# Returns and logreturns differences

I have a time series of stock prices and I tried to calculate simple returns and log returns. However, I end up that simple returns has positive mean, but log returns has negative mean. Is it possible to have something like this on one sample of data?

• Yes, that happens quite a lot. The difference is probably about half the variance of your return series? en.wikipedia.org/wiki/… – demully Oct 14 '19 at 16:19
• yeah, kinda like that, what's the cause of that? – Lukas Tomek Oct 14 '19 at 18:38
• Imagine a 50:50 of halving:doubling. Compounds to zero; but the average is +25%. Imagine a 50:50 of +/- 10%. Average is zero; but compounds to 1.10^0.5*0.9^0.5 = 0.995, ie a 0.5% loss. Linear numbers and logarithms just behave a little differently; and the difference is more generous to the linear, and more taxing on the logs. The half variance is just the mean (first moment) of a lognormal (versus a normal) distribution. – demully Oct 14 '19 at 22:42
• Just checking -- I don't mean to condescend: are you sure you're calculating "logreturn" correctly? The log-return of $x_t$ is $ln(x_t / x_{t-1})$ or $ln(x_t) - ln(x_{t-1})$ (which is the same; it's also called the "log-difference"). It is not $(ln(x_t)-ln(x_{t-1}))/ln(x_{t-1})$ or something. Again, just checking! It's possible to find what you've found, but not that common in data showing modest returns. These two statistics should be very close for small returns, which is why they are typically used interchangeably. – Drew Oct 16 '19 at 19:18

OK, this need have nothing to do with any single sample of data. It's an inherent difference between the behaviour of linear vs logarithmic numbers. Which is what make up your respective simple and log returns, and associated averages.

Imagine I offered you a bet in which you put a pound on the table, I put two down, we flipped a fair coin, and winner takes all. I would reasonably imagine that you wold take that bet; and would continue to take that bet for so long as I continued to offer it. Right? You would almost certainly milk me until I went bust.

Imagine instead that Bill Gates offered you the same game, but with stakes of 100% of your wealth instead of £1. Would you play that game, rinse and repeat? Of course, not. There's a 50% chance of instant bankruptcy, and an almost 100% chance of eventual bankruptcy.

The rules of the game haven't changed ;-) But yet the game is clearly not the same ;-) This is but an extreme (and hopefully elucidating) metaphor for the difference between the linear and the logarithmic, that is the simple vs log return problem you face.

One simpler example: the market halves and doubles with equal probability. In the long run, it's expected return is clearly zero. But for every iteration along the way, the expected return is +25% (50*100%-50%*50%). Another simple example: a market in which the chances of the next 10% are equal up or down. Expected return each iteration is zero. But compound 1.1^0.5*0.9^0.5-1 = 0.995. equals a 0.5% loss over the long haul.

Speak heresy softly, but fair bets represent bad investments; while breakeven investments represent favourable bets [if you don't compound (investment jargon) = double down (betting jargon)]. There's no moral point here. It's just arithmetic vs geometric mathematics!

Normally (no pun intended), the difference between the two tends to be around half the variance of the returns in question. This is simply because the normal distribution (for simple/arithmetic returns) is symmetric. The lognormal distribution (of log/geometric returns) is not (because of the examples above). It's slightly skewed/biased, with a mean of mu - 0.5 * sigma^2.

hope this helps.

• It's called variance drain. Here's a nice paper on it. caepr.indiana.edu/RePEc/inu/caeprp/CAEPR2012-004.pdf – mark leeds Oct 15 '19 at 3:46
• if someone knows how to make the link above be permanent in case it changes on the net, could you tell me how to do that ? thanks. – mark leeds Oct 15 '19 at 3:48
• Also, note that the effect of variance drain is quite small when the returns are small. so, for small returns, the approximation is less harmful. – mark leeds Oct 15 '19 at 3:50

It happens because the log function is concave around 1, which means it returns "more negative" numbers for values less than 1 than the positive values it returns for numbers the same distance greater than 1. What that means in a practical sense is that when simple returns average zero, log returns are negative, since negative returns have a more negative log return than "equal" positive returns. If your arithmetic mean is positive but close to zero, then it's not unusual to have a small negative log return average.

For example, if the 2-period return is +10%, -10%, the log returns would be

ln(1.1) =  0.09531
ln(0.9) = -0.10536
------------------
mean    = -0.00503