Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable
representing annual percent return for the Vanguard Total Stock Index (all Stocks). Let y be a
random variable representing annual return for the Vanguard Balanced Index (60% stock and
40% bond). For the past several years, assume the following data. Compute the coefficient of
variation for each fund. Round your answers to the nearest tenth.
x: 11 0 36 24 35 26 27 –11 –11 –24
y: 9 –4 28 14 25 17 14 –4 –5 –9
To compute the coefficient of variation for each fund, we need to calculate the mean and standard deviation for each random variable.
For random variable x:
Mean (µx) = (11 + 0 + 36 + 24 + 35 + 26 + 27 - 11 - 11 - 24) / 10 = 14.3
To calculate the standard deviation, we first need to calculate the variance:
Variance (σx^2) = [(11 - 14.3)^2 + (0 - 14.3)^2 + (36 - 14.3)^2 + (24 - 14.3)^2 + (35 - 14.3)^2 + (26 - 14.3)^2 + (27 - 14.3)^2 + (-11 - 14.3)^2 + (-11 - 14.3)^2 + (-24 - 14.3)^2] / 10
= [(-3.3)^2 + (-14.3)^2 + (21.7)^2 + (9.7)^2 + (20.7)^2 + (11.7)^2 + (12.7)^2 + (-25.3)^2 + (-25.3)^2 + (-38.3)^2] / 10
= [10.89 + 204.49 + 470.89 + 94.09 + 428.49 + 136.89 + 161.29 + 638.09 + 638.09 + 1470.49] / 10
= 4252.0 / 10
= 425.2
Standard Deviation (σx) = √(425.2) ≈ 20.6
Now, let's calculate the mean and standard deviation for random variable y:
Mean (µy) = (9 - 4 + 28 + 14 + 25 + 17 + 14 - 4 - 5 - 9) / 10 = 11.1
Variance (σy^2) = [(9 - 11.1)^2 + (-4 - 11.1)^2 + (28 - 11.1)^2 + (14 - 11.1)^2 + (25 - 11.1)^2 + (17 - 11.1)^2 + (14 - 11.1)^2 + (-4 - 11.1)^2 + (-5 - 11.1)^2 + (-9 - 11.1)^2] / 10
= [(-2.1)^2 + (-15.1)^2 + (16.9)^2 + (2.9)^2 + (13.9)^2 + (5.9)^2 + (2.9)^2 + (-15.1)^2 + (-16.1)^2 + (-20.1)^2] / 10
= [4.41 + 227.01 + 285.61 + 8.41 + 193.21 + 34.81 + 8.41 + 227.01 + 259.21 + 404.01] / 10
= 1662.29 / 10
= 166.2
Standard Deviation (σy) = √(166.2) ≈ 12.9
Now, let's calculate the coefficient of variation for each fund:
Coefficient of Variation (CVx) = (σx / µx) * 100
= (20.6 / 14.3) * 100
≈ 143.4
Coefficient of Variation (CVy) = (σy / µy) * 100
= (12.9 / 11.1) * 100
≈ 116.2
So, the coefficient of variation for the Vanguard Total Stock Index is approximately 143.4, and the coefficient of variation for the Vanguard Balanced Index is approximately 116.2.