Question

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.

x: 32 0 36 21 38 12 33 −24 −22 −24

y: 26 −10 18 13 9 26 18 −4 −4 −9

(a) Compute Σx, Σx², Σy, Σy².

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.)

(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)\

(d) Compute the coefficient of variation for each fund. (Round your answers to the nearest whole number.)

Answer 1

Solution:

Given:x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks) and y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond)

x: 32 0 36 21 38 12 33 −24 −22 −24

y: 26 −10 18 13 9 26 18 −4 −4 −9

Part (a) Compute Σx, Σx², Σy, Σy².

Thus we need to make following table:

x	x2	y	y2
32	1024	26	676
0	0	-10	100
36	1296	18	324
21	441	13	169
38	1444	9	81
12	144	26	676
33	1089	18	324
-24	576	-4	16
-22	484	-4	16
-24	576	-9	81

Thus we get:

Part (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y.

For variable x:

Mean:

Variance:

and standard deviation is:

For variable y:

Variance:

and standard deviation is:

Part (c) Compute a 75% Chebyshev interval around the mean for x values and also for y values.

According to Chebyshev rule, at least 75% of data lies within 2 standard deviation from the mean.

Thus find interval :  

and

Thus

and

Part (d) Compute the coefficient of variation for each fund.

That is coefficient of variation = CV

and