How to calculate the annualized return of a stock from a series of 60 monthly returns?

How to calculate the annualized return of a stock?


What I mean, is that I have historical data for 5 years i.e. a series of 59 monthly returns in % of some stocks i.e. 3 stocks.How will calculate the annualized return of each stock and of the portfolio? And how will I will calculate the annualized standard deviation when they give me the annualized standard deviation of 200 days? Thank you for your answers.|||The average annualized return is easy, that's the geometric mean. For example if over three years, the returns were 2%, 10% and 18%, many people would think that the average rate was 10% but it's not, the average would be 6.61%. This is because the geometric mean of 1.02, 1.10 and 1.18 is 1.0661. The geometric mean of n numbers would be the nth root of their product. Another way to look at it is that the geometric mean is the arithmetic mean of the logarithms, i.e.: for the 2%, 10% and 18% example, it would be e^( ( ln(1.02) + ln(1.10) + ln(1.18) ) / 3 ). Of course that presumes that the annual rates were properly calculated and were indeed the effective annual rates, this is not assured in finance where abuse of math abound hence the term nominal interest rates (the definition of nominal is wrong but close enough).



Now the standard deviation is problematic in that since investment is compounding i.e.: the investment that the second year begins with is the investment that the previous year ended with, the applicable standard deviation should also be in the logarithmic aka geometric domain. Indeed that's how volatility is expressed in the Black Scholes equation in that they take the growth factor defined as the days closing prices divided by the previous days closing price, take the logarithms of the growth factors and calculate the standard deviation of the logarithms, they also calculate the average of those logarithms and call it mu and as you can see the geometric mean would be e^(mu). Hence the standard deviations that they provide annually are probably of limited use because the probability distribution of the prices is closer to a log normal distribution than a normal distribution. However, I doubt very much that the standard deviation cited in historical data would be the volatility in the logarithmic domain. However in general given standard deviations of three equal periods, it would just be the sum of the standard deviations squared divided by three and then square rooted, it's just that it's pointless to do that with investing unless the standard deviations were of logarithms of the growth factors, not the prices themselves and again would represent a geometric mean.



In finance there's also the issue of how many days there are in a year, although it would seem that there should be 365 days in a year, there are only 260 week days. Also it's common for financial calculations to be based upon 360 day years and of course if you take statutory holidays into account, there are about 250 business days in a year. However if you had a effective rate for 200 days and wished to annualize it to 365 days, it would simply be the 200th root taken to the 365th power, hence 10% effective over 200 days would be e^( ln( 1.10 ) / 200 * 365 ) = 1.1900 or 19.00% However if the rate over 200 days was said to be annualized then you would need to know whether it was annualized to a nominal annual rate or an effective annual rate. If it's an effective annual rate that that's simply the effective annual rate. If it was annualized to a nominal rate, it was most likely calculated by the mathematically flawed technique of dividing the rate by 200 and multiplying it by 365 so you would have to reverse the procedure, divide by 365, multiply by 200 and then take the 200th root and raise it to the 365th power hence for a 10% per annum incorrectly but commonly annualized as a nominal rate from a 200 day period would be e^( ln( 1 + 0.10 * 200 / 365 ) / 200 * 365 ) = 1.1023 therefore it would be 10.23% per annum effective.



Note most methods of annualization in common use are mathematically wrong.