# Generate data for long only portfolio optimization.
srand(9);
n = 10;
mu = abs(randn(n, 1));
Sigma = randn(n, n);
Sigma = Sigma' * Sigma;
# Long only portfolio optimization.
using Convex, SCS, ECOS
set_default_solver(SCSSolver(verbose=0));
w = Variable(n);
ret = sum(mu' * w);
risk = sum(quad_form(w, Sigma));
# Compute trade-off curve.
SAMPLES = 100;
risk_data = zeros(SAMPLES);
ret_data = zeros(SAMPLES);
gamma_vals = logspace(-2, 3, SAMPLES);
for i=1:SAMPLES
gamma = gamma_vals[i];
problem = maximize(ret - gamma*risk, [sum(w) == 1, w >= 0]);
solve!(problem);
risk_data[i] = sqrt(evaluate(risk));
ret_data[i] = evaluate(ret);
end
using Gadfly
markers_on = [29, 40];
labels = [@sprintf("γ = %0.2f", gamma_vals[marker]) for marker in markers_on];
plot(
layer(x=[sqrt(Sigma[i,i]) for i=1:n], y=mu,
Geom.point, Theme(default_color=color("red"))),
layer(x=risk_data, y=ret_data,
Geom.line, Theme(default_color=color("green"))),
layer(x=risk_data[markers_on], y=ret_data[markers_on], label=labels,
Geom.point, Geom.label, Theme(default_color=color("blue"))),
Guide.XLabel("Risk"), Guide.YLabel("Return")
)
# Plot return distributions for two points on the trade-off curve.
using DataFrames, Distributions
xdata = linspace(-2, 5, 1000);
df = DataFrame(x=xdata,
y=pdf(Normal(ret_data[markers_on[1]], risk_data[markers_on[1]]), xdata),
label=@sprintf("γ = %0.2f", gamma_vals[markers_on[1]]));
for i=2:length(markers_on)
m = markers_on[i];
df = vcat(df, DataFrame(x=xdata,
y=pdf(Normal(ret_data[m], risk_data[m]), xdata),
label=@sprintf("γ = %0.2f", gamma_vals[m])));
end
plot(df, x="x", y="y", color="label", Geom.line, Guide.XLabel("Return"), Guide.YLabel("Density"))
# Portfolio optimization with leverage limit.
# Compute trade-off curve for each leverage limit.
L_vals = [1, 2, 4];
SAMPLES = 100;
risk_data = zeros(length(L_vals), SAMPLES);
ret_data = zeros(length(L_vals), SAMPLES);
for k=1:length(L_vals)
for i=1:SAMPLES
Lmax = L_vals[k];
gamma = gamma_vals[i];
problem = maximize(ret - gamma*risk,[sum(w) == 1, norm(w, 1) <= Lmax]);
solve!(problem);
risk_data[k, i] = sqrt(evaluate(risk));
ret_data[k, i] = evaluate(ret);
end
end
df = DataFrame(x=vec(risk_data[1,:]), y=vec(ret_data[1,:]), label=@sprintf("Lmax = %d", L_vals[1]));
for i=2:length(L_vals)
df = vcat(df, DataFrame(x=vec(risk_data[i,:]), y=vec(ret_data[i,:]), label=@sprintf("Lmax = %d", L_vals[i])));
end
plot(df, x="x", y="y", color="label", Geom.line, Guide.XLabel("Return"), Guide.YLabel("Standard deviation"))
# Portfolio optimization with a leverage limit and a bound on risk.
# Compute solution for different leverage limits.
w_vals = zeros(n, length(L_vals));
for i=1:length(L_vals)
problem = maximize(ret, [sum(w) == 1, norm(w, 1) <= L_vals[i], risk <= 2]);
solve!(problem);
w_vals[:, i] = evaluate(w);
end
# Generate data for factor model.
n = 3000;
m = 50;
mu = abs(randn(n, 1));
Sigma_tilde = randn(m, m);
Sigma_tilde = Sigma_tilde' * Sigma_tilde;
D = diagm(0.9 * rand(n));
F = randn(n, m);
# Factor model portfolio optimization.
w = Variable(n);
f = F' * w;
ret = sum(mu' * w);
risk = quad_form(f, Sigma_tilde) + quad_form(w, D);
# Solve the factor model problem.
Lmax = 2
gamma = 0.1
factor_problem = maximize(ret - gamma*risk, [sum(w) == 1, norm(w, 1) <= Lmax]);
solve!(factor_problem, ECOSSolver())
ECOS 1.0.5 - (c) A. Domahidi, ETH Zurich & embotech 2012-14. Support: ecos@embotech.com It pcost dcost gap pres dres k/t mu step IR 0 -7.928e-01 -1.279e+01 +2e+04 3e+01 1e+02 1e+00 3e+00 N/A 4 5 - 1 -1.139e+01 -1.274e+01 +2e+03 4e+00 1e+01 4e-01 4e-01 0.9899 4 3 4 2 -7.570e+00 -7.792e+00 +4e+02 6e-01 2e+00 2e-02 7e-02 0.8105 3 3 3 3 -7.340e+00 -7.529e+00 +4e+02 5e-01 2e+00 2e-02 6e-02 0.2022 4 4 4 4 -6.937e+00 -7.099e+00 +4e+02 4e-01 1e+00 2e-02 6e-02 0.2089 5 4 5 5 -6.565e+00 -6.694e+00 +3e+02 3e-01 1e+00 2e-02 5e-02 0.2735 4 4 4 6 -5.876e+00 -5.948e+00 +2e+02 2e-01 7e-01 1e-02 3e-02 0.5625 5 5 5 7 -5.533e+00 -5.582e+00 +2e+02 1e-01 4e-01 8e-03 3e-02 0.3618 5 5 5 8 -5.313e+00 -5.348e+00 +1e+02 1e-01 3e-01 5e-03 2e-02 0.4704 5 5 5 9 -4.961e+00 -4.975e+00 +5e+01 4e-02 1e-01 2e-03 9e-03 0.7557 5 5 5 10 -4.989e+00 -5.003e+00 +5e+01 4e-02 1e-01 2e-03 8e-03 0.2647 5 5 5 11 -4.857e+00 -4.864e+00 +3e+01 2e-02 6e-02 8e-04 4e-03 0.5682 5 5 5 12 -4.819e+00 -4.824e+00 +2e+01 1e-02 4e-02 5e-04 3e-03 0.5266 5 5 5 13 -4.789e+00 -4.791e+00 +8e+00 5e-03 2e-02 2e-04 1e-03 0.8850 5 5 5 14 -4.774e+00 -4.775e+00 +4e+00 2e-03 8e-03 8e-05 6e-04 0.7023 5 5 5 15 -4.768e+00 -4.768e+00 +2e+00 1e-03 4e-03 4e-05 3e-04 0.6936 5 5 5 16 -4.766e+00 -4.766e+00 +1e+00 6e-04 2e-03 2e-05 2e-04 0.7589 5 5 5 17 -4.765e+00 -4.765e+00 +3e-01 2e-04 6e-04 5e-06 4e-05 0.7661 5 5 5 18 -4.764e+00 -4.764e+00 +2e-02 2e-05 5e-05 5e-07 4e-06 0.9461 5 5 5 19 -4.764e+00 -4.764e+00 +9e-04 6e-07 2e-06 2e-08 2e-07 0.9777 5 5 5 20 -4.764e+00 -4.764e+00 +1e-05 1e-08 3e-08 3e-10 2e-09 0.9845 5 5 5 21 -4.764e+00 -4.764e+00 +3e-07 2e-10 7e-10 6e-12 5e-11 0.9791 5 5 5 OPTIMAL (within feastol=6.7e-10, reltol=6.5e-08, abstol=3.1e-07). Runtime: 1.193969 seconds.
# Standard portfolio optimization with data from factor model.
risk = quad_form(w, F * Sigma_tilde * F' + D);
problem = maximize(ret - gamma*risk, [sum(w) == 1, norm(w, 1) <= Lmax]);
# Warning: This takes a long time.
# solve!(problem, ECOSSolver())