Why Simple and Log Returns Can Have Different Average Signs
Summary
The document explains why a time series can have a positive arithmetic average of simple returns but a negative average of log returns. The difference arises because log returns measure compounded growth: losses and gains of equal size in simple-return terms do not offset after compounding. The responses illustrate this with alternating gains and losses and with a fair-bet analogy. They also note that, under a normal-return approximation, the average log return is often about half the return variance below the arithmetic mean.
The examples clarify the distinction between average one-period returns and long-run compounded growth. The variance relationship is an approximation tied to distributional assumptions, not a universal identity; the sign and size of the gap depend on the return observations and their distribution. The discussion is explanatory and does not present an estimation procedure or investment strategy.
Key ideas
- Arithmetic mean simple returns and mean log returns can have different signs.
- Log returns reflect compounding, so equal-sized simple gains and losses do not cancel geometrically.
- A sequence of returns can have a positive arithmetic average and still lose value when compounded.
- The gap between arithmetic and log-return means is often approximated by half the variance under a normal-return assumption.
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Full text
# Returns and logreturns differences # 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? ## Answer by demully (score 12, accepted) https://quant.stackexchange.com/a/49206 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. ## Answer by D Stanley (score 4) https://quant.stackexchange.com/a/49204 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 ```
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