https://www.investopedia.com/personal-finance/apr-apy-bank-hopes-cant-tell-difference/
https://en.wikipedia.org/wiki/Rate_of_return
Let's describe this logic step-by-step for yield farming strategies.
For example, you have $1000 A and invested it in strategy X
Strategy X puts your asset in a project pX with standard Harvest logic (periodically claims rewards and adds them back to the project)
Project pX has $1M TVL pTVL and $100,000 of rewards pReward that will be provided over a 30 day period pPeriod
Your $1000 is 0.1% of the Total Value Locked (TVL) A / pTVL
This means your asset will receive $100 during the month pReward * (A/pTVL)
In DeFi when comparing profits, most projects show APR, but call it the APY.
For example, you can multiply your monthly profit for 12 months of the year and get an annual profit figure.
Of course, 12 months with the same profit and TVL is an unrealistic situation. Let's call this the "Instant APR" iAPR , because it is the APR for the current situation and will change as TVL and rewards change.
For your $1000 it will be ( pReward * (A/pTVL)* 12 = $1200) ⇒ $1000 / $1200 = 120% APR
But your asset is under the strategy's control. It claims profit every day and adds it back into the project. In this way, it generates additional profit the longer you hold it, due to the principal of compounding returns.
Daily profit will be pReward /pPeriod = $100,000 / 30 = $3333.33.
For your share of 0.1% it will be $3.30 per day.
So after the first day your total stake will be $1003.30.
Your new share % will be $1003.30 / $1M = 0.1003%
For day 2 you will receive $3.44, due to the compounding effect from day 1. Each day thereafter, your daily profit will continue to increase.
Your final compounded annual profit is the APY, but don't forget the project only has rewards for the month!