# Difference between price ratio and spread cumulative return.

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. [1]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%. [2] stock A daily return 25/20 -1= 25%; B = 25/30 -1= -17%. 25+17%=42%. WHY THE DIFFERENCE?

• Which following approaches are in your mind? – callculus May 22 '18 at 22:56
• What is “the cumulative return on the day to day change in the spread/ratio”? – spaceisdarkgreen May 22 '18 at 23:37
• 1 & 2 approaches listed below the colon. – Definiteweird May 23 '18 at 0:54
• The cumulative return on the daily difference btwn the return on stock A vs stock B. The cumulative difference on the daily return ratio between the price on stock A and stock B – Definiteweird May 23 '18 at 0:56