How do we interpret variance?

Seems shocking to ask this question after studying levels 1 and 2, but I guess the more basic the knowledge gap, the faster it should be filled.

How do we interpret variance? I understand that a larger variance means more volatility than a smaller variance – this is an ‘ordinal’ quality (rememeber NOIR? haha) but beyond that, what quantitative meaning does it have? Or how is it used?

Standard deviation seems to tell us a lot more about the probability of deviation from the mean (e.g. 68% of returns should be within ±1SD for gaussian distributions), so why do we still hear about Variance so much? Why not just use SD and be done with it?

Does variance have more value in non-normal distributions? How have I come this far without discovering it?

Just to clarify (for people who don’t read the details question), I AM already clear on:

  • how to calculate variance

  • the ordinal quality of variance (i.e. bigger variance means more volatility / “risk”)

  • the quantitative meaning of standard deviation (e.g. 68% of normally distributed results will within ±1 SD etc.)

Really good question…

What I know is that we cannot interpret variance because it has no sense. For example you can have a variance of 0.5%^2 (percentage squared), so to correct this lack of sense we calculate standard deviation which gives 7.1% in this example. This last value can be compared with other distribution variances.

I think talking about variance and standard deviation, from a qualitative perspective, is the same since both are dispersion metrics and their objectives are the same (to quantify dispersion, so risk).

When people say variance to denote risk, I think they actually mean to say variation.


You just make sure that the balls have correct PSI

Edit: Oh - you are talking about Mr. Variance. Me thought it was Mr. Brady.

I doubt that.

When Harry Markowitz wrote his seminal work on the efficient frontier, everything was couched as risk and reward. Near the end he mentioned that if you substitute “expected return” for “reward” and “standard deviation of returns” for “risk”, all of his analysis still holds.

People latched onto that and have used standard deviation of returns (or, equivalently, variance of returns) as a measure of risk. Not mere variability or variation: actual (statistical) variance.

(What’s worse is that they now generally talk only about standard deviation or variance, leaving out the “of returns” part; this sloppy language leads to sloppy thinking and many erroneous conclusions about portfolio performance.)

I honestly think that people talk about variance more than standard deviation simply because it’s easier to say and write; it’s laziness. Nothing more.

They carry exactly the same information. However, in general, standard deviation is a more useful form for that information simply because the units on standard deviation are the same as the units on the mean (and on the underlying data), whereas the units on variance are the square of the underlying units.

And, clearly, variance is more than an ordinal measure; it’s (the square of) a _ cardinal _ measure. A variance of 6 is not merely _ more _ than a variance of 3; the ratio (2) is significant: it means that the standard deviation of the former is √2 times the standard deviation of the latter, and that ratio can be used in making statistical inferences.

It is quite interesting how late this basic mean-variance relationship was formally introduced into finance theory.

Markowitz wrote his article on portfolio selection in 1952. Although people before where implicitly very much aware of this relationship for thousands of years (like “Do not put all eggs in one basket” or think about insurances which exist since decades: they work on this principles) there was never made an explicit formal presentation.

This is more astonishing as the mathematical/statistical concepts (mean, variance, covariance, etc.) where all known and available since long. This must have been some kind of “Eureka”-moment for Markowitz. For me it was really astonishing that something so basic was discovered so late… As a side note: When Markowitz defended his thesis in Chicago he had difficulties to get through. Milton Friedman one of the examiners said he could not approve his thesis. He argued that although all calculations and thinkings were correct, the thesis was neither a thesis appropriate for management theory, nor economics, nor mathematics,… So, he couldn’t approve it.

Kind of weird if you think of the repercussions of Markowitz’s idea and the Nobel Prize he won years later.

P.S.: It is worth reading the original 1952 article of Markowitz. It is mathematically not so sophisticated that you won’t understand it and it gives you a sense that there was a before and after his article.

Thanks, that’s right. My mistake.

Regarding the sloppy use of terminology, it could be possible that many people never really understood its meaning; they are just submerged in the terminology long enough to soak in and re-spout in conversation. I’m late coming to the finance world because I always thought the skills required were beyond me. It’s only recently I’ve realized that quite a few of the practitioners I meet have less understand of what they’re talking about that even I do!

It must have been quite hard to admit your own personal knowledge and opinions of a company can easily be dominated by unpredictable factors, and admit that the particular outcomes over short time frames are basically unpredictable. Even now it’s hard. Treating a stock as a random variable is a real insult to your own intelligence and takes a lot of humility.

Also, I wonder when people first started treating up and down moves together as variance. It can’t have been easy. Today we want a higher compensation for volatility but try explaining to non-finance friends why sudden strong up moves can directly lead you to demand a higher expected return…

I do want to check the original, is it online?

Anyway, treating a stock as a random variable is a model, right? It’s not an actual truth. And it’s not even close to being as accurate as Newtonian physics or that kind of model. I mean, you’d be pretty shocked if you calculated an expected return R and then, 10 years later, found it averaged R per year. I don’t find it surpising about how late it was discovered, I’m surprised how popular it it now!

As said. The concept of diversification is as old as mankind (see below from Wikipedia), but Markowitz was the first who formalized it. Additionally every bank or insurance business is based on the principles of diversification. For me it was amazing that it took so long to build a coherent theory around it.

Diversification is mentioned in the Bible, in the book of Ecclesiastes which was written in approximately 935 B.C.: But divide your investments among many places, for you do not know what risks might lie ahead.

Diversification is also mentioned in the Talmud. The formula given there is to split one’s assets into thirds: one third in business (buying and selling things), one third kept liquid (e.g. gold coins), and one third in land (real estate).

Diversification is mentioned in Shakespeare (Merchant of Venice):

My ventures are not in one bottom trusted, Nor to one place; nor is my whole estate Upon the fortune of this present year: Therefore, my merchandise makes me not sad.

That’s not exactly what I heard.

I heard that after defending his dissertation, Harry went out into the hall to wait. A few minutes later, Milton Friedman came out and said to Harry, “You know, what you did there isn’t really economics.”

Harry, of course, was completely deflated. “So you mean I won’t get a PhD?”

“Oh, don’t be silly! We won’t deny you a PhD simply because it’s not economics!”

Now, I admit that the story I heard may be incorrect, but I think that the source was reliable: Harry told it to me when I interviewed him on video in 2009 for an online class I was developing: Asset Allocation.

Look here: as of minute 17 (the whole interview is quite interesting)

There he is telling the anecdote a bit different…

in the first 1 second my only thought was this guy has no money. and harry looks even poorer.

It’s interesting how memory works