Log Returns

Log returns are defined as:

$$r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)$$

Where:

• $P_t$ is the price at time $t$,
• $P_{t-1}$ is the price at the prior time step.

They're popular because they sum over time. If you add log returns across days, you get the multi-day log return. To convert back to a simple return, use $\exp(\text{sum}) - 1$.

Why we use them in tail risk

Log returns handle extreme moves better and make it easier to compare tail events across assets. That's why our tail-risk section uses log returns for VaR, Expected Shortfall, skew, and kurtosis.

Are they different from simple returns?

For small daily moves, log and simple returns are almost the same. For big moves, they diverge -- which matters when you're analyzing crashes.