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 hlblacklittermanfunction [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: 0.026728 seconds Execution Time - MEX function : 0.017718 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/. .. _clang-format buildInfo.mat codeInfo.mat codedescriptor.dmr compileInfo.mat examples hlblacklitterman.a hlblacklitterman.c hlblacklitterman.h hlblacklitterman.o hlblacklitterman_data.h hlblacklitterman_initialize.c hlblacklitterman_initialize.h hlblacklitterman_initialize.o hlblacklitterman_rtw.mk hlblacklitterman_terminate.c hlblacklitterman_terminate.h hlblacklitterman_terminate.o hlblacklitterman_types.h interface inv.c inv.h inv.o mtimes.c mtimes.h mtimes.o pinv.c pinv.h pinv.o rtGetInf.c rtGetInf.h rtGetInf.o rtGetNaN.c rtGetNaN.h rtGetNaN.o rt_nonfinite.c rt_nonfinite.h rt_nonfinite.o rtw_proj.tmw rtwtypes.h
Inspect the C Code for the hlblacklitterman.c Function
type codegen/lib/hlblacklitterman/hlblacklitterman.c/*
* Prerelease License - for engineering feedback and testing purposes
* only. Not for sale.
* File: hlblacklitterman.c
*
* MATLAB Coder version : 24.2
* C/C++ source code generated on : 20-Jul-2024 12:20:30
*/
/* Include Files */
#include "hlblacklitterman.h"
#include "inv.h"
#include "mtimes.h"
#include "pinv.h"
#include "rt_nonfinite.h"
#include <emmintrin.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])
{
__m128d r;
__m128d r1;
double b_er_tmp[49];
double ts[49];
double er_tmp[7];
double pi[7];
double b_P;
double b_tmp;
double c_P;
double d;
double d1;
double d2;
int i;
int i1;
int ps_tmp;
/* Reverse optimize and back out the equilibrium returns */
/* This is formula (12) page 6. */
for (i = 0; i < 7; i++) {
d = 0.0;
for (i1 = 0; i1 < 7; i1++) {
d += weq[i1] * sigma[i1 + 7 * i];
}
pi[i] = d * delta;
}
/* We use tau * sigma many places so just compute it once */
for (i = 0; i <= 46; i += 2) {
_mm_storeu_pd(&ts[i],
_mm_mul_pd(_mm_set1_pd(tau), _mm_loadu_pd(&sigma[i])));
}
ts[48] = tau * sigma[48];
/* Compute posterior estimate of the mean */
/* This is a simplified version of formula (8) on page 4. */
b_P = 0.0;
c_P = 0.0;
for (i = 0; i < 7; i++) {
d = 0.0;
d1 = 0.0;
for (i1 = 0; i1 < 7; i1++) {
d2 = P[i1];
d += ts[i + 7 * i1] * d2;
d1 += d2 * ts[i1 + 7 * i];
}
er_tmp[i] = d;
d = P[i];
b_P += d1 * d;
c_P += d * pi[i];
}
b_tmp = 1.0 / (b_P + Omega);
b_P = Q - c_P;
/* We can also do it the long way to illustrate that d1 + d2 = I */
c_P = 1.0 / Omega;
/* Compute posterior estimate of the uncertainty in the mean */
/* This is a simplified and combined version of formulas (9) and (15) */
r = _mm_set1_pd(b_tmp);
for (i = 0; i < 7; i++) {
__m128d r2;
er[i] = pi[i] + er_tmp[i] * b_tmp * b_P;
r1 = _mm_loadu_pd(&er_tmp[0]);
d = P[i];
r2 = _mm_set1_pd(d);
_mm_storeu_pd(&b_er_tmp[7 * i], _mm_mul_pd(_mm_mul_pd(r1, r), r2));
r1 = _mm_loadu_pd(&er_tmp[2]);
_mm_storeu_pd(&b_er_tmp[7 * i + 2], _mm_mul_pd(_mm_mul_pd(r1, r), r2));
r1 = _mm_loadu_pd(&er_tmp[4]);
_mm_storeu_pd(&b_er_tmp[7 * i + 4], _mm_mul_pd(_mm_mul_pd(r1, r), r2));
b_er_tmp[7 * i + 6] = er_tmp[6] * b_tmp * d;
}
for (i = 0; i < 7; i++) {
for (i1 = 0; i1 < 7; i1++) {
d = 0.0;
for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
d += b_er_tmp[i + 7 * ps_tmp] * ts[ps_tmp + 7 * i1];
}
ps_tmp = i + 7 * i1;
ps[ps_tmp] = ts[ps_tmp] - d;
}
}
/* 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, b_er_tmp);
for (i = 0; i < 7; i++) {
for (i1 = 0; i1 < 7; i1++) {
d = 0.0;
for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
d += b_er_tmp[i + 7 * ps_tmp] * ps[ps_tmp + 7 * i1];
}
ts[i + 7 * i1] = d;
}
}
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;
r = _mm_set1_pd(c_P);
for (i = 0; i < 7; i++) {
r1 = _mm_set1_pd(P[i]);
_mm_storeu_pd(&b_er_tmp[7 * i],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[0]), r), r1));
_mm_storeu_pd(&b_er_tmp[7 * i + 2],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[2]), r), r1));
_mm_storeu_pd(&b_er_tmp[7 * i + 4],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[4]), r), r1));
b_er_tmp[7 * i + 6] = P[6] * c_P * P[i];
}
for (i = 0; i < 7; i++) {
for (i1 = 0; i1 < 7; i1++) {
d = 0.0;
for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
d += b_er_tmp[i + 7 * ps_tmp] * ps[ps_tmp + 7 * i1];
}
ts[i + 7 * i1] = d;
}
}
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;
r = _mm_set1_pd(c_P);
for (i = 0; i < 7; i++) {
r1 = _mm_set1_pd(P[i]);
_mm_storeu_pd(&b_er_tmp[7 * i],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[0]), r), r1));
_mm_storeu_pd(&b_er_tmp[7 * i + 2],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[2]), r), r1));
_mm_storeu_pd(&b_er_tmp[7 * i + 4],
_mm_mul_pd(_mm_mul_pd(_mm_loadu_pd(&P[4]), r), r1));
b_er_tmp[7 * i + 6] = P[6] * c_P * P[i];
}
for (i = 0; i < 7; i++) {
for (i1 = 0; i1 < 7; i1++) {
d = 0.0;
for (ps_tmp = 0; ps_tmp < 7; ps_tmp++) {
d += b_er_tmp[i + 7 * ps_tmp] * ps[ps_tmp + 7 * i1];
}
ts[i + 7 * i1] = d;
}
}
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 <= 46; i += 2) {
r = _mm_loadu_pd(&ps[i]);
_mm_storeu_pd(
&b_er_tmp[i],
_mm_mul_pd(_mm_set1_pd(delta), _mm_add_pd(_mm_loadu_pd(&sigma[i]), r)));
}
b_er_tmp[48] = delta * (sigma[48] + ps[48]);
inv(b_er_tmp, ts);
/* Compute posterior weights based on uncertainty in mean and covariance */
/* 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++) {
d = 0.0;
d1 = 0.0;
d2 = 0.0;
for (i1 = 0; i1 < 7; i1++) {
b_P = ts[i1 + 7 * i];
b_tmp = er[i1] * b_P;
d += b_tmp;
d1 += pi[i1] * b_P;
d2 += b_tmp;
}
w[i] = d;
pw[i] = d1;
er_tmp[i] = d2 * (tau + 1.0) - weq[i];
}
pinv(P, pi);
*lambda = mtimes(pi, er_tmp);
}
/*
* File trailer for hlblacklitterman.c
*
* [EOF]
*/
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