Correlation possibly the most important and least understood topic in finance

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1 Correlatio possibly the most importat ad least uderstood topic i fiace 2014 Gary R. Evas. May be used oly for o-profit educatioal purposes oly without permissio of the author.

2 The first exam... Eco 136 First exam breakdow 184+ A A B B B C C C- 1 Below 3 What ca I say... you are a buch of geiuses.

3 Lessos from HW Date Volume Adj Close l DCGR Norm DCGR Duratio Date Volume Adj Close l DCGR Norm DCGR Duratio 1 SPY 2/1/ ,173, [empty] [empty] Oe Year: 1 CSCO 2/1/ ,754, Oe Year: 2 2/4/ ,073, Mea DCGR: /4/ ,053, Mea DCGR: /5/ ,912, Stadard Deviatio: /5/ ,952, Stadard Deviatio: /6/ ,762, Mi DCGR: /6/ ,848, Mi DCGR: /7/ ,490, Max DCGR: /7/ ,442, Max DCGR: /8/ ,133, Mi Norm DCGR: /8/ ,056, Mi Norm DCGR: /11/ ,775, Max Norm DCGR: /11/ ,552, Max Norm DCGR: /12/ ,392, /12/ ,463, /13/ ,322, day: 9 2/13/ ,608, day: 10 2/14/ ,834, Mea DCGR: /14/ ,172, Mea DCGR: /15/ ,226, Stadard Deviatio: /15/ ,439, Stadard Deviatio: /19/ ,105, Mi DCGR: /19/ ,590, Mi DCGR: /20/ ,574, Max DCGR: /20/ ,494, Max DCGR: /21/ ,257, Mi Norm DCGR: /21/ ,007, Mi Norm DCGR: /22/ ,356, Max Norm DCGR: /22/ ,483, Max Norm DCGR: /25/ ,824, /25/ ,391, /26/ ,596, Day: 17 2/26/ ,254, Day: 18 2/27/ ,781, Mea DCGR: /27/ ,190, Mea DCGR: /28/ ,866, Stadard Deviatio: /28/ ,337, Stadard Deviatio: /1/ ,634, Mi DCGR: /1/ ,174, Mi DCGR: /4/ ,010, Max DCGR: /4/ ,634, Max DCGR: /5/ ,431, Mi Norm DCGR: /5/ ,829, Mi Norm DCGR: /6/ ,469, Max Norm DCGR: /6/ ,966, Max Norm DCGR: /7/ ,101, /7/ ,379, /8/ ,477, /8/ ,839, /11/ ,746, Oe year data 26 3/11/ ,120, /12/ ,755, Correlatio: /12/ ,987, /13/ ,550, Ratio of SD: /13/ ,188, /14/ ,329, Beta: /14/ ,479, /15/ ,601, /15/ ,804, /18/ ,704, day data 31 3/18/ ,222, /19/ ,567, Correlatio: /19/ ,264, /20/ ,759, Ratio of SD: /20/ ,571, /21/ ,605, Beta: /21/ ,437, /22/ ,163, /22/ ,902, your spread should look like this above, with the results show o the ext page

4 SPY Oe Year: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: day: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: Day: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: Oe year data Correlatio: Ratio of SD: Beta: day data Correlatio: Ratio of SD: Beta: Oe Year: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: day: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: Day: Mea DCGR: Stadard Deviatio: Mi DCGR: Max DCGR: Mi Norm DCGR: Max Norm DCGR: The results from HW2... CSCO Well, ow, this is very iterestig, is t it? Is there aythig useful here? SD with 3 5-sigma observatios removed: So what are these 7- sigma??

5 From the CSCO mappig... CSCO axis altered from default. Millios What does this tell you about stragle policy? Q: How would you write a scriptig screeer to fid stragle cadidates? Volume CSCO Origial PIT debate (Asaf Berstei ad Jaso Christiaso), does the adjusted SD mea aythig?

6 The Beta iterpretatios... Oe year data Correlatio: Ratio of SD: Beta: day data Correlatio: Ratio of SD: Beta: B = ρ cs σ σ c s 1. We ca see why we eed to separate the Beta ito the two compoets ad leave them that way. 2. The ratio of SD is really idicative of the relative volatilities, BUT 3. The traditioal Beta matters because if you do add a ucorrelated asset to a portfolio, the variace of the portfolio is reduced, which meas the portfolio has less risk!! That is a big issue!

7 Some elemetary startig poits... We are talkig about a series of radom variables ad weighted radom variables that fit a Gaussia distributio as we have defied it. Notatioally... X, etc ~ N(µ x,v x ) ad V x 2 = σ where V x i = 1 ( X µ ) i x 2 ad it turs out that if Y =e X the Y has a logormal distributio.

8 2-asset Portfolio Variace Sums Variace is purely additive if two variables are strictly idepedet: V( x + y) = V( x) + V( y) + 2 COV ( x, y) rememberig that Covariace is equal to the Correlatio Coefficiet (0 if o perfectly idepedet, 1 if perfectly correlated, -1 if perfectly polar) times the product of the stadard deviatios: (, ) = (, ) ( ) ( ) COV x y CORREL x y SD x SD y

9 2-asset weighted portfolio variace V ( ) 2 ( ) 2 ax + by = a V X + b V ( Y) + 2abCOV ( X, Y ) ad i a weights sum to oe, the coefficiets above are restricted to the coditio α =1 i so i the special case of portfolio P cosistig of two completely idepedet stocks with exactly the same variace ad each equally represeted, the the variace of the portfolio will be... ( P) 0.5V ( X) V = so if you oly ivested i oe of the two stocks your volatility would be V ( X ) but if you diversified your portfolio 50/50 your volatility would be V( x)

10 Simple example of diversificatio usig our formula: Suppose you have two ucorrelated stocks, X( µ, σ), X 1 (0.02,0.03) ad X 2 (0.04,0.05). If you are risk-adverse, you may wat to put all of your moey i stock X 1 ad accept the lower 2% yield. But what if you split your portfolio 50/50, givig you a 3% yield? What would your risk be?? V 1 = ad V 2 = ad each alpha equals 1/2. Therefore V 1,2 = 0.25 X ( ) = σ 1,2 = /2 = Therefore, by diversifyig your portfolio you have raised your yield to 0.03, 50% more tha the coservative stock, while lowerig your risk to a level below the most coservative of the two stocks (which was at 0.03).

11 The risk-yield efficiecy frotier Portfolio yield Efficiet trade-off regio Alpha Vol Var X X X1a X2a PVar PVol Palpha Portfolio risk

12 -asset Portfolio Volatility If we have '' assets i the portfolio, the we calculate the variace usig this additive formula: 1 VAR xi = VAR x + COV x x i= 1 i= 1 i= 1 j= ( i+ 1) ( ) ( ) i 2 i, j which is easy to program if* you have the data. What fially matters, of course, is the square root of this term, the stadard deviatio, which is our volatility measure. *this requires the calculatio of all '' stadard deviatios ad all correlatios (15 1 for 6 stocks). i= 1 i paired For referece ad discussio, see

13 Weight-adjusted -asset Portfolio Volatility If you assig weights to your portfolio, represeted here as alphas, which of course you would, the the variace formula is: 1 VAR αix 2 i = α VAR x + α α COV x x i= 1 i= 1 i= 1 j= ( i 1) ( ) 2 (, ) i i i j i j The volatility of this portfolio, the stadard deviatio, is the square root of this expressio. Clearly, the greater the idepedece of your portfolio compoets, the smaller the risk. This shows the beefits of diversificatio ito o-correlated stocks.

14 Help from David Coates '08 Codig the Covariace (prior 2 equatios) For the covariace part of the equatio oly, for I = 1 to (-1) do for J = (i+1) to do COV(I,J) = CORREL(I,J)*SD(I)*SD(J); SUMCOV = SUMCOV + COV(I,J); ed; ed; [ S1 S2 S3 S4] 0 C C C 0 0 C C C S S S S SUMCOV = 2*SUMCOV; ad the weighted portfolio calculatio would be the same except WCOV(I,J) = a(i)*a(j)*correl(i,j)*sd(i)*sd(j); Memo slide for sticklers for accuracy (a desirable trait), those of you who wat to work i fiace, ad you coders who ow a laptop ad wat to retire before age 35 tradig off of ay beach with a wireless setup.

15 Read the quotatios from Whe Geius Failed... ad lets do this i class together.

... possibly the most important and least understood topic in finance

Correlation...... possibly the most important and least understood topic in finance 2017 Gary R. Evans. This lecture is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International