# Log returns vs Relativizing to Portfolio size of $1 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 1. Cost in terms of log returns 2. 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.

Cheers

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

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It might be useful if you add more details about what you're trying to do and what the various calculations are. – John Mar 17 '14 at 18:44
Hey John, I've edited the original question. Does my intuition seem valid? – user2561981 Mar 21 '14 at 17:54
It makes it difficult to recommend something without understanding the context. For instance, if I am performing mean-variance optimization including transaction costs I might think about them differently than if I were trying to evaluate how much transaction costs have impacted my portfolio in the past. In one case, I might think about transaction costs as a function of a change in weights or holdings, while in the other I might think in terms of dollars as a % of AUM. I'm not sure why you would analyze them in terms of log returns. – John Mar 21 '14 at 18:35