optimisation

Consider a portfolio of two assets, x1 and x2, with the expected returns at time t being E(x1) = 4 + 2t^(1/2) and E(x2) = 2 + t^(2/3),respectively.

Assume that their variances and covariance at time t are V ar(x1) = t^2 + 2, V ar(x2) = 2t+ t^(1/2) + 1 and Cov(x1, x2) = −1.

Using the mean-variance portfolio strategy, what are the optimal weights for the two assets at t = 0 and t = 1?

In addition, what is the expected loses/profit, if you use the optimal weights from t = 0 at t = 1?

when t=0
E(x1) = 4 and E(x2) = 2
V ar(x1) = 2, V ar(x2) = 1 and Cov(x1, x2) = −1.

if we invest α in x1 and (1-α) in x2
Then
Var(r)=4α^2-2α(1-α)+(1-α)^2

dVar(r)/dα=8α-2+4α-2(1-α)=0
12α-2-2+2α=0
α=4/14=0.2857

So the optimal weight is invest 28.57% in x1 and 71.43% in x2

when t=1
E(x1) = 6 and E(x2) = 3
V ar(x1) = 3, V ar(x2) = 3 and Cov(x1, x2) = −1.

if we invest α in x1 and (1-α) in x2
Then
Var(r)=9α^2-2α(1-α)+9(1-α)^2

dVar(r)/dα=18α-2+4α-18(1-α)=0
22α-2-18+18α=0
α=20/40=0.5

So the optimal weight is invest 50% in x1 and 50% in x2

The expected profit at t=0
=28.57%*4+71.43%*2
=2.5714

The expected profit at t=1
=4.5