Standard Deviation: some comments on Volatility and Risk and Risk
We consider, at some point in time, the various monthly gains in our portfolio over the past N months ... like so (where the number of occurrences of each gain, out of N = 200, is plotted):

The spread about the average (or mean) gain is measured by the Standard Deviation:
SD2 = (1/N)Σ (Gk - A)2

where there are N portfolio gains, Gk (k=1 to N), and A = (1/N)ΣGk is their Average (or Mean) and SD is the Root Mean Square Deviation between the gains and their average and it can also be computed like so:

SD2 = (1/N)Σ Gk2 - {(1/N)ΣGk}2

Fig. 1 A 10% monthly gain is unusual, but what the heck.
It looks like a Normal Distribution, but that's an accident!
i.e. SD2 is the difference between the average of the squares and the square of the average.

To prove this we first drop the subscripts (for sanitary reasons) and write:
SD2 = (1/N)Σ (G - A)2 = (1/N)Σ{G2 - 2GA + A2} = (1/N)ΣG2 - 2A(1/N)ΣG + (1/N)ΣA2
    = (1/N)ΣG2 - 2AA + (1/N)NA2 = (1/N)ΣG2 - A2
>Why the "squares" of the deviations from the Mean?
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 |Gk - A|

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%
Standard Deviation = 15.8%
Maximum Deviation magnitude = 39.8%     in 1952
Average Deviation magnitude = 13.1%
>I like the last guy.
Pay attention.
Suppose each Gk is increased (decreased) by a factor λ. Then:

SD = {(1/N)Σ (λGk)2 - {(1/N)ΣλGk}2 }1/2 = λ{k2 - {(1/N)ΣGk}2 }1/2

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
Now, suppose the individual gains are changed by ADDING a constant C (rather than multiplying by a constant λ)
Here's a picture of two stocks whose returns differ by a constant

We have, as the new average:

(1/N)Σ(G + C) = (1/N)ΣG + (1/N)ΣC = (1/N)ΣG + (1/N)NC
    = A + C
hence the new Standard Deviation:
(1/N)Σ{(G + C) - (A+C)}2 = (1/N)Σ{G - A}2 = SD2
In other words, the SD is unchanged by the addition of a constant. (Uh ... did I mention that we dropped the subscripts again?)
These results are independent of the type of statistical distribution: Normal, logNormal, MickeyMouse, etc.

One important consequence of these results is that if we have a collection of numbers, say {Gk},
with Mean = 0 and SD = 1, then the collection {λ Gk + C} will have Mean = C and Standard Deviation = λ.

Aah, but what if the individual gains do NOT increase by the same factor?


SD2 = (1/N)Σ Gk2 - {(1/N)ΣGk}2

and consider the effect of modifying a single gain, namely Gi.
Compute d/dGi of each side and get:

SD dSD/dGi = (1/N) {Gi - {(1/N)ΣGk} = (1/N) (Gi - A)

and this is positive (or negative), implying an increase (or decrease), if the gain Gi is greater (or less) than the average of all the gains, A = (1/N)ΣGk.

     Increases in those gains which are LESS than the average gain, will cause the SD to decrease.
     Decreases in those gains which are GREATER than the average gain, will also 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?

Now suppose MY monthly gains are all increased by some positive constant C compared to YOUR monthly gains.(Say C = 30%, that'd be nice.) As we've seen, SD, hence the volatility, doesn't change. Our graph just gets shifted to the right by an amount C. Same spread, same SD, same volatility. But wouldn't you say that the risk has decreased, for my portfolio? After all, my gains are now 30% higher than yours. Less risky, right? Alas, the investment community says the risk is the same for both our portolios because they normally DEFINE risk as the Standard Deviation (and not as the risk of a loss).

For this association of "risk" with "SD", see 1 and 2 and 3 and ... and n. Even William Sharpe defines the Return per unit of Risk as some Return divided by the Standard Deviation!

Fig. 4
Now suppose MY monthly gains are all increased by some positive multiplier F > 1 compared to YOUR monthly gains. If your monthly gains are all positive, then F = 2 would be nice. That'd make my gains twice the size of yours.
As we've seen, SD, hence the volatility, changes by the same factor. All the gains in MY distribution graph get shifted to the right by a factor F, the spread increases by that factor and the SD (and the volatility) change by the same factor. (The height of my graph decreases 'cause the number of returns hasn't changed, so there are fewer in each of the intervals along the horizontal axis - because of the increased spread.) Because of this increased spread, there is a greater probability of Returns far from the mean; that's the extended tail of the distribution chart. For some distributions, this extended tail means a greater chance of disastrous Returns! (See Kurtosis)
See Fig. 3, compared to Fig. 2, or, better still, here's a closeup of the tails:

Fig. 5a The tail of the distribution

Fig. 5

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!

Fig. 6
One other thing: the charts above look like Normal Distributions. Well ...uh, they are, but it's only so I could get pretty pictures. The real, live monthly returns for, say, the S&P 500 look sorta Normal ... but that's not important for what we're discussing.

Fig. 7
Oh yeah ... one other thing: We often (?) hear that the Standard Deviation increases as the time interval increases. In fact, a common statement is that SD varies as the square root of the time: SQRT(time).

This follows (mathematically speaking) from Einstein's 1905 analysis of a random walk (or Brownian Motion) and, earlier, it's associated with Louis Bachelier and even earlier with Jules Regnault (See this.PDF). It assumes the returns are random in this sense. Are they?

Looking at all intervals of length 1 month, 6 months, 12 months, etc., and calculating the SD for each set, we get this chart. Close, but no cigar.

P.S. The argument goes something like this:

If you start at x = 0 and take N steps of length x1, x2, ...., xN then your distance from the origin (x = 0) is R where, to get the Standard Deviation of the set of distances from the origin (over all possible sets x1, x2, ... ), we consider the Mean Square of R ... hence we consider:
R2 = (x1+ x2+...+ xN )2 = x12+ x22+... +xN2 +2x1 x2 +2x1 x3+ ...
If the Mean = 0 (for the steps x1, etc.), then each is equally likely to be positive as negative and, for large N, the cross terms like x1x2 will average to zero and we're left with
R2 = x12+ x22+... +xN2 = N { (x12+ x22+... +xN2)/N } = N {Standard Deviation of the x's}2

leaving us with: R = N1/2 {Standard Deviation of x's} * ... and the square root pops up.
If the number of steps, N, increases with time, then this is proportional to SQRT(time).

* 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:
{Standard Deviation of the x's}2 = (1/N)Σ xk2 - {(1/N)Σxk}2
But, if the average value of the x's is zero, then Σxk = 0 so
(1/N)Σ xk2 = {Standard Deviation of the x's}2

We might also do this is 2 dimensions.
Suppose we start at the origin (0,0) and move left or right in steps of length x1, x2, ...
and, at the same time, move up or down in steps of length y1, y2, ...

Then, after N steps, we're at position (x1+x2+ ...+xN, y1+y2+ ...+yN)

Our distance from the origin is then R where:

R2 = (x1+x2+ ...+xN)2+(y1+y2+ ...+yN)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
R2 = 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)
R2 = N ( Variance[x] + Variance[y] )
We conclude (again!) that the distance from where we started increases as the Square Root of N.

See also Square Root of Time

If we accept this Square-Root-of-Time scenario, then a Standard Deviation of N-month returns will be larger than the Standard Deviation of 12-month (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 M-month returns, we can write:


To annualize "Monthly" returns, we can do the following:

  • Calculate the Standard deviation of Monthly percentage Returns (over, say, 36 months).
  • Annualize this SD of 1-month returns by writing:
SD(1)/ SQRT(1) = SD(12)/SQRT(12)

Hence, the Standard Deviation of annual returns (meaning 12-month returns) will should may be estimated as

SD(1) SQRT(12)

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 3-year 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 get estimate the SD of 1-year returns.

>That's assuming you believe in this Square-Root-of-Time stuff.
Right! But it makes some sense. After all, one would expect the 12-month gains to be 12 times the monthly gains (roughly), so the Mean Squared Deviation would be 122 times larger, but over any given time interval (like N years worth of gains), there would be 1/12 as many terms in the sum (1/N)Σ (Gk - A)2 so we're up by a factor 122/12 = 12 and the Standard Deviation (being the square root of the Mean Squared Deviation) would be SQRT(12) times larger ... roughly.

>I haven't the foggiest idea what you just said.
Here's an example. Over the 51-year period Dec/50 to Dec/00 the 1-year returns for the S&P 500 averaged 10.1% - that's the return for the year ending Dec 31/50, then the return for the year ending Dec 31/51, etc..
The Standard Deviation of these fifty-one annual returns is was 15.8%.

On the other hand, the Average monthly return was 0.79% and the Standard Deviation of these monthly returns was 4.09%
(and there were 12 x 51 = 612 of these monthly returns).

The ratio of SD(Annual)/SD(Monthly) = 15.8/4.09 = 3.86 whereas
SQRT(12) is ... uh ...
>3.46 (roughly)?

>And I notice that the 12-month average return is larger than the average 1-month return by a factor 10.1/0.79 which is ... uh ...
12, roughly.

>12.78, actually.
However, I should mention the following:

  • We consider just monthly returns over N months.
  • We calculate the Standard Deviations of these monthly returns, as N increases.
  • We get an decreasing Standard Deviation!!

>Didn't you just say the Standard Deviation increases?
That's if you consider the Standard Deviation on N-month returns. What I'm talking about now is the Standard Deviation of 1-month returns, over N months. See the difference?

Then just look at this picture. See? The variability decreases as N, the number of 1-month returns, increases.

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)?
risk n. The possibility of suffering harm or loss. -v To expose to the possibility of harm or loss.
             (That's not my definition of risk. It's Webster's.)

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 1-month intervals between Jan 1, 1970 and Jan 1, 2000 (that's thirty years and 360 such 1-month intervals) and count the number of times we would have LOST money. Then we look at all 6-month intervals, then 12-month intervals, ... then 120 month-intervals. (The last is 10 years, right? And there's 240 120-month 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 N-month period was less than 1). (That's our suffering harm or loss.)
We'd get:

Fig. 8
risk HAS decreased with length of investment horizon!
... and, for the NEXT thirty years?

Fig. 9
Showing the average of the N-month gains, annualized.
P.S. Note that the '87 CRASH is just a minor blip!

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 3-year intervals (between Jan/70 and Jan/00), 4% of them would have suffered a loss of at least 20%.

'course, mebbe the time length (30 years) is too short (too small a sample?), so here's 50 years:

Fig. 11 The percentage of times that we would have incurred a Loss of P% (Jan 1,1950 to Jan 1, 2000)

and then there's:

Fig. 12 ... but, what about the NEXT thirty years?

Fig. 13


Fig. 14

Fig. 15

and finally:

Fig. 16

Fig. 17

Fig. 18 There were 64 10-year periods in Oct/84 - Jan/00. None suffered a loss of 5% or more.
Data obtained

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?
A BIG difference in the final result, eh what? (... yet the Distribution of Returns is the same.)

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.
I invest in GICs at 2% and my MER (Management Expense Ratio) is 3%

"One should always change the meaning of a word when practising the art of obfuscation"

Related stuff: Random Walks and Portfolio Growth

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 150-week 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?

Uh ... one more thing:

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.
It may not make you rich ... but you'll sleep more soundly.

And one other thing:
You might have gained sorry the impression that increased Gains means having to suffer increased Volatility.
Here's a chart with thirty points.

Each point identifies a Standard Deviation (or Volatility) and a Gain for the S&P500, for each year from 1970 to 1999. The Gains run from -30% to about +35%. The SD runs from about 1.5% to 8.5% and you'll have noticed that there is little if any indication that they're correlated. Indeed, it would appear that the bigger Gains were associated with the smaller SDs.
>Look closely and you'll see a faint line. That's the "best-fit" regression line ... and it slopes DOWN to the right !

We should also note the effect of increasing the Standard Deviation on your eventual portfolio after 1, 2, 3, ... 15 years.

If we look at the range of possible portfolios (within two standard deviations of the Mean Return), assuming (for example) a Log-normal distribution and SD = 15% and SD = 25% ... we get

       for more on risk ...