Q Use the single index model to compute the covariance between stocks G and H. Home, - Calculate beta for stock G Question - Yearly data on returns are presented below for stock G and the market index (M). Year Stock G Market Index (M) 2014 -1 -2 2015 2 3 2016 6 4 2017 4 -1 2018 0 2 2019 1 3 (i) Calculate beta for stock G. Consider stock H with a beta equal to 1.2. Use the single index model to compute the covariance between stocks G and H. Answer - Stock G Market Index (M) mean return 2.00 1.50 =(-2+3+4-1+2+3)/6 return variance 6.80 5.90 =((-2-1.5)^2+(3-1.5)^2+(4-1.5)^2+(-1-1.5)^2+(2-1.5)^2+(3-1.5)^2)/6 return stdev 2.61 2.43 =(5.9)^(1/2) Correlation 0.41 The covariance between G and the market is 2.60 =0.41*6.8*5.9 =((-2-1.5)(-1-2)+(3-1.5)(2-2)+(4-1.5)(6-2)+(-1-1.5)(4-2)+(2-1.5)(0-2)+(3-1.5)(1-2))/6 The beta of G is 0.44 =2.6/5.9 The beta of H is 1.2 The covariance between G and H is 3.12 =5.9*1.2*0.44 (ii) Consider the following: - a stock Awith expected return 10%and beta 1.0; - a stock B with expected return 14% and beta 1.4; - a stock D with expected return 16% and beta 1.2; - a stock E with expected return 7% and beta 1.4. Describe how to exploit the arbitrage opportunities. Answer - If we consider another portfolio, say portfolio C, which is formed by half of portfolio A and half of portfolio B, the characteristics of this portfolio C, in terms of expected return and beta would be: - Expected return = 0.5×10 + 0.5×14 = 12; - Beta = 0.5×1.0 + 0.5×1.4 = 1.2. Now D has a higher return (16%) than C, but the same beta (risk) of C (1.2). D cannot exist in the market for a long time. This is because an investor could sell short, for example, £100 of portfolio C and buy £100 of portfolio D. In doing this, the investor would set up an arbitrage Portfolio with the following characteristics: - The investment cash Invested is zero; - Beta equal to 0; - Expected return of 16%-12%=4%. This arbitrage portfolio gives: -a positive profit on average, -with a zero systematic risk, -and a zero net investment. An investor will engage in this arbitrage and in doing this he/she will bring down the expected return on D to the equilibrium level of 12%. Then E has a lower expected return (7%) than B, but the same beta as B (1.4). Also E cannot exist for a long time: - an investor could sell short, for example, £100 of E and buy £100 of B. In doing this, the investor would set up an arbitrage Portfolio with the following characteristics: - The investment cash Invested is zero; - Beta equal to 0; - Expected return of 14%-7%=7%. This arbitrage portfolio gives: -a positive profit on average, -with a zero systematic risk, -and a zero net investment. An investor will engage in this arbitrage and in doing this he/she will bring up the expected return on E to the equilibrium level of 14%. Related: Derive the standard version of CAPM What the minimum risk combination of two assets Calculate beta for stock G
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