We start off firslty by reminding the general model for valuing assets. A well know accounting identity expresses the relation between the value of an asset, the income stream it generates and to which the holder of the asset is entitled (\(C_{1}\),\(C_{2}\), . . . \(C_{n}\)) and an endogeneous return \(R\) \[V_{t}=\displaystyle\sum_{i=1}^{K}\frac{C_{t+i}}{(1+R)^i}+\frac{V_{t+K}}{(1+R)^K}\]
In other words, \(R\) is what you earn if you pay \(V_{0}\), receive \(C_{1}\),\(C_{2}\), . . . \(C_{K}\) and sell the asset at \(V_{t+K}\). The value of the asset when you sell it is the terminal value of the asset. This basic principle is at the root of many equity valuation models; the DDM is the exact translation of this accounting principle where dividends are the revenues a shareholder is entitled to.
Applying equation (1) to equities leads to \[P_{t}=\displaystyle\sum_{i=1}^{K}\frac{D_{t+i}}{(1+R)^i}+\frac{P_{t+K}}{(1+R)^K}\] where \(P_{t}\) is the stocks price at \(t\), \(D_{t+i}\) the future dividend at \(t+i\), \(R\) the discount rate and \(P_{t+K}\) the terminal value. Again, \(R\) is necessarely the average total return of the shareholder over one period if he pays \(P_{t}\), receive \(D_{t+1}\),\(D_{t+2}\), . . . \(D_{t+K}\) and sells the stock at \(P_{t+K}\). It is worth mentioning that the dividends are always reinvested and that the total shareholder return is going to be \((1+R)^K-1\) over \(K\) periods.
The Gordon Growth Model is a simple version of the DDM where it is assumed that dividends will grow at a constant rate, duration of equity is infinite so that terminal value is negligeable: \[P_{t}=\displaystyle\sum_{i=1}^{\infty}\frac{D_{t+i}}{(1+R)^i}\approx\frac{D_{t+1}}{R-g}\] where \(g\) is the expected constant dividend growth rate to perpetuity.If we isolate future returns \(R\), we get: \[R=\frac{D_{t+1}}{P_{t}}+g\] This equation highlights the fact that future returns are driven by the current valuation and future growth.
Although the DDM is theoretically correct, it carries some well known caveats. One in particular is its expression of equity valuation purely from a dividend distribution standpoint. Value creation is not apparent in this formula. By injecting the book value of equity in the DDM one can explain how dividends are generated through time and why investment and economic returns are at the basis of dividend growth and value creation. The Residual Income Model (RIM hereafter) makes this possible.
The RIM developed by Ohlson and Felthman (1995) assumes an accounting identity, the clean surplus rule , which states that the change in book value is equal to the difference between earnings and dividends \(B_{t}-B_{t-1}=E_{t}-D_{t}\). Earnings that are not distributed to investors are reinvested in the company. It then appears obvious that if a company’s economic profitability is better than what shareholders expect, the company has an incentive to reinvest profits in order to generate even bigger future earnings and dividends. Residual income, or abnormal earnings, is constructed as the difference between accounting earnings and the previous-period book value mutliplied by the cost of equity (i.e. the cost of equity being what investors expect as future returns) \(A_{t}=E_{t}-RB_{t-1}\). Using these accounting identities allows us to rewrite dividends as \(D_{t}=B_{t-1}(1+R)-B_{t}+A_{t}\). Replacing \(D_{t}\) with this new expression into the DDM formula (2) and operating some simplifications leads to the following RIM equation : \[P_{t}=B_{t}+\displaystyle\sum_{i=1}^{K}\frac{A_{t+i}}{(1+R)^i}-\frac{B_{t+K}}{(1+R)^K}+\frac{P_{t+K}}{(1+R)^K}\] where \(P_{t}\) is the stocks price at \(t\), \(B_{t}\) is the book value at \(t\), \(A_{t+i}\) the future abnormal earnings in \(t+i\), \(R\) the discount rate, \(P_{t+K}\) and \(B_{t+K}\) the market value and book value of equity in \(t+K\) respectively. It is now obvious that a market value of equity superior to its book value necessarely implies that the company generates abnormal earnings i.e. that itsROE is above the shareholder expected return (Cost Of Equity). Abnormal earnings are the ability of the company to generate more earnings than what investors are asking for. Under General Equilibrium Theory assumptions, abnormal earnings do not last indefinitely and tend to fade away. If we assume that at a period sufficiently far out in the future \(t+k\), abnormal earnings have been arbitraged away and disappear, then market value of equity must equal the book value and the formula (3) becomes \[P_{t}=B_{t}+\displaystyle\sum_{i=1}^{K}\frac{A_{t+i}}{(1+R)^i}\] Using similar mathematical simplification tools than the ones used in the Gordon Growth Model, we can simplify Eq. 5 : \[P_{t}=B_{t}+\frac{1}{(1+R-\omega)}A_{t+1}\] Or, \[P_{t}=B_{t}+\frac{1}{(1+R-\omega)}(E_{t+1}-RB_{t})\] Dividing the value by the current book value leads to the following relation between price to book and return on equity : \[\frac{P_{t}}{B_{t}}=1+\frac{1}{(1+R-\omega)}(ROE_{t+1}-R)\] This formula highlights some important facts. First the higher the ROE, the higher the PB, everything else being equal; valuation can increase without destroying shareholder returns as long as the ROE also increases. Moreover given two companies with the same ROE but different PBs, the higher PB will either have a lower discount rate \(R\) and/or have a higher persistence rate \(\omega\) (ability to generate more abnormal earnings). There is a clear similarity between the PB-ROE model and the Gordon Growth Model (GGM) (Equ. 3). Recall that the GGM is : \[P_{t}=\frac{D_{t+1}}{R-g}\] Or, \[P_{t}=\frac{\rho E_{t+1}}{R-g}\] By dividing both terms of the equation by the book value \(B_{t}\) we get : \[\frac{P_{t}}{B_{t}}=\frac{\rho ROE_{t+1}}{R-g}\] Where \(\rho\) is the payout ratio. We can use the clean surplus accounting rule and replace \(\rho ROE_{t+1}\) by \(ROE_{t+1}-g\) where \(g\) is the perpetual growth rate in dividends. We therefore have another version of the GGM based on the price-to-book : \[\frac{P_{t}}{B_{t}}=\frac{ROE_{t+1}-g}{R-g}\] The GGM shows that the \(PB-ROE\) relationship reflects a trade-off between the discount rate and future growth while the RIM shows that the \(PB-ROE\) relationship is a trade-off between the discount rate and the persistence rate. There is, thus, a close relationship between growth and persistence of abnormal earnings. This is very intuitive since future abnormal earnings drive investment which in turn drives growth in dividends.Finally, the term \((ROE_{t+1}-R)\) reflects the ability for the firm to create value. If a firm is able to create value, its PB will be above 1.
As a conclusion to this section, hereafter are the important ideas we wish to highlight before moving on to the cash-flow approach :
The RIM is a derivation of the the DDM using clean surplus accounting and introducing the abnormal earnings concept
The RIM helps us better understand the notion of value creation and the relationship between value creation and valuation
PB-ROE is a simplified version of the RIM the same way the GGM is a simplified version of the DDM.
Valuation multiples are simple versions of multi-period discounting models and as such are very helpful as a starting point for stock selection (through screening for example).
As a consequence, it is advised to build more sophisticated valuation models after the initial screen.