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Quant Systems Lab · Control Systems for Quantitative Finance
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Returns and Log-Returns

Simple returns measure percentage change; log-returns add nicely over time and align with GBM-style models.

Explanation

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.

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.

returnslogcompoundinggbm
Interactive visualisation

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
Time step tPrice Sₜ0123456Sₘₐₓ ≈ 101.60Sₘᵢₙ ≈ 100.00price path SₜSₜ evolves through compounded step returns
Step index t (1…6)Return per step0%-0.5%1.0%123456simple rₜlog ℓₜPer-step returns under the same price path
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.