# What does the average log-return value of a stock mean?

Assuming log-return is defined by the log of the price today divided by the price yesterday.

Does knowing this single piece of information (the average log-return of a stock over a long period of time), indicate something about the stock? Is this a commonly used indicator?

• What "single piece of information" are you asking about? Are you asking about the log-return over a single day, or the average of many daily log-returns over a period of time? Log-return is just another measure of return, with the same relevance (but different underlying mathematicaly assumptions) as linear measures of return. – John Bensin Sep 26 '13 at 16:33
• average of many daily log-returns over a period of time. – Yosi Dahari Sep 26 '13 at 16:34
• There are mathematical differences between linear returns and log returns (and those are probably out of scope for this site), but they're both simply measures of return. Are you asking about the differences, or just the general question of what the average return of a stock tells you? One of this site's users has a good blog post explaining the basic mathematical differences between linear and log returns, so that might be a place to start if that's what you're interested in. – John Bensin Sep 26 '13 at 16:39
• The question doesn't tell nothing about comparison between return and log-return. I am interested to know what the actual value really says, and if/how it's actually used as a factor for selecting a stock – Yosi Dahari Sep 26 '13 at 16:43
• This answer on quant.SE should give you a place to start. Log-return is just another measure of return, so it tells you all of the information that's usually contained in any measure of return. The mathematics are different, however, and more conclusions can be drawn from that. That's very general, but the quant.SE answer should give you a place to start. – John Bensin Sep 26 '13 at 16:46

Probably the best way to investigate this is to look at an example. First, as the commenters above have already said, the log-return from one period is log(price at time t/price at time t-1) which is approximately equal to the percentage change in the price from time t-1 to time t, provided that this percentage change is not big compared to the size of the price. (Note that you have to use the natural log, ie. log to the base e -- ln button on a calculator -- here.)

The main use of the log-return is that is a proxy for the percentage change in the price, which turns out to be mathematically convenient, for various reasons which have mostly already been mentioned in the comments.

But you already know this; your actual question is about the average log-return over a period of time. What does this indicate about the stock?

The answer is: if the stock price is not changing very much, then the average log-return is about equal to the average percentage change in the price, and is very easy and quick to calculate. But if the stock price is very volatile, then the average log-return can be wildly different to the average percentage change in the price.

Here is an example: the closing prices for Pitchfork Oil from last week's trading are: 10, 5, 12, 5, 10, 2, 15. The percentage changes are: -0.5, 1.4, -0.58, 1, -0.8, 6.5 (where -0.5 means -50%, etc.) The average percentage change is 1.17, or 117%. On the other hand, the log-returns for the same period are -0.69, 0.88, -0.88, 0.69, -1.6, 2, and the average log-return is about 0.068. If we used this as a proxy for the average percentage change in the price over the whole seven days, we would get 6.8% instead of 117%, which is wildly wrong. The reason why it is wrong is because the price fluctuated so much.

On the other hand, the closing prices for United Marshmallow over the same period are 10, 11, 12, 11, 12, 13, 15. The average percentage change from day to day is 0.073, and the average log-return is 0.068, so in this case the log-return is very close to the percentage change. And it has the advantage of being computable from just the first and last prices, because the properties of logarithms imply that it simplifies to (log(15)-log(10))/6. Notice that this is exactly the same as for Pitchfork Oil.

So one reason why you might be interested in the average log-return is that it gives a very quick way to estimate the average return, if the stock price is not changing very much.

Another, more subtle reason, is that it actually behaves better than the percentage return. When the price of Pitchfork jumps from 5 to 12 and then crashes back to 5 again, the percentage changes are +140% and -58%, for an average of +82%. That sounds good, but if you had bought it at 5, and then sold it at 5, you would actually have made 0% on your money. The log-returns for the same period do not have this disturbing property, because they do add up to 0%.

What's the real difference in this example? Well, if you had bought \$1 worth of Pitchfork on Tuesday, when it was 5, and sold it on Wednesday, when it was 12, you would have made a profit of \$1.40. If you had then bought another \$1 on Wednesday and sold it on Thursday, you would have made a loss of \$0.58. Overall, your profit would have been \$0.82. This is what the average percentage return is calculating.

On the other hand, if you had been a long-term investor who had bought on Tuesday and hung on until Thursday, then quoting an "average return" of 82% is highly misleading, because it in no way corresponds to the return of 0% which you actually got!

The moral is that it may be better to look at the log-returns if you are a buy-and-hold type of investor, because log-returns cancel out when prices fluctuate, whereas percentage changes in price do not. But the flip-side of this is that your average log-return over a period of time does not give you much information about what the prices have been doing, since it is just (log(final price) - log(initial price))/number of periods. Since it is so easy to calculate from the initial and final prices themselves, you commonly won't see it in the financial pages, as far as I know.

Finally, to answer your question: "Does knowing this single piece of information indicate something about the stock?", I would say: not really. From the point of view of this one indicator, Pitchfork Oil and United Marshmallow look like identical investments, when they are clearly not. Knowing the average log-return is exactly the same as knowing the ratio between the final and initial prices.

Log-returns are very commonly used in financial maths, especially quantitative finance. The important property is that they're symmetrical around 0 with respect to addition.

This property makes it possible to talk about an average return.

For instance, if a stock goes down 20% over a period of time, it has to gain 25% to be back where you started. For the log-return on the other hand the numbers are 0.223 down over a period of time, and 0.223 up to get you back to square 1.

In this sense, you can simply take an arithmetic average and it makes sense. You can freely add up or subtract values on the log-return scale, like log-interest rates or log-inflation rates. Whereas the arithmetic mean of (non-log) returns is simply meaningless: A stock with returns -3% and +3% would have 0% on average, when in fact the stock has declined in price?

The correct approach on direct price-returns would be to take a different mean (e.g. geometric) to get a decent average. And yet it will be hard to incorporate other information, like subtracting the risk-free rate or the inflation rate to get rate-adjusted average returns.

In short: Log-returns are easier to handle computationally, esp. in bulk, but non-log-returns are easier to comprehend/imagine as a number of their own.

Knowing the log return is useful - the log return can help you to work out the annual return over the period it was estimated - and this should be comparable between stocks.

One should just be careful with the calculation so that allowance for dividends is made sensibly.