • Smart contract exploits
   • Rug pulls for altcoin projects
   • Oracle manipulation attacks

Realistic Earnings Scenarios

Conservative Scenario (Bear Market):

• Average daily volume: $50,000
• Position in range: 40% of time
• TVL grows to $66,666
• Your $3,333 position = 5% of pool
• Daily Earnings: $50,000 × 0.003 × 0.4 × 0.05 = $3.00
• Annual Return: $1,095 (33% APR before gas fees and IL)

Moderate Scenario (Normal Conditions):

• Average daily volume: $200,000
• Position in range: 60% of time
• Competition increases TVL to $100,000
• Your $3,333 position = 3.33% of pool
• Daily Earnings: $200,000 × 0.003 × 0.6 × 0.0333 = $12.00
• Annual Return: $4,380 (131% APR before gas fees and IL)

Optimistic Scenario (Early LP in Trending Token):

• Average daily volume: $500,000
• Position in range: 90% of time (active management)
• Limited competition increases TVL to $200,000
• Your $3,333 position = 1.67% of pool
• Daily Earnings: $500,000 × 0.003 × 0.9 × 0.0167 = $22.50
• Annual Return: $8,212 (246% APR before gas fees and IL)

Key Insights on Earnings

• Volume/TVL Ratio is Critical: High ratio creates exceptional fee opportunities
• Early Positioning Matters: Being first to provide liquidity captures highest fees
• Active Management Required: Keeping position in range maximizes earnings
• Risk/Reward Trade-off: Higher returns come with IL risk and potential total loss
• Realistic Expectations: Expect considerably lower APR than theoretical maximums

Understanding Impermanent Loss

Impermanent loss (IL) occurs when the price ratio between paired assets changes from the initial deposit ratio. The loss becomes "permanent" only when you withdraw liquidity. IL is calculated as:

$\text{IL} = \frac{2\sqrt{\text{price\_ratio}}}{1 + \text{price\_ratio}} - 1$

Where $\text{price\_ratio} = \frac{\text{current\_price}}{\text{initial\_price}}$

AMM Rebalancing Causes IL

When providing liquidity to a constant product AMM $(x \times y = k)$, your tokens automatically rebalance as prices change based on the liquidity formula given above. This rebalancing is what creates impermanent loss - you end up with more of the depreciating asset and less of the appreciating one.

With concentrated liquidity, IL is non-linear and amplified, 5-20x higher IL within range can be expected. At interval boundaries, a complete conversion to single asset occurs, as described with concentrated liquidity formulas above. Missing price moves outside range is a form of loss as well. Active management is therefore advised - liquidity should be repositioned before price exits the price range. Tighter price ranges results in higher earnings from swap fees but higher IL as well. This is why concentrated liquidity is often described as "selling options" - a liquidity provider is earning premium (swap fees) but taking on significant directional risk.

Impermanent Loss Scenarios

info

All examples provided here use the constant product AMM for simplicity

Case A: volatile/stable pair

Assume the pair is SYNT/USDC. SYNT is a volatile altcoin, while USDC is a stablecoin which effectively does not change its dollar value, since its value is pegged to the U.S. dollar.

Initial liquidity position: 10,000 SYNT at $0.01 each + $100 USDC = $200 total.

Scenario A.1: SYNT rises against USDC

Scenario: SYNT goes from $0.01 to $0.02 (2x increase)

Initial LP Position: $k = 10{,}000 \times 100 = 1{,}000{,}000$

Calculating New Holdings: When price changes, the AMM rebalances to maintain the constant product while reflecting the new price. For a constant product AMM where $\text{Token}_X \times \text{Token}_X = k$:

• If price of SYNT in USDC = $P$, then: $\frac{\text{USDC}}{\text{SYNT}} = P$
• This means: $\text{USDC} = P \times \text{SYNT}$

Using the constant product formula:

• $\text{SYNT} \times \text{USDC} = k$
• $\text{SYNT} \times (P \times \text{SYNT}) = k$
• $\text{SYNT}^2 \times P = k$
• $\text{SYNT} = \sqrt{\frac{k}{P}}$

Step-by-step calculation:

1. New SYNT amount: $\sqrt{\frac{1{,}000{,}000}{0.02}} = \sqrt{50{,}000{,}000} = 7{,}071.07$ SYNT
2. New USDC amount: $\sqrt{1{,}000{,}000 \times 0.02} = \sqrt{20{,}000} = 141.42$ USDC
3. Verification: $7{,}071.07 \times 141.42 \approx 1{,}000{,}000$ ✓
4. Price check: $\frac{141.42}{7{,}071.07} = 0.02$ USDC per SYNT ✓

Result without providing liquidity (HODL):

• 10,000 SYNT × $0.02 = $200
• 100 USDC = $100
• Total: $300

Result as a liquidity provider (LP):

• New holdings: 7,071.07 SYNT + 141.42 USDC
• Value: 7,071.07 × $0.02 + $141.42 = $282.84
• IL = -$17.16 (-5.72%)

Scenario A.2: SYNT falls against USDC

Scenario: SYNT goes from $0.01 to $0.005 (50% or a reciprocal 1/2x decrease as compared to Scenario A.1)

Initial LP Position: $k = 10{,}000 \times 100 = 1{,}000{,}000$

Calculating New Holdings: Using the same formulas (where $P$ is the price in USDC):

• New amount of SYNT: $\sqrt{\frac{k}{P}}$
• New amount of USDC: $\sqrt{k \times P}$

Step-by-step calculation:

1. New SYNT amount: $\sqrt{\frac{1{,}000{,}000}{0.005}} = \sqrt{200{,}000{,}000} = 14{,}142.14$ SYNT
2. New USDC amount: $\sqrt{1{,}000{,}000 \times 0.005} = \sqrt{5{,}000} = 70.71$ USDC
3. Verification: $14{,}142.14 \times 70.71 \approx 1{,}000{,}000$ ✓
4. Price check: $\frac{70.71}{14{,}142.14} = 0.005$ USDC per SYNT ✓

Result without providing liquidity (HODL):

• 10,000 SYNT × $0.005 =$50
• 100 USDC = $100
• Total: $150

Result as a liquidity provider (LP):

• New holdings: 14,142.14 SYNT + 70.71 USDC
• Value: 14,142.14 × $0.005 + $70.71 = $141.42
• IL = -$8.58 (-5.72%)

Case B: volatile/volatile pair

Assume the pair is SYNT/ETH. SYNT is a volatile altcoin, while ETH is a major cryptocurrency it is very volatile as well.

Initial position: 10,000 SYNT at 0.0000025 ETH each + 0.025 ETH (assume SYNT = $0.01 and ETH = $4,000). Initial value: $100 SYNT + $100 ETH = $200.

Scenario B.1: SYNT rises against ETH, doth depreciate in USD

Initial Position:

• 10,000 SYNT + 0.025 ETH
• Initial liquidity: $k = 10{,}000 \times 0.025 = 250$
• Initial SYNT/ETH price = 0.0000025 (1 SYNT = 0.0000025 ETH)

Scenario:

• SYNT/ETH: 0.0000025 to 0.000005 (SYNT doubles against ETH)
• ETH: $4,000 to $3,000
• SYNT: $0.01 to $0.015

Calculating New Holdings:

1. Since price is now 0.000005 ETH per SYNT, new SYNT amount: $\sqrt{\frac{250}{0.000005}} = \sqrt{50{,}000{,}000} = 7{,}071.07$ SYNT
2. New ETH amount: $\sqrt{250 \times 0.000005} = \sqrt{0.00125} = 0.0354$ ETH
3. Verification: $7{,}071.07 \times 0.0354 \approx 250$ ✓
4. Price check: $\frac{0.0354}{7{,}071.07} = 0.000005$ ETH per SYNT ✓

HODL Results:

• 10,000 SYNT × $0.015 = $150
• 0.025 ETH × $3,000 = $75
• Total: $225

LP Results:

• New holdings: 7,071.07 SYNT + 0.0354 ETH
• Value: 7,071.07 × $0.015 + 0.0354 × $3,000 = $212.12
• IL = -$12.88 (-5.73%)

Scenario B.2: SYNT rises against ETH, both appreciate in USD

Scenario:

• SYNT/ETH: 0.0000025 to 0.000005
• ETH: $4,000 to $6,000
• SYNT: $0.01 to $0.03

HODL Results:

• 10,000 SYNT × $0.03 = $300
• 0.025 ETH × $6,000 = $150
• Total: $450

LP Results:

• New holdings: 7,071.07 SYNT + 0.0354 ETH
• Value: 7,071.07 × $0.03 + 0.0354 × $6,000 = $424.23
• IL = -$25.77 (-5.73%)

Scenario B.3: SYNT falls against ETH, both depreciate in USD

Scenario:

• SYNT/ETH: 0.0000025 to 0.00000125
• ETH: $4,000 to $3,000
• SYNT: $0.01 to $0.00375

Calculating New Holdings:

1. New SYNT amount: $\sqrt{\frac{250}{0.00000125}} = \sqrt{200{,}000{,}000} = 14{,}142.14$ SYNT
2. New ETH amount: $\sqrt{250 \times 0.00000125} = \sqrt{0.0003125} = 0.0177$ ETH

HODL Results:

• 10,000 SYNT × $0.00375 = $37.50
• 0.025 ETH × $3,000 = $75
• Total: $112.50

LP Results:

• New holdings: 14,142.14 SYNT + 0.0177 ETH
• Value: 14,142.14 × $0.00375 + 0.0177 × $3,000 = $106.06
• IL = -$6.44 (-5.73%)

Scenario B.4: SYNT falls against ETH, both appreciate in USD

Scenario:

• SYNT/ETH: 0.0000025 to 0.00000125
• ETH: $4,000 to $6,000
• SYNT: $0.01 to $0.0075

HODL Results:

• 10,000 SYNT × $0.0075 = $75
• 0.025 ETH × $6,000 = $150
• Total: $225

LP Results:

• New holdings: 14,142.14 SYNT + 0.0177 ETH
• Value: 14,142.14 × $0.0075 + 0.0177 × $6,000 = $212.12
• IL = -$12.88 (-5.73%)

Scenario B.5: SYNT and ETH move by same percentage

Scenario:

• SYNT appreciates by 50%: $0.01 to $0.015
• ETH appreciates by 50%: $4,000 to $6,000
• SYNT/ETH ratio unchanged: 0.0000025

Why No Impermanent Loss?

When both assets move by the same percentage, their ratio remains constant. Since the AMM rebalances based on the price ratio, not absolute prices:

• Initial: 1 SYNT = 0.0000025 ETH
• After: 1 SYNT = 0.0000025 ETH (unchanged)
• No rebalancing occurs! You keep your original 10,000 SYNT + 0.025 ETH

HODL Results:

• 10,000 SYNT × $0.015 = $150
• 0.025 ETH × $6,000 = $150
• Total: $300

LP Results:

• Holdings remain: 10,000 SYNT + 0.025 ETH (no rebalancing needed)
• Value: 10,000 × $0.015 + 0.025 × $6,000 = $300
• IL = $0 (0%)

This demonstrates a key principle: IL only occurs when the price ratio between assets changes, not from USD price movements alone.

Key Takeaways

1. Capital Efficiency: Concentrated liquidity can be 10-4000x more capital efficient than constant product AMMs

2. Liquidity Requirements: For $1M daily volume with 10% max slippage on a $0.01 altcoin, you only need ~$33,000 TVL in concentrated positions (±20% range). This explains why small-cap tokens can have high volume/TVL ratios.

3. Earnings Potential: Theoretical APRs can exceed 3,000% but realistic returns range from 100-1,000% after accounting for IL, range management, and competition.

4. Impermanent Loss Patterns:

   • IL is always negative (you never gain from IL)
   • IL is symmetrical for price ratios (2x and 0.5x create same IL%)
   • Maximum IL approaches -100% as price ratio approaches infinity
   • No IL when assets move proportionally

5. Risk Management:

   • Tighter ranges = higher fee earnings but higher IL risk
   • Volatile pairs require wider ranges or more active management
   • Consider fee earnings vs IL when setting ranges
   • Small-cap altcoins often justify wider acceptable slippage

6. Practical Strategies:

   • For stable pairs: Use tight ranges (±0.5-2%)
   • For volatile altcoin pairs: Use wider ranges (±20-50%)
   • Monitor and rebalance positions regularly
   • Factor in gas costs for rebalancing
   • Accept higher slippage for low-liquidity tokens

warning

Remember: Successful LP strategies balance fee generation against IL risk, considering volatility, correlation, and trading volume of the pair. For small-cap altcoins, wider price impact tolerance and broader ranges are often necessary and acceptable.

Learn More

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