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Linear vs Log Returns#

Linear vs Log Returns#

python

X_log = np.log(1 + X_lin)
mu = X_log.mean().values
Sigma = X_log.cov().values

In finance, transforming linear (arithmetic) returns to log returns is a common practice. Let’s break down the reasons and the advantages of computing:

X_log = np.log(1 + X_lin)

🔁 Why compute log returns from linear returns?#

This transformation is based on the formula:

\[ r_{\text{log}} = \ln(1 + r_{\text{linear}}) \]

Where:

  • \(r_{\text{linear}}\): linear (arithmetic) return

  • \(r_{\text{log}}\): continuously compounded (log) return

✅ Advantages of using log returns:#

1. Time Additivity#

  • Log returns are additive over time, which simplifies analysis.

    Example:

    \[ \ln\left(\frac{P_2}{P_0}\right) = \ln\left(\frac{P_2}{P_1} \cdot \frac{P_1}{P_0} \right) = \ln\left(\frac{P_2}{P_1}\right) + \ln\left(\frac{P_1}{P_0}\right) \]

  • This property is useful when aggregating returns over multiple periods.

2. Normality Assumption#

  • Log returns are closer to normally distributed than linear returns, especially for daily/short-term data.

  • This makes them more suitable for statistical modeling, risk management, and portfolio optimization.

3. Better for Multivariate Statistics#

  • When you’re computing:

    mu = X_log.mean().values
Sigma = X_log.cov().values

    you’re likely planning to use them for multivariate analysis (e.g., mean-variance optimization, multivariate forecasting).

  • Log returns have more stable variance-covariance structures.

4. Ease of Derivation in Calculus#

  • Log returns arise naturally in models using calculus (e.g., Black-Scholes, Ito calculus in stochastic processes).

🚫 When not to use log returns?#

  • For very large time intervals or extremely volatile assets, the difference between log and linear returns can be significant.

  • Investors often think in terms of percent gain/loss (linear), so for reporting or user interfaces, linear returns are more intuitive.

Summary Table#

Property

Linear Returns

Log Returns

Additive over time

❌

✅

Approximate normal distribution

❌ (less accurate)

✅

Suitable for statistical modeling

⚠️ (OK)

✅

Intuitive to interpret

✅ (in %)

❌ (log scale)

Final Thought#

So, in your code:

mu = X_log.mean().values
Sigma = X_log.cov().values

This is probably part of a multivariate model (e.g., mean-variance optimization, portfolio simulation), where log returns are a better fit due to their statistical and mathematical properties.

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