Black Litterman Model Application
gretl version 1.9.9
Current session: 2012-11-14 14:01
## Black Litterman model application
## M Boldin Oct 2012
## Stock Return data
#open "C:.gdt"
? smpl --full
Full data range: 2001:01 - 2011:12 (n = 132)
## Compute Excess returns for Market and Alternative portfolios
? series MPORTret= SPYret - RF
Replaced series MPORTret (ID 30)
? series APORTret= (TGTret+IBMret+AMZNret+EBAYret+PDCOret)/5 - RF
Replaced series APORTret (ID 31)
? summary RF MKT MKTxRF SPYret MPORTret APORTret
Summary statistics, using the observations 2001:01 - 2011:12
Mean Median Minimum Maximum
RF 0.0016455 0.0013000 0.00000 0.0054000
MKT 0.0034591 0.011000 -0.18470 0.11530
MKTxRF 0.0018136 0.0082000 -0.18550 0.11530
SPYret 0.0023195 0.0089880 -0.16519 0.10915
MPORTret 0.00067407 0.0063025 -0.16599 0.10915
APORTret 0.012614 0.015517 -0.21784 0.23936
Std. Dev. C.V. Skewness Ex. kurtosis
RF 0.0014739 0.89575 0.60879 -0.89865
MKT 0.049322 14.259 -0.58275 0.89218
MKTxRF 0.049405 27.241 -0.55295 0.87372
SPYret 0.046726 20.145 -0.52343 0.70702
MPORTret 0.046806 69.438 -0.49124 0.69156
APORTret 0.071420 5.6619 0.13396 1.5455
# Set sub-sample period to estimate mean retunr and var-cov parameters
? smpl 2001:01 2007:12
Full data range: 2001:01 - 2011:12 (n = 132)
Current sample: 2001:01 - 2007:12 (n = 84)
? summary RF MKT MKTxRF SPYret MPORTret APORTret
Summary statistics, using the observations 2001:01 - 2007:12
Mean Median Minimum Maximum
RF 0.0023607 0.0021500 0.00060000 0.0054000
MKT 0.0049274 0.011600 -0.10000 0.083800
MKTxRF 0.0025667 0.0089500 -0.10320 0.081800
SPYret 0.0034229 0.0091975 -0.10473 0.084612
MPORTret 0.0010622 0.0063025 -0.10613 0.083612
APORTret 0.014947 0.014790 -0.13758 0.23936
Std. Dev. C.V. Skewness Ex. kurtosis
RF 0.0013332 0.56473 0.27736 -1.3527
MKT 0.039796 8.0764 -0.55301 0.34387
MKTxRF 0.039875 15.536 -0.54489 0.35966
SPYret 0.038062 11.120 -0.50185 0.56956
MPORTret 0.038112 35.881 -0.48887 0.58476
APORTret 0.072875 4.8757 0.62774 1.1135
? scalar rfc= mean(RF)
Replaced scalar rfc = 0.00236071
## Build matrix with return observations
? matrix Rx = MPORTret, APORTret
Replaced matrix Rx
? matrix R = meanc(Rx)
Replaced matrix R
? scalar k= cols(Rx)
Replaced scalar k = 2
? scalar n= rows(Rx)
Replaced scalar n = 84
## Variance-Covariance matrix
? S = mcov(Rx)
Replaced matrix S
? SI = inv(S)
Replaced matrix SI
? sd1 = sdc(Rx)
Replaced matrix sd1
? C1 = mcorr(Rx)
Replaced matrix C1
## Implied mean-variance optimization weights
? w1 = SI*R'
Replaced matrix w1
? w1 = w1/sumc(w1)
Replaced matrix w1
? R2 = R+rfc
Replaced matrix R2
? w2 = inv(S)*R2'
Replaced matrix w2
? w2 = w2/sumc(w2)
Replaced matrix w2
## Check results
? scalar check = sumc(w1)
Replaced scalar check = 1
? scalar check = sumc(w2)
Replaced scalar check = 1
## Two points on Computed Efficient frontier
? scalar rm1= R*w1
Replaced scalar rm1 = -0.0396636
? scalar rm2= R2*w2
Replaced scalar rm2 = -0.356467
? scalar s1= sqrt(w1'*S*w1)
Replaced scalar s1 = 0.138495
? scalar s2= sqrt(w2'*S*w2)
Replaced scalar s2 = 1.28771
## Compute Betas by regressing X on M=MKTxRF
? matrix Rm= MKTxRF
Replaced matrix Rm
? B= inv(Rm'Rm)*(Rm'Rx)
Replaced matrix B
? scalar sdRm= sdc(Rm)
Replaced scalar sdRm = 0.0396372
# CAPM Beta based Expected Returns
? scalar Em= .06/12
Replaced scalar Em = 0.005
? Rb= B'Em
Replaced matrix Rb
## Compute weights based on CAPM returns
? wb = inv(S)*Rb
Replaced matrix wb
? wb = wb/sumc(wb)
Replaced matrix wb
## Compute Expected returns and st dev
? scalar ERb= 12*Rb'wb
Replaced scalar ERb = 0.0575961
? scalar sdb= sqrt(wb'*S*wb)
Replaced scalar sdb = 0.038752
## Compute B-L adjustment for view on Alternative Portfolio
## tau is an adjustment to CAPM based prior var-cov view reflecting relative confidence
## Smaller (larger) value puts more (less) weight on CAPM-based var-cov
## P= Alternative view portfolio, use [0 1] for simple case
## Q= Alternative view Expected return
## Oa= Var-cov matrix for alternative, assumed to have 0's off-diagonal
? scalar tau= 1
Replaced scalar tau = 1
? matrix P= zeros(1,k)
Replaced matrix P
? P[1,2]= 1
Modified matrix P
? matrix Q= zeros(1,1)
Replaced matrix Q
? Q[1,1]= Rb[2] + (.03/12) # add x% per year (1/12 for monthly) for \par alternative
Modified matrix Q
? Oa= 2*S[2,2]
Replaced scalar Oa = 0.0106216
? OaI= inv(Oa)
Replaced scalar OaI = 94.1481
? Ma= P'OaI*P ## Alternative information matrix
Replaced matrix Ma
? A= P'OaI*Q
Replaced matrix A
? M = inv(inv(tau*S)+Ma) ## Combine prior and alternative var-cov views
Replaced matrix M
? ua = M*(inv(tau*S)*Rb+A) ## Combined expected returns
Replaced matrix ua
## Compute combined view weights
## delta is a risk adjustment, higher value lowers acceptable risk
## for one alternative use sdRm for delta to get a ERp = sd type result
? scalar delta = 1
Replaced scalar delta = 1
? S2= S + M
Replaced matrix S2
? ua2= (1/delta)*ua ## risk adjusted expected returns
Replaced matrix ua2
? wa2= (1/delta)*inv(S2)'ua
Replaced matrix wa2
? wa2= wa2/sumc(wa2)
Replaced matrix wa2
## Explore results
? print R
R (1 x 2)
0.0010622 0.014947
? print sd1
sd1 (1 x 2)
0.037885 0.072440
? print C1
C1 (2 x 2)
1.0000 0.76433
0.76433 1.0000
? print w1
w1 (2 x 1)
3.9332
-2.9332
? print B
B (1 x 2)
0.94326 1.4272
? print Rb
Rb (2 x 1)
0.0047163
0.0071360
? print wb
wb (2 x 1)
0.96555
0.034454
? print ua2
ua2 (2 x 1)
0.0050494
0.0079693
? print wa2
wa2 (2 x 1)
0.84475
0.15525
? print Rb
Rb (2 x 1)
0.0047163
0.0071360
? print Q
Q (1 x 1)
0.0096360
? print S
S (2 x 2)
0.0014525 0.0021229
0.0021229 0.0053108
? print M
M (2 x 2)
0.0011697 0.0014152
0.0014152 0.0035405
? print S2
S2 (2 x 2)
0.0026222 0.0035381
0.0035381 0.0088513
## Use period after sub-sample estimation
## to test performance of portfolio weights
? smpl 2008:01 2011:12
Full data range: 2001:01 - 2011:12 (n = 132)
Current sample: 2008:01 - 2011:12 (n = 48)
## Compute Excess returns for Market and Alternative portfolios
? matrix Rx = MPORTret, APORTret
Replaced matrix Rx
? series Port1 = Rx*w1
Replaced series Port1 (ID 32)
? series Portb = Rx*wb
Replaced series Portb (ID 33)
? series Porta2 = Rx*wa2
Replaced series Porta2 (ID 34)
? series Porta2xM= (Porta2-MPORTret)
Replaced series Porta2xM (ID 35)
? series Porta2xb= (Porta2-Portb)
Replaced series Porta2xb (ID 36)
## Port1 based on naive 'optimal' in-sample mean-variance weights
## Portb based on CAPM-Beta means weights
## Porta2 based on Black-Litterman + Alternative view weights
? summary MPORTret APORTret Port1 Portb Porta2 Porta2xM Porta2xb
Summary statistics, using the observations 2008:01 - 2011:12
Mean Median Minimum Maximum
MPORTret -5.1458e-006 0.0043355 -0.16599 0.10915
APORTret 0.0085326 0.016737 -0.21784 0.15141
Port1 -0.025048 -0.015994 -0.37023 0.26717
Portb 0.00028901 0.0051012 -0.16777 0.10775
Porta2 0.0013203 0.0051703 -0.17404 0.10734
Porta2xM 0.0013255 0.0020232 -0.017841 0.013903
Porta2xb 0.0010313 0.0015742 -0.013882 0.010818
Std. Dev. C.V. Skewness Ex. kurtosis
MPORTret 0.059501 11563. -0.42280 -0.096541
APORTret 0.069368 8.1298 -0.90111 2.0755
Port1 0.13494 5.3873 -0.12369 0.086784
Portb 0.059423 205.60 -0.43565 -0.039250
Porta2 0.059402 44.990 -0.48677 0.19703
Porta2xM 0.0061998 4.6774 -0.40325 0.47156
Porta2xb 0.0048239 4.6774 -0.40325 0.47156
? corr MKTxRF Port1 Portb Porta2
Correlation Coefficients, using the observations 2008:01 - 2011:12
5% critical value (two-tailed) = 0.2845 for n = 48
MKTxRF Port1 Portb Porta2
1.0000 0.4724 0.9963 0.9935 MKTxRF
1.0000 0.4798 0.4071 Port1
1.0000 0.9967 Portb
1.0000 Porta2
? plot1 <- gnuplot MPORTret APORTret --with-lines --time-series
plot1 replaced
? plot2 <- gnuplot MPORTret Port1 Portb Porta2 --with-lines --time-series
plot2 replaced
? plot3 <- gnuplot Porta2xM Porta2xb --with-lines --time-series
plot3 replaced
Plot 2 Graph
