# Vol -> Correlation proof

So, in the curriculum I read that increased volatility increases correlation (think this was in economics). Has anyone seen some sort of mathematical proof of this or can explain this logically? Thanks, Jack

Here is simple explanation based on CAPM and a few assumptions. x1,x2 - two stocks, y - market. x1=a1+b1*y+epsilon1 x2=a2+b2*y+epsilon2 Assume cov(y,epsilon1)=0, cov(epsilon1,epsilon2)=0, var(epsilon)=v1, var(epsilon2)=v2 and v1, v2 are constant. Think of epsilons as idiosyncratic risks of the firms. Cor(x1,x2)=Cov(x1,x2)/(sqrt(var(x1)*var(x2))=b1*b2*Var(y)/sqrt[(b1^2*Var(y)+v1)*(b2^2*Var(y)+v2)]. Notice that if Var(y) increases, Cor(x1,x2) decreases. Does that help?

I don’t fully understand your explaination (probably need to take more time reading it), however, isn’t the argument that if vol increases, corr increases? Your last sentence suggests the opposite?

Sorry, my last sentence is incorrect. You can see from the formula, that when Var(y) increases, correlation increases because v1 and v2 become smaller relative to Var(y). If v1 and v2 are close to zero, correlation becomes close to 1 (or negative 1 depending on the sign of b1*b2 which is most likely positive)

I don’t see it. I plugged in a whole bunch of numbers into a spreadsheet. You need to change the inputs, i.e. the returns for asset x and asset y. This changes both the covariance and the covariance. Try it yourself. I found it was possible to increase the standard deviation for both or either assets (e.g. higher volatility) and the correlation could move either way as both the standard deviation of the asset and the covariance moved. I’m looking for a better proof than above, or else I remain unconvinced that this is true. Even logically as vol increases, st dev increases for both assets, the denominator gets bigger, correlation decreases. So you would have to say that covariance (the numerator) gets bigger, but I dont see why. Perhaps I am missing something. Copy paste the figures below and go play. I took it to some extremes, but you get the jist. Date GM return MSFT return 31-Dec-90 -78.00% -90.00% 31-Dec-91 -90.00% -81.76% 31-Dec-92 116.54% 15.11% 31-Dec-93 12.64% 45.56% 30-Dec-94 -21.78% -51.63% 29-Dec-95 28.13% 80.00% 31-Dec-96 88.46% 88.32% 31-Dec-97 89.00% 50.00% 31-Dec-98 140.00% 14.60% 31-Dec-99 121.34% 50.00% Average return, E(rGM) and E(rMSFT) 40.63% 12.02% Variance of return, s2GM and s2MSFT 62.75% 37.72% Standard deviation of return, sGM and sMSFT 79.22% 61.42% Covariance of returns, Cov(rGM,rMSFT) 0.3558

Muddahudda Wrote: > I’m looking for a better proof than above, or else > I remain unconvinced that this is true. Even > logically as vol increases, st dev increases for > both assets, the denominator gets bigger, > correlation decreases. So you would have to say > that covariance (the numerator) gets bigger, but I > dont see why. Assume cov(y,epsilon1)=0, cov(epsilon1,epsilon2)=0, var(epsilon)=v1, var(epsilon2)=v2 and v1, v2 are constant. Think of epsilons as idiosyncratic risks of the firms. x1=a1+b1*y+epsilon1 x2=a2+b2*y+epsilon2 assume for simplicity: a1=a2=0, b1=b2=1, current Var(y)=(10%)^2=0.01, v1=v2=(5%)^2=0.0025 since Cor(x1,x2)=Cov(x1,x2)/(sqrt(var(x1)*var(x2))=b1*b2*Var(y)/sqrt[(b1^2*Var(y)+v1)*(b2^2*Var(y)+v2)]. current correlation = 0.01/sqrt[(0.01+0.0025)*(0.01+0.0025)]=0.01/(0.01+0.025)=0.8 now if Var(y) goes down to (8%)^2=0.0064, then correlation = 0.0064/(0.0064+0.0025)=0.72 now if Var(y) goes up to (20%)^2=0.04, then correlation = 0.04/(0.04+0.0025)=0.94 It is based on some assumptions that can be questioned, but it describes positive relationship between market volatility and correlations pretty well.

Thanks maratikus for responding but you lost me. Help me out, i’m not a quant (and pretty stupid). From first principles, as you alluded: Correlation = Covariance (X, Y) / (sigma X . sigma Y). Sigma = standard deviation. This is the measure of volatility for me (and I assume Jack). Clearly if you were to increase sigma, the correlation would fall. What’s the deal with all your epsilons? I never saw any of that in the CFA curriculum, only the formula I have described. Where do you get what you posted from and what does it mean. I suspect I am not the only one that doesn’t follow.

Muddahudda Wrote: ------------------------------------------------------- > Thanks maratikus for responding but you lost me. > Help me out, i’m not a quant (and pretty stupid). > > From first principles, as you alluded: > > Correlation = Covariance (X, Y) / (sigma X . sigma > Y). > > Sigma = standard deviation. This is the measure of > volatility for me (and I assume Jack). Clearly if > you were to increase sigma, the correlation would > fall. But Covariance would also likely increase. > > What’s the deal with all your epsilons? I never > saw any of that in the CFA curriculum, only the > formula I have described. Where do you get what > you posted from and what does it mean. I suspect I > am not the only one that doesn’t follow. You asked for a proof, which the CFA curriculum doesn’t provide! The epsilons come from the error term in SML: R(stock) = alpha + beta * R(market-riskfree) + error so maratikus’s v1 is the variance of the error term - so the variance of “how far away the actual return is from the return predicted by the SML” Now his big assumption is that this stays constant when general market vol increases. If this is true, then correlation tends to 1 as market vol increase (Var(y) in his notation). However, if this assumption doesn’t hold true (and personally I see no reason why it should), then the effect on overall correlation is uncertain. Going back a bit earlier, the CAPM is a load of arse, so there’s no particular reason to believe any of this Muddahudda Wrote: > Correlation = Covariance (X, Y) / (sigma X . sigma > Y). > > Sigma = standard deviation. This is the measure of > volatility for me (and I assume Jack). I agree with everything up to this point >Clearly if you were to increase sigma, the correlation would fall. Only if covariance doesn’t increase as much as the product of sigmas. Try this: Use your datasets, calculate standard deviations, covariance and correlation. Then try this in Excel. Multiply the first column by 2, your standard deviation will double, your covariance will double and your correlation will stay the same. Date GM return MSFT return 31-Dec-90 -78.00% -90.00% 31-Dec-91 -90.00% -81.76% 31-Dec-92 116.54% 15.11% 31-Dec-93 12.64% 45.56% 30-Dec-94 -21.78% -51.63% 29-Dec-95 28.13% 80.00% 31-Dec-96 88.46% 88.32% 31-Dec-97 89.00% 50.00% 31-Dec-98 140.00% 14.60% 31-Dec-99 121.34% 50.00% > What’s the deal with all your epsilons? I never > saw any of that in the CFA curriculum, only the > formula I have described. Where do you get what > you posted from and what does it mean. I’m using a factor model (regression). I assume that the price of the first stock has beta of b1 with the market, alpha of a1 and residual of epsilon1. Then I calculate covariance using two rules: cov(e+f,g)=cov(e,g)+cov(f,g) cov(ce,g)=c*cov(e,g)

chrismaths > Now his big assumption is that this stays constant > when general market vol increases. > However, if this assumption doesn’t hold true (and > personally I see no reason why it should), then > the effect on overall correlation is uncertain. I absolutely agree with you.

This is good. We’re making progress. If you don’t have children, this will prepare you. It is like painting by colours. Thanks for your patience. It might help make professors out of you. Sorry for hijacking your thread Jack (o’-'o) What I do understand: 1) Covariance increases too. Variance up, standard deviation up, covariance up. Yep, that is logical. Though I struggle to see why it would increase any more that the product of the standard deviations. What I do not understand: 2) Why do you bring the SML into it? I know what it is and dare say I could have quoted that automatically, although I probably would have ignored the error term. The only bit that has some meaning is that the SML has the beta term which is the covariance of the asset and the market. Lastly, I wish it was a proof - I would love to quote in front of clients, higher vol = higher correlation without the quant guy bashing my brain into small pieces. Shame about that.

Muddahudda Wrote: > Lastly, I wish it was a proof - I would love to > quote in front of clients, higher vol = higher > correlation without the quant guy bashing my brain > into small pieces. Actually I should’ve given a reference. I heard the described explanation at my Financial Econometrics class at the University of Chicago (and I questioned it just as chrismaths did). http://faculty.chicagobooth.edu/jeffrey.russell/teaching/finecon/handouts/ Notes4, slide 11.

Thanks for that link. It has some parts that I can follow and some that I can’t - don’t worry i’m not going to start peppering you with questions, but it is very interesting stuff. Maratikus are you studying Financial Econometrics or is that just one class from a broader degree? What about you chrismaths? I’m trying to wonder what it is you are studying that gives you a deeper insight than me. I might chose to do another degree down the road.

Well I did a maths degree many moons ago.

That’ll be it then. Damn my anthropology and media studies degree.

Muddahudda Wrote: > Maratikus are you studying Financial Econometrics > or is that just one class from a broader degree? I’m getting an MBA. I had to take a statistics class and I chose Financial Econometrics. I’m trying to stay away from quant classes because of my quant background.

maratikus, Have you ever done anything with chaos theory? I am thinking of viewing the following course: http://www.teach12.com/ttcx/CourseDescLong2.aspx?cid=1333 Any thoughts? Edit: I have a non quant background, so this would be just a review of the subject to see if I want to delve deeper.

tbh, the implications of chaos theory are more interesting than the maths. I’ve heard good stuff about: http://books.google.co.uk/books?id=9w15j-Ka0vgC&printsec=frontcover&dq=mandelbrot

Thanks for the link chrismaths. Maybe I will check that book out. I have read this one on chaos: http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501/ref=sr_1_3?ie=UTF8&s=books&qid=1245859935&sr=1-3 and this on complexity: http://www.amazon.com/Complexity-Emerging-Science-Order-Chaos/dp/0671872346/ref=pd_sim_b_2 but I am not properly trained to even evaluate the merits of either. I just found them interesting.

mwvt, I’m not an expert in the subject but I know a little about it because of my math background and trying to use it in trading. The theory itself is beautiful but hard to apply in real life in my opinion. Mathematically chaos theory is nothing else but a study of unstable points of deterministic dynamic systems. I would suggest reading a little about phase space as it is an extremely useful concept: http://en.wikipedia.org/wiki/Phase_space You can also access free lectures on Chaos here: http://ocw.mit.edu/OcwWeb/Mathematics/18-385JFall-2004/CourseHome/index.htm If you’d like, we can discuss the subject more in depth. Pretty soon I will not know the answers but, hopefully, I will still be able to point in the right direction.