-
Notifications
You must be signed in to change notification settings - Fork 3
/
Markowitz.m
154 lines (128 loc) · 5.02 KB
/
Markowitz.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
assignment(prices)
% plot the following:
% 1. the boundary of the mean-variance frontier,
% 2. all points corresponding to individual assets,
% 3. the global minimum variance portfolio.
function assignment(prices)
% prices = get_prices();
[mu_annualized, sigma_annualized] = preprocess(prices);
riskFreeRate = 0.005;
hold on
% a
[mu_p, sigma_p, mu_gm, sigma2_gm, psi] = ...
efficient_frontier(mu_annualized, sigma_annualized);
% scatter(sqrt(diag(sigma_annualized)), mu_annaulized, '*');
color = 'blue';
plot(sigma_p, mu_p, color);
scatter(sqrt(sigma2_gm), mu_gm, 'o', color);
[mu_tangency, sigma2_tangency, mus, sigmas] = ...
calculate_tangency_portfolio(mu_gm, sigma2_gm, riskFreeRate, psi);
color = 'red';
scatter(sqrt(sigma2_tangency), mu_tangency, '*', color);
plot(sigmas, mus, color);
n = 1000;
mus = zeros(n, 1);
sigmas = zeros(n, 1);
% b
for gamma = 1:n
[mu, sigma] = eff_frontier_with_no_short_selling(...
mu_annualized, sigma_annualized, gamma * 0.1);
mus(gamma) = mu;
sigmas(gamma) = sigma;
end
color = 'black';
plot(sqrt(sigmas), mus, color)
% c
for gamma = 1 : n
[mu, sigma] = eff_frontier_with_constraints(...
mu_annualized, sigma_annualized, gamma);
mus(gamma) = mu;
sigmas(gamma) = sigma;
end
color = 'green';
plot(sqrt(sigmas), mus, color)
end
function prices = get_prices()
if(~exist('prices','var') || size(prices) ~= [1509, 10])
prices = zeros(1509, 10);
stocks = ["MSFT", "AAPL", "ORCL", "EBAY", "GOOG", "INTC", "BBBY", "MMM", "TEVA", "GE"];
for i = 1:length(stocks)
stock = stocks(i);
data = getMarketData(char(stock), '1-Jan-2012', '31-Dec-2017', '1d', 5);
prices(:, i) = data;
end
end
end
%% arithmatic returns
function [mu_annualized, sigma_annualized, sigma2_gm, mu_gm] = preprocess(prices)
arithmatic_returns = prices(2:end, :) ./ prices(1:end-1, :) - 1;
mu_daily = mean(arithmatic_returns);
sigma_daily_matrix = cov(arithmatic_returns);
rho_daily = corrcov(sigma_daily_matrix);
sigma_daily_vector = sqrt(diag(sigma_daily_matrix));
%% annulaized returns
days = 252;
% µ(a)i = (µ(d)i + 1)N +1 1
mu_annualized = (mu_daily + 1) .^ days - 1;
mu_annualized = mu_annualized';
% ?(a)i,j?(a)i?(a)j=(?(d)i,j?(d)i?(d)j+ (?(d)i+ 1)(?(d)j+ 1))N?(?(d)i+1)N(?(d)j+ 1)N.
sigma_annualized = (sigma_daily_matrix + (mu_daily + 1)' * (mu_daily + 1)).^ days ...
- ((mu_daily + 1) .^ days)' * ((mu_daily + 1) .^ days);
end
function [mu_p, sigma_p, mu_gm, sigma2_gm, psi] = efficient_frontier(mu_annaulized, sigma_annualized)
assets = length(mu_annaulized);
inverseAnnualizedCovariance = inv(sigma_annualized);
B = mu_annaulized' * inverseAnnualizedCovariance * ones(assets, 1);
A = sum(sum(inverseAnnualizedCovariance));
C = mu_annaulized' * inverseAnnualizedCovariance * mu_annaulized;
% solving for the global minimum variance portfolio
mu_gm = B/A; % expected return
sigma2_gm = 1/A;
pi = inv(sigma_annualized) * ones(assets, 1) / A;
psi = A * C - B^2;
% plot the efficient frontier from mu = -3 to mu = 3, that is for many
% gammas
mu_p = linspace(-3, 3, 1000);
sigma_p = sqrt(sigma2_gm + ((mu_p - mu_gm) .^ 2) / (psi * sigma2_gm));
end
function [mu_tangency, sigma2_tangency, mus, sigmas] = ...
calculate_tangency_portfolio(mu_gm, sigma2_gm, riskFreeRate, psi)
sigma2_tangency = ((sigma2_gm) ^ 3 * psi) / (mu_gm - riskFreeRate) ^ 2 + sigma2_gm;
mu_tangency = ((sigma2_gm) ^ 2 * psi) / (mu_gm - riskFreeRate) + mu_gm;
m = (mu_tangency - riskFreeRate) / sqrt (sigma2_tangency); % sharpe ratio
x = linspace(0, 2, 100);
y = m * x + riskFreeRate;
mus = y;
sigmas = x;
end
function [mu, sigma] = eff_frontier_with_no_short_selling(mu_annualized, sigma_annualized, gamma)
% no short selling means the lower bound should be zero
w = optimvar('w', 10, 1, 'LowerBound', 0);
problem = optimproblem('ObjectiveSense','max');
options = optimoptions('quadprog','Display','off');
problem.Objective = w' * mu_annualized - gamma / 2 * w' * sigma_annualized * w;
cons1 = sum(w) == 1;
problem.Constraints.cons1 = cons1;
% showproblem(problem);
pi = solve(problem, 'Options', options);
pi = pi.w;
mu = pi' * mu_annualized;
sigma = pi' * sigma_annualized * pi;
end
function [mu, sigma] = eff_frontier_with_constraints(mu_annualized, sigma_annualized, gamma)
w = optimvar('w', 10, 1);
problem = optimproblem('ObjectiveSense','max');
options = optimoptions('quadprog','Display','off');
problem.Objective = w' * mu_annualized - gamma / 2 * w' * sigma_annualized * w;
cons1 = sum(w) == 1;
cons2 = w(1) + w(2) >= 0.05;
cons3 = w(1) + w(2) <= 0.1;
problem.Constraints.cons1 = cons1;
problem.Constraints.cons2 = cons2;
problem.Constraints.cons3 = cons3;
% showproblem(problem);
pi = solve(problem, 'Options', options);
pi = pi.w;
mu = pi' * mu_annualized;
sigma = pi' * sigma_annualized * pi;
end