I will appreciate If someone help me to understand how the final expansion is made. Specifically, how CF & DR are drived. This model is introduced by Chen et. al. (2013).What Drives Stock Price Movements?
$$P_t=f(c^t,q_t)$$
$$return_t = \frac{P_{t+j}-P_t}{P_t} =\frac{f(c^{t+j}, q_{t+j}) - f(c^t,q_t)}{P_t}= CF_j + DR_j $$
$$CF_j=(\frac{f(c^{t+j},q_{t+j})- f(c^t ,q_{t+j} )}{P_t} +\frac{f(c^{t+j} ,q_t )-f(c^t ,q_t)}{P_t})/2$$
$$DR_j=(\frac{f(c^t,q_{t+j})- f(c^t ,q_t )}{P_t} +\frac{f(c^{t+j} ,q_{t+j} )-f(c^{t+j} ,q_t)}{P_t})/2$$
$P_t$ = Price at time $t$
$c$ = Cash flows
$q$ = Discount rate
$CF$ : It is labeled as $CF$ news because the numerator is calculated by holding the discount rate constant, and $CF_j$ captures the price change driven primarily by the changing CF expectations from $t$ to $t+j$ (page 846).
Best,