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

$$w_{i,t}=\frac{1}{N}(r_{i,t-1}-r_{m,t-1})$$

where $r_{m,t}$ is the equal-weighted market index returns in month $t$. Assume returns have unconditional mean $\mu=E[r_t]$ and autocovariance matrix $\Omega=E[(r_{t-1}-\mu)(r_t-\mu)^T]$. The portfolio return in month t equals:

$$\pi_t=\sum_i w_{i,t}r_{i,t}=\frac{1}{N}\sum_i (r_{i,t-1}-r_{m,t-1})r_{i,t}.$$

Hence, the expected profit is

$$E[\pi_t] = \frac{1}{N}E\Bigg[\sum_i r_{i,t-1}r_{i,t}\Bigg]-\frac{1}{N}E\Bigg[r_{m,t-1}\sum_i r_{i,t}\Bigg] \

= \frac{1}{N} \sum_i (\rho_i+\mu_i^2)-(\rho_m+\mu_m^2),$$

where $\rho_i$ and $\rho_m$ are the autocovariances of the asset i and the equal-weighted index, respectively. Using that fact that average autocovariance equals $tr(\Omega)/N$ and the autocovariance of the market portfolio equals $\varsigma^T\Omega\varsigma/n^2$, where $\varsigma$ is the vector of ones.

$$E[\pi_t]=\frac{1}{N}tr(\Omega)-\frac{1}{N^2}\varsigma^T\Omega\varsigma+\sigma_{\mu}^2=\frac{N-1}{N^2}tr(\Omega)-\frac{1}{N^2}[\varsigma^T\Omega\varsigma-tr(\Omega)]+\sigma_{\mu}^2.$$

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.

$$\pi_t=\sum_i w_{i,t}r_{i,t}=\frac{1}{N}\sum_i (r_{i,t-1}-r_{m,t-1})r_{i,t}.$$

Hence, the expected profit is

$$E[\pi_t] = \frac{1}{N}E\Bigg[\sum_i r_{i,t-1}r_{i,t}\Bigg]-\frac{1}{N}E\Bigg[r_{m,t-1}\sum_i r_{i,t}\Bigg] \

= \frac{1}{N} \sum_i (\rho_i+\mu_i^2)-(\rho_m+\mu_m^2),$$

where $\rho_i$ and $\rho_m$ are the autocovariances of the asset i and the equal-weighted index, respectively. Using that fact that average autocovariance equals $tr(\Omega)/N$ and the autocovariance of the market portfolio equals $\varsigma^T\Omega\varsigma/n^2$, where $\varsigma$ is the vector of ones.

$$E[\pi_t]=\frac{1}{N}tr(\Omega)-\frac{1}{N^2}\varsigma^T\Omega\varsigma+\sigma_{\mu}^2=\frac{N-1}{N^2}tr(\Omega)-\frac{1}{N^2}[\varsigma^T\Omega\varsigma-tr(\Omega)]+\sigma_{\mu}^2.$$

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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