Kerry Back
Suppose returns \(r_{1},\cdots,r_{n}\) are independent draws from a normal \((\mu,\sigma^2)\) distribution.
Let \(m =\) sample mean and \(s =\) sample std dev = \(\sqrt{\sum_{i=1}^n \frac{(r_{i}-m)^2}{n-1}}\)
Then, \(m\) is normal \((\mu,\sigma^2/n)\) and
\((n-1)s^2/\sigma^2\) is \(\chi^2(n-1)\).
\[0.12 ± 1.96 \times 0.06 = [0.013,0.227]\] - A similarly wide confidence interval for \(\sigma\) is implied by the \(\chi^2\) distribution.
If we sample monthly, weekly, \(\ldots\) then we have more data points, so estimates are more accurate.
When we scale to annual parameters, the accuracy gain vanishes for the mean.
To illustrate effect of sampling frequency,
import numpy as np
from scipy.stats import norm
# monthly parameters
mu, sigma = 0.01, 0.3/np.sqrt(12)
mrets = norm.rvs(loc=mu, scale=sigma, size=12*25*5000)
mrets = mrets.reshape(12, 25, 5000)
mmeans = np.mean(mrets, axis=(0,1))
msds = np.std(mrets, axis=(0,1))
arets = np.prod(1+mrets, axis=0) - 1
ameans = np.mean(arets, axis=0)
asds = np.std(arets, axis=0)
Higher frequency is also better for correlations, covariances, and betas.