Continuously Compounded Return

An alternative interpretation of Log-Returns as the continuously compounded rate of return.

Definition

Given log-return $r_t=\log (p_t)-\log (p_{t-1})$, we have:

$$e^{r_t}=\frac{p_t}{p_{t-1}}⟹p_t=e^{r_t}\cdot p_{t-1}$$

This can also be written as:

$$p_t=\underset{n\to \infty }{\lim }{\left(1+\frac{r_t}{n}\right)}^n{p}_{t-1}$$

Interpretation

Interpretation

The continuously compounded return $r_t$ represents the instantaneous growth rate that, if compounded continuously over the period, would transform $p_{t-1}$ into $p_t$.

This is the standard return measure used in:

• Financial econometrics
• Option pricing models (Black-Scholes)
• Portfolio theory
• Risk management (VaR calculations)

Relationship to Log-Returns

Continuously compounded returns are log-returns. The two terms are used interchangeably in financial econometrics:

• Log-return emphasizes the mathematical form ($\log$ difference)
• Continuously compounded return emphasizes the economic interpretation (instantaneous growth rate)

Related

• Log-Returns
• ARCH(m) Model