实证资产定价 Big Problem Set - Suggested Solutions.pdf

Problem Set 1 Doctoral Seminar in Empirical Finance Suggested Solutions Lars A. Lochstoer Fall, 2012 1 Multivariate GRS The GRS test statistic illustrates Rollís point: A one-factor pricing model implies that the (factor-mimicking) portfolio is mean-variance e¢ cient. Thus, a test of the model boils down to a test of the mean-variance e¢ ciency of this portfolio. But, if a one-factor model fails, maybe a K-factor model will work. Assume the stochastic discount factor is 0 M = E [M ] + (b ) F + t+1 t t+1 t+1 t+1 is a Kx1 vector of unobserved where Et Ft+1 = Et t+1 = 0, and b is a Kx1 vector. Ft+1 factors. Finally, Et F t+1 = 0 and that Et t+1Ri;t+1 = 0 8i. Assume a risk free asset t+1 exists, and that there exists traded portfolios that span the unobserved factors: F = A + B F t+1 t t t+1 Here Ft+1 is the excess returns to these K portfolios. 1. Show that a linear multi-factor model of excess returns holds in this economy. What is the role of ? Start with the Law of One Price: i 1 = E M R t t+1 t+1 for an arbitrary asset i. Plugging in the above expression for the SDF, we have 0 i 1 = E E [M ] + (b ) F