Summary
Log returns, defined as
$$
Rlog=ln (PtPt−1),R_{\log} = \ln!\bigl(\tfrac{P_t}{P_{t-1}}\bigr),
$$
are exactly the continuously compounded returns for an asset over the period from t−1t-1 to tt, since they arise as the limit of discrete compounding when the compounding frequency goes to infinity . In practice, this means that if you reinvest gains instantaneously at every moment, the growth factor is
$$
Pt=Pt−1 eRlog,P_t = P_{t-1},e^{R_{\log}},
$$
and hence the log return RlogR_{\log} is the natural rate that delivers the observed price change under continuous compounding . Moreover, log returns coincide with the “force of interest” or instantaneous rate of growth, enjoy time‑additivity (so multi‑period log returns simply sum), and yield symmetric treatment of gains and losses .
1. Simple vs. Continuously Compounded Returns
1.1 Simple (Discrete) Return
- Definition: The simple (or holding‑period) return RR for a single period is
$$
R=Pt−Pt−1Pt−1,R = \frac{P_t - P_{t-1}}{P_{t-1}},
$$
where Pt−1P_{t-1} and PtP_t are the asset prices at the beginning and end of the period, respectively .
- Discrete Compounding: If interest or returns are reinvested mm times per year at a nominal rate rr, the accumulation over one year is
$$
(1+rm)m,(1 + \tfrac{r}{m})^m,
$$
converging to continuous compounding as m→∞m \to \infty .
1.2 Continuously Compounded Return (Log Return)
- Definition: The continuously compounded return (also called the log return or force of interest) is
$$
Rlog=ln (PtPt−1),R_{\log} = \ln!\bigl(\tfrac{P_t}{P_{t-1}}\bigr),
$$
which directly measures the instantaneous rate that transforms Pt−1P_{t-1} into PtP_t under continuous compounding .
- Accumulation Relation: Under continuous compounding, if the constant rate is rlogr_{\log}, then
$$
Pt=Pt−1 e rlogΔt⟹rlog=1Δtln (PtPt−1)P_t = P_{t-1} , e^{,r_{\log} \Delta t}
\quad\Longrightarrow\quad
r_{\log} = \tfrac{1}{\Delta t}\ln!\bigl(\tfrac{P_t}{P_{t-1}}\bigr)
$$
.
2. From Discrete to Continuous: The Limit Argument
Consider discrete compounding over one period t=1t=1 with rate rr and subperiods mm:
$$
(1+rm)m →m→∞ er.\bigl(1 + \tfrac{r}{m}\bigr)^{m}
;\xrightarrow[m\to\infty]{};
e^{r}.
$$
Hence, if you observe a total simple return RR in one period, the equivalent continuous rate rlogr_{\log} satisfying erlog=1+Re^{r_{\log}} = 1+R is
$$
rlog=ln(1+R).r_{\log} = \ln(1 + R).
$$
Applied to prices,
$$
R=PtPt−1−1⟹Rlog=ln (1+R)=ln (PtPt−1).R = \frac{P_t}{P_{t-1}} - 1
\quad\Longrightarrow\quad
R_{\log} = \ln!\bigl(1 + R\bigr) = \ln!\bigl(\tfrac{P_t}{P_{t-1}}\bigr).
$$
This limit is the mathematical foundation of why log returns are continuously compounded returns .
3. Interpretation as the “Force of Interest”
- Force of Interest: In actuarial science, the instantaneous “force of interest” δ(t)\delta(t) for an accumulation function a(t)a(t) is $$
δ(t)=a′(t)a(t)=ddtlna(t).\delta(t) = \frac{a’(t)}{a(t)} = \frac{d}{dt}\ln a(t).
$$
For constant rate rr, a(t)=erta(t)=e^{rt} and δ=r\delta=r. Hence, δ\delta is exactly the continuously compounded rate .
- Connection to Log Returns: Over a short interval Δt\Delta t, δ Δt≈ln (PtPt−Δt)\delta ,\Delta t \approx \ln!\bigl(\tfrac{P_{t}}{P_{t-\Delta t}}\bigr), tying the instantaneous force directly to the log return between two instants .
4. Key Properties of Log Returns
- Time‑Additivity
If an asset experiences successive log returns Rlog,1R_{\log,1} and Rlog,2R_{\log,2}, the total log return over both periods is
$$
Rlog=Rlog,1+Rlog,2;R_{\log} = R_{\log,1} + R_{\log,2}, since
$$
$$
ln (P2P0)=ln (P2P1)+ln (P1P0).\ln!\bigl(\tfrac{P_2}{P_0}\bigr)
= \ln!\bigl(\tfrac{P_2}{P_1}\bigr)
- \ln!\bigl(\tfrac{P_1}{P_0}\bigr).
$$
This additive property simplifies multi‑period analysis .
- Symmetry
Log returns treat percentage gains and losses symmetrically: a +50% move (Rlog=40.5%R_{\log}=40.5%) followed by −40.5% (Rlog=−50%R_{\log}=-50%) brings you back to the start, unlike simple returns .
- Normality Assumption
When asset prices are modeled as following a lognormal distribution, log returns are normally distributed—a cornerstone of the Black–Scholes framework .
5. Practical Usage
- Volatility Estimation: Historical volatility is usually computed from the standard deviation of log returns, because of their additive and normal‑like behavior over time .
- Option Pricing: Continuous compounding underpins risk‑neutral pricing and discounting cash flows in derivatives models, making log returns the natural choice for modeling price evolution .
In essence, log returns are by construction the continuously compounded returns: they emerge from taking the limit of discrete compounding and coincide with the instantaneous rate of growth, offering mathematical elegance and powerful properties for financial modeling.
Volatility
$$ \sigma_{anual} = \sigma_{daily}\sqrt{PT} $$
Usually, $\sqrt{PT}$ can be denoted as $\sqrt{252}$ since 252 is the trading days of a year.