# 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] */ ```