Asset 1 has expected rate of return r1=r1=0.08 and volatility 1=1=0.12, and

asset 2 has expected rate of return r2=r2=0.13 and volatility 2=2=0.2.

The covariance of the rates of return r1r1 and r2r2 is 1,2=1,2=0.0072.

Consider portfolios that combine the two assets with weights w1w1 and w2w2, where w1+w2=1w1+w2=1.

To find the weights for the minimum variance portfolio using Lagrange multipliers, we first define

(1) f(w1,w2)=122P=12var(w1r1+w2r2)f(w1,w2)=12P2=12var(w1r1+w2r2),

Then using the known values for 11, 22, and 1,21,2, equation (1) becomes

(2) f(w1,w2)=f(w1,w2)= (type w1w1 as w1, w2w2 as w2)

We next define

(3) F(w1,w2,)=f(w1,w2)(w1+w21)F(w1,w2,)=f(w1,w2)(w1+w21)

The partial derivative of FF with respect to w1w1 is

(4) Fw1=Fw1= (type as lambda)

The partial derivative of FF with respect to w2w2 is

(5) Fw2=Fw2= (type as lambda)

The partial derivative of FF with respect to is

(6) F=F=

Requiring that Fw1=0Fw1=0, Fw2=0Fw2=0, and F=0F=0, gives algebraic equations for w1w1, w2w2 and .

The minimum variance portfolio corresponds to weights

w1=w1= , w2=w2=

The minimum variance portfolio has expected return rate and volatility

rP=rP= ,

P=P= .

(1 point) Asset 1 has expected rate of return Fi =0.08 and volatility 01 =0.12, and asset 2 has expected rate of return 72 =0.13 and volatility 02 =0.2. The covariance of the rates of return r and r2 is 01,2 =0.0072. Consider portfolios that combine the two assets with weights wi and W2, where wi + w2 = 1. To find the weights for the minimum variance portfolio using Lagrange multipliers, we first define (1) f(W1, wn) = { } = { var(wr + wzr2), Then using the known values for 01, 02, and 01,2, equation (1) becomes (2) f(W1, W2) = (type w1 as w1, w, as w2) We next define (3) F(W1, W2, 1) = f(W1, W,) 1(W1 + W2 - 1) The partial derivative of F with respect to wi is (4) Fw1 (type as lambda) The partial derivative of F with respect to w is (5) Fw2 (type as lambda) The partial derivative of F with respect to . is (6) Fi = Requiring that Fv1 = 0, Fw2 = 0, and Fi = 0, gives algebraic equations for w1, W2 and The minimum variance portfolio corresponds to weights W1 = W2 = The minimum variance portfolio has expected return rate and volatility rp= op= (1 point) Asset 1 has expected rate of return Fi =0.08 and volatility 01 =0.12, and asset 2 has expected rate of return 72 =0.13 and volatility 02 =0.2. The covariance of the rates of return r and r2 is 01,2 =0.0072. Consider portfolios that combine the two assets with weights wi and W2, where wi + w2 = 1. To find the weights for the minimum variance portfolio using Lagrange multipliers, we first define (1) f(W1, wn) = { } = { var(wr + wzr2), Then using the known values for 01, 02, and 01,2, equation (1) becomes (2) f(W1, W2) = (type w1 as w1, w, as w2) We next define (3) F(W1, W2, 1) = f(W1, W,) 1(W1 + W2 - 1) The partial derivative of F with respect to wi is (4) Fw1 (type as lambda) The partial derivative of F with respect to w is (5) Fw2 (type as lambda) The partial derivative of F with respect to . is (6) Fi = Requiring that Fv1 = 0, Fw2 = 0, and Fi = 0, gives algebraic equations for w1, W2 and The minimum variance portfolio corresponds to weights W1 = W2 = The minimum variance portfolio has expected return rate and volatility rp= op=