Input Data

##                     SPY           QQQ          EEM           IWM           EFA
## 2018-01-03  0.006325023  9.716366e-03  0.009581402  0.0010396119  0.0048000859
## 2018-01-04  0.004214892  1.749738e-03  0.004951427  0.0026612984  0.0109597716
## 2018-01-05  0.006663904  1.004317e-02  0.008622480  0.0020717627  0.0055593049
## 2018-01-08  0.001828444  3.890907e-03  0.000000000  0.0015501879 -0.0002764224
## 2018-01-09  0.002263920  6.150098e-05 -0.001628333 -0.0013544728  0.0011061173
## 2018-01-10 -0.001530157 -2.337575e-03 -0.006319983  0.0001290871 -0.0019335361
##                      TLT          IYR           GLD
## 2018-01-03  0.0047813945 -0.001612988 -0.0026368504
## 2018-01-04 -0.0001584050 -0.015658337  0.0051273786
## 2018-01-05 -0.0028555967  0.001010372 -0.0010361649
## 2018-01-08 -0.0006364847  0.005675256 -0.0001596128
## 2018-01-09 -0.0133727926 -0.011537594 -0.0046284751
## 2018-01-10 -0.0012098906 -0.012179500  0.0024051585
##                     SPY         QQQ          EEM          IWM          EFA
## 2018-01-12  0.016458318  0.01587184  0.007734559  0.021771350  0.015894928
## 2018-01-19  0.008959497  0.01124705  0.018784049  0.002781902  0.008571579
## 2018-01-26  0.022003601  0.02759404  0.032514000  0.006305358  0.015108433
## 2018-02-02 -0.038837516 -0.03697425 -0.058179781 -0.036153369 -0.035880477
## 2018-02-09 -0.050644581 -0.05169784 -0.053618822 -0.045894521 -0.054583072
## 2018-02-16  0.044397860  0.05675860  0.067212676  0.044763993  0.041259637
##                     TLT          IYR          GLD
## 2018-01-12 -0.009466439 -0.029385855  0.013005643
## 2018-01-19 -0.011724976  0.006496791 -0.004253315
## 2018-01-26  0.004306806  0.016911979  0.013051805
## 2018-02-02 -0.030431312 -0.029325845 -0.013117887
## 2018-02-09 -0.013714562 -0.041590465 -0.012817492
## 2018-02-16  0.006528368  0.024290466  0.025567064
##                     SPY         QQQ          EEM          IWM          EFA
## 2018-02-28 -0.036360555 -0.01292763 -0.058984870 -0.038436834 -0.048347810
## 2018-03-29 -0.027410447 -0.04078854  0.005414432  0.012175269 -0.008396179
## 2018-04-30  0.005168179  0.00505834 -0.028169096  0.009813426  0.015212394
## 2018-05-31  0.024309053  0.05672928 -0.026214774  0.061635820 -0.018942622
## 2018-06-29  0.005750861  0.01145044 -0.045456771  0.006143869 -0.015841024
## 2018-07-31  0.037046550  0.02796385  0.035310232  0.016490301  0.028520144
##                     TLT          IYR          GLD
## 2018-02-28 -0.030414497 -0.066573196 -0.020759902
## 2018-03-29  0.028596471  0.037719119  0.006320007
## 2018-04-30 -0.020881364  0.002252617 -0.009539745
## 2018-05-31  0.020044199  0.033712267 -0.011959209
## 2018-06-29  0.006457735  0.040594721 -0.036149448
## 2018-07-31 -0.014368601  0.008314823 -0.022418910

GMVP in terms of weekly asset returns

GMVP (weekly return)
Weight
SPY 0.9920195
QQQ -0.4595023
EEM 0.1668409
IWM -0.0741922
EFA -0.1213614
TLT 0.4855613
IYR -0.2326812
GLD 0.2433153

GMVP in terms of monthly asset returns

GMVP (monthly return)
Weight
SPY 1.1924503
QQQ -0.7673217
EEM 0.0189768
IWM 0.1903898
EFA -0.1510400
TLT 0.5703813
IYR -0.3099006
GLD 0.2560640

Q2 Find Portfolio with annual return equal 0.045 with weekly data

Portfolio with minimum risk and annual return at 4.5% (weekly return)
Weight
SPY 1.0759010
QQQ -0.4015910
EEM 0.0884016
IWM -0.0962383
EFA -0.1668181
TLT 0.4265144
IYR -0.2435860
GLD 0.3174164

Annual return equal 0.045 with monthly data

Portfolio with minimum risk and annual return at 4.5% (monthly return)
Weight
SPY 1.3335698
QQQ -0.7064505
EEM -0.0449000
IWM 0.1495396
EFA -0.2471815
TLT 0.4885659
IYR -0.3156011
GLD 0.3424578

Q3 Find Tangency Portfolio with weekly data

Tangency Portfolio with weekly data
Weight
SPY 2.8906003
QQQ 0.8512661
EEM -1.6085612
IWM -0.5731866
EFA -1.1502321
TLT -0.8509111
IYR -0.4795025
GLD 1.9205271

Tangency Portfolio with monthly data

Tangency Portfolio with weekly data
Weight
SPY 4.2187016
QQQ 0.5380367
EEM -1.3508365
IWM -0.6856262
EFA -2.2127570
TLT -1.1841182
IYR -0.4321453
GLD 2.1087448

Q4

Find covariance matrix

Ri = a + B*X + ei, where Ri is the stock return and an is the anticipated excess return, is the single index model. B represents the stock’s market sensitivity. The market excess return is X. (Rm-rf) The company-specific surprise is ei. Finding the return covariance matrix is our aim. We will provide the beta (B) and the covariance of ei in order to accomplish the purpose.

##           CITCRP         CONED      CONTIL      DATGEN         DEC       DELTA
## [1,] -0.08906182 -0.0924325479 -0.08644938 -0.03294113 -0.07166573 -0.01200381
## [2,] -0.02947098 -0.0213220221  0.03962148 -0.10233449 -0.08098502 -0.04367536
## [3,]  0.02222799  0.0001459493 -0.02322229  0.01692842 -0.04144488  0.04005781
## [4,]  0.08137923 -0.0280423719  0.14407347  0.11922277  0.10229784  0.11357564
## [5,] -0.04312626 -0.0373788404  0.02232575 -0.01165793 -0.02788962 -0.06925976
## [6,] -0.00126376  0.0159743906 -0.05395791 -0.02491578 -0.03617703  0.01394154
##            GENMIL       GERBER         IBM        MOBIL       PANAM
## [1,] -0.100413553 -0.028965238 -0.01280576 -0.021675972  0.06312884
## [2,]  0.001973487  0.144900665 -0.05166925 -0.031704502 -0.07498803
## [3,] -0.049582534 -0.075756917 -0.08963000  0.006102306  0.15303258
## [4,]  0.015865982 -0.044052752  0.09732727  0.024616945  0.04828272
## [5,]  0.031860393 -0.004401685 -0.05238371 -0.066068798  0.03852208
## [6,] -0.007140637  0.014969963 -0.01116196 -0.055338490  0.01944396
##              PSNH       TANDY      TEXACO         WEYER
## [1,]  0.008151819 -0.04037246 -0.03121181 -0.0790751380
## [2,] -0.020345243 -0.02681935 -0.02066688 -0.1429243732
## [3,]  0.022349580  0.05968252 -0.01983395  0.0436777086
## [4,] -0.014444627 -0.02327918 -0.04296475  0.0927777034
## [5,]  0.011764204  0.09376762 -0.07426696 -0.0853085904
## [6,]  0.037351739 -0.03333676 -0.03363874 -0.0002054953
##              [,1]        [,2]       [,3]       [,4]        [,5]        [,6]
##  [1,] 0.004526617 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
##  [2,] 0.000000000 0.002511069 0.00000000 0.00000000 0.000000000 0.000000000
##  [3,] 0.000000000 0.000000000 0.02039461 0.00000000 0.000000000 0.000000000
##  [4,] 0.000000000 0.000000000 0.00000000 0.01140139 0.000000000 0.000000000
##  [5,] 0.000000000 0.000000000 0.00000000 0.00000000 0.006537662 0.000000000
##  [6,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.008161795
##  [7,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
##  [8,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
##  [9,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [10,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [11,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [12,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [13,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [14,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [15,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
##             [,7]        [,8]        [,9]       [,10]      [,11]      [,12]
##  [1,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [2,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [3,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [4,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [5,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [6,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [7,] 0.00393017 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##  [8,] 0.00000000 0.005930261 0.000000000 0.000000000 0.00000000 0.00000000
##  [9,] 0.00000000 0.000000000 0.002540858 0.000000000 0.00000000 0.00000000
## [10,] 0.00000000 0.000000000 0.000000000 0.004107716 0.00000000 0.00000000
## [11,] 0.00000000 0.000000000 0.000000000 0.000000000 0.01498815 0.00000000
## [12,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.01187757
## [13,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [14,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [15,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
##            [,13]      [,14]       [,15]
##  [1,] 0.00000000 0.00000000 0.000000000
##  [2,] 0.00000000 0.00000000 0.000000000
##  [3,] 0.00000000 0.00000000 0.000000000
##  [4,] 0.00000000 0.00000000 0.000000000
##  [5,] 0.00000000 0.00000000 0.000000000
##  [6,] 0.00000000 0.00000000 0.000000000
##  [7,] 0.00000000 0.00000000 0.000000000
##  [8,] 0.00000000 0.00000000 0.000000000
##  [9,] 0.00000000 0.00000000 0.000000000
## [10,] 0.00000000 0.00000000 0.000000000
## [11,] 0.00000000 0.00000000 0.000000000
## [12,] 0.00000000 0.00000000 0.000000000
## [13,] 0.01121942 0.00000000 0.000000000
## [14,] 0.00000000 0.00464578 0.000000000
## [15,] 0.00000000 0.00000000 0.004133323

The Global Minimum Variance Portfolio

GMVP from single index model
Weight
CITCRP 0.0443133
CONED 0.3755073
CONTIL 0.0057706
DATGEN -0.0236873
DEC -0.0050185
DELTA 0.0526752
GENMIL 0.1816517
GERBER 0.0428599
IBM 0.1859635
MOBIL 0.0338142
PANAM 0.0075571
PSNH 0.0662851
TANDY -0.0262830
TEXACO 0.0581522
WEYER 0.0004386