Problem
To calculate the rolling annual historical standard deviations of the S&P 500 capital index over the past year at each point in time, you can follow these steps:
A. Gather the daily historical prices of the S&P 500 capital index for the past year. You can obtain this data from various financial data providers, such as Bloomberg, Yahoo Finance, or the S&P Dow Jones Indices website.
B. Calculate the daily returns of the S&P 500 index by taking the percentage change in prices from one day to the next. The formula for daily returns is: (Price_today - Price_yesterday) / Price_yesterday.
C. Determine the number of trading days in a year. In this case, assume 252 trading days, which is the approximate number of trading days in a year.
D. Use a rolling window approach to calculate the standard deviation. For each trading day, consider the past year's worth of daily returns (252 trading days) and calculate the standard deviation of these returns. This will give you the rolling annual historical standard deviation at each point in time.
E. Apply equal weighting to each observation. Since you want to use an equal weight (1/n) for each observation, calculate the standard deviation using the equal weights. In this case, divide the sum of squared deviations by the number of observations (252 in our example).
F. Repeat this calculation for each trading day over the past year, shifting the rolling window by one day each time.