Question

a) Consider an economy with two types of firms, S and I. Type S firms always move
together, but type I firms move independently of each other. For both types of firms, the
expected return of an individual firm is given by 5%, and the return volatility of an individual
firm is given by 25%.

i) (10 points) There is an equally-weighted portfolio made of 10 type S firms and 10
type I firms. What is the expected return and the return volatility of the portfolio?

ii) (7 points) Now, suppose that there are 15 type S firms and 5 type I firms in an
equally-weighted portfolio. What is the expected return and the volatility of the
portfolio? Compare your result with that in (a). Briefly explain the intuition behind your
findings.

Answer 1

a)i) Equally-weighted portfolio made of 10 type S firms

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

Cov(1,2)=Covariance of returns of asset1 and asset2 Portfolio Standard Deviation =Square Root (Portfolio Variance)

Volatility =Standard Deviation=25%=0.25 Variance =Square of Standard Deviation=0.25^2=0.0625

0.4

Correlation(1,2)=1.0

0.3

Covariance(1,2) =Correlation(1,2)* (Standard Deviation1)*(StandardDeviation2)

Covariance(1,2) =0.25*0.25=0.0625

0.048

If there are n shares with equal weight

Weight of each Shares=(1/n) Portfolio Variance=((1/n)^2)*n*0.0625+2*((1/n)^2))*(nC2)*0.0625 Portfolio Variance=((1/n)^2)*n*0.0625+((1/n)^2))*(nC2)*0.125 n=10 Portfolio Variance=((1/10)^2)*10*0.0625+((1/10)^2))*(10C2)*0.125 Portfolio Variance=0.00625+0.05625=0.0625 Equally-weighted portfolio made of 10 type I firms Volatility =Standard Deviation=25%=0.25 Variance =Square of Standard Deviation=0.25^2=0.0625 Correlation(1,2)=0 Covariance(1,2) =Correlation(1,2)* (Standard Deviation1)*(StandardDeviation2) Covariance(1,2) =0 Portfolio Variance=((1/n)^2)*n*0.0625+0 n=10 Portfolio Variance=((1/10)^2)*10*0.0625=0.00625 COMBINED PORTFOLIO OF 10 TYPE S Firms and 10 Type I firms Weight of Type S firms=1/2=0.5 Weight of Type I firms=1/2=0.5 V1=S1^2=Variance of Type S firms=0.0625 V2=S2^2=Variance of Type I firms=0.00625 Variance of Combined portfolio =(0.5^2)*0.0625+(0.5^2)*0.00625=(0.5^2)*(0.0625+0.00625)=0.017188 Standard Deviation of Combined Portfolio of 10 Type S Firms and 10 Type I firms=SQRT(0.017188)=0.131101 Standard Deviation =13.11% Expected Return =(1/20)*20*5%=5% (ii) 15 type S firms and 5 type I firms in an equally-weighted portfolio: Portfolio Variance of 15 Type S firms: n=15 Portfolio Variance=((1/15)^2)*15*0.0625+((1/15)^2))*(15C2)*0.125 Portfolio Variance=0.004167+0.058333=0.0625 Portfolio Variance of 5 Type I firms n=5 Portfolio Variance=((1/5)^2)*5*0.0625=0.0125 COMBINED PORTFOLIO OF 15 TYPE S Firms and 5Type I firms Weight of Type S firms=15/20=0.75 Weight of Type I firms=5/20=0.25 V1=S1^2=Variance of Type S firms=0.0625 V2=S2^2=Variance of Type I firms=0.0125 Variance of Combined portfolio =(0.75^2)*0.0625+(0.25^2)*0.0125=0.035938 Standard Deviation of Combined Portfolio of 15Type S Firms and 5Type I firms=SQRT(0.035938)=0.1896 Standard Deviation =18.96% Expected Return of Type S Portfolio=5% Expected Return of Type I Portfolio=5% Expected Return of Combined Portfolio=0.75*5+0.25*5=5% Standard Deviation of S with 10 or 15 firms are the same, Standards deviation of 5 I firms is higher than 10 I foirms Hence Portfolio having 15 S and 5 I has higher volatility than Portfolio with 10 S and 10 I Diversification beyond some point does not reduce volatility 0.3 0.048