# Cash Flow News and Discount Rate News + Return

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,

• I appreciate the cited link, but it might be a better, self contained question if you posted the definition of the variables. – Attack68 Aug 1 '18 at 20:29
• Thank you for kind attention. I hope the notations' definitions are clear now. – David J. Aug 2 '18 at 21:28

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 $$j$$.

$$CF_j$$ 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

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

• I appreciate your answer. It was helpful. However, I am looking for the expansion the authors did to generate CF and DR from return. What you wrote above is the other way around, i.e. from CF and DR to return. Thanks again. – David J. Aug 20 '18 at 7:25
• There is in fact no explicit derivation for $CF$ and $DR$; both are set up by definition to match the above equation. If you just look at the math formulas, just read my answer the other way. Starting from a true definition for $return$, i expand the definition of $P_t$ and $P_{t+j}$ and than $define$ the resulting summands as $CF$ and $DR$. Besides the mathematics, if you look at the indices and the rearrangement of the terms, it is clear, that each summand catches only the effect of cash flow or return news, i.e. the definition is based on economic reasons. Just comment for additional help. – skoestlmeier Aug 22 '18 at 8:40
• My thoughts was also like yours that they are set up by definition. But I was not sure, and your confirmation is reassuring. Thank you. – David J. Aug 24 '18 at 13:09