Ability      test

You will be required to complete any three of the following four questions within 24 hours. Your work should be done by yourself, do not use gpt or other forms of AI tools, if you are not clear about the meaning of a question, just write your understanding of the question, we will have a professional review teacher to score your calculation results, please remember that some questions may not be calculated correctly. But as long as you write out your own thinking process is also OK, remember never use AI, once found using AI will not work with you for life! Good luck!

Question 1

Consider a stylized two-period portfolio choice problem with a representative agent. The agent’s preference is given by the following:

U(C) = u(C0) + E[u(C1)],                                               (1)

where u(c) = −e −γc, and γ > 0.

The agent faces the following constraints at each period:

C0 + hp ≤ W0,                                                  (2)

C1 ≤ W1 + hX,                                                   (3)

where h is the portfolio choice that the agent chooses at time 0. The per capita endowment at time 0 and time 1 are given by: W0 = 1, W1 = (3, 6, 9).

At time 1, there are three equally likely states.

The payoff matrix X is given by:

(4)

where Xjs denotes the payoff of asset j (row) in state s (column).

Please answer the following questions:

1. Is the market complete? Why or why not?

2. What are the agent’s absolute and relative risk aversion?

3. Describe the agent’s optimization problem carefully (e.g., choice variables, constraints).

4. What are the optimal consumption and portfolio allocation? Explain.

5. What is the pricing kernel?

6. Derive the pricing equation that allows one to price any asset j in this model.

7. What is the riskfree rate?

8. What is the price of asset 2 with payoff (1 2 3)?

Question 2

Consider a stylized asset pricing model with two periods. The representative agent’s preference is given by

V (C) = v(C0) + E[v(C1)],                                                         (5)

where and γ ≠ 1. The agent’s endowment at time 0 is W0 = 1. Assume the market is complete.

1. For a random consumption z, with ln derive the exact risk compensation function ρ(y, z).

2. How does the exact risk compensation function vary with y, σz, and γ? Interpret your results.

3. The distributional assumption for C1 continues to hold as in part 3. For this question, further assume γ = 2 and σc = 1%. There is a risky security j with expected excess return E[rj − rf ] = 5%. What can you say about the volatility σj of this risky security?

Question 3

Consider a single-period binomial tree of firm value Vt, where t = 0, 1. Suppose the firm value Vt evolves from time 0 to time 1 as follows:

• With probability p, the up state of the world is realized, and V1 = uV0

• With probability 1 − p, the down state of the world is realized, and V1 = dV0

Assume that the market is frictionless and dynamically complete. Also, assume u > d. Denote rf as the riskfree rate. For illustration, the binomial tree is given below:

For the following questions, assume V0 = 1, u = 1.6, d = 0.7, p = 0.2, rf = 0.02..

1. Given that the market is complete, we will have two Arrow securities, one corresponding to each state. Compute the prices of these two Arrow securities. Interpret your results.

Now, we extend the binomial tree to two periods, with the same setting.

Similar to setting in the lecture, we extend the state space from period 1 to period 2. That is, there are two states up and down (with same probabilities p and 1 − p) possible from each of the period 1 state realizations. Now, we have four states of the world at the end of period 2: uu, ud, du, and dd. In other words, the time 2 firm value is given by: V2(s) = sV0, where s = uu, ud, du, dd.

Again, for illustration, the binomial tree is given below:

2. What are the four period 0 Arrow prices for the four states at period 2?

3. Consider the manager of the firm has the option to make an initial investment at time 0, with an up-front cost I0 = αV0. The investment only pays off at period 2. In particular, conditional on investment, firm value at time 2 will be: sV0 in uu state, mV0 in ud and du states, and fV0 in dd state. What is the net present value of this investment project at time 0? Under what conditions will the manager choose to make the investment at time 0?

4. Now, assume that the manager chooses not to exercise the option to invest at time 0. However, the manager can learn the prospect of the project as time evolves. In particular, at time 1, the manager re-evaluates the prospect of the investment project. The investment project has the same payoff profile as in the previous part. Conditional on investment, firm value at time 2 will be: sV0 in uu state, mV0 in ud and du states, and fV0 in dd state. Under what conditions should the manager choose to invest at time 1? Note that the context under which the manager makes the decision depends on which state is realized at time 1.

Question 4

Consider a continuum of investors i, with mean-variance preferences:

There are N risky assets with payoff vector and a common initial endowment w0. The supply of the assets is the vector x¯ + x, where x¯ is known and All shocks are independent and normally distributed in this economy. Suppose that investors get a signal vector of the form. s = z + yf + ei where is an N × 1 vector that is independently and identically distributed across investors. z and y are known parameters, common to all investors.

1. Is s an unbiased signal about f? If not, what transformation of s is required to make it unbiased?

2. What is V [f|s]? Is this matrix diagonal? After updating their beliefs with their signal, do agents believe that asset payoffs are conditionally uncorrelated across assets? Why or why not?

3. What is E[f|s]?

4. What is the dispersion (cross-sectional variance) in the beliefs from part 3? In other words, what is E[(E[f|s] − E¯[f|s])2], if E¯[f|s] := R E[f|si ]di is the average agent’s belief?

5. What is the market return in this model?

6. Suppose that the price vector is a linear function of the payoffs f and the supply noise x: p = A + Bf + Cx. Does the CAPM price assets in this economy? Justify your answer.

7. What is E[f|s, p]?

8. What is V [f|s, p]?

9. What is the optimal portfolio choice of investor i? Is this portfolio on the mean-variance frontier?

10. Express the market clearing condition (equate supply and total demand).