SF1811 HW2

% vvvvvvvvvvvvvvv ARBITRARY ASSETS vvvvvvvvvvvvvvv
clc
close all
clear all

% expected return of the portfolio
r = (3:0.25:9)';

mu_ = zeros(size(r, 1), 4);
sigma = zeros(size(r, 1), 4);

[C, mu] = getCorrMatrixAndExpectedReturn();

% minimize      x'Cx
% subject to    mu'*x = r
%               sum(x) = 1
%               x >= 0
% A summarizes the equality constraints, s.t. Ax = b
A = [
    mu'
    ones(size(mu, 1), 1)'
];

Z = null(A);

if rank(Z) == size(A, 2)
    disp("The nullspace of A spans the input space, no solution exists")
    return
end

if min(eig(C) >= 0) == 0
    disp("C is not positive semi-definite")
    return
end

% for each expected return, r, of the portfolio we wish to find its
% volatility (risk)
for i = 1:size(r, 1)
    rr = r(i);
    b = [ rr; 1 ]; % since mu'x = r and sum(x) = 1

    % solve using quadprog and equality constraints
    x = quadprog(2*C, [], [], [], A, b, zeros(size(A, 2), 1), []);
    sigma(i, 1) = sqrt(x' * C * x);
    mu_(i, 1) = mu'*x;

    % let sum(x) <= 1
    x = quadprog(2*C, [], A(2, :), b(2), A(1, :), b(1), zeros(size(A, 2), 1), []);
    sigma(i, 2) = sqrt(x' * C * x);
    mu_(i, 2) = mu'*x;

    % keep sum(x) = 1 but let mu'x >= r, i.e. -mu'x <= -r
    x = quadprog(2*C, [], -A(1, :), -b(1), A(2, :), b(2), zeros(size(A, 2), 1), []);
    sigma(i, 3) = sqrt(x' * C * x);
    mu_(i, 3) = mu'*x;

    % allow short selling but keep mu'x = r
    x = quadprog(2*C, [], [], [], A, b, [], []);
    sigma(i, 4) = sqrt(x' * C * x);
    mu_(i, 4) = mu'*x;
end

xmin = min(min(sigma));
xmax = max(max(sigma));
ymin = min(min(mu_));
ymax = max(max(mu_));

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Portfolios [1]')
title('Portfolios [1], 100% of Capital Invested')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 2), mu_(:, 2), '-o', 'Color', [.85 .325 .098], 'DisplayName', 'Portfolios [2]')
title('Portfolios [2], No Constraint on Portion of Capital Invested')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 3), mu_(:, 3), '-o', 'Color', [.929 .694 .125], 'DisplayName', 'Portfolios [3]')
title('Portfolios [3], 100% of Capital Invested and ER >= r')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 4), mu_(:, 4), '-o', 'Color', [.494 .184 .556], 'DisplayName', 'Portfolios [4]')
title('Portfolios [4], 100% of Capital Invested and Short Selling Allowed')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 4), mu_(:, 4), '-o', 'Color', [.494 .184 .556], 'DisplayName', 'Portfolios [4]')
hold on
plot(sigma(:, 2), mu_(:, 2), '-o', 'Color', [.85 .325 .098], 'DisplayName', 'Portfolios [2]')
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Portfolios [1]')
plot(sigma(:, 3), mu_(:, 3), '-o', 'Color', [.929 .694 .125], 'DisplayName', 'Portfolios [3]')
hold off
title('Portfolios [1], [2], [3] and [4], Comparison')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Portfolios [1]')
hold on
plot(sigma(:, 2), mu_(:, 2), '-o', 'Color', [.85 .325 .098], 'DisplayName', 'Portfolios [2]')
hold off
title('Portfolios [1] and [2], Comparison')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Portfolios [1]')
hold on
plot(sigma(:, 3), mu_(:, 3), '-o', 'Color', [.85 .325 .098], 'DisplayName', 'Portfolios [3]')
hold off
title('Portfolios [1] and [3], Comparison')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Portfolios [1]')
hold on
plot(sigma(:, 4), mu_(:, 4), '-o', 'Color', [.85 .325 .098], 'DisplayName', 'Portfolios [4]')
hold off
title('Portfolios [1] and [4], Comparison')
xlabel('Volatility')
ylabel('Expected Return')
axis([xmin xmax ymin ymax]);
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

function [C, mu] = getCorrMatrixAndExpectedReturn()
    % generate the correlation matrix C and expected return of assets mu
    n = 8;
    Corr = zeros(n, n);

    for i = 1:n
        for j = 1:n
            Corr(i, j) = (-1)^abs(i-j)/(abs(i-j)+1);
        end
    end

    sigma = zeros(n, 1);
    mu = zeros(n, 1);
    sigma(1) = 2;
    mu(1) = 3;

    for i = 1:n-1
        sigma(i+1) = sigma(i) + 2*rand;
        mu(i+1) = mu(i) + 1;
    end

    D = diag(sigma);
    C2 = D * Corr * D;
    C = 0.5 * (C2 + C2');
end


% vvvvvvvvvvvvvvv REAL-WORLD ASSETS vvvvvvvvvvvvvvv
close all
clear all
clc

% AAPL TSLA JPM AMZN WMT T(AT&T) CVX(CHEVRON) BRK.A(Berkshire Hathaway A) UNH(UnitedHealth Group)
CC = [
 1.00 0.10 0.26 0.34 0.18 0.09 0.17 0.20 0.20
 0.10 1.00 0.11 0.05 -.06 -.18 0.00 0.16 0.18
 0.26 0.11 1.00 0.28 0.09 0.15 0.51 0.57 0.32
 0.34 0.05 0.28 1.00 0.08 0.12 0.19 0.29 0.22
 0.18 -.06 0.09 0.08 1.00 0.21 0.09 0.30 0.16
 0.09 -.18 0.15 0.12 0.21 1.00 0.43 0.25 0.00
 0.17 0.00 0.51 0.19 0.09 0.43 1.00 0.42 0.28
 0.20 0.16 0.57 0.29 0.30 0.25 0.42 1.00 0.30
 0.20 0.18 0.32 0.22 0.16 0.00 0.28 0.30 1.00
];

ST_DEV = [
 .0743
 .1591
 .0696
 .0803
 .0481
 .0452
 .0541
 .0386
 .0533
];

mu = ([
 .256
 .322
 .1751
 .3467
 .1287
 .1067
 .1003
 .1136
 .2943
] + 1).^(1/12) - 1;

C = zeros(size(CC, 1));

rr = min(mu):0.0005:max(mu);

A = [
 mu'
 ones(size(mu, 1), 1)'
];

sigma = zeros(size(C, 2), 3);
mu_ = zeros(size(C, 2), 3);

for i=1:size(CC, 1)
    for j=1:size(CC, 2)
        if i == j
            C(i, i) = round(ST_DEV(i)^2, 5);
        else
            C(i, j) = round(CC(i, j) * ST_DEV(i) * ST_DEV(j), 5);
        end
    end
end

for i=1:size(rr, 2)
    r = rr(i);
    b = [ r; 1 ];

    x = quadprog(2*C, [], [], [], A, b, zeros(size(A, 2), 1), []);
    sigma(i, 1) = sqrt(x' * C * x);
    mu_(i, 1) = mu' * x;

    x = quadprog(2*C, [], [ -A(1, :); A(2, :) ], [ -b(1); b(2) ], [], [], zeros(size(A, 2), 1), []);
    sigma(i, 2) = sqrt(x' * C * x);
    mu_(i, 2) = mu' * x;

    x = quadprog(2*C, [], [ -A(1, :); A(2, :) ], [ -b(1); b(2) ], [], [], [], []);
    sigma(i, 3) = sqrt(x' * C * x);
    mu_(i, 3) = mu' * x;
end

mu_ = ((mu_ + 1).^12 - 1) * 100;

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Standard Problem')
hold on
plot(sigma(:, 2), mu_(:, 2), '-o', 'Color', [.85 .325 .098], 'DisplayName', "c'x ≥ r and any portion of capital invested")
hold off
title('Optimal Portfolios')
xlabel('Volatility')
ylabel('Expected Return (CAGR)')
axis([ 0 max(max(sigma)) min(min(mu_))*.9 max(max(mu_))*1.2 ])
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 1), mu_(:, 1), '-o', 'Color', [0 .447 .741], 'DisplayName', 'Standard Problem')
hold on
plot(sigma(:, 3), mu_(:, 3), '-o', 'Color', [.929 .694 .125], 'DisplayName', 'Short Selling Allowed and any portion of capital invested')
hold off
title('Optimal Portfolios')
xlabel('Volatility')
ylabel('Expected Return (CAGR)')
axis([ 0 max(max(sigma)) min(min(mu_))*.9 max(max(mu_))*1.2 ])
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)

figure('color', [1 1 1])
plot(sigma(:, 2), mu_(:, 2), '-o', 'Color', [.85 .325 .098], 'DisplayName', "c'x ≥ r and any portion of capital invested")
hold on
plot(sigma(:, 3), mu_(:, 3), '-o', 'Color', [.929 .694 .125], 'DisplayName', 'Short Selling Allowed and any portion of capital invested')
hold off
title('Optimal Portfolios')
xlabel('Volatility')
ylabel('Expected Return (CAGR)')
axis([ 0 max(max(sigma)) min(min(mu_))*.9 max(max(mu_))*1.2 ])
legend
set(findall(gcf, '-property', 'FontSize'), 'FontSize', 14)