Question

Suppose that the standard deviation of returns from a typical share is about 0.40 (or 40%) a year. The correlation between the returns of each pair of shares is about 0.6.

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" answers to 3 decimal places.)

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.)

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" answers to 3 decimal places.)

Answer 1

a)

Standard deviation=s*sqrt((1/n)+(1-1/n)*corr)

2 shares
=40%*SQRT((1/2)+(1-1/2)*0.6)=35.7770876399966%

3 shares
=40%*SQRT((1/3)+(1-1/3)*0.6)=34.253953543107%

4 shares
=40%*SQRT((1/4)+(1-1/4)*0.6)=33.466401061363%

5 shares
=40%*SQRT((1/5)+(1-1/5)*0.6)=32.9848450049413%

6 shares
=40%*SQRT((1/6)+(1-1/6)*0.6)=32.659863237109%

7 shares
=40%*SQRT((1/7)+(1-1/7)*0.6)=32.4257393351111%

8 shares
=40%*SQRT((1/8)+(1-1/8)*0.6)=32.2490309931942%

9 shares
=40%*SQRT((1/9)+(1-1/9)*0.6)=32.1109188767795%

10 shares
=40%*SQRT((1/10)+(1-1/10)*0.6)=32.00%

Variance=standard deviation^2

2 shares
=(35.7770876399966%)^2=0.128

3 shares
=(34.253953543107%)^2=0.117333333333333

4 shares
=(33.466401061363%)^2=0.112000

5 shares
=(32.9848450049413%)^2=0.108800

6 shares
=(32.659863237109%)^2=0.106666666666666

7 shares
=(32.4257393351111%)^2=0.105142857142857

8 shares
=(32.2490309931942%)^2=0.104000

9 shares
=(32.1109188767795%)^2=0.103111111111111

10 shares
=(32%)^2=0.102400

b)

Underlying market variance that cannot be diversified away=(40%)^2*0.6=0.0960000

c)

2 shares
=40%*SQRT((1/2)+(1-1/2)*0)=0.282842712474619

3 shares
=40%*SQRT((1/3)+(1-1/3)*0)=0.23094010767585

4 shares
=40%*SQRT((1/4)+(1-1/4)*0)=0.200

5 shares
=40%*SQRT((1/5)+(1-1/5)*0)=0.178885438199983

6 shares
=40%*SQRT((1/6)+(1-1/6)*0)=0.163299316185545

7 shares
=40%*SQRT((1/7)+(1-1/7)*0)=0.151185789203691

8 shares
=40%*SQRT((1/8)+(1-1/8)*0)=0.14142135623731

9 shares
=40%*SQRT((1/9)+(1-1/9)*0)=0.133333333333333

10 shares
=40%*SQRT((1/10)+(1-1/10)*0)=0.126491106406735

Variance=standard deviation^2

2 shares
=(0.282842712474619)^2=0.0800000

3 shares
=(0.23094010767585)^2=0.0533333333333332

4 shares
=(0.2)^2=0.0400000

5 shares
=(0.178885438199983)^2=0.0319999999999999

6 shares
=(0.163299316185545)^2=0.0266666666666666

7 shares
=(0.151185789203691)^2=0.0228571428571429

8 shares
=(0.14142135623731)^2=0.0200000000000001

9 shares
=(0.133333333333333)^2=0.0177777777777777

10 shares
=(0.126491106406735)^2=0.016000