WHERE DO ALPHAS COME FROM?: A MEASURE OF THE VALUE OF ACTIVE INVESTMENT MANAGEMENTAndrew W. Lo
JOURNAL OF INVESTMENT MANAGEMENT, Vol. 6, No. 2, (2008), pp. 1–29
Proposition 1 Under assumptions (A1)–(A2), the expected return of any portfolio P satisﬁes the following
Proposition 1 is a simple decomposition of a portfolio’s expected return into two components:
the sum of the covariances between portfolio weights and returns, and the sum of the products of expected portfolio weights and expected returns.
Proposition 2 Under Assumptions (A1)–(A3), the expected return of any portfolio P satisﬁes the following
Proposition 2 provides a more reﬁned decomposition than Proposition 1, thanks to the linear Kfactor structure assumed in (A3). Expected returns are now the sum of three components: a securityselection component (18a) that depends on the αi’s, a factor-timing component (18b) that depends on the covariance between the portfolio betas and factors, and a risk-premia component (18c) that represents the expected return from passive exposures to factor risks.
Proposition 3 Under(A1) and (A2),the active component δp and active ratio θp of any portfolio P may
be estimated consistently by their sample counterparts (31) and (32), and both estimators are asymptotically
normal with variances that may be consistently estimated via the Generalized Method of Moments.