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 .
x: 14 0 36 23 33 25 26 14 14 23
y: 6 5 26 17 24 17 17 5 6 6
4607
4803
5332
4243
4940
4343
x:
15
0
38
25
31
27
28
15
15
25
y:
8
2
29
18
26
16
18
2
3
8
To compute the value, you need to multiply each value of x with the corresponding value of y from the given data and then sum up the products.
Let's calculate step by step:
1. Multiply each value of x with the corresponding value of y:
x * y = (14 * 6) + (0 * 5) + (36 * 26) + (23 * 17) + (33 * 24) + (25 * 17) + (26 * 17) + (14 * 5) + (14 * 6) + (23 * 6) = 84 + 0 + 936 + 391 + 792 + 425 + 442 + 70 + 84 + 138 = 3302
2. Sum up the products:
3302 + 4607 + 4803 + 5332 + 4243 + 4940 = 27327
Therefore, the value is 27327.
To determine whether bonds reduce the overall risk of an investment portfolio, we need to compare the risk of investing solely in stocks (represented by the random variable x) to the risk of investing in a balanced portfolio (represented by the random variable y).
One way to measure risk is by calculating the standard deviation of the returns. The higher the standard deviation, the higher the risk.
To calculate the standard deviation, follow these steps:
Step 1: Calculate the mean (average) return for each random variable.
For random variable x:
Mean of x = (14 + 0 + 36 + 23 + 33 + 25 + 26 + 14 + 14 + 23) / 10 = 20.8
For random variable y:
Mean of y = (6 + 5 + 26 + 17 + 24 + 17 + 17 + 5 + 6 + 6) / 10 = 11.5
Step 2: Calculate the deviation for each return. Deviation is the difference between each return value and the mean.
For random variable x:
Deviation of each value in x = (14 - 20.8, 0 - 20.8, 36 - 20.8, 23 - 20.8, 33 - 20.8, 25 - 20.8, 26 - 20.8, 14 - 20.8, 14 - 20.8, 23 - 20.8) = (-6.8, -20.8, 15.2, 2.2, 12.2, 4.2, 5.2, -6.8, -6.8, 2.2)
For random variable y:
Deviation of each value in y = (6 - 11.5, 5 - 11.5, 26 - 11.5, 17 - 11.5, 24 - 11.5, 17 - 11.5, 17 - 11.5, 5 - 11.5, 6 - 11.5, 6 - 11.5) = (-5.5, -6.5, 14.5, 5.5, 12.5, 5.5, 5.5, -6.5, -5.5, -5.5)
Step 3: Square each deviation to eliminate negative values.
For random variable x:
Squared deviation of each value in x = (-6.8)^2, (-20.8)^2, (15.2)^2, (2.2)^2, (12.2)^2, (4.2)^2, (5.2)^2, (-6.8)^2, (-6.8)^2, (2.2)^2
For random variable y:
Squared deviation of each value in y = (-5.5)^2, (-6.5)^2, (14.5)^2, (5.5)^2, (12.5)^2, (5.5)^2, (5.5)^2, (-6.5)^2, (-5.5)^2, (-5.5)^2
Step 4: Calculate the variance for each random variable.
For random variable x:
Variance of x = ( (-6.8)^2 + (-20.8)^2 + (15.2)^2 + (2.2)^2 + (12.2)^2 + (4.2)^2 + (5.2)^2 + (-6.8)^2 + (-6.8)^2 + (2.2)^2 ) / 10 = 161.12
For random variable y:
Variance of y = ( (-5.5)^2 + (-6.5)^2 + (14.5)^2 + (5.5)^2 + (12.5)^2 + (5.5)^2 + (5.5)^2 + (-6.5)^2 + (-5.5)^2 + (-5.5)^2 ) / 10 = 75.45
Step 5: Calculate the standard deviation for each random variable. It is the square root of the variance.
For random variable x:
Standard deviation of x = sqrt(161.12) ≈ 12.69
For random variable y:
Standard deviation of y = sqrt(75.45) ≈ 8.69
Comparing the standard deviations, we can see that the standard deviation for the stock index (12.69) is higher than the balanced index (8.69). This indicates that investing in a balanced portfolio with bonds (y) reduces the overall risk compared to investing solely in stocks (x).