What does the average log-return value of a stock mean?
"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.
That is absolute rubbish.  Warren Buffet follows simple value and GARP tenants that literally anyone could follow if they had the discipline to do so.  I have never once heard of an investment made by Warren Buffet that wasn't rooted in fundamentals and easy to understand.  The concept is fairly simple as is the math, buying great companies trading at discounts to what they are worth due to market fluctuations, emotionality, or overreactions to key sectors etc.  If I buy ABC corp at $10 knowing it is worth $20, it could go down or trade sideways for FIVE YEARS doing seemingly nothing and then one day catch up with its worth due to any number of factors.  In that case, my 100% return which took five years to actualize accounts for an average 20% return per year.  Also (and this should be obvious), but diversification is a double edged sword.  Every year, hundreds of stocks individually beat the market return.  Owning any one of these stocks as your only holding would mean that YOU beat the market.  As you buy more stocks and diversify your return will get closer and closer to that of an index or mutual fund.  My advice is to stick to fundamentals like value and GARP investing, learn to separate when the market is being silly from when it is responding to a genuine concern, do your own homework and analysis on the stocks you buy, BE PATIENT after buying stock that your analysis gives you confidence in, and don't over diversify.  If you do these things, congrats.  YOU ARE Warren Buffet. "The formula for standard deviation is fairly simple in both the discrete and continuous cases. It's mostly safe to use the discrete case when working with adjusted closing prices. Once you've calculated the standard deviation for a given time period, the next task (in the simplest case) is to calculate the mean of that same period. This allows you to roughly approximate the distribution, which can give you all sorts of testable hypotheses. Two standard deviations (σ) away from the mean (μ) is given by:  It doesn't make any sense to talk about ""two standard deviations away from the price"" unless that price is the mean or some other statistic for a given time period. Normally you would look at how far the price is from the mean, e.g. does the price fall two or three standard deviations away from the mean or some other technical indicator like the Average True Range (an exponential moving average of the True Range), some support level, another security, etc.  For most of this answer, I'll assume we're using the mean for the chosen time period as a base. However, the answer is still more complicated than many people realize. As I said before, to calculate the standard deviation, you need to decide on a time period. For example, you could use S&P 500 data from Yahoo Finance and calculate the standard deviation for all adjusted closing prices since January 3, 1950. Downloading the data into Stata and applying the summarize command gives me: As you can probably see, however, these numbers don't make much sense. Looking at the data, we can see that the S&P 500 hasn't traded close to 424.4896 since November 1992. Clearly, we can't assume that this mean and standard deviation as representative of current market conditions. Furthermore, these numbers would imply that the S&P 500 is currently trading at almost three standard deviations away from its mean, which for many distribution is a highly improbable event. The Great Recession, quantitative easing, etc. may have changed the market significantly, but not to such a great extent.  The problem arises from the fact that security prices are usually non-stationary.. This means that the underlying distribution from which security prices are ""drawn"" shifts through time and space. For example, prices could be normally distributed in the 50's, then gamma distributed in the 60's because of a shock, then normally distributed again in the 70's. This implies that calculating summary statistics, e.g. mean, standard deviation, etc. are essentially meaningless for time periods in which prices could follow multiple distributions.  For this and other reasons, it's standard practice to look at the standard deviation of returns or differences instead of prices. I covered in detail the reasons for this and various procedures to use in another answer. In short, you can calculate the first difference for each period, which is merely the difference between the closing price of that period and the closing price of the previous period. This will usually give you a stationary process, from which you can obtain more meaningful values of the standard deviation, mean, etc.  Let's use the S&P500 as an example again. This time, however, I'm only using data from 1990 onwards, for the sake of simplicity (and to make the graphs a bit more manageable). The summary statistics look like this: and the graph looks like this; the mean is the central horizontal red line, and the top and bottom lines indicate one standard deviation above and below the mean, respectively.   As you can see, the graph seems to indicate that there were long periods in which the index was priced well outside this range. Although this could be the case, the graph definitely exhibits a trend, along with some seemingly exogenous shocks (see my linked answer).  Taking the first difference, however, yields these summary statistics: with a graph like this:  This looks a lot more reasonable. In periods of recession, the price appears much more volatile, and it breaches the +/- one standard deviation lines indicated on the graph.  This is only a simple summary, but using first differencing as part of the wider process of detrending/decomposing a time series is a good first step.  For some technical indicators, however, stationary isn't as relevant. This is the case for some types of moving averages and their associated indicators. Take Bollinger bands for instance. These are technical indicators that show a number of standard deviations above and below a moving average. Like any calculation of standard deviation, moving average, statistic, etc. they require data over a specified time period. The analyst chooses a certain number of historical periods, e.g. 20, and calculates the moving average for that many previous periods and the moving/rolling standard deviation for those same periods as well. The Bollinger bands represent the values a certain number of standard deviations away from the moving average at a given point in time. At this given point, you can calculate the value two standard deviations ""away from the value,"" but doing so still requires the historical stock price (or at least the historical moving average). If you're only given the price in isolation, you're out of luck.  Moving averages can indirectly sidestep some of the issues of stationarity I described above because it's straightforward to estimate a time series with a process built from a moving average (specifically, an auto-regressive moving average process) but the econometrics of time series is a topic for another day.  The Stata code I used to generate the graphs and summary statistics:" "The short answer is the annualised volatility over twenty years should be pretty much the same as the annualised volatility over five years. For independent, identically distributed returns the volatility scales proportionally. So for any number of monthly returns T, setting the annualization factor m = 12 annualises the volatility.  It should be the same for all time scales.  However, note the discussion here: https://quant.stackexchange.com/a/7496/7178 Scaling volatility [like this] only is mathematically correct when the   underlying price model is driven by Geometric Brownian motion which   implies that prices are log normally distributed and returns are   normally distributed. Particularly the comment: ""its a well known fact that volatility is overestimated when scaled over long periods of time without a change of model to estimate such ""long-term"" volatility."" Now, a demonstration.  I have modelled 12,000 monthly returns with mean = 3% and standard deviation = 2, so the annualised volatility should be Sqrt(12) * 2 = 6.9282.   Calculating annualised volatility for return sequences of various lengths (3, 6, 12, 60 months etc.) reveals an inaccuracy for shorter sequences.  The five-year sequence average got closest to the theoretically expected figure (6.9282), and, as the commenter noted ""volatility is [slightly] overestimated when scaled over long periods of time"". Annualised volatility for varying return sequence lengths  Edit re. comment Reinvesting returns does not affect the volatility much.  For instance, comparing some data I have handy, the Dow Jones Industrial Average Capital Returns (CR) versus Net Returns (NR).  The return differences are somewhat smoothed, 0.1% each month, 0.25% every third month.  More erratic dividend reinvestment would increase the volatility." "The answer to your question depends on what you mean when you say ""growth"".  If you mean a literal increase in the aggregate market capitalization of companies, across the entire market, then, no, this sort of growth is not possible without concomitant economic growth.  The reason why is that the market capitalization of each company is proportional to its gross revenue, and the sum of all revenue from selling ""final goods"" (i.e., things purchased and used by consumers) is, apart from a few technicalities, the definition of GDP.  The exact multiplier might fluctuate up or down depending on investors' expectations about how sales will grow or decline going forward, but in a zero-growth economy this multiplier should be stable over the long run.  It might, however, still fluctuate over the short term, but more about that in a minute.   Note that all of this applies to aggregate growth across all firms.  Individual firms can still grow, of course, but as they must do this by gaining market share from other companies such growth would be balanced by a decline for some other firm.  Also, I've assumed zero net exports (that's one of the ""technicalities"" I mentioned above) because obviously you could have export-driven growth even if the domestic economy were stationary. However, often when people talk about ""growth"" in the market, what they really mean is ""return"".  That is, how much does your investment earn for you.  This isn't really the same thing as growth, but people often think of it that way, particularly in the saving phase of their investing career, when they are reinvesting their returns, and therefore their account balances are growing.  It is possible to have a positive return, averaged across the market, even in a stationary economy.  The reason why is that there are really only two things a firm can do with its net profits.  One possibility is that it could invest it in growing the business.  However, there is not much point in doing that in a stationary economy because by assumption no increase in aggregate consumption (and therefore, in the long run, aggregate production) are possible.  Therefore, firms are left with only the second option, which is to pay them out to investors as dividends.  Those dividends provide a return that is independent of economic growth. Would the stock market still be a good investment in such an economy?  Yes.  Well, sort of.  The rate of return from firms' dividend payouts will depend on investors' demand (in aggregate) for returns on their investments.  Stock prices will rise or fall, causing returns to respectively fall or rise, to find that level.  If your personal desire for returns is lower than the average across the investing public, then the stock market would look like a good investment.  If your desired return is higher than the average, then it will look like a poor investment.  The marginal investor will, of course be indifferent.  The practical upshot of this is that the people who invest in the stock market in this scenario will be precisely the ones for whom the stock market is a good investment, given their personal propensity to save and desire for returns, and so forth. Finally, you mentioned that in your scenario the GDP stagnation is due to declining population.  I am less certain what this means for investment, but my first thought is that you would have a large retired population selling its investments to fund late-life consumption, and you would have a comparatively small (relative to history) working population buying those assets.  This would lead to low asset prices, and therefore high rates of return.  However, that's assuming that retirees need to sell assets to fund their retirement consumption.  If the absolute returns on retirees' assets are large enough to fund their retirement consumption then you would wind up with relatively few sellers, resulting in high prices and therefore relatively low rates of return.  It's not obvious to me which effect would dominate, and so it's hard to say whether or not the resulting returns would look attractive to the working-age population." I stock is only worth what someone will pay for it.  If you want to sell it you will get market price which is the bid. The point of that question is to test if the user can connect shares and stock price.  However, that being said yeah, you're right. Probably gives off the impression that it's a bit elementary. I'll look into changing it asap. As said previously, most of the time volume does not affect stock prices, except with penny stocks. These stocks typically have a small volume in the 3 or 4 figure range and because of this they typically experience very sharp rises and drops in stock prices, contrasting normal stocks that go up and down constantly every minute. Volume is not one thing you should be looking at when analyzing a stock in most cases, since it is simply the number of people of trades made in a day. That has no effect on the value of the company, whereas looking at P/E ratios, dividend growth, etc all can be analyzed to see if a company is growing and is doing well in its field. If I buy an iPhone, it doesn't matter if 100 other people or 100,000 other people have bought it as well, since they won't really affect my experience with the product. Whereas the type of iPhone I buy will. There are books on the subject of valuing stocks. P/E ratio has nothing directly to do with the value of a company. It may be an indication that the stock is undervalued or overvalued, but does not indicate the value itself. The direct value of company is what it would fetch if it was liquidated. For example, if you bought a dry cleaner and sold all of the equipment and receivables, how much would you get? To value a living company, you can treat it like a bond. For example, assume the company generates $1 million in profit every year and has a liquidation value of $2 million. Given the risk profile of the business, let's say we would like to make 8% on average per year, then the value of the business is approximately $1/0.08 + $2 = $14.5 million to us. To someone who expects to make more or less the value might be different. If the company has growth potential, you can adjust this figure by estimating the estimated income at different percentage chances of growth and decline, a growth curve so to speak. The value is then the net area under this curve.  Of course, if you do this for NYSE and most NASDAQ stocks you will find that they have a capitalization way over these amounts. That is because they are being used as a store of wealth. People are buying the stocks just as a way to store money, not necessarily make a profit. It's kind of like buying land. Even though the land may never give you a penny of profit, you know you can always sell it and get your money back. Because of this, it is difficult to value high-profile equities. You are dealing with human psychology, not pennies and dollars.