Decomposing Stock Returns into Cash Flow and Discount Rate News
Summary
This document explains a two-part decomposition of stock price changes into cash flow news and discount rate news. A price function depends on both expected cash flows and an implied cost of capital; the decomposition attributes changes to each input by varying one while holding the other fixed. It uses averages across the two possible orders of changing the inputs, which splits their interaction evenly.
The included answer expands the definitions algebraically to show that the two components sum to the total return. It also notes that the decomposition differs somewhat from Campbell–Shiller’s framework and corrects an index typo in the question’s cash flow expression. The material gives definitions and an algebraic identity, but no empirical application or evidence that either component predicts returns. Its interpretation depends on the chosen price function and on treating the implied cost of capital and cash flow forecasts as the two drivers being separated.
Key ideas
- Cash flow news measures price changes while holding the implied cost of capital fixed.
- Discount rate news measures price changes while holding cash flow forecasts fixed.
- Averaging both orders of changing the inputs splits their interaction between the two components.
- Adding the components recovers the total price return through algebraic cancellation.
- The described decomposition is not identical to the Campbell–Shiller return decomposition.
Tags
Full text
# Cash Flow News and Discount Rate News + Return
# 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,
## Answer by skoestlmeier (score 0, accepted)
https://quant.stackexchange.com/a/41239
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
$$r_t =\frac{f(c^{t+j}, q_{t+j}) - f(c^t,q_t)}{P_t} =\frac{P_{t+j}-P_t}{P_t}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.