Midterm_2023_solutions

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Professor John Heaton Midterm Exam Solutions Investments, Winter 2023 1. (15 points total) Short answer and true/false questions. Provide discussion or justi- fication in each case. You are graded on the basis of the quality of your discussion. (a) (5 points) True or False and explain why. You are currently holding a well-diversified portfolio. Call this portfolio “P.” You are approached by an investment manager offering an investment strategy (call it “A”). As part of your evaluation you look at historical performance and run a regression of the form: r A,t − r f,t = α A + β A ( r P,t − r f,t ) + ϵ A,t You find that the value of α A is 5% annually. Based on this result you should switch your investments from “P” to “A.” Solution: You should switch “some” of your investment in P to A. Alpha needs to be balanced against the idiosyncratic risk in the term ϵ A,t (b) (5 points) Short answer You are considering a strategy that invests in stocks with positive news about earnings announcements (announced earnings that beat analyst forecasts). You find that this strategy consistently generates a positive CAPM alpha. Is this evidence against a “semi-strong” form of market efficiency. Explain. Solution: Potentially since the extra return comes from public news other than past price movements. If the CAPM properly corrects for the risk of the position then this is evidence of semi-strong market efficiency. However, the alpha could be due to a correct for risk that we are missing and therefore would reflect that risk and not market inefficiency. (c) (5 points) Short Answer You are looking at the historical performance of the endowment of the Art Institute. Several of the trustees of the Art Institute are pushing to have the endowment invest substantially more in investment classes that have historically performed very well. What advice might you give the trustees as they consider how to use this historical evidence to inform their investment allocation? Solution: Issues advice: 1

In using past data we must consider that we are estimating parameters governing performance. Even if the investment world is stable over time we will be chancing performance that was good randomly. The world may not stay the same over time. We should be therefore be careful of using past performance of an indicating of the future. Look at what other similar institutions are using. Carefully consider the purpose of the endowment. 2

2. (45 points) Consider the following information about two risky securities: E ( r A ) = 8% σ A = 0 . 25 ρ A,B = 0 . 2 E ( r B ) = 12% σ B = 0 . 25 (a) (7 points) Suppose you currently hold a portfolio that invests 50% in security B and 50% in security C . What is the expected return to this portfolio? What is the standard deviation of the return to the portfolio? Solution E ( r p ) = 50% × 8% + 50 × 12% = 10% σ 2 p = 0 . 5 2 × 0 . 25 2 + 0 . 5 2 × 0 . 25 2 + 2 × 0 . 5 × 0 . 5 × 0 . 2 × 0 . 25 × 0 . 25 = 0 . 0375. σ p = √ 0 . 0375 = 19 . 36%. (b) (8 points) i. Calculate the covariance of the portfolio in part (2a) with each of the secu- rities. Compare the two covariances. ii. What does this result imply about the portfolio from part (2a) compared to all other portfolios that combine securities A and B together? In a world without a risk-free security, why might an investor choose this portfolio? Solution The covariance between security A and the portfolio is given by: cov ( r A , r p ) = 0 . 5 × cov ( r A , r A ) + 0 . 5 × cov ( r A , r B ) = 0 . 5 × 0 . 25 2 + 0 . 5 × 0 . 2 × 0 . 25 × 0 . 25 = 0 . 0375 The covariance between security B and the portfolio is given by: cov ( r B , r p ) = 0 . 5 × cov ( r B , r A ) + 0 . 5 × cov ( r B , r B ) = 0 . 5 × 0 . 2 × 0 . 25 × 0 . 25 + 0 . 5 × 0 . 25 2 = 0 . 0375 Since this are equal, this must be the minimum variance portfolio . The investor does not like risk and would like the minimum volatility possible. 3

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(c) (15 points) Now suppose an investor has 500,000 and that the risk-free (security with no volatility) rate is 5%. The investor would like to match the expected return of the position from (2a). Construct a portfolio using the risk-free security, A and B that matches the required expected return but has lowest possible volatility. Determine and report the positions (in ’s) in the risk-free security, in A and in B. What is the Sharpe ratio of this position? Solution: Let w A MV E be the weight on portfolio A in the MVE portfolio for this case. This must satisfy: w MV E A = E (˜ r e A ) σ 2 B − E (˜ r e B ) cov (˜ r A , ˜ r B ) E (˜ r e A ) σ 2 B + E (˜ r e B ) σ 2 A − [ E (˜ r e A ) + E (˜ r e B )] cov (˜ r A , ˜ r B ) = 6% × 0 . 25 2 − 10% × 0 . 2 × 0 . 25 × 0 . 25 6% × 0 . 25 2 + 10% × 0 . 25 2 − [6% + 10%] × 0 . 2 × 0 . 25 × 0 . 25 = 0 . 20 The expected return to this portfolio is: E ( r MV E ) = 0 . 2 × 8% + (1 − 0 . 2) × 12% = 11 . 2% To achieve the expected return from part ( ?? ) we need a position in the risk-free and the MVE portfolio such that: (1 − w MV E ) × 5% + w MV E × 11 . 2% = 10% This implies: w MV E = 10% − 5% 11 . 2% − 5% = 80 . 65% Security Dollar Allocation A 80 . 65% × 20% × $500 , 000 = 80,645.16 B 80 . 65% × (1 − 20%) × $500 , 000 = 322.580.65 r f (1 − 80 . 65%) × $500 , 000 = 96,774.19 The variance of the MVE portfolio is: σ 2 MV E = 0 . 2 2 × 0 . 25 2 + (1 − 0 . 2) 2 × 0 . 25 2 + 2 × 0 . 2 × (1 − 0 . 2) × 0 . 2 × 0 . 25 × 0 . 25 = 0 . 0465 4

The standard deviation of the portfolio return is: σ MV E = √ 0 . 0358 = 21 . 56% The Sharpe ratio of the MVE portfolio and the portfolio constructed to target the expected return from part (2a) is: Sharpe portfolio = 11 . 2% − 5% 21 . 56% = 0 . 29 (d) (5 points) How does this Sharpe ratio of part (2c) compare to the Sharpe ratio of the portfolio of part (2a)? Provide a numerical comparison. Solution The Sharpe ratio of the portfolio from (2a) is: Sharpe parta = 10% − 5% 19 . 36% = 0 . 26 This is much lower than the Sharpe ratio of the MVE portfolio. (e) (7 points) You report and explain the results of part (2d) to the investor currently holding position that is 50% in A and 50% in B. After understanding the tradeoffs the investor would like to take more risk. They would like a position with higher return that increases the standard deviation of their position to 1.5 times the standard deviation you found in part (2a). Again assume this investor has 500,000 to invest. Report the positions in ( ’s) you would allocate to the risk- free security, to A, and to B. Solution The standard deviation to target is: 1 . 5 × 19 . 36% = 29 . 05%. When need a weight in the MVE portfolio to get this: w MV E = 29 . 05% 21 . 56% = 135% The positions in the securities would be: Security Dollar Allocation A 135% × 20% × $500 , 000 = 134,703.98 B 135% × (1 − 20%) × $500 , 000 = 538,815.91 r f (1 − 135%) × $500 , 000 = - 173,519.88 (f) (3 points) What practical issue might you face in actually implemented the portfolio from part (2e)? Solution : Cost of buying stocks on margin would not be the same as the risk- free rate. 5

3. (30 points) Consider the following information about the returns to two securities (“1” and “2”), the market portfolio (“m”) and the risk-free security (“f”): E ( r 1 ) = 12% σ 1 = 30% E ( r 2 ) = 6% σ 2 = 25% E ( r m ) = 10% σ m = 20% ρ 1 , 2 = 0 . 5 ρ 1 ,m = 0 . 5 ρ 2 ,m = 0 . 5 r f = 2% (a) (10 points) You would like to assess whether the CAPM is a good explanation of the returns to securities 1 and 2. What are the CAPM alpha’s and beta’s of securities 1 and 2? Solution: β 1 = cov ( r 1 , r m ) σ 2 m = 0 . 5 × 0 . 3 × 0 . 2 0 . 2 2 = 0 . 75 α 1 = ( E ( r 1 ) − r f ) − β 1 × [ E ( r m ) − r f ] = 0 . 12 − 0 . 02 − 0 . 75 × (0 . 10 − 0 . 02) = 4% β 2 = cov ( r 2 , r m ) σ 2 m = 0 . 5 × 0 . 25 × 0 . 2 0 . 2 2 = 0 . 62 α 2 = ( E ( r 2 ) − r f ) − β 2 × [ E ( r m ) − r f ] = 0 . 06 − 0 . 02 − 0 . 62 × (0 . 10 − 0 . 02) = − 1% (b) (2 points) Based on the results of part (3a), what do you conclude about the CAPM? What does this imply about how you should construct a portfolio with the best Sharpe ratio possible? (Answer this part in words only. ) Solution: Non-zero alpha: CAPM not working. (c) (2 points) You decide to create a long-short position in 1 and 2 which is 50% in security 1, -50% in security 2 and then 100% in the risk-free security. (Note the weights in this position using 1, 2 and the risk-free security add to 100%). Why would this long-short position be the right direction to consider if you are currently holding the market portfolio? Solution: In exploiting non-zero alpha you should shift towards positive and away from negative alpha. This is exactly what this long-short position does. (d) (8 points) What would be the expected return, standard deviation, beta and alpha of the long-short position from part (3c)? Solution E ( r ls ) = 50% × 12% − 50% × 6% + 100% × 2% 6

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= 5% σ 2 ls = 0 . 5 2 × 0 . 3 2 + ( − 0 . 5) 2 × 0 . 25 2 + 2 × 0 . 5 × ( − 0 . 5) × 0 . 5 × 0 . 3 × 0 . 25 = 0 . 0194 σ ls = √ 0 . 0194 = 14% β ls = 0 . 5 × β 1 + ( − 0 . 5) × β 2 = 0 . 5 × 0 . 75 − 0 . 5 × 0 . 62 = 0 . 06 α ls = E ( r ls − r f ) − β ls × [ E ( r m ) − r f ] = 5% − 2% − 0 . 06 × (10% − 2%) = 3% (e) (8 points) What position would you take in the long-short position of part (3c) and the market portfolio to provide the maximum Sharpe ratio possible? Solution The position in the market portfolio in the maximum Sharpe ratio portfolio (MVE) is given by: w MV E m = E (˜ r e m ) σ 2 ls − E (˜ r e ls ) cov (˜ r m , ˜ r ls ) E (˜ r e m ) σ 2 ls + E (˜ r e ls ) σ 2 m − [ E (˜ r e m ) + E (˜ r e ls )] cov (˜ r m , ˜ r ls ) From part (3d) we have all of the inputs except: cov ( r m , r ls ) = β ls × σ 2 m = 0 . 06 × 0 . 2 2 = 0 . 0025 So we have: w MV E m = 5% × 0 . 0194 − 3% × 0 . 0025 5% × 0 . 0194 + 3% × 0 . 2 2 + [5% + 3%] × 0 . 0025 = 59 . 6% Hence the weight in the long-short position would be 40.4%. 7