Simple vs Logarithmic Returns

Most financial reports give us simple returns, but financial modelling uses log returns because they are consistent with the formulation of the model. We can convert from one to the other.

Let R be the simple return and r_t be the log return, where r_t = log(P_t/P_t-1).

R = e^r – 1

m = E[R] = e^μ – 1

s² = Var(R) = e^2μ(e^{σ^2} – 1)

For small values

R ≈ r

m ≈ μ

s² ≈ σ²