Returns Analysis
Calculate daily and 22-day rolling returns. Analyze distributions and autocorrelations.
Returns Calculation
Log returns are calculated from normalized index levels. The 22-day rolling returns approximate monthly returns and are used for covariance estimation to reduce noise while maintaining responsiveness.
Key Steps:- Normalize indices to 1.0 at start date
- Calculate daily log returns
- Compute 22-day rolling sum of returns
- Calculate annualization factors
Log Returns:
$$
r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)
$$
Rolling Returns:
$$
r_{t,22} = \sum_{s=0}^{21} r_{t-s} = \ln\left(\frac{P_t}{P_{t-22}}\right)
$$
Annualization:
$$
\sigma_{\text{annual}} = \sigma_{\text{daily}} \cdot \sqrt{252}
$$