I'm sure there has been a post on this before, but after trying the search on the forum I wasn't able to find an answer to this question. I also cannot for the life of me find a simple explanation online for this... maybe I'm mathematically challenged

I prepared a month by month share price fluctuation spreadsheet from month 0 to month 12, the price in month 0 and month 12 are the same - so essentially the average return/cummulative return is ZERO over the 12 mths.

I then calculated the return for each month using the simple return and the continuous compound return.

The continuous compound return is zero and the simple return is positive - even though the 12 mth rtn is essentially zero. I understand what continuous compound does in a forward looking sense, for example continuously compound interest on a bank balance vs simple return.

I fail to conceptualise this looking back though - in this instance looking back at the share price fluctuations, why cont - comp achieves the correct answer and simple return achieves a biased number. I think it has something to do with the natural log and i'm not 100% clear on what ln does...

any help would be greatly appreciated by the math gurus...