Question 1  问题 1

r = returns.mean().round(6)
print(r)

Question 3A: 问题 3A：

Expected_return = np.dot(x.T, new_shape_r).round(5).item()
print("Expected return " + str(Expected_return))
C = C.to_numpy() if isinstance(C, pd.DataFrame) else C
Risk = np.sqrt(np.matmul(np.matmul(x.T, C), x).item()).round(5)
print("Risk " + str(Risk))

Question 3B: 问题 3B：

def f(x):

def c(x):

return sum(x) - 1

        np.array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]),
        constraints=cons,
        tol=1e-15,
        bounds=[(0, None)] * 10
)
x_MVP = soln.x.reshape(10, 1)
print(x_MVP)

dollars_sum = 100000
dollars_inversment = x_MVP * dollars_sum
dollars_inversment.round(2)

expected_return = np.dot(soln.x, r).round(5)
print("Expected return " + str(float(expected_return)))

Risk 0.00774 风险 0.00774

grad = 2 * C @ soln.x
print("Gradient:", grad)
print("Is Gradient Close to 0:", np.allclose(grad, 0, atol=1e-8))
print("Sum of Weights is 1:", np.isclose(np.sum(soln.x), 1, atol=1e-8))
print("Are All Weights Non-Negative:", np.all(soln.x >= 0))

Gradient: [0.0001199 0.0001199 0.0001199 0.0001199 0.0001199
                                    0.0001199
    0.00013319 0.0001199 0.00013446 0.0001285 ]

1. For a specific investor with a risk aversion parameter of $t=0.06$$t=0.06$t=0.06t=0.06, without shortselling for any asset, their optimal investment proportions are as follows:
   对于风险规避参数为 $t=0.06$$t=0.06$t=0.06t=0.06 的特定投资者来说，在不卖空任何资产的情况下，其最优投资比例如下：

Question 4: 问题 4：

    mu_opt = (b + d * t) / a
    sig_opt = np.sqrt((1 + d * t ** 2) / a)

print(round(mu_opt, 6), round(sig_opt, 6))

print(round(Minimum_risk_mu, 6), round(Minimum_risk_sig, 6))

0.0005980 .0077

ret = []
sig = []
ret1 = []
sig1 = []
vertex_mu = b / a
for i in t_range:
    if (b + d * i) / a >= vertex_mu:
        ret.append((b + d * i) / a)
        sig.append(np.sqrt((1 + d * i**2) / a))
    else:
        ret1.append((b + d * i) / a)
        sig1.append(np.sqrt((1 + d * i**2) / a))
Utility_opt = t * mu_opt - 0.5 * sig_opt**2
sigma_range = np.linspace(0.005, 0.025, 5000)
mu_index = []
for i in sigma_range:
    mu_index.append((0.5 * i**2 + Utility_opt) / t)

plt.plot(sig, ret, label="Efficient Frontier")

• Optimal Portfolio 最佳投资组合
• Minimum Risk Portfolio 最低风险组合

1. The Minimum Variance Frontier (MVF) is represented as a hyperbola, given by the equation:
   最小方差边界（MVF）用双曲线表示，其方程为

4. The lower branch of the hyperbola (( $\mathrm{\setminus }mu<\mathrm{\setminus }frac\{b\}\{a\}))$$\mathrm{\setminus }mu<\mathrm{\setminus }frac\{b\}\{a\}))$\\mu < \\frac{b}{a}))\backslash m u<\backslash f r a c\{b\}\{a\})) forms the inefficient frontier, which corresponds to the critical line with ( $t<0$$t<0$t < 0t<0 ). This is depicted by the yellow dashed curve.
   双曲线 (( $\mathrm{\setminus }mu<\mathrm{\setminus }frac\{b\}\{a\}))$$\mathrm{\setminus }mu<\mathrm{\setminus }frac\{b\}\{a\}))$\\mu < \\frac{b}{a}))\backslash m u<\backslash f r a c\{b\}\{a\})) 的下分支构成无效率边界，与临界线 ( $t<0$$t<0$t < 0t<0 ) 相对应。黄色虚线表示这条曲线。
5. The purple dot on the plane represents the optimal portfolio when the risk aversion parameter ( t ) equals ( 0.06 ).
   平面上的紫色圆点代表风险规避参数 ( t ) 等于 ( 0.06 ) 时的最优投资组合。
6. The ten red dots on the plane denote the average returns and risk volatility of the ten individual stocks.
   平面上的十个红点表示十只个股的平均收益和风险波动。
7. For $(\mathrm{n}=10$$(\mathrm{n}=10$(n=10(\mathrm{n}=10 ), the feasible set ( $F$$F$FF ) is the interior of the hyperbola, also known as the bullet.
   对于 $(\mathrm{n}=10$$(\mathrm{n}=10$(n=10(\mathrm{n}=10 ），可行集 ( $F$$F$FF ) 是双曲线的内部，也称为子弹。

Question 5A: 问题 5A：

Expected Return Rate 预期回报率

Optimal Solution 最佳解决方案

new_optimal_portfolio = t * np.matmul(np.linalg.inv(C), r_bar)
print(new_optimal_portfolio)

[[-0.14790681]
    [ 0.19327912]
    [-0.17317226]
    [-0.05897882]
    [ 0.22757084]
    [-0.01199638]
    [-0.19561854]
    [ 0.27845899]
    [ 0.18774781]
    [ 0.1539242 ]]
    if determine_Value > 0:
        print("Lending portfolio")
    else:
        print("Borrowing portfolio")

0.546692

1. Allocation to Risk-Free Asset
1.无风险资产配置

3. Allocation to Risky Assets
3.风险资产分配

3. Strategic Insights and Risk Management
3.战略洞察和风险管理

4. Conclusion 4.结论

0.012408176479107264

0.0008940 .006684

In [46]: plt.figure(figsize=(60, 30))
在 [46] 中： plt.figure(figsize=(60, 30))
plt.plot(returns.std(), r, “o”, label=“10 Investment Stocks”, markersize=18)
plt.plot(returns.std(), r, "o", label="10 Investment Stocks", markersize=18)
plt.plot(sig_opt, mu_opt, “o”, label=“Optimal Portfolio with only risky asset fo
plt.plot(sig_opt, mu_opt, "o", label="仅有风险资产的最优投资组合")
plt.plot(sig_new, mu_new, “o”, label=“Optimal Portfolio with risk-free asset for
plt.plot(sig_new, mu_new, "o", label="Optimal Portfolio with risk-free asset for
plt.plot(sig, ret, label=“Efficient Frontier”)
plt.plot(sig, ret, label="Efficient Frontier")
plt.plot(sig1, ret1, label=“Inefficient Frontier”, linestyle=”-.”)
plt.plot(sig1, ret1, label="Inefficient Frontier", linestyle="-.")
plt.plot(sigma, r0 + sigma0 * np.sqrt(d) * sigma, label=“Capital Market Line”)
plt.plot(sigma, r0 + sigma0 * np.sqrt(d) * sigma, label="资本市场线")
plt.plot(0, r0, “o”, label=“Risk-free Asset”, markersize=18)
plt.plot(0, r0, "o", label="无风险资产", markersize=18)
plt.plot(sigma_m, mu_m, “*”, label=“Market Portfolio”, markersize=25)
plt.plot(sigma_m, mu_m, "*", label="市场投资组合", markersize=25)
plt.xlabel(r"$\sigma$", fontsize=30)
plt.ylabel(r"$\mu$", fontsize=30)
plt.title("Graph of Risk-free Asset, Market Portfolio, and the Capital Market Li
plt.title("Graph of Risk-free Asset, Market Portfolio, and the Capital Market Li")。
plt.legend(fontsize=30)
plt.show()