BMAN71111 Asset Pricing
Academic Year 2017-2018
Section A
Instructions: Attempt all questions. Each question carries 3 marks. No marks will be deducted for incorrect answers.
FOR EACH QUESTION IN THIS SECTION, PLEASE PLACE A CROSS IN THE BOX NEXT TO THE ANSWER THAT YOU BELIEVE IS CORRECT. IN EACH QUESTION, ONLY ONE ANSWER IS CORRECT.
Question 1. A given portfolio p is mean-variance efficient when:
(A) It is NOT possible to find any other portfolio such that the condition SR[Rp] ≥ SR[Rother] holds, where SR[R] denotes the Sharpe ratio and Rother is the return on any other portfolio different from p.
(B) It is possible to find another portfolio such that the condition E[Rp] ≥ E[Rother] and Var[Rp] ≤ Var[Rother] holds with at least one inequality being strict, where E[R] denotes expected returns, Var[R] denotes the variance of returns, and Rother is the return on a portfolio different from p.
(C) It is NOT possible to find any other portfolio such that the condition E[Rp] ≥ E[Rother] and Var[Rp] ≤ Var[Rother] holds with at least one inequality being strict, where E[R] denotes expected returns, Var[R] denotes the variance of returns, and Rother is the return on any other portfolio different from p.
(D) It is NOT possible to find any other portfolio such that the condition E[Rp] ≤ E[Rother] and Var[Rp] ≥ Var[Rother] holds with at least one inequality being strict, where E[R] denotes expected returns, Var[R] denotes the variance of returns, and Rother is the return on any other portfolio different from p.
Question 2. Which one of the following statements regarding the expectation of short-term interest rates is correct?
(A) Suppose that the pure expectation hypothesis holds for the term structure of interest rates. An upward sloping yield curve clearly indicates that short-term interest rates are expected to decrease.
(B) Suppose that the liquidity premium hypothesis holds for the term structure of interest rates. An upward sloping yield curve must only indicate that short-term interest rates are expected to increase.
(C) Suppose that the liquidity premium hypothesis holds for the term structure of interest rates. An upward sloping yield curve can be consistent with the fact that short-term interest rates are expected to decrease.
(D) Suppose that the liquidity premium hypothesis holds for the term structure of interest rates. The yield curve must only be either monotonically increasing or monotonically decreasing.
Question 3. Suppose two risky assets, X and Y, are both subject to two sources of systematic risk factors: the inflation rate and consumption growth. The factor risk premiums on the inflation rate and consumption growth are 0.1 and 0.2, respectively. The risk-free rate is 0.03. For asset X, the factor loadings on inflation rate and consumption growth are -0.3 and 0.5, respectively. For asset Y, the factor loadings on inflation rate and consumption growth are 0.3 and 0.2, respectively. In the framework of arbitrage pricing theory, which asset has a higher expected return?
(A) X has a higher expected return than Y does.
(B) Y has a higher expected return than X does.
(C) X and Y should have the same expected return.
(D) None of the above: we cannot judge because some information is missing.
Question 4. The variance ratio statistic:
(A) is defined as VR(k)=(12/k)[Var(Rk)/Var(R12)] where Rj denotes the return measured over a period of j months; under the random walk hypothesis, return variances grow linearly with the horizon so that VR(k)=1 for all k; if returns display negative serial correlation, we expect VR(k)<1.
(B) is defined as VR(k)=(12/k)[Var(R12)/Var(Rk)] where Rj denotes the return measured over a period of /k) months; under the random walk hypothesis, return variances grow linearly with the horizon so that VR(k) = 1 for all k; if returns display momentum, we expect VR(k)<1.
(C) is defined as VR(k)=(12/k)[Var(Rk)/Var(R12)] where Rj denotes the return measured over a period of j months; under the random walk hypothesis, return variances grow exponentially with the horizon so that VR(k) = 1 for all k; if returns display negative serial correlation, we expect VR(k) < 1.
(D) is defined as VR(k)=(12/k)[Var(Rk)/Var(R12)] where Rj denotes the return measured over a period of j months; under the random walk hypothesis, return variances grow exponentially with the horizon so that VR(k)<1 for all k; if returns display positive serial correlation, we expect VR(k)<1.
Question 5. Given the following implications of any asset pricing model studied in the lectures, which combination of the implications will be tested when we empirically test the CAPM?
(i) Beta is the only variable that can explain cross-sectional expected stock returns.
(ii) The correlation between stock returns and aggregate consumption growth determines cross-sectional expected stock returns.
(iii) The pricing errors are equal to zero on average.
(iv) Idiosyncratic risks do not matter for cross-sectional expected stock returns.
(v) The factors SMB (small minus big) and HML (high minus low) are the two key determinants of expected asset returns.
(A) (i), (ii) and (iv).
(B) (i), (iii) and (iv).
(C) (iii), (iv) and (v).
(D) (i), (ii) and (iii).
Question 6. Campbell and Shiller’s linearized rational valuation formula,
implies that:
(A) A high price-dividend ratio today must be caused by either forecasts of low dividend growth rates in the future or by forecasts of future expected returns; moreover, given dividends, an increase of the price-dividend ratio today implies an increase in the price and therefore an increase in the current return; then future higher (lower) expected returns are associated with current higher (lower) realized returns.
(B) A high dividend yield today must be caused by either forecasts of high earning growth rates in the future or by forecasts of low future long-term bond returns; moreover, given dividends, an increase of the dividend yield today implies an increase in the price and therefore a decrease in the current return; then future lower (higher) expected returns are associated with current higher (lower) realized returns.
(C) A high price-dividend ratio today must be caused by either forecasts of high dividend growth rates in the future or by forecasts of lower future expected returns; moreover, given dividends, an increase of the price-dividend ratio today implies an increase in the price and therefore an increase in the current return; then future lower (higher) expected returns are associated with current lower (higher) realized returns.
(D) A high price-dividend ratio today must be caused by either forecasts of high dividend growth rates in the future or by forecasts of lower future expected returns; moreover, given dividends, an increase of the price-dividend ratio today implies an increase in the price and therefore an increase in the current return; then future lower (higher) expected returns are associated with current higher (lower) realized returns.
Question 7. Which one of the following statements regarding the zero-beta CAPM is correct?
(A) In the zero-beta CAPM, the efficient frontier portfolios lie on a straight line.
(B) The minimum variance zero-beta portfolio must be on the inefficient segment of the minimum variance frontier.
(C) The variance of returns on the minimum variance zero-beta portfolio must be lower than the variance of returns on the global minimum variance portfolio.
(D) There no longer exists a linear relationship between expected returns and betas.
Question 8. Which one of the following observations is NOT among the main findings of Fama and French (1992) "The Cross-Section of Expected Stock Returns", Fama and French (1993) "Common risk factors in the returns on stocks and bonds" and Fama and French (1996) "Multifactor explanations of asset pricing anomalies"?
(A) Fama and French find that when one allows for variation in betas that is unrelated to size, the relationship between betas and average returns is almost flat, even when beta is the only explanatory variable.
(B) Fama and French find that the estimate of the risk premium on book-to-market equity is positive and statistically significant when book-to-market equity is the only explanatory variable. This finding implies that portfolios with low book-to-market ratios tend to have low average returns while portfolios with high book-to-market ratios tend to have high average returns.
(C) In a three-factor time series regression with the market factor, small-minus-big (SMB) factor and high-minus-low (HML) factor, Fama and French find that for 25 portfolios sorted according to size and book-to-market, the slopes on HML increase monotonically from high book-to-market to low book-to-market quintiles. This captures the book-to-market effect on cross-sectional expected returns.
(D) In a three-factor time series regression with the market factor, small-minus-big (SMB) factor and high-minus-low (HML) factor, Fama and French find that for 25 portfolios sorted according to size and book-to-market, the slopes on SMB increase monotonically from big size to small size quintiles. This captures the size effect on cross-sectional expected returns.
Question 9. You use the CAPM to determine the required (equilibrium) return of an investment. Suppose the market portfolio gives an expected return of 8% per year. We also know that in the standard CAPM, investing in the market portfolio generates the best risk-return trade-off, i.e., the highest Sharpe ratio. Which of the following phenomena could most likely amount to evidence against the efficient market hypothesis when the CAPM is used to determine the required (equilibrium) return of an investment?
(A) Hening Hedge Funds has developed a new trading strategy based on some trend factor. This trading strategy involves rebalancing portfolios every week. The CAPM beta of this investment strategy is always equal to 1. During the past three months, this new strategy has consistently earned annualized returns of about 6%.
(B) Hening Hedge Funds has developed a new trading strategy based on weather conditions. This trading strategy involves rebalancing portfolios every week. The CAPM beta of this investment strategy is always equal to 1. During the past three months, this new strategy has consistently earned annualized returns of about 10%, net of transaction costs.
(C) Hening Hedge Funds has developed a new trading strategy based on investors' mood. This trading strategy involves rebalancing portfolios every week. The CAPM beta of this investment strategy is always equal to 1. During the past three months, this new strategy has consistently earned annualized returns of about 10%. But the average return net of transaction costs is approximately 8%.
(D) Hening Hedge Funds has developed a new trading strategy based on information about firms' profitability. This trading strategy involves rebalancing portfolios every week. The CAPM beta of this investment strategy is always equal to 1. During the past three months, this new strategy has consistently earned annualized returns of about 10%. But the average return net of transaction costs is approximately 6%.
Question 10. With reference to the excess volatility puzzle, the following plot can be interpreted to show that:
(A) The actual, historical real stock price has been persistently more volatile than the prices computed from the rational valuation formula under perfect foresight (on future dividends) and both constant and time- varying real discount rates; therefore actual real stock prices appear to have been excessively volatile.
(B) The prices computed from the rational valuation formula under perfect foresight (on future dividends) and both constant and time-varying real discount rates have been persistently more volatile than the actual, historical real stock price; therefore actual real stock prices do not appear to have been excessively volatile.
(C) The actual, historical real stock price has been negatively correlated with the prices computed from the rational valuation formula under perfect foresight (on future dividends) and both constant and time-varying real discount rates; therefore actual real stock prices appear to have been excessively volatile.
(D) The efficient market hypothesis does not hold because actual, real stock prices have significantly drifted away from the prices computed under perfect foresight of future dividends and a constant discount rate.
Section B
Instructions: Answer ONE question in all its parts. All parts together are worth 35 marks. Marks for all parts are shown below.
PLEASE USE THE SPACE PROVIDED ON THE QUESTION PAPER TO ANSWER ALL SECTIONS. The space provided is sufficient to give a concise and pertinent answer to each question and/or sub-point of a question. Extra answer booklets will not be provided – please use the spaces provided only. Answers written outside the spaces or on other booklets will not be marked.
Question 11 (Mean-Variance Algebra)
Suppose that your investment menu has two risky assets and a risk-free asset. The risk-free rate is 0.01. The first risky asset has a mean return of 0.06 and a standard deviation of 0.12. The second risky asset has a mean return of 0.05 and a standard deviation of 0.18. The correlation coefficient between returns on the two risky assets is 0.2.
11a. (8 marks) Assume (for this part only) that only the two risky assets are available for investment. Find the global minimum variance portfolio, i.e., the fractions of wealth invested in the two risky assets that compose the global minimum variance portfolio. Also calculate the mean and standard deviation of the global minimum variance portfolio return.
11b. (15 marks) Assume that the investor wants to maximize the mean-variance objective function
by choosing the fractions of wealth to invest in the three assets. Here μ and σ are the mean and standard deviation of the investor’s portfolio, and Y is the risk aversion coefficient. Of the optimal amount of investment in the two risky assets, what fraction is optimal to invest in the first risky asset? Equivalently, find the optimal risky portfolio.
11c. (12 marks) The optimal risky portfolio obtained in part (b) is the tangency portfolio. (i) Illustrate the unique property of the tangency portfolio in the presence of a risk-free asset. (ii) Formulate an alternative optimization problem from which you can obtain the same optimal risky portfolio as in part (b). (iii) Solve the problem and verify that the resulting optimal portfolio is the same as the optimal risky portfolio in part (b).
Question 12. (The Arrow-Debreu economy)
Consider a one-period Arrow-Debreu economy. Suppose that the structure of the state-contingent dividend is given by
in which πi denotes the probability of state i.
Assume that in this economy a representative agent exists and has a constant relative risk aversion utility function
with the coefficient of relative risk aversion Y = 2 . At date t = 0 , her consumption c0 = 1 . At date t = 1 , her consumption in three states are
c1 = 0.9, c2 = 1.2, c3 = 0.5
respectively. The subjective discount factor is β = 0.95 .
12a. (15 marks) Suppose that the endowment at date t = 0 is w0 and the endowment in each state at date t = 1 is wi , i = 1,2,3 . Write down the Lagrangian and first-order conditions for the agent's problem. Calculate state prices and expected returns for the three Arrow-Debreu assets. Discuss the economic intuition of your results.
12b. (8 marks) Suppose that another asset (asset 4) has the state contingent dividend:
and its current price is 4.5. Discuss whether arbitrage opportunities exist or not. Create a portfolio strategy that generates arbitrage profits if such opportunities exist.
12c. (12 marks) (For this part only: the information given above is not relevant for solving this part) Suppose there are only two possible states i = 1 and 2 at t = 1 , which occur with probabilities π1 and π2 respectively. There are two agents (indexed by j, j=1,2) who have the same utility function
uj (c) = ln(c), j = 1,2
so that uj, (c) = c/1 , j = 1,2 . The two agents have different endowments (w0(1); w1(1) , w2(1) ) and
(w0(2); w1(2) , w2(2) ) where w0(j) , j = 1,2 denotes the date 0 endowment for agent j, and wi(j) , j = 1,2; i = 1,2
denotes the endowment in state i for agent j. The subjective discount factor is β . A competitive equilibrium is defined by the following three requirements:
1) Agent 1 takes state prices as given and chooses consumption plans optimally.
2) Agent 2 takes state prices as given and chooses consumption plans optimally.
3) All markets clear (for date 0 and all states in date 1).
Characterize the competitive equilibrium by the first-order conditions and market clearing conditions.