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In an NBIM paper I read the following:

"... one can break down the total equity return into the dividend yield (the starting valuation), the change in the P/E ratio (the change in valuation) and the growth in dividends (or earnings) per share."

This breakdown is claimed to be an accounting exercise.

I do not see how these three components together form the equity return. In additon, how does this breakdown relates to this formula capturing equity return: $\frac{P_{1}-P_{0} + D}{P_0}$, with $P_1 - P_0$ the stock price increase and $D$ the dividend.

http://www.nbim.no/en/transparency/discussion-notes/2012/economic-growth-and-equity-returns/

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This is really the Campbell-Shiller (1988) decomposition: one of the key contributions leading to the 2013 Nobel Prize. The idea is very simple. By definition, the return between today and tomorrow is $$R_{t+1}=\frac{P_{t+1}+D_t}{P_t}$$ You can invert this: $$P_t = \frac{P_{t+1}+D_t}{R_{t+1}}$$ Take logs ($log P_t = p_t$ , $logR_{t+1}=r_{t+1}$ , $log D_t=d_t$, $\Delta d_{t+1}=d_{t+1}-d_t$ , $\rho$ is just a constant), take a first order Taylor expansion (details here): $$r_{t+1}= \rho(p_{t+1}-d_{t+1}) - (p_t - d_t) + \Delta d_{t+1}$$

So $-(p_t - d_t)=log \left(\frac{D_t}{P_t} \right)$ is related to the dividend yield, $\rho(p_{t+1}-d_{t+1}) - (p_t - d_t)= log \left(\frac{P_{t+1}}{D_{t+1}}\right)^{\rho} - log \left(\frac{P_{t}}{D_{t}}\right)$ is related to the change price-earnings (proxied by price-dividends ratios), and $\Delta d_{t+1}$ the growth in dividends. Notice that there is zero economic assumption here (just accounting) and that the only bit of math is the Taylor approximation.

By the way, this is very similar to Gordon’s growth model where $P_0 = \frac{D_1}{r-g}$ and therefore $$r=\frac{D_1}{P_0} + g $$ i.e. the return is dividend yield plus its growth

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