Question

Q4. Suppose you develop a mutual fund that includes 500 NASDAQ stocks, all with equal weights in the fund's portfolio. The average return standard deviation of the stocks is 44 percent, and the average pairwise correlation among the stocks is 0.30. What is your estimate of the standard deviation of the fund's portfolio?

Answer 1

If w1, w2 , w3 …wn are weight in the portfolio for assets 1, 2,3 ….n

Then,w1+w2+w3+……………………+wn=1

S1, S2, S3……Sn are the standard deviation of the assets 1, 2, 3 …n

Portfolio Variance=(w1^2)*(S1^2)+(w2^2)(S2^2)+………….(wn^2)*(Sn^2)+2w1w2*Cov(1,2)+2w1w3*Cov(1,3)+………+w(n-1)wn*Cov(n,(n-1)

Cov(1,2)=Covariance of returns of asset1 and asset2

Portfolio Standard Deviation =Square root of Portfolio variance

In this Case ,

n=500

w1=w2=w3……….=w500=1/500

Average Si=44%

Average Variance =(Si^2)=(44^2)=1936

Average Corrli,j(Correlation between Stock iand j)=0.3

Average Covi,j(Covariance between stock i and j)=Si*Sj*Corrli,j=44*44*0.3=580.8

Portfolio Variance=(w1^2)*(S1^2)+(w2^2)(S2^2)+………….(wn^2)*(Sn^2)+2w1w2*Cov(1,2)+2w1w3*Cov(1,3)+………+w(n-1)wn*Cov(n,(n-1)

Portfolio Variance=(1/500)^2)*((S1^2)+(S2^2)+(S3^2)+……..(S500^2))+2*(1/500)*(1/500)(Sum of Covariances)

Portfolio Variance=(1/500)^2)*500*Average Variances+2*(1/500)*(1/500)*500*Average Covariance

Portfolio Variance=(1/500)^2)*500*1936+2*(1/500)*(1/500)*500*580.8

Portfolio Variance=(1/500)*1936+2*(1/500)*580.8

Portfolio Variance=(1936/500)+2*(580.8/500)

Portfolio Variance=3.872+2.323=6.1952

Standard Deviation of the Portfolio =Square root (6.1952)=2.49%

Estimate of Standard deviation of Fund's Portfolio

2.49%