Simple vs. Log Returns

Log returns are additive over time and better suited to compounding math; simple returns are the raw period-over-period percentage change.

1. The idea

A simple return over a period is $(P_{1} - P_{0}) / P_{0}$ ; a log return is $ln (P_{1} / P_{0})$ . For small returns they're nearly equal (via $ln (1 + x) \approx x$ ), but they diverge for large moves and behave differently under compounding.

Log returns are additive across time: the log return over two periods is the sum of the log returns of each period, making them convenient for statistical modeling and for computing multi-period compounded growth by simple addition.

Simple returns are the ones actually realized for P&L purposes (they're what your portfolio value grows by), and they are the ones that must be compounded multiplicatively, not added, across periods.

2. Why it matters

Interviewers test whether you know why returns can't just be averaged arithmetically across time, and this distinction underlies Sharpe ratio computations and compounding questions.

3. Core mathematics

Simple return

R_{simple} = \frac{P _{1} - P _{0}}{P _{0}}

Log return

r = ln (\frac{P _{1}}{P _{0}}) = ln (1 + R_{simple})

Multi-period compounding (simple returns)

1 + R_{total} = i = 1 \prod n (1 + R_{i})

Multi-period compounding (log returns)

r_{total} = i = 1 \sum n r_{i}

4. How to recognize it

Think of this when…

Asks to compound returns over multiple periods
Distinguishes arithmetic mean return from geometric mean return
Involves converting between price ratios and percentage returns

5. Worked example

Worked example

A stock goes from $100 to $110, then from $110 to $99. Find the simple return and log return for each period, and verify the total two-period simple return using both methods.

1
Period 1 simple return.
$R_{1} = \frac{110 - 100}{100} = 0.10$
2
Period 2 simple return.
$R_{2} = \frac{99 - 110}{110} = - 0.10$
3
Total return via compounding simple returns.
$(1.10) (0.90) - 1 = 0.99 - 1 = - 0.01$
4
Check via log returns: $r_{1} = ln (1.10)$ , $r_{2} = ln (0.90)$ , sum then exponentiate.
$e^{l n (1.10) + l n (0.90)} - 1 = e^{l n (0.99)} - 1 = - 0.01$
Answer: Total two-period simple return is -1% (price falls from 100 to 99), even though the arithmetic average of +10% and -10% is 0% — illustrating why simple returns must be compounded, not averaged.

6. Common mistakes

Averaging simple returns arithmetically to estimate multi-period growth instead of compounding multiplicatively
Adding simple returns across periods instead of log returns
Forgetting that log returns can be summed but must be exponentiated back to get the actual price ratio

7. Quick check

Which type of return is additive across time periods?

A stock returns +50% then -50% (simple returns). The net simple return is:

8. Practice

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9. Related methods

This lesson is conceptual; see the topic hub for methods.

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