Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Stylized Facts of Financial Returns › Linear Returns vs. Log Returns

Michael BrenndoerferNovember 21, 20257 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 25Stylized Facts of Financial Returns › Linear Returns vs. Log Returns

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Before examining distributional properties, we need to clarify how returns are measured. Two definitions dominate financial analysis, each with distinct mathematical properties. The choice between them affects how we interpret results and which statistical tools we can legitimately apply. Understanding both definitions and the relationship between them provides a solid foundation for everything that follows.

Simple (arithmetic) returns measure the percentage change in price. This is the most intuitive definition because it directly answers the question: "What fraction of my investment did I gain or lose?" When you say a stock "went up 5%," you typically mean the simple return was 0.05.

R_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}} - 1

where:

$R_t$ : simple return at time $t$
$P_t$ : price of the asset at time $t$
$P_{t-1}$ : price of the asset at time $t-1$

The numerator captures the dollar change in price, while dividing by the initial price normalizes this change to a percentage. This formula can equivalently be written as the price ratio minus one, which will prove useful when connecting simple returns to log returns.

Log returns (continuously compounded returns) use the natural logarithm of the price ratio. While less intuitive at first glance, log returns possess mathematical properties that make them invaluable for statistical modeling. The main point is that taking the logarithm turns multiplication into addition, which dramatically simplifies multi-period analysis.

r_t = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(P_t) - \ln(P_{t-1})

where:

$r_t$ : log return at time $t$
$\ln$ : natural logarithm function
$P_t$ : price of the asset at time $t$
$P_{t-1}$ : price of the asset at time $t-1$

The second form of this equation reveals an elegant interpretation: the log return equals the change in the logarithm of price. If we think of log-price as a transformed variable that evolves over time, then log returns represent the increments in this transformed space.

These two measures are intimately related, and understanding the connection helps clarify when each is appropriate. Since the simple return satisfies $1 + R_t = P_t / P_{t-1}$ , we can derive the relationship by taking the natural logarithm of both sides. This algebraic manipulation reveals that log returns are simply a nonlinear transformation of simple returns.

\begin{aligned} r_t &= \ln\left(\frac{P_t}{P_{t-1}}\right) \\ &= \ln(1 + R_t) && \text{(substitution)} \end{aligned}

where:

$r_t$ : log return at time $t$
$P_t$ : price of the asset at time $t$
$P_{t-1}$ : price of the asset at time $t-1$
$R_t$ : simple return at time $t$
$\ln$ : natural logarithm function

For small returns (typical of daily data), they are approximately equal since $\ln(1 + x) \approx x$ when $x$ is close to zero. This approximation, derived from the Taylor series expansion of the logarithm, explains why the choice between simple and log returns often matters little for daily analysis but becomes increasingly important over longer horizons or during volatile periods when returns are large.

Log returns have several properties that make them preferred for modeling. Each of these properties addresses a specific challenge that arises when working with financial data, and together they explain why we overwhelmingly favor log returns for statistical analysis.

Time additivity: Multi-period log returns are the sum of single-period log returns. If $r_1, r_2, \ldots, r_n$ are daily log returns, the total return over $n$ days is simply $r_1 + r_2 + \cdots + r_n$ . Simple returns require compounding: $(1 + R_1)(1 + R_2) \cdots (1 + R_n) - 1$ . This additive property emerges directly from the logarithm's transformation of multiplication into addition and simplifies both analytical derivations and numerical calculations.
Statistical convenience: If log returns are independent and identically distributed, the Central Limit Theorem applies directly to multi-period returns. Because multi-period log returns are sums of independent random variables, the CLT tells us that sufficiently long-horizon returns will be approximately normally distributed regardless of the single-period distribution. This theoretical guarantee does not apply to simple returns because they aggregate through multiplication.
Symmetry: A +10% log return followed by -10% log return returns exactly to the starting price. With simple returns, +10% followed by -10% leaves you with a net loss. This symmetry property makes log returns more natural for modeling because equal-magnitude positive and negative shocks have symmetric effects on wealth.
No lower bound violation: Log returns can range from $-\infty$ to $+\infty$ , avoiding the awkward lower bound of -100% that simple returns face. Since prices cannot go negative, simple returns have a hard floor at -1 (total loss), but no ceiling. Log returns, being unbounded in both directions, are more compatible with symmetric probability distributions like the normal.

Throughout this chapter and the textbook, we primarily work with log returns unless otherwise specified. Let's load historical data and compute both return types to see them in practice.

In[2]:

Code

import warnings

import numpy as np
import pandas as pd

warnings.filterwarnings("ignore")

# Generate synthetic data with realistic properties
np.random.seed(42)
dates = pd.date_range(start="2000-01-01", end="2024-01-01", freq="B")
# Simulate prices using geometric Brownian motion with realistic parameters
n_days = len(dates)
daily_returns = np.random.normal(
    0.0003, 0.012, n_days
)  # ~7.5% annual return, ~19% annual vol
# Add some fat tails by mixing with t-distribution
t_shocks = np.random.standard_t(df=4, size=n_days) * 0.008
mixed_returns = 0.7 * daily_returns + 0.3 * t_shocks
price_path = 1000 * np.exp(np.cumsum(mixed_returns))
prices = pd.Series(price_path, index=dates, name="Adj Close")

# Calculate both types of returns
simple_returns = prices.pct_change().dropna()
log_returns = np.log(prices / prices.shift(1)).dropna()

# Calculate summary statistics for display
n_obs = len(prices)
start_date = prices.index[0].strftime("%Y-%m-%d")
end_date = prices.index[-1].strftime("%Y-%m-%d")
mean_simple = simple_returns.mean()
std_simple = simple_returns.std()
mean_log = log_returns.mean()
std_log = log_returns.std()
corr_simple_log = np.corrcoef(simple_returns, log_returns)[0, 1]

Out[3]:

Console

S&P 500 Data: 6261 daily observations
Period: 2000-01-03 to 2024-01-01

Simple Returns - Mean: 0.000227, Std: 0.009006
Log Returns    - Mean: 0.000186, Std: 0.009002

Correlation between simple and log returns: 0.999978

The correlation between simple and log returns is essentially 1.0 for daily data because the daily changes are small enough that the linearization holds. The differences become more pronounced over longer horizons or during extreme market events.

Out[4]:

Visualization

Scatter plot showing a near-perfect linear relationship between simple and log returns along the 45-degree line. — Scatter plot of simple versus log returns. The near-perfect linear relationship demonstrates that for typical daily returns, the two measures are practically interchangeable.

Scatter plot showing the approximation error between simple and log returns growing quadratically for larger return magnitudes. — Difference between simple and log returns versus simple returns. The approximation error grows quadratically with return magnitude, becoming significant only at extreme values.

The scatter plot confirms that for typical daily returns (within a few percent), the two measures are nearly identical. However, the right panel reveals that the difference grows quadratically with return magnitude, becoming meaningful during market crashes or rallies.