Stocks and Bonds Do bonds reduce the overall risk of an investment portfolio? Le
ID: 3231446 • Letter: S
Question
Stocks and Bonds Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data (Reference: Morningstar Research Group, Chicago). x: 11 0 36 21 31 23 24 -11 - 11 - 21 y: 10 - 2 29 14 22 18 14 - 2 - 3 - 10 (a) Compute sigma x, 2, sigma x^2, sigma y, and sigma y^2. (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (c) Compute a 75% Chebyshev interval about the mean for x values and also for y values. Use the intervals to compare the two funds. (d) Compute the coefficient of variation for each fund. Use the coefficients of variation to compare the two funds. If s represents risk and x represents expected return, then s/x can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.Explanation / Answer
Part a
The computation table is given as below:
No.
x
y
x^2
y^2
1
11
10
121
100
2
0
-2
0
4
3
36
29
1296
841
4
21
14
441
196
5
31
22
961
484
6
23
18
529
324
7
24
14
576
196
8
-11
-2
121
4
9
-11
-3
121
9
10
-21
-10
441
100
Total
103
90
4607
2258
x = 103
y = 90
x^2 = 4607
y^2
Part b
Sample mean = x/n
Sample variance = (x – mean)^2/(n – 1)
Calculation table is given as below:
No.
x
y
x^2
y^2
(X - mean)^2
(Y - mean)^2
1
11
10
121
100
0.49
1
2
0
-2
0
4
106.09
121
3
36
29
1296
841
660.49
400
4
21
14
441
196
114.49
25
5
31
22
961
484
428.49
169
6
23
18
529
324
161.29
81
7
24
14
576
196
187.69
25
8
-11
-2
121
4
453.69
121
9
-11
-3
121
9
453.69
144
10
-21
-10
441
100
979.69
361
Total
103
90
4607
2258
3546.1
1448
Mean
10.3
9
Sample mean for x = 10.3
Sample mean for y = 9
Sample variance = (x – mean)^2/(n – 1)
Sample variance for x = 3546.1/(10 – 1) = 394.0111111
Sample standard deviation for x = sqrt(394.0111111) = 19.84971312
Sample variance for y = 1448/9 = 160.8888889
Sample standard deviation for y = sqrt(160.8888889) = 12.68419839
Part c
Here, we have to find 75% Chebyshev interval.
According to Chebyshev rule, at least 75% of the observations fall between -2 and +2 standard deviations from mean.
That’s, for x, interval is given as below:
Lower limit = Mean – 2*SD = 10.3 - 2*19.84971312 = -29.3994
Upper limit = Mean + 2*SD = 10.3 + 2*19.84971312 = 49.99943
For y, interval is given as below:
Lower limit = Mean – 2*SD = 9 - 2*12.68419839 = -16.3684
Upper limit = Mean + 2*SD = 9 + 2*12.68419839 =34.3684
Width for the interval for x is more than width of the interval y.
Part d
Here, we have to find coefficient of variation for x and y.
For x,
CV = S/Xbar = 19.84971312/10.3 = 1.927157
For y,
CV = S/Ybar = 12.68419839/9 = 1.409355
Coefficient of variation is low for y, this means y is better than x.
Smaller CV’s are better because it indicate less variation in the observations.
No.
x
y
x^2
y^2
1
11
10
121
100
2
0
-2
0
4
3
36
29
1296
841
4
21
14
441
196
5
31
22
961
484
6
23
18
529
324
7
24
14
576
196
8
-11
-2
121
4
9
-11
-3
121
9
10
-21
-10
441
100
Total
103
90
4607
2258