P Dybvig Investments: Lecture 1, Slide 10

Volatility of returns

The variance is the average of the squared deviation from the mean, and the standard deviation is the square root of the variance. In a sample of returns, we measure the variance and standard deviation of returns as

$variance = (1/(T-1))sum_t=1,T (R_t - mean)^2$

and

$standard deviation = sqrt(variance)$

Sometimes we divide by $T-1$ instead of $T$ in computing variance, to adjust for a tendency to understate the variance when the mean is unknown. If $T$ is large, it obviously doesn't make much difference whether we divide by $T$ or $T-1$ . For daily returns, the mean is much smaller than a typical $R_t$ , and it doesn't make much difference if we don't subtract the mean.

Practitioners commonly refer to volatility as ``vol'' for short.

Phil Dybvig

	The variance and standard deviation of return are both measures of return volatility. The standard deviation is perhaps more intuitive, because it gives a ``typical'' distance that a return would be from the mean return, measured in the same units as the return itself.
	Although the standard deviation may be more intuitive, the variance is of great conceptual importance, because it is the variance of returns that adds up when we take the sum of independent draws. This property is the mathematical foundation of the value of diversification, as we will see in the next lecture.