Module 3 — Statistics for Markets

Simple vs Log Returns

Returns, volatility, covariance — and the tail behaviour that breaks naïve models.

Module lessons

Simple vs Log ReturnsVolatility — Measuring UncertaintyCorrelation and Covariance

Learning objectives

  • ▸Compute both and reconcile small differences.
  • ▸Know when each is the right choice.

FORMULA

Definitions

Simple: r_t   = (P_t - P_{t-1}) / P_{t-1}
Log:    r_log = ln(P_t / P_{t-1}) = ln(1 + r_t)

TEXT

When to use which

Use SIMPLE returns when aggregating across assets at a single point in time — a portfolio's simple return is the weighted sum of its constituents' simple returns. Use LOG returns when aggregating across time for a single asset — log returns add across periods, simple returns don't. Also, log returns are approximately normal for short horizons, which is convenient for option pricing.

EXAMPLE

The Jensen gap

If r = 5%, then ln(1.05) ≈ 0.0488. The gap is small for daily returns (~1%) but grows quickly: at 100% gain, simple = 1.0, log = 0.693. Never compare a simple-return Sharpe to a log-return Sharpe and pretend they're the same.

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