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: 17 0 17 28 28 27 29 −12 −12 −13
y: 14 −2 27 18 20 11 14 −2 −3 −10
A) Compute ∑x, ∑x2, ∑y, ∑y2
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 around 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 risks and image from custom entry tool represents expected return, then image from custom entry tool can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.

a.

∑x=109, ∑x²=4173, ∑y=87, ∑y²=2083.

b.

sample mean for x=10.9.

variance for x=331.66.

standard  deviation for x=18.21.

sample mean for y=8.7.

variance for y=147.34.

standard  deviation for y=12.14.

c.

75% Chebyshev interval around the mean for x values

-25.52,47.32

75% Chebyshev interval around the mean for y values

-15.58,32.98

The interval shows that Vanguard Balanced index has smaller spread.

d.

coefficient of variation for x=167.06%

coefficient of variation for y=139.54%

Smaller CV means lower risk so, smaller CV is better.

Step-by-step explanation:

Note: ALL ANSWERS ARE ROUNDED TO TWO DECIMAL PLACES

a.

∑x=17+0+17+28+28+27+29−12−12−13=109,

∑x²=17²+0²+17²+28²+28²+27²+29²+(−12)²+(−12)²+(−13)²=4173,

∑y=14−2+27+18+20+11+14−2−3−10=87,

∑y²=14²+(−2)²+(27)²+(18)²+(20)²+(11)²+(14)²(−2)²+(−3)²+(−10)²=2083.

b.

sample mean for x=∑x/n=109/10=10.9.

x-mean

6.10

-10.90

6.10

17.10

17.10

16.10

18.10

-22.90

-22.90

-23.90

sum(x-mean)²=6.10 ²+(-10.90 )²+6.10 ²+17.10² +17.10²+ 16.10² +18.10² +(-22.90 )²+(-22.90 )²+(-23.90 )²=2984.9

Variance for x=331.66.

standard  deviation for x=√variance for x

standard  deviation for x=√331.66

standard  deviation for x=18.21.

sample mean for y=∑y/n=87/10=8.7.

y-mean

5.30

-10.70

18.30

9.30

11.30

2.30

5.30

-10.70

-11.70

-18.70

sum(y-mean)²=5.30 ²+(-10.70 )²+18.30² +9.30² +11.30² +2.30² +5.30² +(-10.70 )²+(-11.70 )²+(-18.70 )²=1326.1

variance for y=147.34.

standard  deviation for y=√variance for y

standard  deviation for y=√147.34

standard  deviation for y=12.14.

c.

75% Chebyshev Interval

mean±k standard deviation

We have to find value of k.

k²=4

k=2

so, 75% Chebyshev Interval for x

10.9±2*(18.21)

[-25.52,47.32]

75% Chebyshev Interval for y

8.7±2*(12.14)

[-15.58,32.98]

The interval shows that Vanguard Balanced index has smaller spread.

d.

CV for x=167.06%

CV for y

Cv for y=139.54%

The smaller CV shows the smaller amount of risk so, smaller CV is better.

not enough information

step-by-step explanation:

we cannot know this, because the other students could have handed out 10 flyers, or they could have handed out 100. one way, the mean would go up, and another way, the mean would go down.