UNIVERSITY OF EAST ANGLIA School of Economics Main Series PGT Examination 2017-18 FINANCIAL MARKETS ECO-7012A Time allowed: 2 hours Answer FOUR questions out of the following FIVE. Each question carries 25 Marks. A standard normal table is attached to the examination paper. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. ECO-7012A Module Contact: Dr A Jackson, ECO Copyright of the University of East Anglia Version 1
Page 2 Question 1 [25 marks] a) Write down the asset-pricing equation for the Fama-French-Carhart 4-Factor model. Define all of your variables. b) A researcher comments that the returns to holding a portfolio of small companies are already captured in the tendency for these companies to have high CAPM betas. Discuss the strengths and weaknesses of this argument. [7 Marks] c) The model in part a) adds an additional risk factor to the original Fama-French model. What class of behavioural hypotheses helps explain the returns to this additional risk factor? Explain your reasoning. [8 Marks] Question 2 [25 marks] An investment offers the following annual returns in each of three equally-likely states of the world: Your utility function is defined as: +12%, 0%, -3% uu(μμ,σσ) = μμ 1 2 σσ2, where μμ is annual expected return and σσ 2 is annual variance, a) Calculate the Certainty Equivalent Return (CER) of the investment. b) Assume that the returns may be approximated by the Normal distribution. What is the 99% 10-day Value-at-Risk of the investment? c) Utility Theory is based on the use of ordinal rankings. Define this term and contrast it with cardinal rankings. [5 Marks] ECO-7012A Version 1
Page 3 Question 3 [25 marks] a) What is a market anomaly? How does this relate to the Efficient Markets Hypothesis? b) Write down the equation for the variance of a portfolio that consists of two assets. Derive this result using matrix multiplication. c) You plot a scatter-diagram of the returns of the first asset against the returns of the market portfolio, and draw the line of best fit using Ordinary Least Squares. What key change will you need to make to the equation of this line to make it consistent with the Efficient Markets Hypothesis? [5 Marks] Question 4 [25 marks] a) By deriving an expression for the variance of the returns of a portfolio of N assets, and by using a limiting argument, show that specific risk is diversifiable in large portfolios. [12 Marks] b) Describe the procedure used to calculate the off-diagonal elements of the covariance matrix in the Single Index Model. [9 Marks] c) Write down 2 assumptions that are needed to derive the Single Index Model. [4 Marks] TURN OVER ECO-7012A Version 1
Page 4 Question 5 [25 marks] a) An asset has an annual expected return of 5% and an annual standard deviation of 15%. If the risk-free rate is 2%, the annual expected return of the Market Portfolio is 8%, and the annual standard deviation of the Market Portfolio is 12%, what is the asset s systematic risk? b) Using the information provided in part a), draw a diagram of the Efficient Frontier with borrowing and lending. How would an investor determine their own individual optimal portfolio? Sketch this concept on your diagram. c) Use your diagram to demonstrate why rational investors prefer the frontier with borrowing and lending to the frontier that consists only of risky assets. [5 Marks] END OF PAPER ECO-7012A Version 1
Page 5 Table 1: The cumulative standard normal distribution 0 0.01.02 0.03 0.04 0.05 0.06 0.07 0.08.09 0 0.5 0.504 0.508 0.512 0.516 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.591 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.648 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.67 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.695 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.719 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.758 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.791 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.834 0.8365 0.8389 1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.877 0.879 0.881 0.883 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.898 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.937 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.975 0.9756 0.9761 0.9767 2 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.983 0.9834 0.9838 0.9842 0.9846 0.985 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.989 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.992 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.994 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.996 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.997 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.998 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.999 0.999 3.1 0.999 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 Procedure for finding ( z) P( Z z) Φ = < : Look down the first column for the first decimal place of z, look along the top row for Φ z from the middle of the table. the second decimal place of z, and read ( ) END OF MATERIALS ECO-7012A Version 1
ECO-7012A: 2017/18 Main Series Feedback and Suggested Solutions Feedback The mean mark for the examination was 61.81%, and the median mark was 62.5%. The lower quartile mark was 56.5%, and the upper quartile mark was 70.5%. Please find below suggested solutions. Question 1a R i = r f + β i,hml R HML + β i,smb R SMB + β i,p R1Y R R P R1Y R + β i,cap M R CAP M + e i The variables in the model are defined as follows: R i the return to asset i r f the risk-free rate β i,hml the factor sensitivity of asset i with respect to the high-minus-low risk factor R HML the return to the high-minus-low risk factor β i,smb the factor sensitivity of asset i with respect to the small-minus-big risk factor R SMB the return to the small-minus-big risk factor β i,p R1Y R the factor sensitivity of asset i with respect to the momentum risk factor R P R1Y R the return to the momentum risk factor β i,cap M the factor sensitivity of asset i with respect to the CAPM risk factor R CAP M the return to the CAPM risk factor e i a zero-mean disturbance
Question 1b It could well be true that, on average, small companies have a higher CAPM beta than large companies. The justification for this argument is that small companies outperform the general stock market during the upswing of the economic cycle, but suffer disproportionately during recessions. However, the SMB factor measures the excess returns to the size factor after controlling for the CAPM risk factor, so the observed returns to the SMB factor are in addition to those already explained by the CAPM beta. Question 1c Momentum is the additional risk factor. The most common cited reasons for observed momentum are the under-reaction and over-reaction hypotheses, in which stocks either fail to respond to, or overreact to, unexpected news. Eventually, arbitrageurs bring share prices back to equilibrium, thus creating trends in share prices. Question 2a Let E [R] be expected return, and E [ R 2] be expected return-squared. Then, E [R] = 0.03 E [ R 2] = 0.0051 It follows that σ 2 = E [ R 2] (E [R]) 2 = 0.0042 u (µ, σ) = 0.03 0.5 0.0042 = 0.0279 The Certainty Equivalent Return is obtained by setting σ = 0 in the utility function, which implies that 2
the CER is 0.0279, or 2.79%. (Note: there is no ambiguity as to the inputs into the utility function, as one cannot mix % and %-squared in the same equation). Question 2b µ 10 day = 10 252 µ annual µ 10 day = 10 0.03 = 0.00119 252 σ 10 day = 10 252 σ annual σ 10 day = 10 252 0.0042 = 0.012910 10-day 99% Value-at-Risk = (µ 10 day 2.33 σ 10 day ) 10-day 99% Value-at-Risk = (0.00119 2.33 0.012910) = 0.02889 Therefore, 10-day 99% Value-at- Risk = 2.89% Question 2c Two objects may be compared on an ordinal scale if their dimension has no intrinsic meaning, but they can still be ranked against one another. Two objects may be compared on a cardinal scale if their dimension has a real-life meaning, such as percentages, miles-per-hour, US dollars, etc. In utility theory, we often use functions in which the outputs may be negative (such as the natural logarithm function). This does not mean, however, that investors are deriving negative utility; rather, utility functions may only be used to rank lotteries against one another. 3
Question 3a A market anomaly is a risk factor that appears to earn excess returns beyond those already explained by the current consensus asset-pricing model, as according to both academics and practitioners. Once anomalies have longstanding convincing evidence, and a firm economic foundation, they tend to become known as risk factors, rather than anomalies. The Weak Form says that it is not possible to earn excess returns by processing historical prices. An example of a violation of the Weak Form is a trader making excess returns from a technical trading system. We would obtain the monthly returns to the trading strategy, regress them on the factors of an appropriate asset-pricing model, and test whether alpha is significantly different from the risk-free rate. The Semi-String Form says that it is not possible to earn excess returns by processing publicly available information. An example of a violation of the Semi-Strong Form is a portfolio manager earning excess returns by constructing portfolios of stocks with positive earnings surprises. Again, we test whether the intercept term is significantly different from the risk-free rate. The Strong From says that it is not possible to earn excess returns from non-public information. Violations of the Strong Form occur when insiders illegally take advantage of commercially sensitive information. This form of the Efficient Markets Hypothesis is difficult to test in practice, because we are unlikely to obtain data on events where insider information has been acted upon, and the individuals involved escaped prosecution. Question 3b σ 2 p = w 2 Aσ 2 A + w 2 Bσ 2 B + 2w A w B ρ A,B σ A σ B Alternatively, σ 2 p = w V w Let w = [w A w B ] w V = σ 2 A V = ρ A,B σ A σ B [ w A σ 2 A + w Bρ A,B σ A σ B ρ A,B σ A σ B σ 2 B ] w A ρ A,B σ A σ B + w B σb 2 4
w V w = w 2 Aσ 2 A + w A w B ρ A,B σ A σ B + w A w B ρ A,B σ A σ B + w 2 Bσ 2 B w V w = w 2 Aσ 2 A + w 2 Bσ 2 B + 2w A w B ρ A,B σ A σ B Question 3c For the equation to make sense in terms of being an equilibrium asset-pricing model, the intercept must equal the risk-free rate. In practice, this means keeping the slope estimate from the OLS regression, but changing the intercept estimate. Question 4a We consider an equally-weighted portfolio with n assets. Each asset has an error term e i, with corresponding variance σ 2 e i. The variance of the portfolio s error terms is defined as σ 2 e P ( n ) = Var w i e i = = 1 n i=1 n ( 1 n n i=1 i=1 ) 2 σ 2 e i σ 2 e i n = 1 n σ2 e i, where σ 2 e i is the average variance of the error term across the n assets. Clearly, as n, σ 2 e P 0. Question 4b First we estimate the beta of each asset with respect to the Index. Then we estimate the variance of the returns of the Index. Each element of the upper-diagonal of the covariance matrix then has the value ij = β i β j σ 2 M 5
The elements of the lower-diagonal can be obtained in the same way, or by recognizing that, by symmetry, the elements in ij equal those in ji. Question 4c Any two from E [e i ] = 0 E [e i (R m µ m )] = 0 E [e i e j ] = 0 Question 5A Individual asset expected return is defined in the CAPM by E [R i ] = r f + β i (E [R m r f ]) Substituting in the data provided in the question 0.05 = 0.02 + β i (0.08 0.02) which implies that β i = 0.5 Systematic risk is defined in the CAPM by (Systematic Risk) 2 = β 2 i σ 2 M Therefore, (Systematic Risk) 2 = (0.5) 2 (0.12) 2 = 0.0036 Finally, take the square-root to obtain the asset s systematic risk. Systematic risk = 0.06, or 6 percent. 6
Question 5b The investor attains the highest utility, subject to her wealth constraint, in expected return / standard deviation space. The point of tangency between the highest attainable indifference curve and the efficient frontier with borrowing and lending marks her optimal portfolio. Question 5c The efficient frontier with borrowing and lending lies above that of the risky assets frontier for all points except the tangency point. 7
8