# Convert arithmetic returns to log returns [closed]

I have a series of arithmetic returns and I need log returns. I do not have the underlying prices. How do I convert?

All the posts I have found explain why using one versus the other is appropriate but how do I get from one to the other without the underlying data?

Thanks

• Arithmetic returns is $a_t = S_t/S_{t-1}-1$. Log-return is $r_t = \ln(S_t/S_{t-1})$. So $r_t = \ln(a_t+1)$ – Quantuple Sep 6 at 16:18

$$R = exp(r) - 1$$
$$r = log(R+1)$$