Can anyone explain under which conditions/assumptions the mean return of a portfolio is 0 (when returns are normally distributed)?Specifically, is anyone familiar with Kotz and Johnson (1970)? They discuss normal distributions and note that if returns are normally distributed, then an unbiased estimator of standard deviation is the mean deviation multiplied by bn=sqrt((pi/2)*n/(n-1)). Then if returns are normally distributed, an unbiased estimator of standard deviation would be | R – r | * bn, where n is the number of observations; R is the individual stock returns and r is the mean return of the portfolio. When can I assume r=0? Thanks.

I don't understand what you are getting at. The assumption that the mean is zero is a financial one, it has nothing to do with the distribution you assume. And since the Normal distribution has only one scale parameter, any measure of scale is an unbiased estimator of standard deviation if multiplied by the correct constant.The simple answer is for many risky securities, like equities, over short time intervals, like one day, the standard deviation is so much greater than the expected return that the return minus the risk-free rate can be assumed to have zero mean for most practical purposes. For less volatile securities over short periods, you can sometimes assume the price minus the forward price has close to zero mean.

Aaron, I do not know a practical standard of normal interpretation of the stock could you please to explain: let h, l be a highest and lowest for the day and o and c be an open and close prices. Under normality assumption what is commonly assumed to be mean and stdv? Sorry if it is well known fact.

Espen Haug is the guy with all the answers.Don't apologize for asking.

Thanks Aaron for the reference though I hoped you will tell in couple words how one usually interprets mean and stdv for better understanding the sense of the phrase " the standard deviation is so much greater than the expected return"

Sorry, I thought you were asking a more technical question.Say a typical stock has an expected return of 8% per year with a standard deviation of 32%. Over a day, 1/252 of a year, the mean return is 0.08/252 = 0.000317 and the standard deviation is 0.32/252^0.5 = 0.020158. The standard deviation is more than 63 times the mean.

Was there we assume for the next day a stock follows the average day statistics drawn from the previous year history data?It looks more reliable to use say last say 50 day history to predict next day dynamics. In the base of the statistics there exists an assumption that a sample represents fairly an experiment. But say this year history representing DJ , FX and others indicators is different from the previous. On the other hand if we need to predict what might be over the next year one could take into account the previous year?

For expected return, you need to take a very long history and include data from similar securities, or use some theory.For volatility, a short term, say 30 days, is usually about right.Of course, it depends on the precise security and the purpose.

Let me explain a little better…A paper I am reading notes that an unbiased estimator of standard deviation is |R| *sqrt(pi/2) and references Kotz and Johnson (1970), who note that when returns are normally distributed, an unbiased estimator of standard deviation is the mean deviation (|R-r|) multiplied by bn = sqrt((pi/2)*n/(n-1)). When n is large, it drops out, making the estimator |R-r| * sqrt(pi/2). Where R is the individual stock return and r is the mean portfolio return. To get |R| *sqrt(pi/2), you have to assume r=0 and my question is under what conditions can that assumption be made and why is that assumption made?Thanks again.QuoteOriginally posted by: AaronI don't understand what you are getting at. The assumption that the mean is zero is a financial one, it has nothing to do with the distribution you assume. And since the Normal distribution has only one scale parameter, any measure of scale is an unbiased estimator of standard deviation if multiplied by the correct constant.The simple answer is for many risky securities, like equities, over short time intervals, like one day, the standard deviation is so much greater than the expected return that the return minus the risk-free rate can be assumed to have zero mean for most practical purposes. For less volatile securities over short periods, you can sometimes assume the price minus the forward price has close to zero mean.

We assume a mean expected return of zero for daily equity returns under normality because the standard error of the volatility of daily returns is statistically insignificant. For longer holding periods this is obviously no longer true.

I agree with purbani. In any event, the real problem with this estimator is equity returns are far from Normal.

Here the GE data for 14 April 0914-Apr-09 open=12.49 , high= 12.50, low= 11.42, closed= 11.51 could you please teach me how you randomise data presenting its normality, what are mean and stdv.What is in this example "the standard error of the volatility of daily returns "?

Quotestandard error of the volatility of daily returns is statistically insignificant.Can you please clarify more on that? How the S.E. of volatility and expected mean return are related? I thought ppl use mean return as zero because of Efficient market hypothesis wherein for a liquid market no-loss-no-profit is surely expected. Thanks.

the calculations in finance are quite subjective. for example look at derivatives theory for common calculations related to 1 day they use a number close or open price for stock. In this case they eliminate completely one day variability. if they used say 30 days historical data then there are 2 possibilities. if coefficients of the log normal model are assumed to be constant then you have a sample and could a statistical inference regarding mean and stdv and the time moment is a month not a day. if they interpret a month observations as the observation along the path of the random price then conclusions theoretically looks not accurate.