Log-Returns

The log-difference between $t$ and $t-1$

Definition

Let $p_t$ be the price of a financial asset at time $t$ (e.g., daily stock price, currency exchange rate, commodity price).

The log-return at time $t$ is defined as:

$$r_t=\log (p_t)-\log (p_{t-1})=\log \left(\frac{p_t}{p_{t-1}}\right)$$

Approximation to Relative Returns

By Taylor expansion, for small $\frac{p_t-p_{t-1}}{p_{t-1}}$:

$$r_t=\log \left(1+\frac{p_t-p_{t-1}}{p_{t-1}}\right)\approx \frac{p_t-p_{t-1}}{p_{t-1}}$$

So log-returns are approximately equal to relative returns (percentage changes).

Why Use Log-Returns?

Key Advantage: Additivity

Log-returns have the additivity property that relative returns lack:

• Multi-period return: $r_{t,t+k}=\log (p_{t+k}/p_t)={\sum }_{i=1}^k{r}_{t+i}$
• This makes statistical analysis much simpler

Other advantages:

• Log-returns are time-additive (compound naturally)
• Symmetric treatment of gains and losses
• Better statistical properties (closer to normality)
• Mathematical convenience in continuous-time models

Related

• Continuously Compounded Return
• Volatility Clustering