Assume I have a $100%$ long position in Apple vs $100%$ short position in Microsoft. I want to calculate the cumulative return to date. In my mind the following approaches should arrive at the same outcome but they do not (and I have a view why not) but I defer to the experts on a more mathematical explanation?
(1) Take the cumulative return on the day on day change on price ratio between Apple and Microsoft (2) Take the cumulative return on the day on day change on the return spread between Apple and Microsoft
To make it easier to follow, here are some numbers. On day 1, stock A = 20, B = 30; day 2, stock A = 25, Stock B = 25. On day 1, ratio A & B = 20 / 30 = 0.67. On day 2, 25/25 = 1. Ratio represents the degree to which the spread between the stocks has moved. So, 1/0.67 - 1 = 49%.  stock A daily return 25/20 -1= 25%; B = 25/30 -1= -17%. 25+17%=42%. WHY THE DIFFERENCE?