Why Log Prices Beat Raw Spreads

A pair has a log-price β of 1.5. You go long $2,000 of A. How big is the short B leg?

• $1,333, which is $2,000 ÷ 1.5
• $3,000, which is 1.5 × $2,000
• $2,000, so the legs match

Why is a z-score on a log gap comparable across years when one on a raw dollar gap is not?

• The log gap measures percentage differences, which do not grow with the price level
• The log gap is always smaller than the raw gap
• Log prices never trend