Liquidity Provider Fundamentals: Optimizing Token Allocation for LP Contributions (Part 3)

Why This Matters

Most guides overlook a critical question: How should you split your tokens when joining a liquidity pool?

If you only hold Token X, becoming a liquidity provider (LP) requires two steps:

Swap Token X for Token – This changes the pool’s reserves and token prices
Deposit both tokens proportionally – Based on the newly updated reserves

This process introduces complexity – let’s break it down.

Before we proceed, take a deep breath. The math will get intense in the middle – like navigating a complex maze of variables. But stay with me! By the end of this derivation, everything will simplify beautifully, like reaching the clearing after a challenging hike. The reward for your patience will be crystal-clear understanding.

Step-by-Step Derivation

Initial Setup:

x₁: current reserves of Token X
y₁: current reserves of Token Y
f: the DEX fee

Alice has x_a of Token X and she wants to become a liquidity provider.

What quantities shall she allocate into Token X and Token Y?

Let’s assume that she needs to allocate:

x_s: quantity of Token X that needs to be swapped for y_s of Token Y
x_ns: quantity of Token X kept for LP contribution

Naturally, the initial balance of x_a tokens is split between x_s and x_ns.

$\displaystyle x_a = x_s + x_{ns}$

Before Swap:

$\displaystyle P(Y)_1 = \frac{x_1}{y_1}$

After Swap:

$\displaystyle P(Y)_2 = \frac{x_1 + x_s}{y_1 - y_s}$

The new reserves after the swap, of course, are:

x₁+ x_s: new reserves of Token X
y₁– y_s: new reserves of Token Y

LP Contribution Rule

To add liquidity, Alice’s remaining tokens must match the new reserve ratio

$\displaystyle \frac{x_{ns}}{y_s} = \frac{x_1 + x_s}{y_1 - y_s}$

Substitute x_ns:

$\displaystyle x_{ns} = x_a - x_s$

And now let’s solve for x_s:

$\displaystyle \frac{x_a - x_s}{y_s} = \frac{x_1 + x_s}{y_1 - y_s}$

Cross-multiply and simplify:

$\displaystyle (x_a - x_s) \times (y_1 - y_s) = y_s \times (x_1 + x_s)$

$\displaystyle x_a y_1 - x_a y_s - x_s y_1 + x_s y_s = x_1 y_s + x_s y_s$

$\displaystyle y_1 - x_a y_s - x_s y_1 = x_1 y_s$

Group and factor y_s:

$\displaystyle x_a y_1 - x_s y_1 = x_1 y_s + x_a y_s$

$\displaystyle x_a y_1 - x_s y_1 = y_s (x_1 + x_a)$

This is where it gets all crazy. Let’s introduce y_s in terms of x_s.

Remember the swap formula of y_s:

$\displaystyle y_s = \frac{x_s \times (1 - f) \times y_1}{x_1 + (1 - f) \times x_s}$

Plugging x_s into the above:

$\displaystyle x_a y_1 - x_s y_1 = \frac{x_s \times (1 - f) \times y_1}{x_1 + (1 - f) \times x_s} \times (x_1 + x_a)$

$\displaystyle (x_a y_1 - x_s y_1) \times (x_1 + (1 - f) \times x_s) = (x_s \times (1 - f) \times y_1) \times (x_1 + x_a)$

$\displaystyle (x_a y_1 - x_s y_1) \times (x_1 + x_s - f x_s) = (x_s y_1 - x_s y_1 f) \times (x_1 + x_a)$

$\displaystyle x_1 x_a y_1 + x_a x_s y_1 - x_a x_s y_1 f - x_1 x_s y_1 - x_s^2 y_1 + x_s^2 y_1 f = x_1 x_s y_1 + x_a x_s y_1 - x_1 x_s y_1 f - x_a x_s y_1 f$

Dividing the entire equation by y₁ and simplifying:

$\displaystyle x_1 x_a - x_1 x_s - x_s^2 + x_s^2 f = x_1 x_s - x_1 x_s f$

$\displaystyle x_s^2 - x_s^2 f + 2x_1 x_s - x_1 x_s f - x_1 x_a = 0$

Solving the quadratic equation using the quadratic equation formula:

$\displaystyle x_s^2 \times (1 - f) + x_s \times x_1 \times (2 - f) - x_1 x_a = 0$

$\displaystyle D = (x_1 \times (2 - f))^2 + 4x_a x_1$

Negative solution is rejected, while the positive solution is:

$\displaystyle x_s = \frac{-x_1 \times (2 - f) + \sqrt{(x_1 \times (2 - f))^2 + 4x_a x_1 \times (1 - f)}}{2 \times (1 - f)}$

x_ns can then be calculated by plugging x_s into the following equation:

$\displaystyle x_{ns} = x_a - x_s$

Approximation (For Simplicity):

Let’s try to also approximate x_s.

Using Taylor Series Expansion where:

$\displaystyle \sqrt{p + q} \approx \sqrt{p} + \frac{q}{2\sqrt{p}}$

Approximating the square root term in the x_s formula:

$\displaystyle \sqrt{(x_1 \times (2-f))^2 + 4x_a x_1 \times (1-f)} = \sqrt{(x_1 \times (2-f))^2} + \frac{4x_a x_1 \times (1-f)}{2\sqrt{(x_1 \times (2-f))^2}} =$

$\displaystyle x_1(2 - f) + \frac{4x_a x_1(1 - f)}{2x_1(2 - f)} = x_1(2 - f) + \frac{2x_a(1 - f)}{2 - f}$

Let’s plug it into the x_s formula:

$\displaystyle x_{s,approx} = \frac{-x_1 \times (2-f) + x_1 \times (2-f) + \frac{2x_a \times (1-f)}{(2-f)}}{2 \times (1-f)} = \frac{2x_a \times (1-f)}{(2-f)} \div \frac{2 \times (1-f)}{1} =$

$\displaystyle \frac{2x_a \times (1 - f)}{(2 - f)} \div \frac{1}{2 \times (1 - f)} = x_a \times \frac{1}{2 - f}$

x_ns can then be calculated by plugging x_s into the following equation:

$\displaystyle x_{ns,approx} = x_a - x_s = \frac{x_a}{1} - \frac{x_a}{2 - f} = \frac{(2 - f) \times x_a - x_a}{2 - f} = \frac{2x_a - f x_a - x_a}{2 - f} =$

$\displaystyle \frac{x_a - f x_a}{2 - f} = x_a \times \frac{1 - f}{2 - f}$

Practical Example

Consider a pool:

QD(DAI) = 100,000 DAI
QD(eGHST) = 10,000 eGHST
DF = 0.3%

Alice wants to allocate 5,000 DAI into the pool. To do this optimally:

Current price of eGHST:

$\displaystyle P(\text{eGHST}) = \frac{QD(\text{DAI})}{QD(\text{eGHST})} = \frac{100,\!000}{10,\!000} = 10~\text{DAI}$

Exact Calculation

How much DAI should Alice swap?

Using the precise formula:

$\displaystyle x_s = \frac{-QD(\text{DAI}) \times (2 - DF) + \sqrt{(QD(\text{DAI}) \times (2 - DF))^2 + 4 \times Q(\text{DAI})_a \times QD(\text{DAI}) \times (1 - DF)}}{2 \times (1 - DF)} =$

$\displaystyle \frac{-100,\!000 \times (2 - 0.003) + \sqrt{(100,\!000 \times (2 - 0.003))^2 + 4 \times 5,\!000 \times 100,\!000 \times (1 - 0.003)}}{2 \times (1 - 0.003)}$

$\displaystyle x_s = 2,\!473.22~\text{DAI}$

Resulting Swap & LP Allocation

Swapped DAI for eGHST:

$\displaystyle y_s = \frac{x_s \times (1 - DF) \times QD(\text{eGHST})}{QD(\text{DAI}) + (1 - DF) \times x_s} = \frac{2,\!473.22 \times (1 - 0.003) \times 10,\!000}{100,\!000 + (1 - 0.003) \times 2,\!473.22}$

$\displaystyle y_s = 240.65~\text{eGHST}$

Remaining DAI for LP:

$\displaystyle x_{ns} = Q(\text{DAI}) - x_s = 5,\!000 - 2,\!473.22 = 2,\!526.78~\text{DAI}$

New reserves after swap:

QD(DAI)_new = 100,000 + 2,473.22 = 102,473.22 DAI
QD(eGHST)_new = 10,000 – 240.65 = 9,759.35 eGHST

New eGHST price:

$\displaystyle P(\text{eGHST})_{\text{new}} = \frac{QD(\text{DAI})_{\text{new}}}{QD(\text{eGHST})_{\text{new}}} = \frac{102,\!473.22}{9,\!759.35} = 10.50~\text{DAI}$

eGHST price change after swap:

$\displaystyle \frac{P(\text{eGHST})_{\text{new}}}{P(\text{eGHST})} - 1 = \frac{10.50}{10.00} - 1 = 0.05$ or 5%

Approximation Method

For a quicker estimate:

$\displaystyle x_{s,approx} = Q(\text{DAI}) \times \frac{1}{2 - DF} = 5,\!000 \times \frac{1}{2 - 0.003} = 2,\!503.76~\text{DAI}$

Received eGHST from the swap:

$\displaystyle y_{s,approx} = \frac{x_{s,approx} \times (1 - DF) \times QD(\text{eGHST})}{QD(\text{DAI}) + (1 - DF) \times x_{s,approx}} = \frac{2,\!503.76 \times (1 - 0.003) \times 10,\!000}{100,\!000 + (1 - 0.003) \times 2,\!503.76}$

$\displaystyle y_{s,approx} = 243.54~\text{eGHST}$

Remaining DAI:

$\displaystyle x_{ns,approx} = Q(\text{DAI}) - x_{s,approx} = 5,\!000 - 2,\!503.76 = 2,\!496.24~\text{DAI}$

New reserves after swap:

QD(DAI)_new,approx = 100,000 + 2,503.76 = 102,503.76 DAI
QD(eGHST)_new,approx = 10,000 – 243.54 = 9,756.46 eGHST

New eGHST price:

$\displaystyle P(\text{eGHST})_{\text{new,approx}} = \frac{QD(\text{DAI})_{\text{new,approx}}}{QD(\text{eGHST})_{\text{new,approx}}} = \frac{102,\!503.76}{9,\!756.46} = 10.51~\text{DAI}$

eGHST price change after swap:

$\displaystyle \frac{P(\text{eGHST})_{\text{new,approx}}}{P(\text{eGHST})} - 1 = \frac{10.51}{10.00} - 1 = 0.0506$ or 5.06%

Key Insight

The approximation is close (5.06% vs. 5% price impact) but not exact. For precise capital efficiency, use the full formula – especially in large swaps or low-liquidity pools.

Trade-off:

Exact method – Maximizes LP value, but computationally heavy.
Approximation – Good for quick estimates, but slight slippage.

This example highlights why proper modelling matters in DeFi.

Why This Matters

Step-by-Step Derivation

Approximation (For Simplicity):

Practical Example

Exact Calculation

Approximation Method

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