Liquidity Provider Fundamentals: How DEX Fees Offset Impermanent Loss (Part 8)

You are currently viewing Liquidity Provider Fundamentals: How DEX Fees Offset Impermanent Loss (Part 8)

We now arrive at a crucial concept: how DEX trading fees compensate for Impermanent Loss (IL). While most sources vaguely mention this relationship, few explain the exact mechanics.

Let’s revisit our previous IL formula:

\displaystyle IL (\%) = \frac{2}{d + 1} \times \frac{y_2}{y_1} - 1

Now, we’ll adjust for price changes (d) and DEX fees.

Price Ratio (d) with DEX Fees

Define d as the price change ratio:
\displaystyle d = \frac{x_1 y_2}{x_2 y_1}

From this, we derive:
\displaystyle x_1 y_2 = d \times x_2 y_1

Multiply both sides by \displaystyle y_1 y_2:
\displaystyle x_1 y_2 \times y_1 y_2 = d \times x_2 y_1 \times y_1 y_2

\displaystyle x_1 y_1 y_2^2 = d \times x_2 y_2 y_1^2

Since DEX fees increase k (without liquidity additions/withdrawals, only swaps), we use:

\displaystyle k_1 = x_1 y_1 \quad \text{and} \quad k_2 = x_2 y_2

Substituting k1 and k2:
\displaystyle k_1 \times y_2^2 = d \times k_2 \times y_1^2

Solving for \displaystyle \frac{y_2}{y_1}:
\displaystyle \frac{y_2^2}{y_1^2} = d \times \frac{k_2}{k_1}

\displaystyle \frac{y_2}{y_1} = \sqrt{d \times \frac{k_2}{k_1}}

Plugging back into the IL equation:
\displaystyle IL (\%) = \frac{2}{d + 1} \times \sqrt{d \times \frac{k_2}{k_1}} - 1 = \sqrt{\frac{k_2}{k_1}} \times \frac{2\sqrt{d}}{d + 1} - 1

Key Implications:

  • If no swaps occur, \displaystyle k_1 = k_2
  • If swaps occur, \displaystyle k_1 > k_2  (fees accumulate)
  • The term \displaystyle \sqrt{\frac{k_2}{k_1}} > 0 (DEX Liquidity Ratio, DLR) reduces IL impact

Numerical Examples: IL vs. DEX Fees

Define DLR (DEX Liquidity Ratio):
\displaystyle DLR = \sqrt{\frac{k_2}{k_1}}

Figure 1. Impermanent Loss vs. DEX Fees.

When DLR = 1.05

  • d = 25% (Token X drops by 75%) → IL = -16.0%
  • d = 50% (Token X drops by 50%) → IL = -1.0%
  • d = 75% (Token X drops by 25%) → IL = 3.9%
  • d = 125% (Token X rises by 25%) → IL = 4.3%
  • d = 150% (Token X rises by 50%) → IL = 2.9%
  • d = 200% (Token X rises by 100%) → IL = -1.0%
  • d = 300% (Token X rises by 200%) → IL = -9.1%

✅ Profit Zone: d = 53% to 188%

When DLR = 1.10

  • d = 25% (Token X drops by 75%) → IL = -12.0%
  • d = 50% (Token X drops by 50%) → IL = 3.7%
  • d = 75% (Token X drops by 25%) → IL = 8.9%
  • d = 125% (Token X rises by 25%) → IL = 9.3%
  • d = 150% (Token X rises by 50%) → IL = 7.8%
  • d = 200% (Token X rises by 100%) → IL = 3.7%
  • d = 300% (Token X rises by 200%) → IL = -4.7%

✅ Profit Zone: d = 41% to 243%

When DLR = 1.20

  • d = 25% (Token X drops by 75%) → IL = -4.0%
  • d = 50% (Token X drops by 50%) → IL = 13.1%
  • d = 75% (Token X drops by 25%) → IL = 18.8%
  • d = 125% (Token X rises by 25%) → IL = 19.3%
  • d = 150% (Token X rises by 50%) → IL = 17.6%
  • d = 200% (Token X rises by 100%) → IL = 13.1%
  • d = 300% (Token X rises by 200%) → IL = 3.9%

✅ Profit Zone: d = 29% to 347%

Key Takeaways

  • DEX fees reduce IL impact – Higher DLR (more fees) expands the profitable price range.
  • Fee accumulation matters – Even with large price swings, sufficient fees can turn IL into gains.
  • Practical insight – liquidity providers should monitor DLR to assess whether fees outweigh IL.

What is ghostDAO?

ghostDAO is the future of interoperability and governance in web3 by allowing anyone to transfer assets from one chain to another all while being chain and DApp agnostic.

ghostAirdrop | ghostAirdrop Bot | Litepaper | Medium | Telegram | Twitter | Website
Tags: , , ,