Portfolio optimization 投资组合优化 cvx

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Portfolio optimization

https://github.com/cvxgrp/cvx_short_course/blob/master/book/docs/applications/notebooks/portfolio_optimization.ipynb

https://github.com/mathnovel/DayCollection/blob/main/cvx/portfolio_potimization.ipynb

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在这个例子中，我们将展示如何使用CVXPY进行投资组合优化。首先介绍基本定义。在投资组合优化中，我们可以将一定金额的资金投入n种不同的资产中。选择每种资产i的投资比例wi，
i=1,...,n。

我们称w∈R^n为投资组合分配向量。当然有约束条件1^Tw=1。分配w_i<0表示短期头寸，即我们在资产i上借入股份出售，然后必须稍后替换。分配w≥0为长期投资组合。数量

||w||1=1^Tw+ + 1^Tw_-

被称为杠杆。

Asset returns
资产回报

我们只建模持有期为一个周期的投资。初始价格为p_i>0。期末价格为p_i^+>0。资产（分数）回报率为r_i=(p_i^+-p_i)/p_i。投资组合（分数）回报率是R=r^Tw。

常见的模型是r是一个随机变量，其均值E[r]=μ和协方差E[(r-μ)(r-μ)^T]=Σ。这意味着R是一个随机变量，其均值ER=μ^Tw，方差var(R)=w^TΣw。ER是投资组合的（平均）回报。var(R)是风险。风险也有时给出为std(R)=√(var(R))。

投资组合优化有两个相互竞争的目标：高回报和低风险。

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In the following code we compute and plot the optimal risk-return trade-off for 10 assets, restricting ourselves to a long only portfolio.

# Generate data for long only portfolio optimization.
import numpy as np
import scipy.sparse as sp

np.random.seed(1)
n = 10
mu = np.abs(np.random.randn(n, 1))
Sigma = np.random.randn(n, n)
Sigma = Sigma.T.dot(Sigma)

# Long only portfolio optimization.
import cvxpy as cp


w = cp.Variable(n)
gamma = cp.Parameter(nonneg=True)
ret = mu.T @ w
risk = cp.quad_form(w, Sigma)
prob = cp.Problem(cp.Maximize(ret - gamma * risk), [cp.sum(w) == 1, w >= 0])

# Compute trade-off curve.
SAMPLES = 100
risk_data = np.zeros(SAMPLES)
ret_data = np.zeros(SAMPLES)
gamma_vals = np.logspace(-2, 3, num=SAMPLES)
for i in range(SAMPLES):
    gamma.value = gamma_vals[i]
    prob.solve()
    risk_data[i] = cp.sqrt(risk).value
    ret_data[i] = ret.value

# Plot long only trade-off curve.
import matplotlib.pyplot as plt

%matplotlib inline
%config InlineBackend.figure_format = 'svg'

markers_on = [29, 40]
fig = plt.figure()
ax = fig.add_subplot(111)
plt.plot(risk_data, ret_data, "g-")
for marker in markers_on:
    plt.plot(risk_data[marker], ret_data[marker], "bs")
    ax.annotate(
        r"$\gamma = %.2f$" % gamma_vals[marker],
        xy=(risk_data[marker] + 0.08, ret_data[marker] - 0.03),
    )
for i in range(n):
    plt.plot(cp.sqrt(Sigma[i, i]).value, mu[i], "ro")
plt.xlabel("Standard deviation")
plt.ylabel("Return")
plt.show()

[laniakeawhite]

We plot below the return distributions for the two risk aversion values marked on the trade-off curve. Notice that the probability of a loss is near 0 for the low risk value and far above 0 for the high risk value.

# Plot return distributions for two points on the trade-off curve.
import scipy.stats as spstats


plt.figure()
for midx, idx in enumerate(markers_on):
    gamma.value = gamma_vals[idx]
    prob.solve()
    x = np.linspace(-2, 5, 1000)
    plt.plot(
        x,
        spstats.norm.pdf(x, ret.value, risk.value),
        label=r"$\gamma = %.2f$" % gamma.value,
    )

plt.xlabel("Return")
plt.ylabel("Density")
plt.legend(loc="upper right")
plt.show()

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# Portfolio optimization with leverage limit.
Lmax = cp.Parameter()
prob = cp.Problem(
    cp.Maximize(ret - gamma * risk), [cp.sum(w) == 1, cp.norm(w, 1) <= Lmax]
)

# Compute trade-off curve for each leverage limit.
L_vals = [1, 2, 4]
SAMPLES = 100
risk_data = np.zeros((len(L_vals), SAMPLES))
ret_data = np.zeros((len(L_vals), SAMPLES))
gamma_vals = np.logspace(-2, 3, num=SAMPLES)
w_vals = []
for k, L_val in enumerate(L_vals):
    for i in range(SAMPLES):
        Lmax.value = L_val
        gamma.value = gamma_vals[i]
        prob.solve(solver=cp.SCS)
        risk_data[k, i] = cp.sqrt(risk).value
        ret_data[k, i] = ret.value

# Plot trade-off curves for each leverage limit.
for idx, L_val in enumerate(L_vals):
    plt.plot(risk_data[idx, :], ret_data[idx, :], label=r"$L^{\max}$ = %d" % L_val)
for w_val in w_vals:
    w.value = w_val
    plt.plot(cp.sqrt(risk).value, ret.value, "bs")
plt.xlabel("Standard deviation")
plt.ylabel("Return")
plt.legend(loc="lower right")
plt.show()

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# Portfolio optimization with a leverage limit and a bound on risk.
prob = cp.Problem(cp.Maximize(ret), [cp.sum(w) == 1, cp.norm(w, 1) <= Lmax, risk <= 2])

# Compute solution for different leverage limits.
for k, L_val in enumerate(L_vals):
    Lmax.value = L_val
    prob.solve()
    w_vals.append(w.value)

# Plot bar graph of holdings for different leverage limits.
colors = ["b", "g", "r"]
indices = np.argsort(mu.flatten())
for idx, L_val in enumerate(L_vals):
    plt.bar(
        np.arange(1, n + 1) + 0.25 * idx - 0.375,
        w_vals[idx][indices],
        color=colors[idx],
        label=r"$L^{\max}$ = %d" % L_val,
        width=0.25,
    )
plt.ylabel(r"$w_i$", fontsize=16)
plt.xlabel(r"$i$", fontsize=16)
plt.xlim([1 - 0.375, 10 + 0.375])
plt.xticks(np.arange(1, n + 1))
plt.show()

[laniakeawhite]

# Generate data for factor model.
n = 600
m = 30
np.random.seed(1)
mu = np.abs(np.random.randn(n, 1))
Sigma_tilde = np.random.randn(m, m)
Sigma_tilde = Sigma_tilde.T.dot(Sigma_tilde)
D = sp.diags(np.random.uniform(0, 0.9, size=n))
F = np.random.randn(n, m)

# Factor model portfolio optimization.
w = cp.Variable(n)
f = cp.Variable(m)
gamma = cp.Parameter(nonneg=True)
Lmax = cp.Parameter()
ret = mu.T @ w
risk = cp.quad_form(f, Sigma_tilde) + cp.sum_squares(np.sqrt(D) @ w)
prob_factor = cp.Problem(
    cp.Maximize(ret - gamma * risk),
    [cp.sum(w) == 1, f == F.T @ w, cp.norm(w, 1) <= Lmax],
)

# Solve the factor model problem.
Lmax.value = 2
gamma.value = 0.1
prob_factor.solve(verbose=True)

from cvxpy.atoms.affine.wraps import psd_wrap

# Standard portfolio optimization with data from factor model.
risk = cp.quad_form(w, psd_wrap(F.dot(Sigma_tilde).dot(F.T) + D))
prob = cp.Problem(
    cp.Maximize(ret - gamma * risk), [cp.sum(w) == 1, cp.norm(w, 1) <= Lmax]
)

prob.solve(verbose=True, max_iter=100_000)

print("Factor model solve time = {}".format(prob_factor.solver_stats.solve_time))
print("Single model solve time = {}".format(prob.solver_stats.solve_time))