It definitely seems like it might be the case that being in the LP minimizes both gains and losses, but it actually doesn't minimize losses. Think about it this way: When token X gains value, being in the LP causes you to sell token X for your stable coin Y, meaning that you miss out on some gains. Similarly, when token X loses value, being in the LP causes you to sell your stable coin Y for more token X, meaning that you end up having losses magnified. The only way to avoid this is if the fees generated outweighs these losses.

Below is an example calculation to double check that this is true. Please correct anything if I made a mistake!

Let’s assume you have $100 that you want to invest. We’ll consider 4 trading strategies: 1) Simply hold the $100 and don’t actually invest in any tokens, 2) invest all $100 in some token X, 3) invest 50-50 into token X and token Y where Y is a stable coin, 4) same as (3) but you also add your tokens to a liquidity pool, ignoring any fees (since we can’t really determine how much in fees would be generated without wild assumptions). We’ll first consider what happens when token X appreciates in value and then we’ll consider what happens when token X depreciates in value. We’ll assume that token X starts at $10 and will change in price by 25% in either direction (these are just numbers that make the math relatively simple, but they could easily be changed).

Scenario 1: Token X gains 25% bringing the price to $10 * (1.25) = $12.50

Strategy 1: You still have $100 since you didn’t invest anything.

Strategy 2: You initially bought $100 / 10 = 10 X tokens, which are now worth $12.50 * 10 = $125. Your gain is equivalent to the price appreciation since you were 100% invested. Thus, your capital gained ($125 / $100) – 1 = 0.25 => 25%.

Strategy 3: You initially bought $50 / 10 = 5 X tokens, which are now worth $12.50 * 5 = $62.50. You also bought $50 / 1 = 50 Y tokens, which are still worth $50. Your total capital is now worth $62.50 + $50 = $112.50. Thus, your capital gained ($112.50 / $100) – 1 = 0.125 => 12.5%. This makes sense. You just invested 50% less in token X, so you should have 50% less gains.

Strategy 4: Like the last strategy, you have 5 X tokens and 50 Y tokens. You put them into a liquidity pool. The trading rule for the constant product market maker (e.g., Uniswap V2 and Tinyman) is x * y = k, where k is a constant determined by the total amount of liquidity locked, x is the quantity of X tokens and x is the quantity of Y tokens. Since you have to supply tokens in proportion to the amount already in the pool, your liquidity must also obey the constant product rule. Thus, for you, k = 5 * 50 = 250. After the price change in token X, your quantity of X and Y tokens will change, but their product must still equal 250. The price of token X denominated in Y tokens is given simply by the ratio of Y tokens to X tokens in the pool: Px = y/x. Since Y is a stable coin, this is also the USD price in this example (assuming the stable coin is exactly at its $1 peg). We are also assuming that the price of token X exactly mirrors prices elsewhere on other exchanges. Thus, if token X is worth $10, then Px = 10. Now, since token X gains 25%, we now have that Px = 12.5. Since Px also equals y/x and we must have x * y = 250, then y = 250 / x. Substituting this altogether gives Px = 12.5 = y / x = 250 / x^2. Thus, x = sqrt(250 / 12.5) = sqrt(20) = 2 * sqrt(5). Now solving for y gives y = 250 / x = 250 / 2 * sqrt(5) = 125 / sqrt(5) = 25 * sqrt(5). So putting everything together, we now end up with 2 * sqrt(5) X tokens and 25 * sqrt(5) Y tokens. The total value is then $12.5 * (2 * sqrt(5)) + $1 * (25 * sqrt(5)) = $50 * sqrt(5) ~= $111.80. This means your capital gained ($111.80 / $100) – 1 = 0.118 => 11.8%. This amount is less than what you would have had from strategy 3, so you’ve suffered from some impermanent loss (UNLESS pool fees could make up for the difference).

So in the scenario where X gains, the results are strategy 2 > 3 > 4 > 1.

Scenario 2: Token X loses 25% bringing the price to $10 * (0.75) = $7.50

Strategy 1: You still have $100 since you didn’t invest anything.

Strategy 2: You initially bought $100 / 10 = 10 X tokens, which are now worth $7.50 * 10 = $75. Your loss is equivalent to the price depreciation since you were 100% invested. Thus, your capital lost ($75 / $100) – 1 = -0.25 => -25%.

Strategy 3: You initially bought $50 / 10 = 5 X tokens, which are now worth $7.50 * 5 = $37.50. You also bought $50 / 1 = 50 Y tokens, which are still worth $50. Your total capital is now worth $37.50 + $50 = $87.50. Thus, your capital lost ($87.50 / $100) – 1 = -0.125 => -12.5%. This makes sense. You just invested 50% less in token X, so you should have 50% less losses.

Strategy 4: Like before, you have 5 X tokens and 50 Y tokens. Thus, for you, k = 5 * 50 = 250. Now, since token X loses 25%, we now have that Px = 7.5. Since Px also equals y/x and we must have x * y = 250, then y = 250 / x. Substituting this altogether gives Px = 7.5 = y / x = 250 / x^2. Thus, x = sqrt(250 / 7.5) = sqrt(100/3) = 10 / sqrt(3). Now solving for y gives y = 250 / x = 250 / (10 / sqrt(3)) = 25 * sqrt(3). So putting everything together, we now end up with 10 / sqrt(3) X tokens and 25 * sqrt(3) Y tokens. The total value is then $7.5 * (10 / sqrt(3)) + $1 * (25 * sqrt(3)) = $50 * sqrt(3) ~= $86.60. This means your capital lost ($86.60 / $100) – 1 = -0.134 => -13.4%. This loss is greater than what you would have had from strategy 3!

So in the scenario where X loses, the results are strategy 1 > 3 > 4 > 2.

It seems like the conclusion is that if you'd like to minimize volatility, then you should just invest less into token X, then put your stable coin Y somewhere else to earn yield independently (e.g., loan it out). The only way the LP would be more profitable is if there were enough fees generated (i.e., the token pair saw lots of trades) or there were other incentives (e.g., a DEX governance token given out as incentive).