There’s a nice blog post here by Quantivity which explains why we choose to define market returns using the log function:

where denotes price on day .

I mentioned this question briefly in this post, when I was explaining how people compute market volatility. I encourage anyone who is interested in this technical question to read that post, it really explains the reasoning well.

I wanted to add two remarks to the discussion, however, which actually argue for not using log returns, but instead using percentage returns in some situations.

The first is that the assumption of a log-normal distribution of returns, especially over a longer term than daily (say weekly or monthly) is unsatisfactory, because the skew of log-normal distribution is positive, whereas actual market returns for, say, S&P is negatively skewed (because we see bigger jumps down in times of panic). You can get lots of free market data here and try this out yourself empirically, but it also makes sense. Therefore when you approximate returns as log normal, you should probably stick to daily returns.

Second, it’s difficult to logically combine log returns with fat-tailed distributional assumptions, even for daily returns, although it’s very tempting to do so because assuming “fat tails” sometimes gives you more reasonable estimates of risk because of the added kurtosis. (I know some of you will ask why not just use no parametric family at all and just bootstrap or something from the empirical data you have- the answer is that you don’t ever have enough to feel like that will be representative of rough market conditions, even when you pool your data with other similar instruments. So instead you try different parametric families and compare.)

Mathematically there’s a problem: when you assume a student-t distribution (a standard choice) of log returns, then you are automatically assuming that the expected value of any such stock in one day is infinity! This is usually not what people expect about the market, especially considering that there does not exist an infinite amount of money (yet!). I guess it’s technically up for debate whether this is an okay assumption but let me stipulate that it’s not what people usually intend.

This happens even at small scale, so for daily returns, and it’s because the moment generating function is undefined for student-t distributions (the moment generating function’s value at 1 is the expected return, in terms of money, when you use log returns). We actually saw this problem occur at Riskmetrics, where of course we didn’t see “infinity” show up as a risk number but we saw, every now and then, ridiculously large numbers when we let people combine “log returns” with “student-t distributions.” A solution to this is to use percentage returns when you want to assume fat tails.



