Index Model

1. common factors: affect all stocksSingle-index (Single Factor) model: only one common factorMultifactor index model - many common factors

2. firm-specific (idiosyncratic) factor: affects only one firmthese factors are assumed to be uncorrelated

Implicationscommon factors affect all stock prices, and hence the stock prices are correlated.that is, covariances among stock prices are determined by the effects of common factors

Benefits of Index Modelidentify two sources of risk

market risk caused by common factors- nondiversifiable firm-specific risk - diversifiable

reduces the number of parameters substantially - easier to forecast means and covariances

Specification of One Factor Index ModelThe factor=broad based index such as S&P 500 index

return of risk-free asset

Excess return (risk premium) of the factor (such as S&P 500 index)

firm specific factor

Model

Assumptions

market index and firm-specific events are not correlated

firm-specific random events are not correlated each otherInterpretations

Idea: There are two types of factors that determine the returns (or prices) of stocks

ri return of risky asset i

rf

Ri Excess return (risk premium) of risky asset i

Rm

fi

fi=ai+ei firm specific factor as a sum of its mean ai and idiosyncratic shock ei

mi E(Ri)

mm E(Rm)

ai E(fi)

si risk(standard deviation) of Ri

sm risk(standard deviation) of Rm

sei risk(standard deviation) of fi standard deviation of idiosyncratic shock ei

sij covariance between asset i and asset j

rij correlation coefficient between asset i and asset j

ai and bi are constant parameters

ei and Rm are not correlated: cov(ei, Rm )=0

ei and ej are not correlated each other: cov(ei, ej )=0

imiii eRR ba

represents the responsiveness of excess return of asset i to changes in the excess return of the market index

Macroeconomic factors affect the market's excess returns

represents the expected excess return of stock i beyond any return induced by market’s excess returnexpected excess return of stock i if the market factor is neutral, that is, if market’s excess return is zero.

It is currently underpriced

It is currently overpriced

component of excess return due to movements in the overall market

The risk attributable to the uncertainty common to the entire marketdepends on both the volatility of the market and sensitivity of the stock to the market movement

This is called the "unsystematic risk" or "firm specific risk", or "diversifiable risk", or ,"unique risk", or "residual risk" risk attributable to firm-specific risk factors“diverifiable risk” means that this risk can be reduced through diversification of a portfolio,

while the nondiversifiable systematic risk cannot be reduced by diversification.Covariances of Excess Returns

Estimation of the One Factor Index Model See the next sheet

bi This is called the beta of asset i

if market’s excess return changes by 1%, the excess return of asset i changes by bi%

bi=1 stock i has the same sensitivity to the macrofactors as the market

bi>1 stock i has a greater sensitivity to the macrofactors than the market (cyclical stocks)

bi<1 stock i has a less sensitivity to the macrofactors than the market (defensive stocks)

bi<0 excess return of stock i moves in the opposite direction of the market (counter cyclical stocks)

ai This is called the alpha of asset i

a positive alpha implies expected excess return of stock i is greater than the return determined by the common factor

a negative alpha implies expected excess return of stock i is lower than the return determined by the common factor

Expected Excess Return of stock i

biE(Rm)Risk of excess return of stock i

bi2sm

2 This is called the "systematic risk" or "market risk" or "non-diversifiable risk" of stock i

sei2

)()( miii RERE ba miii mbam

2222

iemii ssbs

2222222

2

)()(

),cov(),(

ii emi

mji

memi

mji

mi

ij

mi

mimiim RSDRSD

RRRRcorr

ssb

sbb

sssb

sbbsss

r

2),cov( mjijiij RR sbbs 2),cov( mimiim RR sbs

market index and firm-specific events are not correlated

firm-specific random events are not correlated each other

represents the responsiveness of excess return of asset i to changes in the excess return of the market index

represents the expected excess return of stock i beyond any return induced by market’s excess returnexpected excess return of stock i if the market factor is neutral, that is, if market’s excess return is zero.

depends on both the volatility of the market and sensitivity of the stock to the market movement

This is called the "unsystematic risk" or "firm specific risk", or "diversifiable risk", or ,"unique risk", or "residual risk"

“diverifiable risk” means that this risk can be reduced through diversification of a portfolio, while the nondiversifiable systematic risk cannot be reduced by diversification.

has a greater sensitivity to the macrofactors than the market (cyclical stocks)

has a less sensitivity to the macrofactors than the market (defensive stocks)

moves in the opposite direction of the market (counter cyclical stocks)

is greater than the return determined by the common factor

is lower than the return determined by the common factor

This is called the "systematic risk" or "market risk" or "non-diversifiable risk" of stock i

2222222

2

)()(

),cov(),(

ii emi

mji

memi

mji

mi

ij

mi

mimiim RSDRSD

RRRRcorr

ssb

sbb

sssb

sbbsss

r

Estimation of One Factor Index ModelModel

Data Requirement: excess returns of stocks and market index (S&P 500 index)

Estimation method: Least Squares Estimation methodIntuitive IdeaFormulaExcel functions

Intercept( ) for alphaSlope( ) for betaSumsq( ) for the estimation of variance of residuals

Use Regression functionThis is an add-on. You need to download it unless your computer has already installed it in "Data Analysis" under "Data"Estimate both Alpha and BetaClick on Data Analysis - select regression - OK. This pops up a small windowInput: specify Y range and X range. If the names of variables are included, check 'labels'output: choose 'new work sheet'residuals: choose 'line fit plots'Click OKThis opens up the sheet that contains regression results (see the next sheet regression results)

imiii eRR ba

This is an add-on. You need to download it unless your computer has already installed it in "Data Analysis" under "Data"

Click on Data Analysis - select regression - OK. This pops up a small windowInput: specify Y range and X range. If the names of variables are included, check 'labels'

This opens up the sheet that contains regression results (see the next sheet regression results)

Excess Returns (%)=returns-Tbill3

date IBM McDonald S&P5001/2/2004 IBM McDonald S&P5002/2/2004 -32.0515 118.0329 13.720843/1/2004 -58.93952 10.71587 -20.570724/1/2004 -48.87951 -57.55933 -21.0895/3/2004 7.171126 -37.36615 13.480146/1/2004 -7.215241 -19.71323 20.316897/1/2004 -16.10987 68.1564 -42.478638/2/2004 -31.72675 -22.61279 1.2647999/1/2004 13.21119 43.69884 9.586688

10/1/2004 54.37658 45.85905 15.057111/1/2004 60.34171 86.83885 44.2439312/1/2004 53.18109 49.48224 36.75975

1/3/2005 -65.20048 10.17501 -32.678542/1/2005 -11.11037 22.68808 20.144063/1/2005 -18.21811 -73.13627 -25.681184/1/2005 -199.8399 -73.10604 -26.910315/2/2005 -12.58143 63.9728 33.102436/1/2005 -24.43105 -126.558 -3.1412137/1/2005 146.4096 144.5578 39.941848/1/2005 -41.47513 45.5295 -16.906439/1/2005 -9.371537 35.07505 4.918728

10/3/2005 21.24016 -71.39995 -24.9988911/1/2005 102.091 107.8727 38.3433512/1/2005 -94.33994 -8.354286 -5.032875

1/3/2006 -17.38294 41.46575 26.320212/1/2006 -17.04564 -7.883237 -3.8862843/1/2006 28.80917 -23.12472 8.8050094/3/2006 -6.584838 2.435544 9.9867925/1/2006 -36.03299 -53.24459 -41.820286/1/2006 -51.06162 10.69975 -4.686077/3/2006 4.250826 59.36784 1.1529768/1/2006 55.26827 12.11577 20.569129/1/2006 9.569267 102.9417 24.66953

10/2/2006 147.2485 80.82187 32.8896311/1/2006 -6.406276 25.69983 14.8199312/1/2006 63.41523 62.63682 10.2889

1/3/2007 19.74508 -4.314443 11.89092/1/2007 -76.75508 -22.99009 -31.245373/1/2007 11.99607 32.5389 7.0359464/2/2007 96.33138 81.24187 47.078825/1/2007 51.74214 51.78731 34.329076/1/2007 -19.82215 0.349883 -25.989577/2/2007 56.80988 -73.39143 -43.198298/1/2007 65.68292 30.6228 11.236319/4/2007 7.554921 123.0878 39.0628

10/1/2007 -21.01856 112.5523 13.886811/1/2007 -112.4132 1.914118 -56.1221112/3/2007 30.27359 6.094633 -13.35419

1/2/2008 -13.70609 -111.3087 -76.146172/1/2008 78.38523 17.99623 -43.833393/3/2008 12.24538 35.4074 -8.41154/1/2008 56.62039 80.87724 55.766025/1/2008 90.30734 -7.000513 11.078986/2/2008 -102.9481 -57.20389 -105.01497/1/2008 94.05365 74.55247 -13.461258/1/2008 -55.88094 50.22703 12.90869/2/2008 -48.12739 -7.04133 -110.0797

10/1/2008 -246.8094 -74.01349 -203.979411/3/2008 -141.0836 27.79621 -90.0088412/1/2008 37.63529 70.12458 9.355879

1/2/2009 106.6299 -80.56438 -102.91882/2/2009 11.25904 -109.6317 -132.21753/2/2009 63.2529 53.04679 102.27544/1/2009 78.10379 -28.32068 112.55015/1/2009 41.912 128.1595 63.517716/1/2009 -21.12156 -21.06793 0.0550027/1/2009 154.9625 -50.81694 88.790118/3/2009 6.742948 36.48835 40.102219/1/2009 15.69962 17.54485 42.74806

10/1/2009 9.944358 32.36243 -23.7843811/2/2009 62.74256 106.0646 68.786812/1/2009 43.18044 -15.41 21.27472

1/4/2010 -78.1271 -0.276099 -44.429112/1/2010 52.27345 37.49807 34.106433/1/2010 10.08925 53.91916 70.405644/1/2010 6.907021 69.42556 17.551195/3/2010 -29.0542 -53.99036 -98.53116/1/2010 -17.15422 -17.9801 -64.778857/1/2010 47.6672 70.14384 82.37348/2/2010 -43.68127 66.63353 -57.0999/1/2010 107.1028 23.6445 104.9113

10/1/2010 84.56812 52.36669 44.0971311/1/2010 -12.81234 17.47219 -2.88833912/1/2010 44.86445 -23.67268 78.22009

1/3/2011 124.4107 -48.49728 27.024712/1/2011 3.753244 42.61945 38.21793/1/2011 8.725237 6.408707 -1.3567624/1/2011 55.19456 35.0337 34.134435/2/2011 -6.358109 58.71862 -16.241116/1/2011 18.57152 40.82785 -21.949017/1/2011 72.03273 30.78401 -25.809328/1/2011 -60.66833 62.91082 -68.169329/1/2011 20.65598 -34.36034 -86.12442

10/3/2011 66.96633 68.66785 129.247611/1/2011 26.7673 43.89915 -6.08037412/1/2011 -26.26864 60.40828 10.2293

1/3/2012 56.8245 -15.32709 52.269622/1/2012 30.42392 11.05075 48.617343/1/2012 72.65297 -14.29019 37.518854/2/2012 -9.117212 -8.040625 -9.0769975/1/2012 -77.57847 -91.34377 -75.270816/1/2012 16.57932 -11.00163 47.375917/2/2012 2.347315 11.19412 15.017178/1/2012 -1.916094 11.08881 23.616039/4/2012 77.47323 30.23642 28.97331

10/1/2012 -74.79809 -64.83702 -23.8472811/1/2012 -22.51697 14.05286 3.32604412/3/2012 9.276492 16.02601 8.411973

1/2/2013 72.09135 96.20308 60.443682/1/2013 -8.308918 17.31187 13.172723/1/2013 74.4198 47.33028 43.095364/1/2013 -60.61385 29.41411 21.642925/1/2013 38.24705 -56.50076 24.875346/3/2013 -97.61993 30.13789 -18.049197/1/2013 24.63396 -11.28122 59.314538/1/2013 -73.00336 -36.17938 -37.597629/3/2013 19.13839 23.50692 35.67937

use average(8.187651 16.17155 3.629636use stdev.s( 64.68591 54.49275 51.30625

154.9625 144.5578 129.2476-246.8094 -126.558 -203.9794

Sample means and sample covariances of annualized monthly excess returnsVariances and covariances

IBM McDonald S&P500 IBM McDonald S&P500

8.187651 16.17155 3.6296 4148.19624 1082.377237 1975.205289 IBM

64.40649 54.25736 51.0846 1082.377237 2943.860622 1311.65125 McDonald1975.205289 1311.651250 2609.638307 S&P500

number of sample= 116Estimation of alpha and beta using excel functions intercept and slope

IBM McDonaldalpha 5.440421 14.34723 use 'intercept' functionbeta 0.756889 0.502618 use 'slope' function

compute predicted means and error variance (variance of firm-specific shock) using the estimates of alpha and betameans 8.187651 16.17155 Note that the predicted mean is the same as the mean of the dataerror var 2699.733 2324.682 NOTE: to compute the error variances, you need to press shift+ctrl+enter together

compute variance and covariance using the estimates of alpha and beta, and error varianceIBM McDonald S&P500

IBM 4194.743 992.7738 1975.2053McDonald 992.7738 2983.941 1311.6513S&P500 1975.205 1311.651 2609.6383

Estimation with Excel's Regression( ) function in Data Analysis

SUMMARY OUTPUT IBM

Regression StatisticsMultiple R 0.600333R Square 0.3604Adjusted R 0.35479Standard E 51.95896Observatio 116

ANOVAdf SS MS F Significance F

Regression 1 173421.2 173421.185 64.23642 1.05806E-12Residual 114 307769.6 2699.73314Total 115 481190.8

mean mi

SD si

-250 -200 -150 -100 -50 0 50 100 150

-300

-250

-200

-150

-100

-50

0

50

100

150

200

IBM

Predicted IBM

S&P500

IBM

miii RR ba ˆˆˆ

n

iiie RR

ni1

22 )ˆ(2

1s

222)var(iemiiR ssb

jiRR mjiji ,),cov( 2sbb

2),cov( mimi RR sb

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%Intercept 5.440421 4.83643 1.12488361 0.263001 -4.1405096 15.02135097 -4.1405096 15.02135S&P500 0.756889 0.094437 8.01476247 1.058E-12 0.56980995 0.9439671 0.56980995 0.943967

RESIDUAL OUTPUT

ObservationPredicted IBMResiduals1 15.82556 -47.877062 -10.12932 -48.81023 -10.5216 -38.357914 15.64338 -8.4722545 20.81804 -28.033296 -26.71116 10.601297 6.397733 -38.124488 12.69647 0.5147129 16.83696 37.53961

10 38.92814 21.4135711 33.26346 19.9176412 -19.29359 -45.9068913 20.68723 -31.797614 -13.99737 -4.22074315 -14.92768 -184.912216 30.49527 -43.076717 3.062873 -27.4939218 35.67194 110.737719 -7.355863 -34.1192620 9.16335 -18.5348921 -13.48095 34.7211122 34.46206 67.6289223 1.631095 -95.9710424 25.36188 -42.7448225 2.498937 -19.5445826 12.10483 16.7043427 12.99931 -19.5841528 -26.21287 -9.82012129 1.893588 -52.9552130 6.313095 -2.06226931 21.00895 34.2593232 24.1125 -14.5432433 30.33421 116.914334 16.65746 -23.0637335 13.22797 50.1872636 14.44051 5.304576

37 -18.20884 -58.5462338 10.76585 1.2302239 41.07384 55.2575440 31.4237 20.3184341 -14.23079 -5.59136542 -27.25587 84.0657543 13.94506 51.7378744 35.00661 -27.4516945 15.95118 -36.9697546 -37.03776 -75.3754347 -4.66721 34.940848 -52.19374 38.4876549 -27.73657 106.121850 -0.926147 13.1715351 47.64908 8.97130852 13.82598 76.4813753 -74.04412 -28.9039554 -4.748245 98.801955 15.21079 -71.0917356 -77.87767 29.7502957 -148.9493 -97.8601558 -62.68624 -78.3973259 12.52178 25.1135160 -72.45765 179.087661 -94.63347 105.892562 82.8515 -19.5986163 90.62829 -12.524564 53.51625 -11.6042565 5.482051 -26.6036166 72.64463 82.3178767 35.79332 -29.0503768 37.79594 -22.0963269 -12.56171 22.5060670 57.50436 5.23820371 21.54301 21.6374372 -28.18747 -49.9396373 31.25519 21.0182674 58.72964 -48.6403975 18.72472 -11.817776 -69.13664 40.0824377 -43.58995 26.4357378 67.7879 -20.120779 -37.77716 -5.90411180 84.8466 22.2562281 38.81703 45.75109

82 3.25427 -16.0666183 64.64431 -19.7798684 25.89511 98.5155485 34.36711 -30.6138786 4.413503 4.31173487 31.27638 23.9181888 -6.852292 0.49418289 -11.17253 29.7440590 -14.09436 86.1270991 -46.15615 -14.5121892 -59.74616 80.4021593 103.2665 -36.3001694 0.838255 25.9290595 13.18286 -39.451596 45.00269 11.821897 42.23833 -11.8144198 33.83801 38.8149699 -1.429854 -7.687358

100 -51.53119 -26.04728101 41.2987 -24.71938102 16.80674 -14.45943103 23.31513 -25.23122104 27.36999 50.10325105 -12.60932 -62.18877106 7.957865 -30.47484107 11.80735 -2.530854108 51.18955 20.9018109 15.4107 -23.71962110 38.0588 36.361111 21.8217 -82.43554112 24.26828 13.97877113 -8.220805 -89.39913114 50.33491 -25.70095115 -23.01678 -49.98658116 32.44573 -13.30733

compute predicted means and error variance (variance of firm-specific shock) using the estimates of alpha and betaNote that the predicted mean is the same as the mean of the dataNOTE: to compute the error variances, you need to press shift+ctrl+enter together

SUMMARY OUTPUT McDonald

Regression StatisticsMultiple R 0.473227R Square 0.223944Adjusted R 0.217136Standard E 48.21495Observatio 116

ANOVAdf SS MS

Regression 1 76474.11 76474.11Residual 114 265013.7 2324.682Total 115 341487.8

-250 -200 -150 -100 -50 0 50 100 150

-300

-250

-200

-150

-100

-50

0

50

100

150

200

IBM

Predicted IBM

S&P500

IBM

Upper 95.0% CoefficientsStandard Error t StatIntercept 14.34723 4.487932 3.196846S&P500 0.502618 0.087632 5.735555

F Significance F32.89659 8.114E-08

P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%0.001798 5.456669 23.23778 5.456669 23.23778

8.114E-08 0.32902 0.676216 0.32902 0.676216

PPC and CALSample Variances and covariances

IBM McDonald S&P500 IBM McDonald S&P500mean 8.187650755 16.17155 3.62963631 4148.19624 1082.37724 1975.205SD 64.40649222 54.25736 51.0846191 1082.37724 2943.86062 1311.651

1975.20529 1311.65125 2609.638

Index Model IBM McDonald S&P500means 8.187650755 16.17155 3.62963631 4194.74336 992.77378 1975.205SD 64.76683845 54.62546 51.0846191 992.77378 2983.94134 1311.651

1975.20529 1311.65125 2609.638

Opportunity set (portfolio possibility curve PPC)Use Excel's SOLVER

target mean Using Sample Moments (earlier results)

3 0.044688585 -0.066443 1.02175489 3 52.3598761 0.0412234 0.06282789 0.006697 0.93047498 4 50.2402905 0.0614125 0.080967171 0.079838 0.83919526 5 48.3938569 0.0816016 0.09910641 0.152978 0.74791569 6 46.852877 0.101797 0.117245682 0.226118 0.65663598 7 45.6482906 0.1219798 0.135384952 0.299259 0.56535627 8 44.8072392 0.1421689 0.153524226 0.372399 0.47407655 9 44.3504097 0.162357

10 0.171663523 0.44554 0.38279669 10 44.2896932 0.18254611 0.189802767 0.51868 0.29151712 11 44.6267059 0.20273512 0.207942035 0.591821 0.20023741 12 45.3525827 0.22292413 0.226081306 0.664961 0.1089577 13 46.4490962 0.24311314 0.244220585 0.738101 0.01767798 14 47.8907948 0.26330215 0.262359843 0.811242 -0.0736017 15 49.6476165 0.28349116 0.280499155 0.884382 -0.1648816 16 51.6874421 0.3036817 0.298638384 0.957523 -0.2561611 17 53.9781875 0.32386918 0.316777656 1.030663 -0.3474409 18 56.4893445 0.34405819 0.33491693 1.103804 -0.4387206 19 59.1928649 0.36424720 0.353056167 1.176944 -0.5300003 20 62.0636156 0.38443621 0.371195489 1.250085 -0.62128 21 65.07947 0.40462522 0.389334773 1.323225 -0.7125599 22 68.2211933 0.42481423 0.407473877 1.396365 -0.8038393 23 71.4721707 0.445004

The PPC's are almost identical

Compute MVP Using sample moments Using Index Model Moments

0.165355677 0.420106 0.4145388 9.652256 44.2655585 0.176049

mpt w1 w2 w3 mp sp w1

w1 w2 w3 mp sp w1

Optimal Portfolio Using sample moments Using Index Model Moments

Sharpe0.37370395 1.2602 -0.6339043 21.13831 65.5068882 0.32268828 0.407066

Draw CAL 0 0

w1 w2 w3 mp sp w1

40 45 50 55 60 65 70 750

5

10

15

20

25

sampleindexCAL sampleCAL index

IBMMcDonaldS&P500

IBMMcDonaldS&P500

Using Index Model Moments

-0.065184 1.023961 3 52.367190.007212 0.931377 4 50.241330.079607 0.838792 5 48.387670.152003 0.746208 6 46.838540.224398 0.653623 7 45.624960.296794 0.561038 8 44.774230.369189 0.468454 9 44.307260.441585 0.375869 10 44.23620.51398 0.283285 11 44.56294

0.586376 0.1907 12 45.278880.658771 0.098116 13 46.365980.731167 0.005531 14 47.798930.803562 -0.087053 15 49.547740.875958 -0.179638 16 51.580280.948353 -0.272223 17 53.864451.020749 -0.364807 18 56.369661.093144 -0.457392 19 59.06781.16554 -0.549976 20 61.93366

1.237935 -0.642561 21 64.945031.310331 -0.735145 22 68.082621.382726 -0.82773 23 71.32977

Using Index Model Moments

0.418289 0.405662 9.678213 44.21553

w2 w3 mp sp

w2 w3 mp sp

40 45 50 55 60 65 70 750

5

10

15

20

25

sampleindex

Using Index Model Moments

Sharpe1.246699 -0.653766 21.12104 65.31845 0.323355

0.00 0.00

w2 w3 mp sp

40 45 50 55 60 65 70 750

5

10

15

20

25

sampleindexCAL sampleCAL index

40 45 50 55 60 65 70 750

5

10

15

20

25

sampleindex

Textbook Table 5.2

Active Portfolio BenchmarkPortfolio Google Dell

A. Input dataRisk premium 0.7 2.2 1.74SD 4.31 11.39 10.49Sharpe ratio 0.16alpha 1.04 0.75Beta 1.65 1.41Residual SD 9.01 8.55Info ratio 0.115427 0.087719alpha/residual variance 0.012811 0.01026

Portfolio constructionB. Optimal portfolio with Google only in active portfolio

0.33997 0.272261

0.56359287 0.436407sharpe ratio 0.19729

C. Optimal Portfolio with Google and Dell in active portfolio

active portfolio weights 0.555297 0.444703alpha_a of active portfolio 0.911036beta_a of active portfolio 1.543271residual SD of Active Portfolio 6.284033Information ratio 0.144976

Complete portfolioweight of active portfolio 0.917347weight of benchmark 0.082653

weight on Google 0.5094weight on Dell 0.407947

Sharpe ratio 0.215912

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Textbook Table 6.2

Active Portfolio BenchmarkPortfolio Google Dell

A. Input dataRisk premium 0.7 2.2 1.74SD 4.31 11.39 10.49Sharpe ratio 0.16alpha 1.04 0.75Beta 1.65 1.41Residual SD 9.01 8.55Info ratio 0.115427 0.087719alpha/residual variance 0.012811 0.01026

Covariance matrix based on the single index modelBenchmark Google Dell

Benchmark 18.5761 30.65056 26.1923Google 30.650565 131.7535 43.2173Dell 26.192301 43.2173 110.0336

B. Optimal portfolio with Google only in active portfolioweights 0.5608959253 0.439104mean of portfolio 1.3586561121SD of portfolio 6.8077743587Sharpe ratio 0.1995741986

weights 0.0761002601 0.510435 0.408448mean 1.8869272071SD 8.649969142Sharpe ratio 0.2181426519

Theoretical Solution t

0.7 1 0.001919 0.047586 0.0250922.2 1 0.012873

1.74 1 0.010301

mean 1.896440747SD 8.693580697Sharpe rati 0.218142652weights 0.0764839403 0.513009 0.410507

C. Optimal Portfolio with Google and Dell in active portfolio: Use SOLVER

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