MATLAB Implementation of Black Litterman Model
This code implements the Black and Litterman Model As given in the paper He & Litterman: The intuition Behind Black- Litterman
Model Portfolios.
% First The Inputs:
% Correlation Matrix: Page 18
Corrmat = ...
[1,0.4880,0.4780,0.5150,0.4390,0.5120,0.4910;
0.4880,1,0.6640,0.6550,0.3100,0.6080,0.7790;
0.4780,0.6640,1,0.8610,0.3550,0.7830,0.6680;
0.5150,0.6550,0.8610,1,0.3540,0.7770,0.6530;
0.4390,0.3100,0.3550,0.3540,1,0.4050,0.3060;
0.5120,0.6080,0.7830,0.7770,0.4050,1,0.6520;
0.4910,0.7790,0.6680,0.6530,0.3060,0.6520,1;];
% Risk Aversion Parameter---Page 10
RiskAversion = 2.5;
% Standard Deviations and Market Capitalization Weights--Table 2 Page 19
stdevs = ...
[16.0000
20.3000
24.8000
27.1000
21.0000
20.0000
18.7000]./100;
MktWeight = ...
[ 1.6000
2.2000
5.2000
5.5000
11.6000
12.4000
61.5000]./100;
tau = 0.05;
Market Equilibrium Risk Premiums : THE PI
Equation 2 Page 3 PI = RiskAversion * Covariance * MktWeight
% Need to Convert Correlation into Covariance
Covmat = Corrmat .* (stdevs * stdevs');
% Now we have everything to implement the Equation
%EqRiskPrem = RiskAversion * Covmat * MktWeight;
EqRiskPrem = RiskAversion * Covmat * MktWeight;
% Print Out Table 2 in Page 19
AssetNames = {'Australia','Canada','France','Germany','Japan',...
'UK','USA'};
Table2 = [{'Assets' 'Std Dev' 'Weq' 'PI'};
{'------' '------' '---' '--'};
AssetNames' num2cell([stdevs MktWeight EqRiskPrem]*100)]
Table2 =
'Assets' 'Std Dev' 'Weq' 'PI'
'------' '------' '---' '--'
'Australia' [ 16] [ 1.6000] [3.9376]
'Canada' [20.3000] [ 2.2000] [6.9152]
'France' [24.8000] [ 5.2000] [8.3581]
'Germany' [27.1000] [ 5.5000] [9.0272]
'Japan' [ 21] [11.6000] [4.3028]
'UK' [ 20] [12.4000] [6.7677]
'USA' [18.7000] [61.5000] [7.5600]
Views Based Optimal Weights
%View1 is The German Equity Market Will Outperform the rest of European
% Markets by 5% a year.
P = [ 0 %Australia
0 %Canada
-29.5 %France
100 %Germany
0 %Japan
-70.5 %UK
0]'./100; %USA
Q = 5/100;
% The Black Litterman Expected Returns are Calculated
% By Equation 8 and the Optimal Portfolio Weights are
% Calculated By
%RiskAversion * Covariance * ExpectedReturns(MU)
% Lambda = 0.302; Page 11
Omega = diag(diag(P*tau*Covmat*P'));
% Equation 8--Expected Returns : MU
PostCov = inv(inv(tau*Covmat) + (P' * inv(Omega) * P));
SigmaP = Covmat + PostCov;
ExpRet=inv(inv(tau*SigmaP)+P'*inv(Omega)*P)* ...
(inv(tau*SigmaP)*EqRiskPrem +P'*inv(Omega)*Q);
% ExpRet = inv(inv(tau*Covmat) + P' * inv(Omega) * P) * ...
%(inv(tau*Covmat) * EqRiskPrem + P' * inv(Omega) * Q);
% Optimal Weights
OptimalWeights = (1/RiskAversion)* inv(SigmaP) * ExpRet;
Tab4Col4 = OptimalWeights - (MktWeight)/(1+tau);
Table4 = [{'Assets' 'P' 'MU' 'W' 'W - Weq/1+tau'};
{'------' '--' '---' '--' '-------------'};
AssetNames' num2cell([P' ExpRet OptimalWeights ...
round(Tab4Col4 * 1000)./1000]*100)]
Table4 =
'Assets' 'P' 'MU' 'W' 'W - Weq/1+tau'
'------' '--' '---' '--' '-------------'
'Australia' [ 0] [ 4.3328] [ 1.5238] [ 0]
'Canada' [ 0] [ 7.5838] [ 2.0952] [ 0]
'France' [-29.5000] [ 9.2991] [-4.0555] [ -9]
'Germany' [ 100] [11.0615] [35.7733] [ 30.5000]
'Japan' [ 0] [ 4.5087] [11.0476] [ 0]
'UK' [-70.5000] [ 6.9550] [-9.7178] [ -21.5000]
'USA' [ 0] [ 8.0756] [58.5714] [ 0]
