# Absolute vs. Proportional Returns

It will be a safe assumption to make that people who read my blogs work with data. In finance, the data is often in form of asset prices or other market indicators like implied volatility. Analyzing price data often requires calculating returns (aka. moves). Very often we work with proportional returns or log returns. Proportional returns are calculated relative to the price level. For example, given any two historical prices $x_{t}$ and $x_{t+h}$, the proportional change is:
$m_{t,prop} = \frac{x_{t+h}-x_{t}}{x_t}$
The above can be shortened as $m_{t, prop} = \frac{x_{t+h}}{x_t}-1$. In contrast, absolute moves are defined simply as the difference between two historical price observations: $m_{t,abs} = x_{t+h}-x_{t}$.
Essentially, we need to look for evidence of dependency of price returns on price levels. In FX, liquid options on G21 currency pairs do not exhibit such dependency, while emerging market pairs do. I have not been able to locate a free source of implied FX volatility, but I have found two instruments that are good enough to demonstrate the concept. CBOE LOVOL Index is a low volatility index and can be downloaded for free from Quandl. For this example I took the close of day prices from 2012-2017. After plotting $log_{10}(ABS(x_{t}))$ vs. $log_{10}(ABS(m_{t,abs}))$ we look for the value of the slope of the fitted linear line. A slope closer to zero indicates no dependency, while a positive or negative slope shows that the two variables are dependent.