Perfect Correlation? Risk = Standard Deviation? Huh?

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gummy
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Perfect Correlation? Risk = Standard Deviation? Huh?

Post by gummy »

It says here:
Perfect positive correlation ... implies that as one security moves, either up or down, the other security will move in lockstep, in the same direction.
Does anybuddy actually believe that?

One also reads that Standard Deviation (or stock "volatility") measures the risk associated with a stock.
Does anybuddy actually believe that?

Image

Sorry ... but it still bugs me.
Image

See?
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augustabound
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by augustabound »

Uncle Gummy's back for Christmas! :beer:
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newguy
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by newguy »

Correlation of returns is 1 but the correlation of prices is -.57 (in my example).

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tidal
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by tidal »

Stats love.

Unless you understand the subject that you are analyzing, you are prone to doing good stats whilst nonetheless deriving nonsense results.

Here's an example. Supposed you are observing the speed of some objects, and measuring in 1,000's of miles/second.

Here are your 5 measurements:

182
186
183
181
185
184

The mean is 183.5, the standard deviation is 3.7, and based on our sample size, we can estimate that in a larger population about 95% of the observations would be between 179.8k and 187.2k mps. Or, about 2.5% of the observed speeds would be expected to be above 187.2k mps.

Can you see a rather obvious physical problem here? And yet the stats are "correct".

By the way, there is a lot of this kind of nonsense "stats" being flung at climate science these days. Elaborate statistical "Gotchas" by economists or amateur mathmeticians that, upon examination, usually turn out to be "your results are physically impossible or meaningless".
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ghariton
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by ghariton »

gummy wrote:Sorry ... but it still bugs me.
Glancing at the numbers, it seems that the return for Stock A are always exactly 10% more than returns for Stock B. If I know one, I know the other exactly. So yes, I'd say there is perfect correlation of returns.

As to stock price, it is not immediately apparent to me what process is determining stock prices. I note that prices start about $10 different (for a 10% return difference) and end up about $40 different (for the same 10% spread in return). In real life, that means that valuation of $1 odf return is changing drastically over a very short period of time. Not sure what else one could say about the stock prices.

Both these stocks are risky. Are they of equal risk? Clearly not, given the time-varying pricing of returns.

But I may have misinterpreted your example.

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ghariton
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by ghariton »

tidal wrote:The mean is 183.5, the standard deviation is 3.7, and based on our sample size, we can estimate that in a larger population about 95% of the observations would be between 179.8k and 187.2k mps. Or, about 2.5% of the observed speeds would be expected to be above 187.2k mps.

Can you see a rather obvious physical problem here? And yet the stats are "correct".
A nitpick and a more serious concern.

First, you say that based on your sample, you can draw inferences about a larger population. That is incorrect. You can draw inferences about the population from which you drew your sample, assuming that (1) your sampling frame was an adequate representation of the target population and (2) the sample adequately represented the sampling frame. But extrapolating to a different population is very dangerous indeed.

Second, your use of the standard deviation to obtain the probability of speeds above a certain limit is based on the assumption that your observations are drawn from a Gaussian probability distribution. But Gaussians have infinite tails and are symmetric. It is unlikely that a Gaussian is a good model for your physical phenomenon. You should be using another probability distribution -- which one you should choose depends on the physical characteristics of what you are trying to describe.

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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by George$ »

Gummy: Welcome back. I think all of us miss your presence here.

Just by looking at the tables and charts for stocks X and Y I don't understand how they can be perfectly correlated. Is that what the correlation calculation gives? If so, is there a "gummy tutorial" that explains why, what the source of the problem is? :?
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by MaxFax »

gummy wrote: One reads that Standard Deviation (or stock "volatility") measures the risk associated with a stock. Does anybuddy actually believe that?
I do. I changed your bold to include the 'the' because I think that reflects your point, and also my argument. It has become 'accepted for convenience' that the word 'risk' is synonymous with SD when used in finance circles. The sentence is simply a tautology.

I don't think anyone really believes that SD is the risk though. E.g. if you know what SD is, then you know that it measures the upsides as well as the downsides. OF COURSE no investor worries about excess upsides. SD is still important, for various reasons, for understanding risks associated with investing.
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by deaddog »

Lies ; Damn lies & Statistics.

Ponder this;
Stock X has gained 300% and stock Y has lost 90%.
If these stock have a reversal of fortune and over the next 3 years Y gains 300% and X loses 90% both stocks are now near $4.00.
Calculate your risk. :lol:
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by IdOp »

The extent of my view is:

Uncertainty is a non-delta probability distribution.

Risk is a probability computed from a probability distribution.

Not all probability distributions are characterized by mean and standard deviation.

Whether or not all physical and economic theories that employ probability distributions are correct is a debatable. :)
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by gummy »

Aah, wouldn't it be wunnerful if'n financial writers were clear about what "correlation" they were speaking of: correlation between prices or returns.

Here, for example, they write about beta and mean the relationship between returns.
Then it says:
... an asset's beta is equal to the product of its correlation coefficient, R, and its standard deviation divided by the market's standard deviation.
'course, that ain't true unless they're talking about the correlation between returns, right?

Anyway, if you click on the picture in my earlier post, you can download an Excel spreadsheet.
It's got a chart and, if you move that chart, you'll see the correlation of prices has also been calculated
... and it ain't the same as the correlation of returns (which is +1).

Anyway, "perfect" correlation between two sets of data (correlation = +1) means one set of numbers is a linear function of the other set. In the graphical example I gave:
y = ax + b where the values of a and b are hidden beneath the chart (in the spreadsheet). Image
(Note: If'n y'all change a to a negative number, you generate data sets with correlation = -1.)

P.S.
There's some bumpf here:
http://www.gummy-stuff.org/perfect-correlation.htm
and here
http://www.gummy-stuff.org/correlation-stuff.htm

P.P.S.
I reckon using "Standard Deviation" as a proxy for "risk" is old hat (I hope).
It's sad that gurus of the past have adopted the clearly understood word 'risk" and changed its meaning.
A wise man once said:
One should always change the meaning of a word when practising the art of obfuscation.

Even reinterpreting Standard Deviation to mean "uncertainty" ain't much use, either:
http://www.gummy-stuff.org/risk-is-what.htm

Oh well ...
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Re: Perfect Correlation? Risk = Standard Deviation? Huh?

Post by optionable68 »

gummy wrote:Aah, wouldn't it be wunnerful if'n financial writers were clear about what "correlation" they were speaking of: correlation between prices or returns.

Here, for example, they write about beta and mean the relationship between returns.
Then it says:
... an asset's beta is equal to the product of its correlation coefficient, R, and its standard deviation divided by the market's standard deviation.
'course, that ain't true unless they're talking about the correlation between returns, right?

Anyway, if you click on the picture in my earlier post, you can download an Excel spreadsheet.
It's got a chart and, if you move that chart, you'll see the correlation of prices has also been calculated
... and it ain't the same as the correlation of returns (which is +1).

Anyway, "perfect" correlation between two sets of data (correlation = +1) means one set of numbers is a linear function of the other set. In the graphical example I gave:
y = ax + b where the values of a and b are hidden beneath the chart (in the spreadsheet). Image
(Note: If'n y'all change a to a negative number, you generate data sets with correlation = -1.)

P.S.
There's some bumpf here:
http://www.gummy-stuff.org/perfect-correlation.htm
and here
http://www.gummy-stuff.org/correlation-stuff.htm

P.P.S.
I reckon using "Standard Deviation" as a proxy for "risk" is old hat (I hope).
It's sad that gurus of the past have adopted the clearly understood word 'risk" and changed its meaning.
A wise man once said:
One should always change the meaning of a word when practising the art of obfuscation.

Even reinterpreting Standard Deviation to mean "uncertainty" ain't much use, either:
http://www.gummy-stuff.org/risk-is-what.htm

Oh well ...
Gummy, its been 2 years.... time for your return.... :beer:
3-time winner of FWF Annual Stock Market Predictions contest
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