Standard Deviation shows that a price series is riskier but annualized volatility computed with the log of returns shows the opposite

Question

I apologize if this is not a smart question, but it seems contradictory that the standard deviations of two price series show that series A is riskier, but when I plot the annualized volatilities (using the log returns), series B seems riskier. Is it normal? In the volatility plot below, the pink series is the one with the lowest standard deviation for its prices (30.00 vs 47.4 for the blue one)

Answer 1

It's not really a contradiction. Computing the standard deviation of the returns rather than the prices discards information about the scale of the prices.

Take a price series $X_t$ and define the returns:

$$r_t = \log\left(\frac{X_t}{X_{t-1}}\right)$$

Then, any rescaling $Y_t = a X_t$ has the same returns, for any $a \neq 0$:

$$\log\left(\frac{aX_t}{aX_{t-1}}\right) = \log\left(\frac{X_t}{X_{t-1}}\right) = r_t$$

The standard deviation of the returns of $X_t$ and of $Y_t$ are therefore identical.

However, if you compare the standard deviation of the prices, if $a > 1$ then $Y_t$ will have a greater standard deviation, and if $a < 1$, it will be smaller.

More generally, for any two price series, if the first has smaller standard deviation for returns, the second could have larger, smaller, or the same standard deviation for prices, depending on the relative scale of prices.

In a financial context, the variance of the returns is typically the one that is considered, for a number of reasons.