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BKM_Sol_Ch_5 - Chapter 05 Risk and Return Past and Prologue...

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Chapter 05 - Risk and Return: Past and Prologue Chapter 5 Risk and Return: Past and Prologue 1. V(12/31/2007) = V(1/1/1991) × (1 + g) 7 = $100,000 × (1.05) 7 = $140,710.04 2. i and ii. The standard deviation is non-negative. 3. c. Determines most of the portfolio’s return and volatility over time. 4. E(r) = [0.3 × 44%] + [0.4 × 14%] + [0.3 × (–16%)] = 14% σ 2 = [0.3 × (44 – 14) 2 ] + [0.4 × (14 – 14) 2 ] + [0.3 × (–16 – 14) 2 ] = 540 σ = 23.24% The mean is unchanged, but the standard deviation has increased. 5. a. The holding period returns for the three scenarios are: Boom: (50 – 40 + 2)/40 = 0.30 = 30.00% Normal: (43 – 40 + 1)/40 = 0.10 = 10.00% Recession: (34 – 40 + 0.50)/40 = –0.1375 = –13.75% E(HPR) = [(1/3) × 30%] + [(1/3) × 10%] + [(1/3) × (–13.75%)] = 8.75% σ 2 (HPR) = [(1/3) × (30 – 8.75) 2 ] + [(1/3) × (10 – 8.75) 2 ] + [(1/3) × (–13.75 – 8.75) 2 ] = 319.79 σ = 79 . 319 = 17.88% b. E(r) = (0.5 × 8.75%) + (0.5 × 4%) = 6.375% σ = 0.5 × 17.88% = 8.94% 5-1
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Chapter 05 - Risk and Return: Past and Prologue 6. Investment 3. For each portfolio: Utility = E(r) – (0.5 × 4 × σ 2 ) Investment E(r) σ U 1 0.12 0.30 -0.0600 2 0.15 0.50 -0.3500 3 0.21 0.16 0.1588 4 0.24 0.21 0.1518 We choose the portfolio with the highest utility value. 7. Investment 4. When an investor is risk neutral, A = 0 so that the portfolio with the highest utility is the portfolio with the highest expected return. 8. b 9. E(r X ) = [0.2 × (–20%)] + [0.5 × 18%] + [0.3 × 50%)] = 20% E(r Y ) = [0.2 × (–15%)] + [0.5 × 20%] + [0.3 × 10%)] = 10% 10. σ X 2 = [0.2 × (–20 – 20) 2 ] + [0.5 × (18 – 20) 2 ] + [0.3 × (50 – 20) 2 ] = 592 σ X = 24.33% σ Y = [0.2 × (–15 – 10) 2 ] + [0.5 × (20 – 10) 2 ] + [0.3 × (10 – 10) 2 ] = 175 σ Y = 13.23% 11. E(r) = (0.9 × 20%) + (0.1 × 10%) = 19% 12. The probability is 0.50 that the state of the economy is neutral. Given a neutral economy, the probability that the performance of the stock will be poor is 0.30, and the probability of both a neutral economy and poor stock performance is: 0.30 × 0.50 = 0.15 13. E(r) = [0.1 × 15%] + [0.6 × 13%] + [0.3 × 7%)] = 11.4% 5-2
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Chapter 05 - Risk and Return: Past and Prologue 14. a. Time-weighted average returns are based on year-by-year rates of return. Year Return = [(capital gains + dividend)/price] 2005-2006 (110 – 100 + 4)/100 = 14.00% 2006-2007 (90 – 110 + 4)/110 = –14.55% 2007-2008 (95 – 90 + 4)/90 = 10.00% Arithmetic mean: 3.15% Geometric mean: 2.33% b. Time Cash flow Explanation 0 -300 Purchase of three shares at $100 per share 1 -208 Purchase of two shares at $110, plus dividend income on three shares held 2 110 Dividends on five shares, plus sale of one share at $90 3 396
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