A logarithmic return measures a compounded proportional price change by taking the natural logarithm of an asset’s current closing price divided by its previous closing price: ln(Pt / Pt−1). It supports multi-period compounding and provides a useful statistical representation of price movements. section 9.5.1 section 9.5.3
Compute and assess a daily log-return series as follows:
- Align each closing price with the preceding price using shift(1), calculate np.log(Close / shifted Close), and remove the first missing value with dropna(). section 9.5.3
- Inspect the distribution: values commonly cluster near zero, while longer tails indicate occasional larger positive or negative price changes. section 9.5.3
- Estimate the mean as the average daily return and the standard deviation as daily volatility, then use both to generate simulated future returns. section 9.5.4
GM daily returns
For GM closing prices from July through December 2023, the estimated mean log return is approximately −0.00061 and the standard deviation is approximately 0.02005, indicating a slight downward historical trend and moderate daily volatility. section 9.5.4
