1. In class we saw that if an investor splits his/her wealth between just two risky assets, then the expected return and the variance of the newly formed portfolio can be found in the following way: E(rp) = E(r1) + (1 )E(r2) (1) var(rp) = 2 var(r1) + (1 ) 2 var(r2) + 2(1 )cov(r1, r2) (2) where and 1 are fractions of wealth invested in assets 1 and 2, respectively; E(r1) and E(r2) are expected returns on assets 1 and 2, respectively; var (r1) and var (r2) are variances of returns on assets 1 and 2 , respectively; E (rp) is the expected return on the portfolio and var (rp) is its variance. At the same time we saw that if the investor can split his/her wealth between N risky assets, then the expected value and the variance of the portfolio return rp can be found in the following way: E(rp) = wTE(r) (3) var(rp) = (wT)w (4) where w = w1 w2 ... wN is the Nx1 matrix of weights such that PN i=1 wi = 1; E(r) = E(r1) E(r2) ... E(rN ) is the Nx1 matrix of expected returns on individual assets; is the NxN variance-covariance matrix. See my lecture note "Portfolio Models" for details. Show that when N = 2 the two approaches provide identical results (i.e. formula (3) gives the same answer as formula (1), and formula (4) gives 1 the same answer as formula (2)). To show that, rst write a 2x1 matrix of weights, where is invested in asset 1 and 1 is invested in asset 2, w = 1 . Then write a matrix of expected returns: E(r) = E(r1) E(r2) and the variance-covariance matrix = var(r1) cov(r1, r2) cov(r2, r1) var(r2) . Finally, by hand (you don't need Excel) carefully carry out matrix multiplications (and transpositions) described in formulas (3) and (4). To do that, you will have to remember the algorithm for matrix multiplication that I showed in class. See my lecture note "Matrix Arithmetic" for details. Do not substitute any numbers in place of , E(r1), E(r2), var(r1), var(r2), or cov(r1, r2)