It's convenient, mathematically speaking. We could, however, consider other things to measure how far the returns are, from their Mean. For example, we could pick the largest deviation magnitude, or the average of the deviation magnitudes: (1/N)Σ G_{k}  A >Example? For the S&P 500, if we consider the annual returns from Jan 1 to Dec 31, starting in Jan/50 (and ending Jan/00), we'd get: Mean = 10.1%>I like the last guy. Pay attention. Suppose each G_{k} is increased (decreased) by a factor λ. Then:
Conclusion? SD also increases (decreases) by the factor λ. (If all gains double, then SD will double.) Fig. 2 & 3 Distribution of 200 Normally Distributed monthly returns where each has doubled
Aah, but what if the individual gains do NOT increase by the same factor?
Write: and consider the effect of modifying a single gain, namely G_{i}. Compute d/dG_{i} of each side and get:
Increases in those gains which are LESS than the average gain, will cause the SD to decrease. That's sort of obvious since SD measures the spread of gains about the average. Increasing the smaller gains and/or decreasing the larger gains will reduce this spread, hence the SD. Indeed, for a Normal Distribution (the infamous "Bell Curve"), about 2/3 of the returns lie between Mean  SD and Mean + SD. Stare at the above graphs and convince yourself that this is true. (For the first graph, this range is from 10%  20% =  10% to 10% + 20% = 30%.) When SD decreases, these 2/3 crowd closer to the Mean. Now, it's reasonable to measure the volatility of the gains in terms of their spread: widely varying gains means high volatility, right? So, investment gurus DEFINE volatility as the Standard Deviation. Who can argue with that?
But, if all YOUR gains were positive, wouldn't you say that the risk has decreased, for my portfolio? After all, my monthly gains are now higher than yours by the factor F. Less risky, right? Alas, the investment community says the risk is higher for my portfolio because (have we said this before?) they usually DEFINE risk as the Standard Deviation ... and that's increased!
P.S. The argument goes something like this:
^{*} Note that {Standard Deviation of the x's}^{2} is the difference between the average of the squares and the square of the average. That is: We might also do this is 2 dimensions.
Then, after N steps, we're at position (x_{1}+x_{2}+ ...+x_{N}, y_{1}+y_{2}+ ...+y_{N}) Our distance from the origin is then R where: R^{2} = (x_{1}+x_{2}+ ...+x_{N})^{2}+(y_{1}+y_{2}+ ...+y_{N})^{2}As above (assuming that the Mean of the x's and y's is zero and there is no correlation between successive x's or y's), we'd get R^{2} = N [{Standard Deviation of the x's}^{2} + {Standard Deviation of the y's}^{2}] or (since the Variance is the square of the Standard Deviation) R^{2} = N ( Variance[x] + Variance[y] )We conclude (again!) that the distance from where we started increases as the Square Root of N. If we accept this SquareRootofTime scenario, then a Standard Deviation of Nmonth returns will be larger than the Standard Deviation of 12month (that is, annual) Returns ... larger if N is greater than 12. In general, if SD(N) and SD(M) are the Standard Deviations of N and Mmonth returns, we can write:
To annualize "Monthly" returns, we can do the following:
Hence, the Standard Deviation of annual returns (meaning
12month returns)
so y'all just multiply the Standard Deviation of monthly returns by the square root of 12 in order to estimate the Standard Deviation of annual returns.
If we wanted the SD of
annual returns and observed these annual returns over
a 36 month period, we recognize that there would only be
three returns to consider (in this 3year period)! So we can take the SD value for
monthly returns (there's 36 of them!) and multiply by SQRT(12) = 3.46 (roughly)
to
>That's assuming you believe in this SquareRootofTime stuff.
>I haven't the foggiest idea what you just said.
>12.78, actually.
>Didn't you just say the Standard Deviation increases?
>No!
Okay, if the risk (in the investment guru sense, meaning Standard Deviation) goes UP when the time period increases, what about the risk (in the dictionary sense)?
We want to investigate the possibility of suffering harm or loss, so here's what we'll do (for investing in the S&P 500, for example): We look at all 1month intervals between Jan 1, 1970 and Jan 1, 2000 (that's thirty years and 360 such 1month intervals) and count the number of times we would have LOST money. Then we look at all 6month intervals, then 12month intervals, ... then 120 monthintervals. (The last is 10 years, right? And there's 240 120month intervals between Jan 1/70 and Jan 1/00.) For each we plot the percentage of intervals when we would have lost money (i.e. the gain over the given Nmonth period was less than 1). (That's our suffering harm or loss.)
Of course, in addition to the risk of loss, we should consider the degree of risk ... maybe like so:
Fig. 10 The percentage of times that we would have incurred a Loss of P% (Jan 1,1970 to Jan 1, 2000) For example, in all the 3year intervals (between Jan/70 and Jan/00), 4% of them would have suffered a loss of at least 20%.
Fig. 18 There were 64 10year periods in Oct/84  Jan/00. None suffered a loss of 5% or more. Uh ... one more thing. The set of monthly gains for, say the S&P 500, may be rearranged and the rearranged set will have identical Distribution, Mean and Standard Deviation. For example, if we were to rearrange the gains so they occur from Minimum to Maximum (over the past thirty years) the S&P would look quite different (though it'd end up at the same place). If the ordering were from Maximum to Minimum ... well, a picture is worth a $1000 words:
Fig. 19 The best of times, the worst of times ...
And if we invest in the S&P 500, but make monthly withdrawals? What'd our portfolio look like
in each reordered case?
In all fairness, investment gurus equate risk with "uncertainty" so, when the spread of monthly gains increases, there is greater uncertainty so it's reasonable to say there is greater risk. Or is it? If I invested in a stock that decreased by a fixed amount each month^{*}, there would be no uncertainty (SD = 0) and ... uh, no risk ... even tho' I lose every single month! No risk, eh? Strange world, this world of investing. They've got a language of their own.
^{*} An investment with a constant
negative return? Yeah,
that's my mutual fund. You might like to invest in it. 
and A discussion of risk and risk
I forgot to mention how the Standard Variation varies, depending upon the time interval over which it's computed. Normally, for investment purposes, the SD is computed using monthly returns over three years. If (for a change of pace) we compute the weekly percentage gains (instead of monthly) for, say, the Dow Jones Industrial Average, we get:
For the crash of '87, the high volatility hung around for nearly three years, for the 150week Standard Deviation! > Aha! Weekly? Why weekly? Why not Monthly? Okay, here's monthly: Notice that the monthly SD is roughly 2 times the weekly SD because the time interval is about 4 times longer and the SD varies (roughly) as the square root of the time interval and the square root of 4 is 2. Neat, eh?
If the lousy definition of risk is disagreeable, perhaps because it doesn't distinguish between POSITIVE deviations from the mean and NEGATIVE deviations  the POSITIVE ones are good, eh? ... and the NEGATIVE ones ain't good  then there is something called downside risk where we set some goal (say 12% per year or maybe 1% per month) and ignore every monthly gain above this goal (because we feel there's no risk associated with gains above our goal) then compute the Standard Deviation of just those below our goal. (There are variations on this theme, but it means, for example, that we ignore the blue gains in the following chart and use only the green gains when computing the SD):
In case y'all are wondering, the Standard Deviation (SD) for ALL monthly gains, both POSITIVE (above the goal) and NEGATIVE (below the goal), is 4.2% One last thing (I promise): Suppose you invest a certain fraction of your monies in an Equity Mutual Fund (say 75%) and the balance in a Bond Mutual Fund (that'd be 25%), then you'd expect less volatility ... compared to investing everything in Equities, right? Here's a chart which shows two Funds: an Equity and a Bond fund. It also shows the growth of a portfolio which invests certain fractions in each ... along with the Standard Deviation (SD). Note that the SD increases as the percentage devoted to Equities increases ... and the overall gain decreases.
Moral? If volatility scares y'all ... try some Bond Fund.
