Price Standard Deviation and Log-Return Volatility Measure Different Things
Summary
The document clarifies why two calculations on a rising price series produce very different values: one measures dispersion of price levels around their mean, while the other uses changes in log prices. These are different data series and have different units. Price standard deviation describes how far observed prices are spread around their average; log-return dispersion describes variation in proportional changes between observations and is commonly used as a measure of return volatility.
The example uses steadily increasing prices and nearly constant log returns, so the return-based calculation is small even though the prices span a wider range. There is also a statistical detail in the calculation: taking the square root of the average squared log returns is a root-mean-square, not a standard deviation around the mean unless the returns have zero mean. For volatility, returns are generally centered by subtracting their sample or population mean, with the denominator chosen for the intended estimate.
Key ideas
- Standard deviation of price levels measures dispersion in prices, not volatility of returns.
- Log returns are differences in log prices and describe proportional price changes.
- The two standard deviations have different units and should not be compared as though they measure the same quantity.
- The square root of average squared returns is a root-mean-square unless returns are centered at zero.
- A volatility estimate should specify its mean treatment and whether it uses a sample or population denominator.
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Full text
# Why are these two methods to calculate standard deviation gives very different answers?
# Why are these two methods to calculate standard deviation gives very different answers?
Values=[100, 101, 102.01, 103.03]
Method 1: Sum of the squared differences from the mean
Mean = 101.51
std = sqrt(((100 - 101.51)^2 + (101 - 101.51)^2 + (102 - 101.51)^2 + (103 - 101.51)^2) / 4) = 1.13
Method 2: Average of the natural log returns
returns = [1.01, 1.01, 1.01]
log_returns = [0.00995, 0.00995, 0.00995]
log_returns_squared = [0.000099009, 0.000099009, 0.000099009]
average = 0.000099009
sqrt average = 0.00995 which is 1 percent
One method gives 1.13 std and the other giver 1.
I am probably confusing two different things so could you please help me clarify this?
## Answer by Pleb (score 4, accepted)
https://quant.stackexchange.com/a/61152
As far as I understand, you're calculating the standard deviation on two different things (prices and log-returns). Assume that the values (eg. stock prices) are defined by $X_t$, for $t=1,\ldots,T$. Then the first method described above, can be formulated as:
\begin{equation} \bar{\sigma} = \sqrt{\frac{1}{T}\sum_{i=t}^T (X_t - \bar{X})^2}, \end{equation} which is the standard deviation of the stock prices, $X_t$, and is different from your second formulation. To see that, let $r_t = \ln(X_{t}) - \ln(X_{t-1})$ be your log-return at time $t$, then the second method can be described as: \begin{equation} \tilde{\sigma} = \sqrt{\frac{1}{T}\sum_{t=1}^T r_t^2}, \end{equation} and calculates the standard deviation of the log-returns, $r_t$. The process $(r_t)_{t \geq 0}$ is recovered from differencing the log-transformation (log-prices) of the price process $(X_t)_{t\geq0}$ and therefore they are fundamentally different processes, hence giving you different standard deviations.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.