1. (a) The investment opportunity set comprises N risky assets and one riskfree asset with rate of return R_f.

i. Define the notion of "mean-variance efficient" port-folio in this context. Represent graphically the mean-variance frontier corresponding to the entire investment opportunity set, and identify the efficient part of the frontier.

ii. The vector of expected returns on the N risky assets is μ and their covariance is Σ. Characterize the efficiency of a portfolio in terms of risky portfolio weights and in terms of Sharpe ratio.

iii. Using historical data collected over a short sample period, you run time-series regressions of the N asset excess returns on a constant and the excess returns of a portfolio P of the N assets, and you find that the in-sample estimates of some of the intercepts (alphas) are different from zero. What can you conclude regarding the efficiency of portfolio P?

(b) Discuss the relationship between mean-variance efficiency and the notion of "multifactor efficiency" defined in the lecture along the lines of Fama (JFQA, 1996).

(c) The paper by Kan and Zhou (JFQA, 2007) discussed in the lecture sheds light on the problem of estimation risk in the context of portfolio optimization. Explain their approach and summarize their findings.

2. (a) Securities A, B and C have the following characteristics:

In this question, you do not need to calculate optimal portfolio weights. You should only give intuitive answers.

i. Consider the mean-variance frontier portfolio with expected return 10%. Explain why asset A receives a positive weight in this portfolio.

ii. Which of assets B and C receives a greater weight in the portfolio referred to in (i)? Explain.

iii. Now consider the mean-variance frontier portfolio with expected return 5%. Which of assets B and C receives a greater weight in this portfolio? Explain.

(b) Explain the logic of the Treynor-Black model of active portfolio management.

(c) The Fama-French three-factor model does a good job at describing the cross-section of average excess returns on portfolios of stocks sorted along size and book-to-market ratio. After giving a brief explanation of how the factors are constructed, discuss this finding in light of the Treynor-Black model.

(d) Suppose you have historical data on monthly returns for a set of N international stocks.

i. Explain how you would estimate a characteristic-based factor model featuring country- and industry-specific factors.

ii. Explain how you could use this model to estimate the covariance matrix of the N stock returns.

3. (a) Suppose the expected excess return on the aggregate stock market (known as the market risk premium) can take two values, 4% or 8%. The unconditional market risk premium is 7%. The aggregate price-dividend ratio itself can take two values, either 40 or 20, and the following table gives the distribution of the price-dividend ratio P/D conditional on the one-period ahead risk premium μ^e_M:

The price-dividend ratio is currently equal to 40. What is the conditional expectation of the one-period ahead risk premium?

(b) Explain the construction of "market-implied" expected excess returns in the Black-Litterman asset allocation model.

(c) Explain the notions of prior, posterior and predictive distributions in the context of the Black-Litterman model.

(d) Discuss the appealing properties of the portfolios obtained by implementing portfolio optimization according to the Black-Litterman model.

4. (a) Consider a two-factor model of asset returns

R_it = α_i+ β_i1f_1t + β_i2f_2t + ε_it

for a large number of securities indexed by i. Assume the factors have zero mean (i.e., E(f_it) = 0) and E(ε_it) = 0. A riskfree bond is available for borrowing and saving at rate R_f = 5%.

i. What is the defining property of a "well-diversified portfolio"?

ii. Assume a well-diversified portfolio A has a beta of 1.1 on factor 1 (β_A1 = 1.1) and a beta of 0.95 on factor 2 (β_A2 = 0.95). The loadings of another well-diversified port-folio B on factors 1 and 2 are β_B1 = 0.75 and β_B2 = 1.5, respectively. Find a portfolio of portfolios A and B that has no exposure to factor 2. What is its exposure to factor 1?

iii. Use your answer to part (ii) to show how to form a well-diversified portfolio with a unit beta on factor 1 and a zero beta on factor 2. What is its expected rate of return as a function of mean returns E(R_A) and E(R_B)?

iv. Suppose the expected excess rates of return on well-diversified mimicking-factor portfolios for factors 1 and 2 are -3% and 6%, respectively. Consider a security with beta 0.5 on factor 1 and 1.1 on factor 2. What does APT imply for the expected return on this security? Justify your answer carefully.

(b) Some researchers have documented, for different countries and sample periods, that equally-weighted portfolios of stocks (i.e., constructed according to a simple 1/N rule) tend to perform better than portfolios formed according to more sophisticated optimization methods. Discuss this finding. Could it be related to some well-known asset pricing anomalies?

(c) Give several types of constraints that can be imposed on portfolio weights in the implementation of Markowitz's mean-variance portfolio optimization, and discuss how you would design a simulation exercise to assess the impact on portfolio performance of introducing such constraints.

5. (a) Under what conditions is expected utility maximization consistent with a mean-variance objective? Under what additional assumption(s) do we obtain a linear mean-variance tradeoff? No formal proof is required in answering this question.

(b) What was the point made by Richard Roll in his "critique"?

(c) Summarize the analysis of the paper by Barberis (2000), discussed in the lecture, on long-term asset allocation in the presence of return predictability.

(d) Describe the tests of market timing ability suggested by Treynor and Mazuy (1966) and Henriksson and Merton (1981). Provide intuition for the regression specifications they use.

6. Suppose there are N risky securities and the riskfree rate is R_f. The price of security i at time t is P_it and P_t = (P_1t, ..., P_Nt)' denotes the vector or security prices. There are K investors. Let W_k,t denote the wealth of investor k at time t, and let x_ki denote the quantity of security i bought by investor k at time t. The variance-covariance matrix of Pt₊₁ is denoted by ?. Investor k chooses x_k = (x_k1, ..., x_kN)' in order to maximize E(W_k,t+1) - (γk/2)var(W_k,t+1) under the constraint x'_kP_t ≤ θ_kW_k,t, where θ_k > 0.

(a) Interpret the constraint for θ_k = 1 and θ_k = 1.3.

(b) Write E(Wk,t+1) and var(W_k,t+1) as a function of x_k, E(P_t+1) and ?.

 (c) Setup the Lagrangian corresponding to the optimization problem of investor k. Derive the first-order condition and express xk as a function of the Lagrange multiplier on the constraint ψ_k.

(d) The total supply of risky securities is X = (X₁, ..., X_N)'.

Show that in equilibrium

P_t = (1/1 + R_f + ψ) [E(P_t+1) - γ?X],

where γ =(_k=1Σ^K 1/ γk)^-1  and ψ = _k=1Σ^K (γ/γk) ψ_k.

(e) Let R_i,t+1 = (P_i,t+1/P_it) - 1 and let R_m,t+1 denote the return on the market portfolio. Show that in equilibrium

E(R_i,t+1) = Rf+ ψ + (γP'_tX)cov (R_i,t+1, R_m,t+1),     I = 1, ..., N.

Show that furthermore

γP'_tX = (E(R_m,t+1)- R_f - ψ/var(R_m,t+1))

(f) In this context, consider a portfolio strategy S that involves taking positions in a portfolio of high-beta securities with market beta β_H and in a portfolio of low-beta securities with market beta β_L, where 0 < β_L < β_H. The rate of return on this strategy between t and t + 1 can be written

R_S,t+1 = (1/β_L)(R_L,t+1 - R_f) + (1/β_H) (R_f - R_H,t+1),

where R_H,t+1 and R_L,t+1 denote the returns on the high- and low-beta portfolios, respectively.

i. What are the trades involved in implementing strategy S at time t? What is the cost involved at time t? What is the market beta of the strategy?

ii. Compute the expected rate of return on the strategy, E(R_S,t+1). Comment on your finding.

(g) How do the implications of this model for the cross-section of expected excess returns differ from the Security Market Line obtained in the context of an unconstrained CAPM? Provide some intuitive explanation.