Main Content

Generate Code to Optimize Portfolio by Using Black Litterman Approach

This example shows how to generate a MEX function and C source code from MATLAB® code that performs portfolio optimization using the Black Litterman approach.

Prerequisites

There are no prerequisites for this example.

About the hlblacklitterman Function

The hlblacklitterman.m function reads in financial information regarding a portfolio and performs portfolio optimization using the Black Litterman approach.

type hlblacklitterman
function [er, ps, w, pw, lambda, theta] = hlblacklitterman(delta, weq, sigma, tau, P, Q, Omega)%#codegen
% 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.

% 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

% Black-Litterman example code for MatLab (hlblacklitterman.m)
% Copyright (c) Jay Walters, blacklitterman.org, 2008.
%
% Redistribution and use in source and binary forms, 
% with or without modification, are permitted provided 
% that the following conditions are met:
%
% Redistributions of source code must retain the above 
% copyright notice, this list of conditions and the following 
% disclaimer.
% 
% Redistributions in binary form must reproduce the above 
% copyright notice, this list of conditions and the following 
% disclaimer in the documentation and/or other materials 
% provided with the distribution.
%  
% Neither the name of blacklitterman.org nor the names of its
% contributors may be used to endorse or promote products 
% derived from this software without specific prior written
% permission.
%  
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND 
% CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, 
% INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF 
% MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE 
% DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR 
% CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 
% SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, 
% BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR 
% SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 
% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, 
% WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING 
% NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 
% OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH 
% DAMAGE.
%
% 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
%

The %#codegen directive indicates that the MATLAB code is intended for code generation.

Generate the MEX Function for Testing

Generate a MEX function using the codegen command.

codegen hlblacklitterman -args {0, zeros(1, 7), zeros(7,7), 0, zeros(1, 7), 0, 0}
Code generation successful.

Before generating C code, you should first test the MEX function in MATLAB to ensure that it is functionally equivalent to the original MATLAB code and that no run-time errors occur. By default, codegen generates a MEX function named hlblacklitterman_mex in the current folder. This allows you to test the MATLAB code and MEX function and compare the results.

Run the MEX Function

Call the generated MEX function

testMex();
View 1
Country        P       mu      w*
Australia	     0	 4.328	 1.524
Canada   	     0	 7.576	 2.095
France   	 -29.5	 9.288	-3.948
Germany  	   100	 11.04	 35.41
Japan    	     0	 4.506	 11.05
UK       	 -70.5	 6.953	-9.462
USA      	     0	 8.069	 58.57
q        	     5
omega/tau	     0.0213
lambda   	     0.317
theta   	     0.0714
pr theta  	     0.929


View 1
Country        P       mu      w*
Australia	     0	 4.328	 1.524
Canada   	     0	 7.576	 2.095
France   	 -29.5	 9.288	-3.948
Germany  	   100	 11.04	 35.41
Japan    	     0	 4.506	 11.05
UK       	 -70.5	 6.953	-9.462
USA      	     0	 8.069	 58.57
q        	     5
omega/tau	     0.0213
lambda   	     0.317
theta   	     0.0714
pr theta  	     0.929

Execution Time - MATLAB function: 0.018 seconds
Execution Time - MEX function   : 0.008279 seconds

Generate C Code

cfg = coder.config('lib');
codegen -config cfg hlblacklitterman  -args {0, zeros(1, 7), zeros(7,7), 0, zeros(1, 7), 0, 0}
Code generation successful.

Using codegen with the specified -config cfg option produces a standalone C library.

Inspect the Generated Code

By default, the code generated for the library is in the folder codegen/lib/hbblacklitterman/.

The files are:

dir codegen/lib/hlblacklitterman/
.                              hlblacklitterman_terminate.c   
..                             hlblacklitterman_terminate.h   
.gitignore                     hlblacklitterman_terminate.o   
_clang-format                  hlblacklitterman_types.h       
buildInfo.mat                  interface                      
codeInfo.mat                   inv.c                          
codedescriptor.dmr             inv.h                          
compileInfo.mat                inv.o                          
examples                       rtGetInf.c                     
hlblacklitterman.a             rtGetInf.h                     
hlblacklitterman.c             rtGetInf.o                     
hlblacklitterman.h             rtGetNaN.c                     
hlblacklitterman.o             rtGetNaN.h                     
hlblacklitterman_data.c        rtGetNaN.o                     
hlblacklitterman_data.h        rt_nonfinite.c                 
hlblacklitterman_data.o        rt_nonfinite.h                 
hlblacklitterman_initialize.c  rt_nonfinite.o                 
hlblacklitterman_initialize.h  rtw_proj.tmw                   
hlblacklitterman_initialize.o  rtwtypes.h                     
hlblacklitterman_rtw.mk        

Inspect the C Code for the hlblacklitterman.c Function

type codegen/lib/hlblacklitterman/hlblacklitterman.c
/*
 * File: hlblacklitterman.c
 *
 * MATLAB Coder version            : 5.5
 * C/C++ source code generated on  : 31-Aug-2022 01:30:18
 */

/* Include Files */
#include "hlblacklitterman.h"
#include "inv.h"
#include "rt_nonfinite.h"
#include "rt_nonfinite.h"
#include <math.h>

/* Function Definitions */
/*
 * 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.
 *
 * Arguments    : double delta
 *                const double weq[7]
 *                const double sigma[49]
 *                double tau
 *                const double P[7]
 *                double Q
 *                double Omega
 *                double er[7]
 *                double ps[49]
 *                double w[7]
 *                double pw[7]
 *                double *lambda
 *                double theta[3]
 * Return Type  : void
 */
void hlblacklitterman(double delta, const double weq[7], const double sigma[49],
                      double tau, const double P[7], double Q, double Omega,
                      double er[7], double ps[49], double w[7], double pw[7],
                      double *lambda, double theta[3])
{
  double b_er_tmp[49];
  double dv[49];
  double posteriorSigma[49];
  double ts[49];
  double er_tmp[7];
  double pi[7];
  double y_tmp[7];
  double absxk;
  double b_P;
  double b_y_tmp;
  double nrm;
  double scale;
  int br;
  int i;
  int ib;
  int ic;
  int ps_tmp;
  boolean_T p;
  /*  Reverse optimize and back out the equilibrium returns */
  /*  This is formula (12) page 6. */
  for (i = 0; i < 7; i++) {
    nrm = 0.0;
    for (ic = 0; ic < 7; ic++) {
      nrm += weq[ic] * sigma[ic + 7 * i];
    }
    pi[i] = nrm * delta;
  }
  /*  We use tau * sigma many places so just compute it once */
  for (i = 0; i < 49; i++) {
    ts[i] = tau * sigma[i];
  }
  /*  Compute posterior estimate of the mean */
  /*  This is a simplified version of formula (8) on page 4. */
  b_y_tmp = 0.0;
  b_P = 0.0;
  for (i = 0; i < 7; i++) {
    nrm = 0.0;
    scale = 0.0;
    for (ic = 0; ic < 7; ic++) {
      absxk = P[ic];
      nrm += ts[i + 7 * ic] * absxk;
      scale += absxk * ts[ic + 7 * i];
    }
    y_tmp[i] = scale;
    er_tmp[i] = nrm;
    nrm = P[i];
    b_y_tmp += scale * nrm;
    b_P += nrm * pi[i];
  }
  absxk = 1.0 / (b_y_tmp + Omega);
  scale = Q - b_P;
  /*  We can also do it the long way to illustrate that d1 + d2 = I */
  b_y_tmp = 1.0 / Omega;
  /*  Compute posterior estimate of the uncertainty in the mean */
  /*  This is a simplified and combined version of formulas (9) and (15) */
  nrm = 0.0;
  for (i = 0; i < 7; i++) {
    er[i] = pi[i] + er_tmp[i] * absxk * scale;
    nrm += y_tmp[i] * P[i];
  }
  absxk = 1.0 / (nrm + Omega);
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      b_er_tmp[ic + 7 * i] = er_tmp[ic] * absxk * P[i];
    }
  }
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      nrm = 0.0;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        nrm += b_er_tmp[i + 7 * ps_tmp] * ts[ps_tmp + 7 * ic];
      }
      ps_tmp = i + 7 * ic;
      ps[ps_tmp] = ts[ps_tmp] - nrm;
    }
  }
  for (i = 0; i < 49; i++) {
    posteriorSigma[i] = sigma[i] + ps[i];
  }
  /*  Compute the share of the posterior precision from prior and views, */
  /*  then for each individual view so we can compare it with lambda */
  inv(ts, dv);
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      nrm = 0.0;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        nrm += dv[i + 7 * ps_tmp] * ps[ps_tmp + 7 * ic];
      }
      ts[i + 7 * ic] = nrm;
    }
  }
  b_P = 0.0;
  for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
    b_P += ts[ps_tmp + 7 * ps_tmp];
  }
  theta[0] = b_P / 7.0;
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      b_er_tmp[ic + 7 * i] = P[ic] * b_y_tmp * P[i];
    }
  }
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      nrm = 0.0;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        nrm += b_er_tmp[i + 7 * ps_tmp] * ps[ps_tmp + 7 * ic];
      }
      ts[i + 7 * ic] = nrm;
    }
  }
  b_P = 0.0;
  for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
    b_P += ts[ps_tmp + 7 * ps_tmp];
  }
  theta[1] = b_P / 7.0;
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      b_er_tmp[ic + 7 * i] = P[ic] * b_y_tmp * P[i];
    }
  }
  for (i = 0; i < 7; i++) {
    for (ic = 0; ic < 7; ic++) {
      nrm = 0.0;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        nrm += b_er_tmp[i + 7 * ps_tmp] * ps[ps_tmp + 7 * ic];
      }
      ts[i + 7 * ic] = nrm;
    }
  }
  b_P = 0.0;
  for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
    b_P += ts[ps_tmp + 7 * ps_tmp];
  }
  theta[2] = b_P / 7.0;
  /*  Compute posterior weights based solely on changed covariance */
  for (i = 0; i < 49; i++) {
    b_er_tmp[i] = delta * posteriorSigma[i];
  }
  inv(b_er_tmp, dv);
  for (i = 0; i < 7; i++) {
    nrm = 0.0;
    for (ic = 0; ic < 7; ic++) {
      nrm += er[ic] * dv[ic + 7 * i];
    }
    w[i] = nrm;
  }
  /*  Compute posterior weights based on uncertainty in mean and covariance */
  for (i = 0; i < 49; i++) {
    posteriorSigma[i] *= delta;
  }
  inv(posteriorSigma, dv);
  /*  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*. */
  for (i = 0; i < 7; i++) {
    nrm = 0.0;
    for (ic = 0; ic < 7; ic++) {
      nrm += pi[ic] * dv[ic + 7 * i];
    }
    pw[i] = nrm;
    er_tmp[i] = P[i];
  }
  p = true;
  for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
    pi[ps_tmp] = 0.0;
    if (p) {
      nrm = P[ps_tmp];
      if (rtIsInf(nrm) || rtIsNaN(nrm)) {
        p = false;
      }
    } else {
      p = false;
    }
  }
  if (!p) {
    for (i = 0; i < 7; i++) {
      pi[i] = rtNaN;
    }
  } else {
    nrm = 0.0;
    scale = 3.3121686421112381E-170;
    for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
      absxk = fabs(P[ps_tmp]);
      if (absxk > scale) {
        b_P = scale / absxk;
        nrm = nrm * b_P * b_P + 1.0;
        scale = absxk;
      } else {
        b_P = absxk / scale;
        nrm += b_P * b_P;
      }
    }
    nrm = scale * sqrt(nrm);
    if (nrm > 0.0) {
      if (P[0] < 0.0) {
        absxk = -nrm;
      } else {
        absxk = nrm;
      }
      if (fabs(absxk) >= 1.0020841800044864E-292) {
        scale = 1.0 / absxk;
        for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
          er_tmp[ps_tmp] *= scale;
        }
      } else {
        for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
          er_tmp[ps_tmp] /= absxk;
        }
      }
      er_tmp[0]++;
      absxk = -absxk;
    } else {
      absxk = 0.0;
    }
    for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
      y_tmp[ps_tmp] = er_tmp[ps_tmp];
    }
    if (absxk != 0.0) {
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        y_tmp[ps_tmp] = -y_tmp[ps_tmp];
      }
      y_tmp[0]++;
      nrm = fabs(absxk);
      scale = absxk / nrm;
      absxk = nrm;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        y_tmp[ps_tmp] *= scale;
      }
    } else {
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        y_tmp[ps_tmp] = 0.0;
      }
      y_tmp[0] = 1.0;
    }
    if (rtIsInf(absxk)) {
      scale = rtNaN;
    } else if (absxk < 4.4501477170144028E-308) {
      scale = 4.94065645841247E-324;
    } else {
      frexp(absxk, &br);
      scale = ldexp(1.0, br - 53);
    }
    if (absxk > 7.0 * scale) {
      scale = 1.0 / absxk;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        i = ps_tmp + 1;
        for (ic = i; ic <= i; ic++) {
          pi[ic - 1] = 0.0;
        }
      }
      br = 0;
      for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
        br++;
        for (ib = br; ib <= br; ib += 7) {
          i = ps_tmp + 1;
          for (ic = i; ic <= i; ic++) {
            pi[ic - 1] += y_tmp[ib - 1] * scale;
          }
        }
      }
    }
  }
  *lambda = 0.0;
  for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
    *lambda += pi[ps_tmp] * (w[ps_tmp] * (tau + 1.0) - weq[ps_tmp]);
  }
}

/*
 * File trailer for hlblacklitterman.c
 *
 * [EOF]
 */