Please take a look and let me know how many you can correctly solve.

Question 1:

A group of researchers want to understand the effect of firm-level anti-takeover provisions (for example, poison pills') on firm performance.

The standard cross section OLS regressions illustrate that anti-takeover provisions are positively affecting firm performance. How would you in¬terpret these findings? What are your concerns regarding causal inter-pretation? Write down the model specification and derive the potential bias to motivate your explanation.

2. The researchers decide to control for firm fixed effects to identify the effect of anti- takeover provisions. Consequently, the estimates become negative.

(a) Write down the empirical specification. How is the interpretation of these ester- mates different from previous question? Explain why.

(b) Which concerns are alleviated with the inclusion of firm fixed ef-fects and what are the remaining concerns?

(c) Propose an economic interpretation for the change in the sign of the coefficient following the inclusion of the fixed effects (from positive to negative).

3. Bertrand and Mullainathan (2003)² use the introduction of state-level Business Combination (BC) laws to estimate the effect of anti-takeover provisions on firm performance.

(a) What is the advantage of this approach, relative to the approaches discussed in parts (1) and (2) above?

(b) The laws used by the authors affect only incorporated firms. Why is it important in order to identify the causal effect of anti-takeover provisions on firm performance? Which problems does it help to alleviate?

(c) Write down the model specification. What are the identifying requirements to estimate the causal effect of anti-takeover provi¬sions on firm performance? Specify potential concerns about the validity of these assumptions.

(d) The laws are being passed in a staggered manner over time. What is the advantage of this setting? What concerns can he alleviated?

Question 2:

Suppose you estimate a simple regression model for the performance of a mutual fund

r_t = α + βr_mt + ∈_t,

where r_t is the fund's return, measured in excess of a risk-free rate, r_t is the excess return on a market proxy, and ∈_t is the residual.

You are interested in conducting inference about the fund's risk-adjusted performance, as mea¬sured by a. For the purpose of the analysis, you can assume that there is a single risk factor (r_mt).

1. Suppose that β in equation (1) is time-varying and depends linearly on an observable factor, z_t, thus taking the form

β(z₁) = β₀ + β₁z₁,

so (1) is misspecihed and the correct model specification is

r_t = α₀ + β₀r_mt + β₁z_tr_mt + u_t, (2)

where E[z_t] = 0 and E[u_tz_t] = 0.

Demonstrate the properties of the OLS estimate of a in equation (1) under the assumptoin that the true data generating process is equation (2). Under what conditions will the OLS estimate, α, be biased?

2. Next, supppose that you estimate the simple performance model in (1) but that, in fact, the fund can market time so that the true data generating process for the fund's returns is

r_t = α₀+ β₀r_mt + β₁r²_mt + u_t.

Show the conditions under which the OLS estimate of a from equation (1) will be a biased estimate of no in equation (3). Also explain when the OLS estimate of a is unbiased.

3. Now suppose that equation (1) is the correct specification for the fund's returns. Moreover, assume that ε_t ~ iidN(0, σ²), where σ = 1, β = 1, α = -0.2.

How large a sample. i.e., how many data observations (n), is needed to have a 10%, 25%, 50%, 75% probability of correctly identifying the fund as an underperformer, assuming that you use a two-sided test?

Question 3:

Consider an two-date, pure endowment economy with a representative in-vestor who has CRRA preferences over consumption in the current and next period, i.e., she maximizes

u(c₀) + βE(μ(c₁))    (1)

where u (c) = 1/1-γ,C/1-y. Suppose aggregate consumption is log-normally distributed i.e., log (c₁/c_o) ≡ N (μ_c, σ²_c,).

1. Calculate the log risk-free rate in the economy. How does it change mean consumption growth if, and volatility of aggregate consumption (Yr.? Explain briefly.

2. Calculate the log price-dividend ratio and the log risk-premium on a security that is a claim to the aggregate endowment in date 1. How do these quantities change with risk-aversion -y, mean consumption growth pr., and volatility of aggregate consumption cr,:? Explain briefly

3. Consider a security that pays D₁ in the next period, where log (D₁) ≡ d ~ N(μ_d, σ²_d) and the correlation between m and d is given by p_d. Calculate the price and the log expected return on the security. How does the log expected return on the security depend on p_d. Explain briefly.

4. Calculate the price of a binary option that pays $1 it aggregate con-sumption at date 1 is greater than K, and zero otherwise. (Hint: It is sufficient to characterize the price in terms of the CDF of a standard normal.)

Question 4:

Please read the article "Rare Disasters and Asset Markets in the Twentieth Century" by Robert Barro3. The representative investor (in an endowment economy) has a standard first-order condition. The author specifies funda-mentals that are subject to rare events, i.e. low probability of a very negative outcome. Using the Burro (2006) setup:

1. Derive the equilibrium price of the one-period equity claim (equation 8) and its expected return (equation 9).

2. Derive the expected return of the risk-free asset (equation 11) and the expected return of bonds, including the probability of default (equation 12).

3. Derive the unconditional equity premium (equation 13) and the equity premium, conditional on there being no disasters (equation 14).

4. Based on your derivations above, explain how you would go about estimating and calibrating the parameters in the model. Make sure to explain the difference between calibration and estimation.

5. Derive the expected return on equity and T-bills over discrete periods of length T (equations 25 and 26). [Hint: This is non-trivial. You have to start from the first order condition. Shortcuts will get you in trouble.]

Question 5:

Consider a market with two stocks (X and Y), mutual fund managers and investors (to be described later). Mutual fund managers are price-takers and are long-only equity holders (i.e. they must hold the entirety of their portfolio in long positions in stock X, stock Y or some combination of the two) who can adjust their portfolios quarterly. Stocks X and Y announce quarterly earnings and pay out all of earnings as dividends each period. Earnings behave as follows:

Earnings for stock X at quarter t: X_t = X_t-1 + α_t where α_t is i.i.d. and a_t ~ N(0, s^N) for all t

Earnings for stock Y at quarter t: Y_t = Y_t-1 + b_t where b_t is i.i.d. b_t ~ N(0, s^Y) for all t and a is independent of b for all t.

Let δ and be the same quarterly discount rate for stock X and Y, and X₀ and Y₀ be the just-announced earnings for stock X and Y.

1. Find expressions for prices and quarter-ahead expected returns of stock X, stock Y and the market.

2. Derive expressions for the distribution of returns for both stock X and stock Y for the quarterly period beginning at I and ending at t+1.

All mutual fund managers understand the earnings dynamics given above, however, 1`X of all mutual fund managers are also "informed." Informed mutual fund managers know the quarter-ahead realization of a (i.e., at time t they know what atnwill be). Further suppose that every mutual fund manager's objective each quarter is to "beat the market," which means have a return which is higher than that of the market. (In the case of ties, assume a fair coin is tossed to determine whether the manager beat the market.)

3. Describe an informed mutual fund manager's optimal strategy (as a function of atti) if his objective is to maximize the probability of beating the market during the quarterly period t to t+1.

4. Given the strategy you described in part 3, find the probability that an informed manager beats the market.

Now suppose investors observe the performance of mutual fund managers and try to infer whether they are informed or not. However, investors have the following limitation: they only observe whether a mutual fund manager has or has not beaten the market each quarter (i.e. they don't observe the manager's holdings or each stock's performance).

5. Suppose an investor with the correct prior (that 1% of managers are informed) observes his mutual fund manager beating the market two quarters in a row. Given the probability you found in part 4, what chance will he give that his mutual fund manager is informed?

6. Now suppose mutual fund managers are evaluated over annual hori-zons, i.e. their objective is to beat the market over a 4-quarter period, and consider a mutual fund manager that has beaten the market over the past two quarters. Describe the optimal strategy of both informed and uninformed mutual fund managers in this scenario if their objecive is to maximize the probability of beating the market for the year.

7. Suppose you had a panel dataset (mutual fund managers, time) with the quarterly performance and holdings of thousands of mutual fund man-agers over several decades. Describe how you would empirically detect the strategy you suggested in part 6.

Related Reading: Brown, Keith C., W. Van Harlow. and Laura T. Starks, 1996, Of Tournaments and Temptations: An Analysis of Managerial Incen-tives in the Mutual Fund Industry. Journal of Finance 51, 85-110.