Question

In: Finance

Given two random variables x and y State of Nature Probability variable x variable y I...

Given two random variables x and y

State of Nature Probability variable x variable y

I 0.2 18 0

II 0. 2 5 -3

III 0.2 12 15

IV 0.2 4 12

V 0.2 6 1

(i) Calculate the mean and variance of each of these variables and the covariance between them

(ii) Suppose x and y represent the returns from two assets. Calculate the mean and variance for the following part folios.

% in x 125 100 75 50 25 0 -25

% in y -25 0 25 50 75 100 125

(iii)Find the portfolio that has the minimum variance.

(iv)Let portfolio A have 75% in x and portfolio B has 25% in x. Calculate the covariance between the two portfolios.

(v) Calculate the covariance between the minimum variance portfolio and portfolio A.

Solutions

Expert Solution

1. Mean of x(x')= sum of values of x/ number of values of x=(18+5+12+4+6)/5=9

Variance of x= =(81+16+9+25+9)/4=35

Mean of y(y')= sum of values of y/ number of values of y=(0-3+15+12+1)/5=5

Variance of y= sum of =(25+64+100+49+9+16)/4=65.75

Covariance of x and y=

=[(9*-5)+(-4*-8)+(3*10)+(-5*7)+(-3*-4)]/4= -1.5

2. Mean and Variance of each portfolio

a. mean= 1.25*9+(-.25*5)=10

Variance=*35+()*65.75+2*1.25*-0.25*-1.5=59.744

b. mean=1*9+0*5=9

Variance=*35+0*65.75+2*1*0*-1.5=35

c.mean= 0.75*9+(0.25*5)=8

Variance=*35+()*65.75+2*0.75*0.25*-1.5=23.23

d.mean= 0.5*9+(0.5*5)=7

Variance=*35+()*65.75+2*0.5*0.5*-1.5=24.44

e.mean= 0.75*5+(0.25*9)=6

Variance=*65.75+()*35+2*0.75*0.25*-1.5=38.6

f. mean=1*5+0*9=5

Variance=*65.75+0*35+2*1*0*-1.5=65.75

g. mean= 1.25*5+(-.25*9)=4

Variance=*65.75+()*35+2*1.25*-0.25*-1.5=105.86

3. Minimum variance portfolio is the one with 75% allocated to x and 25% allocated to y

jeff jeffy answered 1 year ago

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