An investor is endowed with a certain level of wealth equal to e = 2. Consumption is taken as the numeraire. So, e = 2 means an endowment of two units of consumer goods. He must allocate his current...

An investor is endowed with a certain level of wealth equal to e = 2. Consumption is taken as the numeraire. So, e = 2 means an endowment of two units of consumer goods. He must allocate his current wealth across current consumption c0 and 3 assets, where one unit of asset k ∈ 0, 1, 2 is currently valued at pk and will be worth Zk tomorrow. Asset 0 is a safe asset with p0 =1 and z0 =R>0 is the safe return on this riskless asset. Hence, r≡R−1 is the risk-less interest rate. Assets 1 and 2, however, are contingent assets, so their returns are subject to contingent states, denoted by s. The investor’s portfolio optimization problem is

(1) max u(c0) + E [(1/1+r) x u(c1(s)]

subject to
(2) Σ(k=0, 2) p_k*ω_k + c0 − e = 0

where c1(s) = Σ(k=0, 2) ω_k * z_k(s) is the investor’s state-s consumption tomorrow, 1/(1+r) is the discount rate, the investor plans to purchase ω_k units of asset k today, and (2) is an equality constraint, implying that current consumption c0 plus current investment Σ(k=0, 2) p_k*ω_k must be equal to today’s endowment e = 2. We assume that assets 1 and 2 confronts 4 contingent states, so that the final value (or gross return) of a risky asset has 4 possible values. Since all asset returns are independent of one another, the investor’s investment portfolio confronts 16 (= 4^2) contingent states. Further, assume that all these 16 contingent states are uniformly distributed over the state space s = {sij | i, j, ∈ {1, 2, 3, 4}}, as described as follows:

s11 s12 s13 s14 s21 s22 s23 s24 s31 s32 s33 s34 s41 s42 s43 s44

where sij means that subscript i is the ith return on asset 1 and subscript j is jth return on asset 2, and state sij occurs with probability π(sij) = 1/16. In state sij, the contingent return on asset k ∈ {0,1} is denoted by Zk = zk(sij). This permits to reduce (1) to the following finite, computable optimization problem

max [c0, wi] u(c_0) + E (1/(1+r)) ΣΣ π(sij)u(Σ wk*zk*sij)

subject to: Σ p_k*w_k + c_0 - e = 0
We further parametrized this problem using the following assumptions and parameter values:
• u(c) = −e−ac, a ∈ {0.5, 1.0, 5.0}. This implies the coefficient of absolute risk aversion equal
to parameter a.
• p0 = p1 = p2 = 1.
• Z0 = R = 1.0 in whatever state, implying r = 0%.
• Z1 is uniformly distributed over {0.72, 0.92, 1.12, 1.32}.
• Z2 is uniformly distributed over {0.86, 0.96, 1.06, 1.16}.
• Z1 and Z2 are independent random variables.
• e = 2 (aggregate wealth endowment).
Given these parameters, we can make more sense of the above state-space matrix s. For instance, as state s12 occurs, the investment portfolio delivers a return equal to ω0 × 1 + ω1 × 0.72 + ω2 × 0.96, and as state s34 occurs, the return is given by ω0 ×1+ω1 ×1.12+ω2 ×1.16.
Write a Python program to solve the above constrained maximization problem based on the Penalty Function Method. For such a program, the initial penalty parameter is set at P = 10 and you are free to determine the way to raise penalty in iterations. Your Python program must use each of the Newton algorithm discussed in class.

For each of these algorithms, the initial guess is set at c0 = 1.0 and ω1 = ω2 = ω3 = 0.33, the error tolerance is set at tol = 10−6. You are free to determine whether to use relative or absolute errors for checks on convergence.

Note that your computations must be aimed at three scenarios, which correspond to a ∈ {0.5, 1.0, 5.0} in the utility function.

Your program must be able to report results about c0, c1, ω1, ω2, ω3 in the iteration process, and they must be presented in a nice and neat output format.

Apr 15, 2023

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An investor is endowed with a certain level of wealth equal to e = 2. Consumption is taken as the numeraire. So, e = 2 means an endowment of two units of consumer goods. He must allocate his current...

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