Advanced Investment Theory

Please treat this online exam like any other exam. You are expected to work individually, and please do not discuss the exam with anyone until after 12:00 pm December 15. Furthermore, the questions in this exam are copyrighted and not to be shared outside of the class.^[1] The exam is open book, and you may use Excel to do any calculations as necessary, but you are not permitted to use outside resources. Please type your responses below the corresponding question when possible. You can insert a picture or scan if necessary. Part marks are available for partially correct answers. You have 3 hours to complete and submit your exam, so budget a sufficient amount of time at the end to save as PDF and submit through Canvas.

Part I: Short Answer (24 marks)

1. How does an efficient financial system improve the growth rate of an economy’s capital stock?
2. How does the Allais paradox relate to von Neumann-Morgenstern utility theory?
3. What is the key distinction between first and second order stochastic dominance?
4. A common rule of thumb for personal financial planners is to recommend that their clients reduce their allocation of risky assets as they approach retirement. Is this advice consistent with expected utility theory?
5. The VIX index measures the implied volatility of options on the S&P 500 index. On October 3, 2018 the VIX closed around 11.6 points and on October 4, 2018 it peaked at around 15.7 points. How might this change affect an investor’s saving behaviour?
6. Generally, mean-variance utility is only approximately related to vNM utility. Under which circumstances does mean-variance utility correspond exactly with vNM utility?
7. If the portfolio problem has a solution, then must there exist a risk-neutral probability measure for the economy? Why?
8. According to the consumption CAPM, what should be the primary driver of the expected returns on financial securities?
9. Describe the Mehra-Prescott puzzle. Why is it a puzzle?
10. What is a stochastic discount factor and how does it relate to the concept of a martingale for asset pricing?
11. How do restrictions on short selling affect the Pareto optimality of a firm’s capital structure choice?
12. Strong-form efficiency implies security prices reflect all available information. Why is strong-form efficiency unlikely to hold?

Part II: Problem Solving (24 marks)

1. Suppose the risky asset provides a return of (i.e. it provides a 50% return with probability 0.6, or a -30% return with probability 0.4) and the risk-free rate is r_f = 0.05. Consider an investor with constant relative risk aversion utility of wealth having coefficient  = 1 and initial wealth Y₀ = 100. How would she allocate her wealth between the risky and risk-free assets?
2. Suppose that fundamental security prices are as follows:

What is the present value of a risky cash flow that pays $8.5, $11.5, or $14.5 in states ₁, ₂, or ₃ respectively? Is there a portfolio of fundamental securities that replicates the risky cash flow? How do you know?

1. A futures contract has the following three properties: 1) it costs nothing to enter the contract at any time (so the value of the contract is zero at each time, V_t ≡ 0), 2) it makes payments at each time (called marking to market) based on the difference between the current and previous futures prices, G_t – G_{t – 1}, and 3) the futures price at the maturity time is equal to the spot price of the underlying asset at that time, G_T = S_T. Show that the futures price is the risk-neutral expected value of the spot price at maturity.
2. Show that the risk premium for security i is

where M is an arbitrary stochastic discount factor, R_m is the return on the market portfolio, and r_f is the risk-free rate. Hint: recall that E[M × (1 + R_j)] = 1 + r_f for every asset j.

1. Consider an economy with two dates (0 and 1), two equally likely states of nature (q₁ and q₂), two investors (k = 1, 2), and an entrepreneur who has a project that needs financing. The investors both have utility functions U(c₀, C₁) = 0.25c₀ + E[ln(C₁)], where C₁ consumption depends on the state of the economy at time 1. The project is risk free and pays 2 units of consumption in each state. Investor endowments are given in the following table:

What is the value of the firm, if the entrepreneur issues two Arrow-Debreu securities for each state?

1. A market maker believes that the future stock price can be either $20 or $40, she asses the current probability of it being $40 as 0.32, she assesses the likelihood of encountering an informed trader who knows the future value with 95% certainty as being 0.02, and she can’t distinguish between informed or uninformed traders. What should be her bid price?

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