8. (General tilting o) A general model for information about expected returns can be ex- pressed in vector-matrix form as p = Pr+e In the model P is an mxn matrix, F is an n-dimensional vector, and p and e are m-dimensional vectors. The vector p is a set of observation values and e is a vector of errors having zero mean The error vector has a covariance matrix Q The best (minimum- variance) estimate of F is =(PQP) PQ'p (8.12)

(a) Suppose there is a single asset and just one measurement of the form p =7+e Show that according to (8 12), we have F = p.

(b) Suppose there are two uncorrelated measurements with values p and p2. having variances and Show that -1 (+) (+)

(c) Consider Example 8.5 There are measurements of the form F = P+2 F3 = P3+03 F = P + e F = 11+ B21 F2 = 11+ B21 F3 = 1 + BM FA === where the e,'s are uncorrelated, but where cov

(e,

f) = 250 Using the data of the example, and assuming the B's are known exactly, find the best estimates of the F; 's [Note: You should only need to invert 2 x 2 matrices]