Question

Suppose that the standard deviation of returns from a typical share is about .36 (or 36%) a year. The correlation between the returns of each pair of shares is about .4.

a. Calculate the variance and standard deviation of the returns on a portfolio that has equal investments in 2 shares, 3 shares, and so on, up to 10 shares. (Use decimal values, not percents, in your calculations. Do not round intermediate calculations. Round the "Variance" answers to 6 decimal places. Round the "Standard Deviation" to 3 decimal places.)

No. of		Standard
Shares	Variance	Deviation
1
2
3
4
5
6
7
8
9
10

b. How large is the underlying market variance that cannot be diversified away? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

Market risk                        

c. Now assume that the correlation between each pair of stocks is zero. Calculate the variance and standard deviation of the returns on a portfolio that has equal investments in 2 shares, 3 shares, and so on, up to 10 shares. (Use decimal values, not percents, in your calculations. Do not round intermediate calculations. Round the "Variance" answers to 6 decimal places. Round the "Standard Deviation" to 3 decimal places.)

No. of		Standard
Shares	Variance	Deviation
1
2
3
4
5
6
7
8
9
10

Answer 1

A)

• For each different portfolio ,the relative weight of each share is ( one divided by the number of shares (n) in the portfolio ,
• And the standard deviation of each share is .36 ,and the correlation between each pair of share is .40.
• Thus for each portfolio ,the diagonal terms are the same and the off diagonal terms are the same.
• There are (n) diagonal terms and (n²-n) off diagonal terms.
• In general we have:-

Variance = n(1/n)² (.36)² + (n²-n)(1/n)²(0.4)(.3)(.3)

For one share = variances = 1(1)²(.36)²+0=0.129

For two share variance = 2(.5)²(0.36)²+ 2(0.5)²(0.4)(.36)(.36)= 0.089

B)

The underlying market risk that can not be diversified away is the second term in the formula for variance above :

Underlying market risk = ( n²-n)(1/n)²(.4)(.36)(.36)

As n increase [(n²-n)(1/n²]= [(n-1)/n] become close to 1 ,so that the underlying market risk is : [(.4)(.36)(.36)] = 0.051

C)

This is same as part(a) ,except that all of the off - diagonal terms are now equal to zero.