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Monday, July 27, 2015

Momentum and Autocorrelation in Stock Returns - Lewellen

Role of size and BM factors on stock momentum. Both are negatively auto-correlated and cross-serially correlated over intermediate horizons. The excess covariance of stocks with each other, and not under-reaction, explains momentum in the portfolios.

Firm specific returns and investors under-reaction and belated overreaction does not explain a significant component of momentum. Size and BM factor based momentum is strong and distinct, showing that momentum can't be attributed solely to firm-specific returns - there must be multiple sources of momentum. Momentum shows up in individual stocks and size quintiles, but vanishes at the market level.

Sources of Momentum

Profits depend on both auto-correlations and the lead-lag relationship. The portfolio weight of asset i in month t is
wi,t=1N(ri,t1rm,t1)
where rm,t is the equal-weighted market index returns in month t. Assume returns have unconditional mean μ=E[rt] and autocovariance matrix Ω=E[(rt1μ)(rtμ)T]. The portfolio return in month t equals:
πt=iwi,tri,t=1Ni(ri,t1rm,t1)ri,t.
Hence, the expected profit is
E[πt]=1NE[iri,t1ri,t]1NE[rm,t1iri,t] =1Ni(ρi+μ2i)(ρm+μ2m),
where ρi and ρm are the autocovariances of the asset i and the equal-weighted index, respectively. Using that fact that average autocovariance equals tr(Ω)/N and the autocovariance of the market portfolio equals ςTΩς/n2, where ς is the vector of ones.
E[πt]=1Ntr(Ω)1N2ςTΩς+σ2μ=N1N2tr(Ω)1N2[ςTΩςtr(Ω)]+σ2μ.
This decomposition says that momentum can arise in three ways:
1) stocks might be positively autocorrelated (first term) - meaning stocks with high returns today are expected to have higher returns tomorrow.
2) Cross-serial correlations might be negative - meaning firm with high return today predicts that other firms will have low returns in the future. This is related to excess covariance among stocks.
3) High unconditional mean stocks.

This decomposition is not unique. 

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