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.
The first exam... Eco 136 First exam breakdow 184+ A 24 180-184 A- 5 175-179 B+ 4 165-174 B 4 160-164 B- 1 155-159 C+ 1 145-154 C 1 140-144 C- 1 Below 3 What ca I say... you are a buch of geiuses.
Lessos from HW2... 252 60 30 Date Volume Adj Close l DCGR Norm DCGR Duratio 252 60 30 Date Volume Adj Close l DCGR Norm DCGR Duratio 1 SPY 2/1/2013 131,173,000 148.25 4.99890 [empty] [empty] Oe Year: 1 CSCO 2/1/2013 25,754,100 20.22 3.00667 Oe Year: 2 2/4/2013 159,073,600 146.58 4.98757-0.01133-1.69092 Mea DCGR: 0.00073 2 2/4/2013 32,053,300 20.21 3.00618-0.00049-0.05172 Mea DCGR: 0.00032 3 2/5/2013 113,912,400 148.06 4.99762 0.01005 1.30570 Stadard Deviatio: 0.00713 3 2/5/2013 24,952,600 20.46 3.01847 0.01229 0.76036 Stadard Deviatio: 0.01575 4 2/6/2013 138,762,800 148.17 4.99836 0.00074 0.00140 Mi DCGR: - 0.02511 4 2/6/2013 27,848,400 20.56 3.02335 0.00488 0.28929 Mi DCGR: - 0.11605 5 2/7/2013 162,490,000 147.97 4.99701-0.00135-0.29207 Max DCGR: 0.02132 5 2/7/2013 34,442,900 20.48 3.01945-0.00390-0.26787 Max DCGR: 0.11891 6 2/8/2013 103,133,700 148.80 5.00260 0.00559 0.68147 Mi Norm DCGR: - 3.62239 6 2/8/2013 23,056,900 20.54 3.02237 0.00293 0.16545 Mi Norm DCGR: - 7.38955 7 2/11/2013 73,775,000 148.77 5.00240-0.00020-0.13098 Max Norm DCGR: 2.88677 7 2/11/2013 33,552,000 20.64 3.02723 0.00486 0.28809 Max Norm DCGR: 7.53017 8 2/12/2013 65,392,700 149.01 5.00401 0.00161 0.12327 8 2/12/2013 46,463,500 20.35 3.01308-0.01415-0.91882 9 2/13/2013 82,322,600 149.14 5.00489 0.00087 0.01954 60 day: 9 2/13/2013 63,608,700 20.52 3.02140 0.00832 0.50795 60 day: 10 2/14/2013 80,834,300 149.28 5.00582 0.00094 0.02883 Mea DCGR: 0.00022 10 2/14/2013 67,172,100 20.37 3.01406-0.00734-0.48619 Mea DCGR: - 0.00038 11 2/15/2013 215,226,500 149.10 5.00462-0.00121-0.27186 Stadard Deviatio: 0.00679 11 2/15/2013 44,439,700 20.37 3.01406 0.00000-0.02031 Stadard Deviatio: 0.01828 12 2/19/2013 95,105,400 150.22 5.01210 0.00748 0.94645 Mi DCGR: - 0.02157 12 2/19/2013 45,590,500 20.83 3.03639 0.02233 1.39769 Mi DCGR: - 0.11605 13 2/20/2013 160,574,800 148.34 4.99951-0.01259-1.86830 Max DCGR: 0.01691 13 2/20/2013 47,494,700 20.49 3.01994-0.01646-1.06533 Max DCGR: 0.02163 14 2/21/2013 183,257,000 147.44 4.99342-0.00609-0.95587 Mi Norm DCGR: - 3.20994 14 2/21/2013 33,007,100 20.15 3.00320-0.01673-1.08282 Mi Norm DCGR: - 6.32713 15 2/22/2013 106,356,600 148.88 5.00314 0.00972 1.25987 Max Norm DCGR: 2.45967 15 2/22/2013 20,483,300 20.28 3.00964 0.00643 0.38805 Max Norm DCGR: 1.20364 16 2/25/2013 245,824,800 146.05 4.98395-0.01919-2.79324 16 2/25/2013 37,391,400 20.05 2.99823-0.01141-0.74458 17 2/26/2013 186,596,200 147.05 4.99077 0.00682 0.85392 30 Day: 17 2/26/2013 34,254,700 20.00 2.99573-0.00250-0.17886 30 Day: 18 2/27/2013 150,781,900 148.90 5.00327 0.01250 1.65002 Mea DCGR: 0.00009 18 2/27/2013 25,190,900 20.27 3.00914 0.01341 0.83119 Mea DCGR: 0.00180 19 2/28/2013 126,866,000 148.61 5.00133-0.00195-0.37602 Stadard Deviatio: 0.00773 19 2/28/2013 30,337,500 20.24 3.00766-0.00148-0.11436 Stadard Deviatio: 0.00912 20 3/1/2013 170,634,800 149.10 5.00462 0.00329 0.35878 Mi DCGR: - 0.02157 20 3/1/2013 24,174,800 20.22 3.00667-0.00099-0.08308 Mi DCGR: - 0.01609 21 3/4/2013 99,010,200 149.89 5.00990 0.00528 0.63813 Max DCGR: 0.01691 21 3/4/2013 22,634,600 20.13 3.00221-0.00446-0.30357 Max DCGR: 0.02077 22 3/5/2013 121,431,900 151.24 5.01887 0.00897 1.15430 Mi Norm DCGR: - 2.80085 22 3/5/2013 32,829,500 20.59 3.02481 0.02259 1.41441 Mi Norm DCGR: - 1.96189 23 3/6/2013 94,469,900 151.44 5.02019 0.00132 0.08256 Max Norm DCGR: 2.17469 23 3/6/2013 45,966,600 21.08 3.04832 0.02352 1.47314 Max Norm DCGR: 2.08090 24 3/7/2013 86,101,400 151.72 5.02204 0.00185 0.15625 24 3/7/2013 37,379,500 21.16 3.05211 0.00379 0.22022 25 3/8/2013 123,477,800 152.36 5.02625 0.00421 0.48742 25 3/8/2013 23,839,100 21.19 3.05353 0.00142 0.06966 26 3/11/2013 83,746,800 152.94 5.03005 0.00380 0.42996 Oe year data 26 3/11/2013 29,120,000 21.23 3.05542 0.00189 0.09945 27 3/12/2013 105,755,800 152.60 5.02782-0.00223-0.41472 Correlatio: 0.39446 27 3/12/2013 25,987,300 21.06 3.04738-0.00804-0.53082 28 3/13/2013 92,550,900 152.81 5.02920 0.00138 0.09008 Ratio of SD: 2.20780 28 3/13/2013 30,188,200 20.94 3.04166-0.00571-0.38316 29 3/14/2013 126,329,900 153.63 5.03455 0.00535 0.64757 Beta: 0.87090 29 3/14/2013 40,479,600 20.95 3.04214 0.00048 0.01001 30 3/15/2013 138,601,100 153.42 5.03318-0.00137-0.29448 30 3/15/2013 59,804,200 21.28 3.05777 0.01563 0.97212 31 3/18/2013 126,704,300 152.58 5.02769-0.00549-0.87240 30 day data 31 3/18/2013 30,222,300 21.03 3.04595-0.01182-0.77072 32 3/19/2013 167,567,300 152.22 5.02533-0.00236-0.43388 Correlatio: 0.65438 32 3/19/2013 27,264,100 20.89 3.03927-0.00668-0.44444 33 3/20/2013 113,759,300 153.29 5.03233 0.00700 0.87930 Ratio of SD: 1.17878 33 3/20/2013 24,571,600 21.03 3.04595 0.00668 0.40383 34 3/21/2013 128,605,000 151.98 5.02375-0.00858-1.30594 Beta: 0.77137 34 3/21/2013 64,437,700 20.23 3.00717-0.03878-2.48301 35 3/22/2013 111,163,600 153.20 5.03174 0.00800 1.01818 35 3/22/2013 39,902,900 20.14 3.00271-0.00446-0.30343... your spread should look like this above, with the results show o the ext page
SPY Oe Year: Mea DCGR: 0.00073 Stadard Deviatio: 0.00713 Mi DCGR: - 0.02511 Max DCGR: 0.02132 Mi Norm DCGR: - 3.62239 Max Norm DCGR: 2.88677 60 day: Mea DCGR: 0.00022 Stadard Deviatio: 0.00679 Mi DCGR: - 0.02157 Max DCGR: 0.01691 Mi Norm DCGR: - 3.20994 Max Norm DCGR: 2.45967 30 Day: Mea DCGR: 0.00009 Stadard Deviatio: 0.00773 Mi DCGR: - 0.02157 Max DCGR: 0.01691 Mi Norm DCGR: - 2.80085 Max Norm DCGR: 2.17469 Oe year data Correlatio: 0.39446 Ratio of SD: 2.20780 Beta: 0.87090 30 day data Correlatio: 0.65438 Ratio of SD: 1.17878 Beta: 0.77137 Oe Year: Mea DCGR: 0.00032 Stadard Deviatio: 0.01575 Mi DCGR: - 0.11605 Max DCGR: 0.11891 Mi Norm DCGR: - 7.38955 Max Norm DCGR: 7.53017 60 day: Mea DCGR: - 0.00038 Stadard Deviatio: 0.01828 Mi DCGR: - 0.11605 Max DCGR: 0.02163 Mi Norm DCGR: - 6.32713 Max Norm DCGR: 1.20364 30 Day: Mea DCGR: 0.00180 Stadard Deviatio: 0.00912 Mi DCGR: - 0.01609 Max DCGR: 0.02077 Mi Norm DCGR: - 1.96189 Max Norm DCGR: 2.08090 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: 0.010789 So what are these 7- sigma??
From the CSCO mappig... CSCO axis altered from default. Millios 300 250 200 What does this tell you about stragle policy? 27.00 25.00 23.00 Q: How would you write a scriptig screeer to fid stragle cadidates? 150 21.00 Volume CSCO 100 50 0 19.00 17.00 15.00 Origial PIT debate (Asaf Berstei ad Jaso Christiaso), does the adjusted SD mea aythig?
The Beta iterpretatios... Oe year data Correlatio: 0.39446 Ratio of SD: 2.20780 Beta: 0.87090 30 day data Correlatio: 0.65438 Ratio of SD: 1.17878 Beta: 0.77137 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!
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.
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
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 0.707 V( x)
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 = 0.0009 ad V 2 = 0.0025 ad each alpha equals 1/2. Therefore V 1,2 = 0.25 X (0.0009 + 0.0025) = 0.00085 σ 1,2 = 0.00085 1/2 = 0.0291. 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).
The risk-yield efficiecy frotier Portfolio yield 0.045 0.040 Efficiet trade-off regio 0.035 0.030 0.025 0.020 0.015 0.010 0.005 Alpha Vol Var X1 0.02 0.03 0.0009 X2 0.04 0.05 0.0025 X1a X2a PVar PVol Palpha 0 1 0.0025 0.0500 0.0400 0.1 0.9 0.0020 0.0451 0.0380 0.2 0.8 0.0016 0.0404 0.0360 0.3 0.7 0.0013 0.0361 0.0340 0.4 0.6 0.0010 0.0323 0.0320 0.5 0.5 0.0009 0.0292 0.0300 0.6 0.4 0.0007 0.0269 0.0280 0.7 0.3 0.0007 0.0258 0.0260 0.8 0.2 0.0007 0.0260 0.0240 0.9 0.1 0.0008 0.0275 0.0220 1 0 0.0009 0.0300 0.0200 0.000 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 Portfolio risk
-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 http://mathworld.wolfram.com/variace.html
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.
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 0 0 0 C 0 0 0 0 12 13 14 23 24 34 S S S S 1 2 3 4 2 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.
Read the quotatios from Whe Geius Failed... ad lets do this i class together.