How does this return decomposition work?

http://image-src.bcg.com/Images/BCG-Value-Creators-2017-Appendix-July-2017_tcm9-166061.pdf

The paper here decomposes total shareholder return into different components. Here is my derivation of the decomposition, price $P$ and dividend $D$ are in per share basis, other variables ($S$ for shares outstanding) are on firm level:

$$\begin{eqnarray} TSR_t &=& \frac{P_t + D_t}{P_{t-1}} \\ &=& (\frac{P_t}{P_{t-1}} + \frac{D_t}{P_{t-1}})\\ &=& (\frac{P_t/(EBIT_t/S_t)}{P_{t-1}/(EBIT_{t-1}/S_{t-1})}\frac{EBIT_t/S_t}{EBIT_{t-1}/S_{t-1}} + \frac{D_t}{P_{t-1}})\\ &=& (\frac{P_t/EPS_t}{P_{t-1}/EPS_{t-1}}\frac{EBIT_t}{EBIT_{t-1}} \frac{S_{t-1}}{S_t} + \frac{D_t}{P_{t-1}})\\ &=& (\frac{P_t/EPS_t}{P_{t-1}/EPS_{t-1}}\frac{Sales_t}{Sales_{t-1}} \frac{S_{t-1}}{S_t}\frac{EBITMargin_t}{EBITMargin_{t-1}} + \frac{D_t}{P_{t-1}})\\ &\approx& g_{multiple} + g_{sales} + g_{EBITMargin} + g_{shares} + DivYield \\ \end{eqnarray}$$ The last step comes from taking log on both sides of the equation.

However, where does that net debt growth part come from?

Update

I have done a similar simple decomposition with Amazon, and quickly find that the definition just doesn't work when Earning is negative. One simply cannot take log of a negative number, One can only multiply those terms instead of adding. BCG paper always use sum which doesn' seem like possible. The decomposition that I do as listed above works well when Earning is not negative. As @DavidAddison mentioned, the TRS doesn't match up with my calculation based on my data (Also CaptalIQ).

I think there is still some merits to this decomposition, because you can see what factors drive the return more clearly, at least historically.

• What are $S_t$ and $\mathit{TSR}_t$? – Matthew Gunn Jan 26 '18 at 16:08
• @MatthewGunn $S_t$ represents the number of shares outstanding and $TSR_t$ is the total shareholder return. – zsljulius Jan 26 '18 at 16:09
• So why is $\frac{S_t}{S_{t-1}}$ there? Issuing new shares at the current price doesn't generate positive returns. Retiring shares at the current price doesn't generate negative returns. – Matthew Gunn Jan 26 '18 at 16:16
• @MatthewGunn at time t-1, we pay total P(t-1)S(t-1) for the whole equity, if the firm issues new shares between t-1 and t at t price of P(t-1), then S(t) is greater than S(t-1), and P(t) and D(t) should be adjusted automatically downwards since people who buys new shares know that each share value is diluted. The numerator just represents Total Firm Equity Value at time t and denomintor represents the total firm equity value at time t-1. – zsljulius Jan 26 '18 at 16:21
• Can you find any references where they precisely define terms? – Matthew Gunn Jan 26 '18 at 17:46