# 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: 0.026728 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] */ ```