In a current empirical research project, I am tracking a non-parametric measure of a transaction cost. To this extent, I track this cost in two ways
- Cost in terms of log returns
- Cost in terms of portfolio size of $1
One thing I have noticed is that while tracking these two metrics over time, there are very noticeable discrepancies. For example, the cost in terms of (1) is 5% in a certain year while the cost in terms of (2) reaches upward of 12%.
I was hoping if someone can explain the theoretical intuition which might lead to such drastic differences.
edit: I believe I have found a solution to account for the discrepancy. if we take the log(x/y), as the ratio of x/y diverges from 1, the natural log approximation for relative change becomes less credible via understating the actual relative change. I believe this is the reason why high frequency data extensively utilizes the log-return as a proxy for relative change; intuitively, changes in price are less drastic at higher frequencies.
So if there is a large discrepancy between our log return and percentage return, we can most likely attribute this to relatively larger price movements