Cash Flow News and Discount Rate News + Return

Question

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,

Answer 1

Preliminary remarks:

• CF news is defined as the price change holding the implied cost of capital (ICC) constant
• DR news is defined as the price change holding the cash flow forecasts constant

The authors explicitly emphasize, that their model is slightly different from Campbell and Shiller's (1998) return decomposition.

There is a typo in the definition of your $CF_j$ formula: The index for $q_j$ in the second fraction should be $t$ instead of $j$.

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$CF_j$ and $DR_j$ are properly defined to match the equation with $r_t$ as the return at time $t$:

$$r_t = CF_j + DR_j$$

So let's start:

$$r_t = (\frac{f(c^{t+j},q_{t+j})- f(c^t ,q_{t+j} )}{P_t} +\frac{f(c^{t+j} ,q_t )-f(c^t ,q_t)}{P_t})/2 + (\frac{f(c^t,q_{t+j})- f(c^t ,q_t )}{P_t} +\frac{f(c^{t+j} ,q_{t+j} )-f(c^{t+j} ,q_t)}{P_t})/2$$

Multiplying out and rearranging results in: $$r_t = \frac{2\cdot f(c^{t+j},q_{t+j})}{2\cdot P_t} -\frac{f(c^t ,q_{t+j} )}{2\cdot P_t} +\frac{f(c^t ,q_{t+j} )}{2\cdot P_t}+\frac{f(c^{t+j} ,q_t)}{2\cdot P_t} - \frac{f(c^{t+j} ,q_t)}{2\cdot P_t} - \frac{2\cdot f(c^t,q_{t})}{2\cdot P_t}$$

and finally you get

$$r_t =\frac{f(c^{t+j}, q_{t+j}) - f(c^t,q_t)}{P_t} =\frac{P_{t+j}-P_t}{P_t}$$