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By Petr Adamek, John Y. Campbell, Andrew W. Lo, A. Craig Mackinlay, Luis M. Viceira
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Extra info for A Solution Manual To The Econometrics Of Financial Markets
9) ! e( + 2 )T which is the moment-generating function for a normal random variable with mean T and variance 2 T . 3 Consider n observations in the interval 0 T ] equally spaced at intervals h T=n, and let p(0) = 0 to simplify the algebra. Let pk p(kh). 1) kh;h e; (kh;s) dB (s) : Now let = e; h . 4) by the Principle of Invariance (see Zehna 1966]). 3 47 From here we will assume that the trend is known exactly. 12) n(^ 2 ; 2 ) a N 0 2 4 1 + 2e 2 h2 1 ; e;2 h (1 + 2 h)]2 : To derive the continuous-record asymptotics of ^ and ^ 2 , we let n !
2 except that 2 = 0:006 instead of 0:003. Using this value for 2 , we have 1 = 1:162 and 2 = 1:225 giving P1 = 21:3% and P2 = 23:3%. 4) 0 = a ; ( aop op + ap p ) since Cov Rp Rop ] = 0. The result 2 = ap is immediate, thus we need to show that 1 = 1 ; ap and 0 = 0 to complete the solution. Let r be the minimum variance portfolio with expected return equal to that of portfolio a, a = r . 5) Rr = (1 ; )Rop + Rp where = ( r ; op )=( p ; op ). 8) r = rop op + rp p : Portfolio a can be expressed as portfolio r plus an arbitrage (zero-investment) portfolio a composed of portfolio a minus portfolio r (long a and short r).
The calculations for three selected stocks are left to the reader. 4 Let Zt be a (N +1 1) vector of excess asset returns with mean and covariance matrix . Designate asset N + 1 as the market portfolio m. Assume that is full rank. 1) Using straight forward algebra we have 2 (! 5) 2 m 0 m 2 2 m 3 5 0 m 2 + 2 m 0 m 2 2 m 3 5 where = 0 m2 + is substituted. 3). 1 Let the number of portfolios in the set be K and let RKt be the (K 1) vector of time period t returns for the portfolios. Since the entire minimum variance boundary can be generated from the K portfolios, for any value of the constant y, there exists a combination of the portfolios with expected return y which is minimum variance with respect to the K portfolios plus the N assets.