Simple return
R_t = P_t / P_(t-1) - 1The simple return is the percentage gain relative to starting value.
Foundations
Returns and Log-Returns
Simple returns measure percentage change; log-returns add nicely over time and align with GBM-style models.
Mental model
Simple returns measure percentage change, while log returns make sequential changes additive.
Simple return is (S_t / S_{t-1}) − 1; log-return is ln(S_t / S_{t-1}). Both describe how much a price has moved between two times.
Log-returns are additive across time steps (they sum), which makes them convenient for modelling and statistical analysis.
Explore the mechanism
How does the choice between simple and log returns change aggregation and interpretation?
We follow a short price path St and compare simple returns rt = St/St−1 − 1 with log-returns ℓt = ln(St/St−1). The sliders let you change the volatility scale and a per-step drift and see how the two definitions behave under compounding.
Low-volatility base pattern · steps: 6 · S₀ = 100
Aggregates over the path
Σ simple returns Σ rₜ ≈ 1.30%
Total simple return (S_T / S₀ − 1) ≈ 1.30%
Σ log-returns Σ ℓₜ ≈ 1.29%
exp(Σ ℓₜ) − 1 ≈ 1.30% (matches total growth)
Key variables
rₜ: simple return in step t; ℓₜ: log-return in step t
Σ ℓₜ is the log of the gross return S_T / S₀.
Interpretation
Simple returns rₜ are intuitive percentage moves but do not add cleanly: Σ rₜ is generally different from the total compounded growth. Log-returns ℓₜ are constructed so that their sum is additive: Σ ℓₜ = ln(S_T / S₀).
Increase the volatility scale: large positive and negative simple returns push the green line away from zero, and the gap to the orange log-return line becomes obvious on big moves. Adjust the per-step drift: the whole pattern shifts, but exp(Σ ℓₜ) − 1 stays equal to the total price change, showing why many models prefer log space for time-additive reasoning, while practitioners still quote simple percentage returns.
UNDERSTAND
What to notice
For small moves, simple and log-returns are almost identical; for large or volatile moves, the difference compounds and matters.
In GBM and many continuous-time models, log-returns are assumed to be normally distributed, which simplifies both theory and estimation.
The quantitative structure
Notation
P_t- price at time t
R_t- simple return
r_t- log return
Log return
r_t = ln(P_t / P_(t-1)) = ln(1 + R_t)Log returns add across consecutive periods for one asset.
A small example
A rise followed by a fall
A price moves from 100 to 120, then back to 100.
- First simple return: 120/100 - 1 = 20%.
- Second simple return: 100/120 - 1 = -16.67%.
- The log returns are ln(1.2) and ln(1/1.2), which sum to zero.
ResultThe two simple percentages do not add to zero, but the two log returns do.
Where it appears
- building return series
- estimating volatility and covariance
- compounding portfolio performance
Where the intuition breaks
What this view leaves out
- Log returns are additive through time for a single asset but do not aggregate across portfolio holdings in the same simple way.
Common mistakes
“Simple and log returns are interchangeable at every horizon.”
Not quite.They are close only for small moves; large losses and gains produce meaningful differences.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Returns and Log-Returns
Returns and Log-ReturnsYOU ARE HERE
Foundation ofCovariance Estimation and CorrelationCovariance and correlation are estimated from joint movements in return or change series.
Foundation ofGeometric Brownian Motion (GBM)GBM implies normally distributed log-returns over fixed horizons under its classical assumptions.
Foundation ofVolatility (Realised vs Implied)Realised volatility is estimated from return series, while log-returns are the natural scale for many continuous-time models.
Foundation ofLinear Regression (OLS) in Factor ModelsFactor regressions typically explain asset returns using market or factor-return series.
Foundation of · Covariance Estimation and Correlation
Covariance and correlation are estimated from joint movements in return or change series.
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