log returns

Lynn K. recently asked about "logarithmic returns". That's when, if the stock price goes from P0 to P1, you set the log-return as: log(P1 / P0). Here, "logs" are "natural" logs, to the base e = 2.71828... But why? If the stock goes from $10 to $11 in 1 year, isn't the return 10%? That'd be: P1 / P0 - 1 = 11/10 - 1 = 0.1 or 10%. Nope. It's log(1.1) = 0.0953 or 9.53%. Okay, but math types just love log-returns. Consider a 10% annual return which is taken to be a monthly return of: 10/12% or 0.833% (per month). As a fraction, that's 0.00833. In a year (made up of 12 months!), that'd make $1 grow to $1.0083312 = $1.1047, so that's an annual return of 10.47%. Forging ahead, we now divide a year into n time periods with a return of 10/n% . As a fraction, that's 0.10/n in each time period. In a year, that'd make $1 grow to $(1+0.10/n)n. Now (as mathematicians are wont to do), we let n∞. Guess what happens to (1+0.10/n)n? It turns into a magical e0.10. And e0.10 = 1.1052 so that'd be an annual return of 10.52%. So what happened to the garden-variety, understood-by-everybody 10%? It's vanished beneath the mathematical snow job. The once-only compounding has turned into a continuously compounded return. Of course, one might ask: "What continuously compounded return would change $10 into $11?" Why, that'd be r where er = P1 / P0. Guess what that gives? Surprise! r = log(P1 / P0). Ain't math wunnerful? Who among us wouldn't want a nice smooooth growth, eh?