Question

Two securities, Ames, Inc., and Gilbert, Inc., are to be combined in a portfolio in equal proportions. Ames has an expected return of 8% and Gilbert has an expected return of 14%. Ames has a standard deviation of 10% and Gilbert has a standard deviation of 20%. Find the expected return and standard deviation of the portfolio when the correlation coefficient between the two securities is (a) 1, (b) 0, and (c) -1.

Answer 1

Expected return of portfolio = Weight 1* expected return 1+ Weight2* expected return2

=0.50*0.08+0.50*0.14= 11%

Standard deviation of the portfolio= Square root of variance,

so to calculate variance we need to calculate Covariance

Covariance= correlation coefficient*?1*?2

covariance when correlation coefficient is1= 1*0.10*0.20= 0.02

covariance when correlation coefficient is0= 0*0.10*0.20=0

covariance when correlation coefficient is-1= -1*0.10*0.20= -0.02

Now variance when correlation coefficient is 1=  w²_A*?²(R_A) + w²_B*?²(R_B) + 2*(w_A)*(w_B)*Cov(R_A, R_B)

=0.50²*.10²+ 0.50²*0.20² + 2*0.50*0.50*0.02 =0.0225

Standard deviation = square root of deviation= 0.15

Now variance when correlation coefficient is 0=  w²_A*?²(R_A) + w²_B*?²(R_B) + 2*(w_A)*(w_B)*Cov(R_A, R_B)

=0.50²*.10²+ 0.50²*0.20² + 2*0.50*0.50*0.00 =0.0125

Standard deviation = square root of deviation= 0.11

Now variance when correlation coefficient is -1=  w²_A*?²(R_A) + w²_B*?²(R_B) + 2*(w_A)*(w_B)*Cov(R_A, R_B)

=0.50²*.10²+ 0.50²*0.20² + 2*0.50*0.50*-0.02 =0.0025

Standard deviation = square root of deviation= 0.05