One can just as easily say that R = (Pt+1)/(Pt) - 1 is only an approximation that only works for infinitesimally small returns. If I get a R1 = (P1/P0) - 1 in time period 1 and R2 = (P2/P1) - 1 in time period 2, then total return over both periods is equal to R1+R2 only if R1 and R2 are infinitesimally small. So it all depends on how you define "returns" and what properties you want your returns variable to have.The advantage of log-returns is that they are additive for sequential returns. If Pt = 100, Pt+1 = 50, and Pt+2 = 100, then the log returns are ln( Pt+1 / Pt ) = ln(50/100) = ln(1/2) = -0.69 and then ln( Pt+2 / Pt+1 ) = ln(100/50) = ln(2) = +0.69 which adds to 0 and accurately reflects the fact that Pt = Pt+2. In contrast the linear model would assert that the investor lost 50% in the first timeperiod but then gained 100% in the second which sounds great until you do the math.Having sequentially additive returns helps in discussing returns models on different timescales (quantmeh's point) and in certain mathematical operations (e.g., calculus, differential equations, etc.). The log model also has the advantage that no matter how negative or positive the return, the end price is always above zero.The larger point is that we can define returns any way we want as long as we're consistent. That provides the freedom to choose a calculation of returns that has useful numerical or mathematical properties.
Last edited by Traden4Alpha
on June 7th, 2012, 10:00 pm, edited 1 time in total.