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07%20risk%20and%20return%20-%20lecture%20problems%20solutions

# 07%20risk%20and%20return%20-%20lecture%20problems%20solutions

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Lecture problems – risk and return What were the arithmetic and geometric average returns for a stock that had the following annual returns over the past 4 years: 12% (4 years ago), 15% (3 years ago), -23% (2 years ago), and -49% (last year)? Arithmetic average return = [12% + 15% + (-23%) + (-49%)] / 4 = -45% / 4 = -11.25% = [0.12 + 0.15 + (-0.23) + (-0.49)] / 4 = -0.45 / 4 = -0.1125 = -11.25% = (0.25)(0.12) + (0.25)(0.15) + (0.25)(-0.23) + (0.25)(-0.49) = 0.0300 + 0.0375 + (-0.0575) + (-0.1225) = -0.1125 = -11.25% Geometric average return = [(1 + .12)(1 + .15)(1 + (-.23))(1 + (-0.49))] 1/4 – 1 = [(1.12)(1.15)(0.77)(0.51)] 1/4 – 1 = [0.5057976] 1/4 – 1 = 0.8433 – 1 = -.1567 = -15.67% 1

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Lecture problems – risk and return If we expect a 9.5% real return and we expect inflation to be 2.5%, what is the nominal rate that we expect? (1 + nominal rate) = (1 + real rate) × (1 + inflation rate) (1 + nominal rate) = (1 + .095) × (1 + .025) = 1.095 × 1.025 = 1.1224 So 1 + nominal rate = 1.095 × 1.025 = 1.1224 Nominal rate = 1.1224 – 1 = .1224 = 12.24% If we expect a 9.5% nominal return and we expect inflation to be 2.5%, what is the real rate that we expect? (1 + real rate) = (1 + nominal rate) ÷ (1 + inflation rate) (1 + real rate) = (1 + .095) ÷ (1 + .025) = 1.095 ÷ 1.025 = 1.0683 Real rate = 1.0683 – 1 = .0683 = 6.83% Last year, if the real rate was 12.3% and the nominal rate was 16.7%, what was inflation? (1+inflation rate) = (1+nominal rate) ÷ (1+real rate) (1+inflation rate) = (1 + .167) ÷ (1 + .123) = 1.167 ÷ 1.123 = 1.0392 Inflation rate = 1.0392 – 1 = .0392 = 3.92% 2
Lecture problems – risk and return What were the variance and standard deviation of a bond’s returns if it had the following annual returns over the past 3 years: -9%, 6%, and 12%? Annual return Weight for mean [1/n] = 1/3 An return × wt for mean An ret – mean [An ret – mean] 2 Weight for var [1/(n – 1)] = 1/2 Wt for var × [an ret – mean] 2 -0.09 0.33333 -0.03 -0.12 0.0144 0.5 0.0072 0.06 0.33333 0.02 0.03 0.0009 0.5 0.00045 0.12 0.33333 0.04 0.09 0.0081 0.5 0.00405 Mean = 0.03 Var = 0.0117 SD = 0.1082 Calculate the mean of the actual returns Mean = [-9% + 6% + 12%] / 3 = [-.09 + .06 + .12] / 3 = .09 / 3 = .03 = 3% Calculate the sample variance Var(R) = {[1/2] × [-.09 – .03] 2 } + {[1/2] × [.06 – .03] 2 } + {[1/2] × [.12 – .03] 2 } = {[1/2] × [-.12] 2 } + {[1/2] × [.03] 2 } + {[1/2] × [.09] 2 } = {[1/2] × .0144} + {[1/2] × .0009} + {[1/2] × .0081} = .0117 Calculate the sample standard deviation SD(R) = √Var(R) = √.0117 = .1082 = 10.82% 3

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Lecture problems – risk and return What are the expected return, variance, and standard deviation of a stock with the following returns and probabilities for each of three possible outcomes (bad, medium, and good)? Outcome Return Probability Bad -32.5% 20% Medium 5.0% 50% Good 30.0% 30% E(R) = [p(bad) × R(bad)] + [p(medium) × R(medium)] + [p(good) × R(good)] = [0.20 × (-.325)] + [0.50 × .05] + [0.30 × .30] = -.065 + .025 + .090 = .050 = 5.0% Outcome R(s) Weight for mean p(s) p(s) × R(s) R(s) – E(R) [R(s) – E(R)] 2 Weight for var p(s) p(s) × [R(s) – E(R)] 2 Bad -0.325 0.20 -0.065 -0.375 0.14063 0.20 0.028125 Medium 0.050 0.50 0.025 0.000 0.00000 0.50 0.000000 Good 0.300 0.30 0.090 0.250 0.06250 0.30 0.018750 Total E(R) = 0.050 Var(R) = 0.046875 SD(R) = 0.2165 Calculate the expected variance Var(R) = {.20 × [-.325 – .05] 2 } + {.50 × [.05 – .05] 2 } + {.30 × [.30 – .05] 2 } = {.20 × [-.375] 2 } + {.50 × [.0] 2 } + {.30 × [.25] 2 } = {.20 × .140625} + {.50 × 0} + {.30 × .0625} = .046875 Calculate the expected standard deviation SD(R) = √Var(R) = √.046875 = .2165 = 21.65% 4
Lecture problems – risk and return What is the expected return for a portfolio that consists of 1,000 shares of stock A priced at \$24 each with an expected return of 16.0% and 500 shares of stock B if priced at \$32 each with an expected return of 10.0%?

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07%20risk%20and%20return%20-%20lecture%20problems%20solutions

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