%// Black-Litterman example code for MatLab (hlblacklitterman.m)
%// Copyright (c) Jay Walters, blacklitterman.org, 2008.
%//
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%// This program uses the examples from the paper "The Intuition 
%// Behind Black-Litterman Model  Portfolios", by He and Litterman,
%// 1999.  You can find a copy of this  paper at the following url.
%//     http:%//papers.ssrn.com/sol3/papers.cfm?abstract_id=334304
%//
%// For more details on the Black-Litterman model you can also view
%// "The BlackLitterman Model: A Detailed Exploration", by this author
%// at the following url.
%//     http:%//www.blacklitterman.org/Black-Litterman.pdf
%//

%// hlblacklitterman
%//   This function performs the Black-Litterman blending of the prior
%//   and the views into a new posterior estimate of the returns as
%//   described in the paper by He and Litterman.
%// Inputs
%//   delta  - Risk tolerance from the equilibrium portfolio
%//   weq    - Weights of the assets in the equilibrium portfolio
%//   sigma  - Prior covariance matrix
%//   tau    - Coefficiet of uncertainty in the prior estimate of the mean (pi)
%//   P      - Pick matrix for the view(s)
%//   Q      - Vector of view returns
%//   Omega  - Matrix of variance of the views (diagonal)
%// Outputs
%//   Er     - Posterior estimate of the mean returns
%//   w      - Unconstrained weights computed given the Posterior estimates
%//            of the mean and covariance of returns.
%//   lambda - A measure of the impact of each view on the posterior estimates.
%//   theta  - A measure of the share of the prior and sample information in the
%//            posterior precision.
%//
function [er, ps, w, pw, lambda, theta] = hlblacklitterman(delta, weq, sigma, tau, P, Q, Omega)
  %// Reverse optimize and back out the equilibrium returns
  %// This is formula (12) page 6.
  pi = weq * sigma * delta;
  %// We use tau * sigma many places so just compute it once
  ts = tau * sigma;
  %// Compute posterior estimate of the mean
  %// This is a simplified version of formula (8) on page 4.
  er = pi' + ts * P' * inv(P * ts * P' + Omega) * (Q - P * pi');
  %// We can also do it the long way to illustrate that d1 + d2 = I
  d = inv(inv(ts) + P' * inv(Omega) * P);
  d1 = d * inv(ts);
  d2 = d * P' * inv(Omega) * P;
  er2 = d1 * pi' + d2 * pinv(P) * Q;
  %// Compute posterior estimate of the uncertainty in the mean
  %// This is a simplified and combined version of formulas (9) and (15)
  ps = ts - ts * P' * inv(P * ts * P' + Omega) * P * ts;
  posteriorSigma = sigma + ps;
  %// Compute the share of the posterior precision from prior and views,
  %// then for each individual view so we can compare it with lambda
  theta=zeros(1,2+size(P,1));
  theta(1,1) = (trace(inv(ts) * ps) / size(ts,1));
  theta(1,2) = (trace(P'*inv(Omega)*P* ps) / size(ts,1));
  for i=1:size(P,1)
    theta(1,2+i) = (trace(P(i,:)'*inv(Omega(i,i))*P(i,:)* ps) / size(ts,1));
  end
  %// Compute posterior weights based solely on changed covariance
  w = (er' * inv(delta * posteriorSigma))';
  %// Compute posterior weights based on uncertainty in mean and covariance
  pw = (pi * inv(delta * posteriorSigma))';
  %// Compute lambda value
  %// We solve for lambda from formula (17) page 7, rather than formula (18)
  %// just because it is less to type, and we've already computed w*.
  lambda = pinv(P)' * (w'*(1+tau) - weq)';
end