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Statistics for finance · level 3

Log vs simple returns

Log returns add across time, which is why volatility work uses them. They sit below simple returns, and the gap widens fast once moves exceed 10%.

Theory: Derivatives & Pricing

Worked example: A 30% simple return. What is it as a log return, in %?

simple30%
log26.2%
gap3.8 pts

Step by step

  1. 1

    Definition

    ln(1 + 30%) = ln(1.3)

  2. 2

    Value

    26.2%

  3. 3

    Rule of thumb

    Below about ±10% the two are nearly the same; beyond that the log return is always the smaller number.

A 30% simple return. What is it as a log return, in %? = 26.2

The theory behind it

Intuition

Log returns add across time and are close to simple returns for small moves, which is why they are used for multi-period analysis.

Common pitfalls

  • ×Averaging simple returns across periods and calling it a growth rate.
  • ×Using log returns when the question asks for actual money.

In the interview

Multi-year growth arithmetic and CAGR checks.