Multi-period Example

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import numpy as np
import pandas as pd
import cvxpy as cp
import numpy as np
import pandas as pd
import cvxpy as cp

Suppose $W_0=\$ 500$. Determine the optimal amount of money $x_0$ (in $\$$ ) to invest in the risky asset at time 0 to maximize the expected logarithmic utility of your wealth $W_1$ at time 1 (end of the investment period).

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p = np.array([0.5,0.5])
r = np.array([0.25, -0.2])
W0 = 500
L1 = 0
x0 = cp.Variable(1)

obj = cp.Maximize(p.T@cp.log(W0 + r*x0 - L1))
constraints = [x0 >= 0, x0 <= W0]

prob = cp.Problem(obj,constraints)
# prob.solve(solver=cp.GUROBI)
prob.solve()
print("x0 = ", x0.value[0])
print("Ratio = ", x0.value[0]/(W0 - L1))
p = np.array([0.5,0.5])
r = np.array([0.25, -0.2])
W0 = 500
L1 = 0
x0 = cp.Variable(1)

obj = cp.Maximize(p.T@cp.log(W0 + r*x0 - L1))
constraints = [x0 >= 0, x0 <= W0]

prob = cp.Problem(obj,constraints)
# prob.solve(solver=cp.GUROBI)
prob.solve()
print("x0 = ", x0.value[0])
print("Ratio = ", x0.value[0]/(W0 - L1))

x0 =  249.9449220288116
Ratio =  0.49988984405762316

Suppose $W_0=\$ 500$ and $L_2=\$ 100$. Determine the optimal amount of money $x_0$ (in $\$$ ) to invest in the risky asset at time 0 to maximize the expected logarithmic utility of your wealth $W_2$ at time $t=2$.

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p = np.array([0.25,0.25,0.25,0.25])
r = np.array([[0.25, 0.25],
              [0.25, -0.2],
              [-0.2, 0.25],
              [-0.2, -0.2]])

W0 = 500
L2 = 100
x = cp.Variable(2)

obj = cp.Maximize(p.T@cp.log(W0 + r@x - L2))
constraints = [x >= 0]

prob = cp.Problem(obj,constraints)
prob.solve()
print(x.value)
p = np.array([0.25,0.25,0.25,0.25])
r = np.array([[0.25, 0.25],
              [0.25, -0.2],
              [-0.2, 0.25],
              [-0.2, -0.2]])

W0 = 500
L2 = 100
x = cp.Variable(2)

obj = cp.Maximize(p.T@cp.log(W0 + r@x - L2))
constraints = [x >= 0]

prob = cp.Problem(obj,constraints)
prob.solve()
print(x.value)

[197.555372 197.555372]