Why should I worry about the variance of the return?

I have this doubt (I hope that this is not a completely silly question). If my growth rate depends on the long period average return, why should I worry about the variance of the return? i.e. over the long period, two portfolios with the same average return will grow at the same rate.

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• Thanx for the reply, but i have one comment. In my opinion there is a subtle error in your examples. Of course you are performing a mean, but i think that is pathologically chosen. What i means is that with two only elements, an average is perfectly defined by math, but don't have a full statistical meaning. In other words statistical average say you what is the most probable scenario. – emanuele Mar 30 '12 at 10:12

As Dilip stated, the return (CAGR) will not be the same as the average. For the simple mathematical explanation - (R+x)(R-x) will return R^2-x^2 over the two periods, not the average R^2 result. The higher the variance, the worse the return over time. The issue may be less about this in the accumulation phase then when a retiree is withdrawing funds. I'd prefer the first 10 years of my retirement produce extraordinary returns followed by a below average decade than the reverse.

You do not have to worry about the variance of the return if you do not want to, or if you choose not to. But average long-term growth rate is known only after the fact, not before. Right now, all you can say is that you expect a portfolio to grow at, say, an average rate of 8% per annum for the next 10 years. Whether your expectations are achieved will not be known till 2022. The word average is important implying that the growth rate during each of the ten years might be smaller or larger or the same as 8%, and some idea of how much the annual growth rate might vary from the nominal 8% during the ten years is of some interest to some people. One measure used to describe this variation is the variance (or standard deviation which is the square root of the variance).

Here are some questions for you to ponder.

1. If you have to choose between two investments, both projecting an average growth rate of 8% per annum over 10 years, but one is more volatile so that the actual growth rate in any particular year might be anywhere from 2% to 14%, while the other is more stable with actual growth rate in a year ranging from 7.5% to 8.5%, which would you prefer?

2. You invest in something extremely risky for a two-year period (no backsies) and the investment has a 50% loss the first year and a 50% gain next year. Is the average gain (-50+50)/2% = 0%? Will you get all your money at the end of the two-year period?

• If you say that you can't estimate average return until 2022, then you can't estimate average variance until 2022. p.s: the mean of 50% loss and 50% gain is not 0. you must compound the return in order to get mean return. – emanuele Mar 26 '12 at 16:48
• You can estimate that a certain investment will grow at x% per annum with a standard deviation of y% (which might be large enough to have negative growth rates in some years). You have to make the decision now and you won't know the actual average annual growth rate till 2022. So, if you think that in my first example, both investments are equally good, then you have just answered your own question: you do not need to worry about the variance of your return. In 2022, it you discover that both investments actually achieved a growth rate of 8% per annum over 10 years, (continued) – Dilip Sarwate Mar 26 '12 at 16:56
• ...then it does not matter what the variance was; both investments would have earned you 8% per annum. But you might have a few more gray hairs if you had invested in the more volatile investment. – Dilip Sarwate Mar 26 '12 at 16:58
• :) i agree. but my question is at more basic level. If i cannot estimate the return, why i should trust my estimation of variance? i.e. if the estimations are goods, then the return is the most important (Kelly criterion), but if the estimations are not goods then i cannot trust neither on the estimation of the variance. – emanuele Mar 26 '12 at 20:07
• Well, then; you have answered the question for yourself; the variance provides no useful information to you, and so there is no need for you to worry about it. – Dilip Sarwate Mar 27 '12 at 0:28

I was gonna post this as a comment to Dilip's answer but decided it was worth expanding on, even though some of this has already been mentioned.

If an investment goes down 50% one year, and up 50% the next, even though on average your rate of return is 0%, you're down 25%.

The math:

\$100,000 - 50% = \$50,000
\$50,000 + 50% = \$75,000  <-- ouch!

Some more math:

Both Portfolios start at \$100,000. Both have an average annual rate of return of 8%.

Portfolio A - High Variance:
Year 1: \$100,000 down 50% = \$50,000
Year 2: \$50,000 up 58% = \$79,000 <-- Yay! Average annual gain of 8%!
Ca-ching! Er... except you're down 21% on your original investment.

Portfolio B - Low Variance
Year 1: \$100,000 + 6% = \$106,000
Year 2: \$106,000 + 10% = \$116,600 <-- This also has an average annual gain
of 8%, but instead of being down 21%, you're up 16.6% on your investment!

Which portfolio would you prefer to invest in?

Now, it's true, that when the investment is done, you've sold it, and you have the cash in hand, it doesn't matter anymore what the variance was over the time that you held it. It's all hindsight then, and nothing more to do except brag at neighborhood parties.

Expected variance is useful to figure out if you should invest to begin with. For any given level of expected return, you want the lowest variance. The lower the variance, the more reliable that expected rate of return is.

So yes, you should worry about variance of return.

Correct, assuming that one never has to liquidate an asset, its variance is not a risk but an opportunity since foreknowledge of variance would present optimal times when to purchase.

Variance is not the mean, so variance will not impact the long-run return. This would be the gambler's fallacy, that future random future results will depend upon past random results.

Variance is also geometric, just like the return, so all losses from variance will equal all gains over the long run, leaving the mean for the long run expected value. In other words, a 50% loss is balanced by a 100% gain which is a simple difference in the geometric scale, so all such losses and gains are equal due to variance. Since variance is geometric like the mean, a 50% gain is not balanced by a 50% loss.

All of that said, generally speaking across asset classes, the higher the variance the lower the price yet the higher the return. Variance is a risk because no one is immune from selling at disadvantageous times, and risk is a cost, so that risk must be compensated with high returns.