After Ohlson 1995(O95) and Yeh (2001), literature has been published numerous papers on accounting data and value. A review of this literature reveals that many themes and insights recur across the papers. This paper addresses the essence of O95 and tries to integrate the O95 literatures, and thereby making the O95 literatures more accessible to the average reader. Only with a systemized overview of the O95 literatures can a reader compare how models differ in their key characteristics (like their reliance on certain measures of growth). The paper considers situations in which price equal capitalized forward net income add growth in net income and book values.

As a preliminary to deal with growth, the paper lays out the framework for (i) Book Value plus Residual Income (BVRI) valuation model and (ii) Capitalized Earning & Increments in Residual Income (CE&IRI) valuation model. The derivations of the (i) and the (ii) are from present value of expected dividends (PVED), but the (i) emphasizes book values while the (ii) emphasizes earning (earning also called net income). The (I) represents PVED via book value plus the present value (PV) of residual net income. Residual net income can be thought of as simply the change in book values with an adjustment for dividends. This manner of treating at residual net income causes an emphasis of growth of book values. On the other hand, the (II) formula derivation does not introduce book values. The (II) formula represents PVED via capitalized both forward earning and the PV of capitalized increments in residual net income. Both (I) and (II) are valuation tools; the goal of this paper is to show how growth influences valuation.

The more detailed study of growth assumes that the residual net income variable satisfies a standard growth/decay dynamic The parameter γ identifies the long-term growth rate. The (I) formula then causes the market-to-book(P/B)¹ model familiar from many textbooks (e.g. Penman 2006), and the (II) formula causes the so-called OJ model (Ohlson and Jeuttner-Nauroth 2005). Both (I) and (II) models can explain the price-to- forward-earning (P/E)2 ratio, but they do so with different dependent variables. In case of (I) formula, the return on equity (ROE)³ explains the P/E ratio

In case of (II) formula, the growth in expected net income explains the P/E ratio

The g₂ defines the short-term growth (STG) in expected net income). Therefore one obtains two distinct ways of explaining the P/E ratio with transforming the mathematics. With growth being expected, it follows that the firm's price equates book values (or capitalized forward net income) add growth.

The central variables of this paper are value, earnings (wealth creation), book values (wealth accumulated) and dividends (wealth distribution).

This paper uses the following notation:

V_t= Value (price) of equity, V = P

D_t = (net) Dividends

B_t= Book value

I_t = (net) Income (NI=EAT=earning after tax), I = EPS, the P/E ratio

= Residual (abnormal) Income (RI), R = 1+ r= the discount factor, an exogenous constant equals the risk-free rate. RI is defined as current earnings minus the risk-free rate times the beginning of period book value, that is, earnings minus a charge for the use of capital. (RI =earning after tax and cost of capital (cc), . If the ROE > r () then the firm has the positive RI.

Clean surplus relation (CSR), B_t - B_t-1= I_t - D_t. Regardless one considers growth in expected Income or book values, such growth depends on dividends and the retention of earning. The more the retention, the more the firm will grow.

As is normal, this paper assumes that PVED determines value:

The Et [] is an expectation operator.

We build on the dividend discount model (PVED) using the following unified valuation framework. The framework emphasizes that from a mathematical point of view, one can use dividends, earnings, residual earnings, or free cash flows for valuation. To simplify the mathematical expressions, hereafter date 0 (NOT t) specifies the valuation date.

Mathematical zero-sum series equality provides the analytical starting point. For any series of numbers x₀,x₁…

If x (numerator) grows slower than R-t (denomenator), then

That  makes sense, due to no firm will grow forever. Adding PVED to zero-sum series produces the first equation

(1)

Certainly, the analysis is valid for any x-series. The idea is that one can represent value in respect of two parts: a starting point, x₀ and a complement defined by the present value of a generic series, which  implants the series of dividends.

One starts with the book value of equity as follows.

By putting x_t = B_t and combining it with residual income (RI) and clean surplus relation (C.S.R.).

We get and the second equation (2) and call it as

(I) Book Value plus Residual Income (BVRI) valuation model.

(2)

The price (V₀) is explained by the initial book value (B₀) and the subsequent growth in book value. If the firm has no growth (R.I.), then price (V₀) equate the initial book value (B₀). The model fastens value on book value and plus a premium for the present value of residual income (super growth in book value). The super growth is the growth beyond what could be achieved by making the normal rate of return on book value. Book value growth is explained by the I_t^R due to the subsequent book value t increase in I_t^R ().

The subsequent book value increase in I_t^R, due to

The means fixed dividends policy and k = 0 means zero dividends policy and the means growth in book value.

The market-to-book ratio ( ) increases as subsequent I_t^R increases

Replace I_t in R.I. with in C.S.R., we get

If x_t is book value, is residual income
Replace x_t in (1) with

Book value and its growth are famous valuation models only in banking industries. Analysts in many other industries focus on earnings and earnings growth. In the following, we show that instead of starting the valuation with book value, one can start with capitalized forward earnings and then plus a premium for abnormal earnings growth. We call this model the Capitalized Earning & Increments in Residual Income (CE&IRI) valuation model.

Instead of starting with book value of equity, one can start with the Capitalized earning as follows. In a similar spirit, by putting we get the third equation (3) and call it as (II) Capitalized Earning & Increments in Residual Income (CE&IRI) valuation model.

(3)

The price (V₀) is explained by capitalization of next period's expected earning ( ) and the subsequent growth (increments) in residual net income.

Residual earning (income) growth is explained by the

If the firm has no growth ( ), then price (V_o) equate the capitalization of next period's expected earning (). That means time value of money. The V₀ is the beginning investment, the I₁ is income. Each dollar of stock price (V₀) forecasts r dollars of next period earnings. In a certainty setting (referred to as the "savings account"), where V_t is the amount deposited in the savings account; x, is earnings (the dollar amount of the interest on the savings account deposit) for period t; r is the rate of interest on the savings account deposit; and D_t is the dividend paid to the owner of the savings account (the amount that the depositor chooses to withdraw) at time t.

In CSR we replace with and get

In the third equation

One interprets the item  as the increase in expected earnings in excess of the increase due to reinvestment of wealth () during the period. The equity value equals capitalized forthcoming earnings add a premium for growth in expected earnings in excess of the growth that could be realized by simply retaining the wealth generated in a fixed return () instead of paying dividends. In a fixed rate savings account, the item (-D_t) is the withdrawal and remaining amount (I_t - D_t) increases the savings account balance and yields additional interest income: r (I_t - D_t). For a savings account, the item r (I_t - D_t) equates ΔI_t+1 and the item () is zero. Thus, there is no premium over capitalized forthcoming earnings. That is, the marginal investment in a savings account has zero NPV.

In a similar spirit, by putting  we get the fourth equation (4) and call it as capitalized dividends& increments in dividends valuation model (CD&IDVM).

(4)

The price (V₀) is explained by capitalization of next period's expected dividends  and the subsequent growth (increments) in dividends.

Dividends growth is explained by the

due to

Those derivations (equations (2), (3) and (4)) do not depend on any conceptual restrictions on accounting data. There is, for example, no clear distinction between the distribution (D) and creation (I) of wealth.

If net income means creation of wealth, then dividends mean distribution of wealth.

After those derivations (equation (2), (3) and (4)), the reader will see why the author want to introduce equation (1). Because equation (1) has item   then   mean growth.

If  in the  then  means growth in book value.

If  in the  then  means growth in earning.

If  in the  then  means growth in dividend.

That  in equation (1) means zero growth in all accounting variables.

Moreover, this growth item  goes beyond the call of duty due to retained net income. In other words, this growth item  is, in fact, a dividend-adjusted growth. Now the question arises whether one can find some useful, additional assumption that parameterizes this growth item . All parameterizes are known.

Model (I) BVRIVM develop market value as book value plus a premium above book value for expected growth in book value, model (I) anchors valuation on book values. Such focus on book values is justified when book values near market values, for example, financial instruments are marked to market. So, model (I) is good to value financial institutions. But, the emphasis on book values is lost when firms are conservative accounting rules. E.g., the most important assets of knowledge intensive corporations are not shown on their books as investments in intellectual assets are not shown on their financial statements. Thus, their book values are underestimated and the ROE is over-estimated. Analysts valuing these firms usually do not use book value of equity as the starting point in their valuation. Thus, model (II) CE&IRIVM focus on the earnings and earnings growth expected from these “off-balance sheet” properties. Analysts focus on capitalized earning & increments in residual income (CE&IRIVM) because future earnings and earnings growth are less affected by conservatism than are book values. Thus, model (II) are heavily used for valuation.

In model (I) BVRIVM

In what follows the authors do not initially attach any economic interpretation to this growth item . One can derive V₀ as a function of x₀ and x₀ + x₁ alone, given suitable assumptions on the x-series. Unsurprisingly, the assumption requires the y_t -series to grow (or decay) geometrically. By imposing some structure on the pattern on y_t we get a short formula. To make the model more realistic yet simple we do not restrict on one-year ahead y_t allowing y₁ being any positive number. After year 1 we assume y_t grows at a constant rate. Specifically,  . Where 0 < γ is some presumed growth t parameter and 0 < y_t. If y_t = 0, then .

Given PVED and consider any series satisfying  and a related series  such that

Then

(5)

And

(5.1)

Where all parameterizes (ω, γ) are constants.

Given(1),

Armed with these analytical results, I next identify the two cases that explain the P/E ratio. The first approach, (a), refers to the growth in book value, and the second, (b), refers to the growth in net income.

(I) Setting x = B in (5) simply changes the notation and the (5) decreases to (6)

(6)

One reads  in (6) as the forthcoming growth in the expected book value, adjusted for dividends. The numerator adjustment for dividends is important: it reflects dividend policy irrelevancy (DPI). That is, the numerator does not depend on the next period's dividend decision since the cum-dividend book value, , does not depend on the dividend. Thus the choice of date-one dividends does not affect V₀.

In CSR we replace with

Standard derivations of the model assume CSR. It causes the textbook expression:

Given (6),

(7)

Where

equals the forthcoming expected return on equity. It follows that  increases as ROE₁ increases. With respect to γ - where now

The market-to-book ratio () increases as γ increases when . These conclusions are reasonable because they decrease to the idea that "growth in residual net income is good assuming they are initially positive". Relating the ratio to ROE has some attraction, certainly. But it needs a convincing real-world (For real-world reference purpose, we put a sample in endnote 4) motivation. Investors tend to ask "What factors explain the P/E ratio?" rather than "What factors explain the  ratio?" More important for our purposes, the last question is of interest because it has already been established that   is valid for time value of money

A model resting on book values ought not to rule out an explanation of the P/E ratio ().

Shifting the focus to the P/E ratio, simple manipulations of the last equation causes

(8)

Where

Given (7),

(8)

Q.E.D.

The left-hand-side variable of interest, the P/E ratio (), depends only on the right-hand-side variable ROE₁ in addition to the parameters γ and R.

An evaluation of how ROE₁ influences the P/E ratio (), depends on the sign of K₂. Signing K₂ sequentially lays the duty on the sign of 1 - γ. Is γ greater or less than one? It makes sense to require γ to exceed 1 if residual net income (I_t^R) is positive (i.e., ROE₁ > r), just as γ should be less than one if I_t^R is negative (ROE₁ < r). The first claim is based on the idea that if the firm is profitable, then, in expectation, the dollar amount of I_t^R should expand with time. Such an expected situation occurs if conservative accounting is combined with growth in the firm (if ROE₁ > r but γ < 1, then I_t^R and V_t - B_t decline with t and both go to zero, which is conflicting with conservative accounting). As the second possibility, if the firm is unprofitable (ROE₁ < r), then one should expect that the gap, ROE versus r, to be slowly closed in the future. And given ROE₁ < r (or I_t^R < 0), I_t^R goes to zero as t → ∞ if and only if γ < 1. Thus ROE₁ < r causes the condition y < 1.

Given the above restrictions — summarized by I_t^R (γ - 1) > 0 - it follows that if ROE₁ is less than r, and then the price-to- forward net income ratio, the P/E ratio () decreases as ROE₁ increases. For ROE₁ greater than r, the P/E ratio now increases as ROE₁ increases. In other words, as an empirical matter one should expect the function  on ROE₁ to be U-shaped.

Despite the fact that the analysis may seem somewhat self-important and mechanical, it makes more intuitive sense than one might think initially. Consider a firm with ROE₁ of, say, 20 percent when r = 10 percent. Such a firm is profitable, and the setting corresponds to γ > 1. Now it is clear that if the firm remains about equally profitable some day, then a computation shows that the growth in expected net income will be superior. Hence the P/E ratio ought to show a premium (exceed 1/r). Next consider when ROE₁ is poor, say, 5 percent, which is the setting when y < 1. Now there is reason to expect that ROE improves with the passage of time. A computation now shows that even a modest improvement in ROE to, say, 6 percent implies a considerable growth, in expected net income. Again, the growth principle causes the conclusion that the P/E ratio reflects a premium. (As an empirical matter, it is easily verified, looking at real-world data, that the huge firms with subpar ROE₁ such as than 7 percent, indeed have relatively large P/E ratios. Yet, as any textbook will note, such firms will also have below-average  ratios.)

The modeling allows for the case when the capitalized forward net income alone decides value: γ = 1 or ROE₁ = r give the necessary and sufficient conditions. The case ROE₁ = r is rather stale since now . The idea behind γ = 1 is more clever; now the conclusion follows even though V₀ - B₀ ≠ 0 (the sign depends on the sign of I_t^R, of course). However, γ = 1 implies V₀ - B₀ = V₁ - B₁ (γ = 0 implies V_t = B_t) and hence the expected balance sheet valuation "error" in the next year cancels with the one in the current year. And such canceling of error causes a perfect measure of expected net income so that  . That  means no growth. Profitability, as a concept, refers to the dividend-adjusted growth in book value, and  explains whether one believes CSR or not.

In x = B setting and also  that means no growth and price decay to

The equation

implies

Hence the more general model

admits for a growth in book values

to explain the  ratio less than . In the equation (5.2) that P₀ increases as B₀ decreases, holding (B₁ + D₁) fixed.

In x_t = B_t setting and also . Thus

this setting thereby interlock between x_t = B_t and

(II) Set . This setting interlocks with the (CE&IRIVM:

model framework. A geometric growth (decay) assumption now causes the equation (9).

(9)

Where

Given (5),

Put In(5), get

The item  defines the short-term growth (STG) in expected net income, adjusted for dividends (For realworld reference purpose, we put a sample in endnote 5). Like the book value model, DPI constructs in an adjustment for dividends. In the present case, the idea is that  does not depend on dividends because I₂ depends directly on D₁. A savings account illuminates the idea as it shows that D₁ gives up net income in the subsequent period such that  is independent of D₁.

Thus the dependent variable (V₀) increases as g₂ (STG) increases, as ought to be the case. With respect to γ, the dependent variable increases in γ if one assumes that g₂ exceeds the level of cost-of-capital ( r). It, too, makes sense as γ represents long-term growth (LTG). For g₂ equal cost-of-capital, LTG is now immaterial as the current growth is neutral to value. If the item g₂ < r, a case of inferior growth, then it makes sense to think that γ is less than one. Now the P/E ratio () decreases as γ increases. But this limit does not affect whether the P/E ratio () increases as STG increases (given fixed γ and r): it forever increases as g₂ increases.

The special case when the P/E ratio () suffice to decide value occurs if and only if g₂ =r (equally, y₁ = 0). The pricing (V₀) reflects no premium unless there is growth in expected net income. The g₂ measures the short-term growth in earnings (adjusted for earnings foregone in period t+2 due to next-year dividends). Hence, the model has two measures of growth in earnings, g₂ and γ, explain the price to forward-earnings ratio.

Simple as the analysis is, one gets the following conclusion: the idea that the growth in net income explains the P/E ratio is a principle of generality.

This section affirms the two information dynamics that support the parameterized the (I) model and the (II) model framework, thus the modeling extends the benchmarks by growth. This generality implies that the P/E ratio exceeds 1/r. Simplicity and symmetry will be piece of the modeling: having thanked one of the dynamics, it becomes more or less repeated what the second must be. Each of the two dynamic have two parts: (a) a starting point as decided by either book value or net income, and (b) information that allows on the coming growth. With those concepts in mind, one can next expand the dynamic via periodidiosyncratic information, which makes the practical content of the model more realistic. Finally, I show the ways in which the modeling explains accounting conservatism.

The first case extends the (I) Book Value plus Residual Income (BVRI) valuation model such that it causes the parameterized the BVRIVM. Consider the dynamic equations

(ID1)

The disturbance terms have zero mean in the usual fashion. With respect to the incremental notation, one thinks of 0_1,t as general "other" information. It can depend on almost anything inside and outside the financial statements.

The second case generalizes (II) Capitalized earning& increments in residual income valuation model (CE&IRIVM) framework. Consider the dynamic equations

(ID2)

Each of the two dynamics implies a valuation answer which expresses how the price depends on the information and the parameters γ.

Assume PVED and consider two information dynamics. For ID1 value equals

(10)

Given (2)

For ID2 value equals

(11)

The equation (11) tells us that next period expected earnings, scaled by the inverse of the risk-free rate, decide value, if other information is zero.

Given (3)

Q. E. D.

For real-world reference purpose, we put a sample in endnote 6 about equation 11.

It is obvious that ID1 implies the  model. ID2, on the other hand, gives a more robust valuation setting. This watching suggests, speaking in broad terms, that an earnings point of view provides greater flexibility than a book value view. That said, for both models one easily explains the value via a growth construct. When the setting is certain, one put money a savings account. The dynamic (ID2) can be decreased to (ID2.1), like I_t+1 = R x I_t - r x D_t ... (ID2.1).

Expressing the dynamic this way makes it apparent that the change in earnings, I_t+1 - I_t, depends only on the earnings r x  (I_t - D_t) retained in the current period. t t Accordingly, the dividend policy alone explains the growth in earnings. Zero growth corresponds to a 100 percent payout; a growth that equals the discount factor corresponds to a zero payout. Common sense suggests that this growth effect due to retained earnings should influence any model of earnings and dividends. More general models, however, ought to also allow for growth that goes beyond this dividend policy effect, unlike a savings account. The more the retention holds, the more it grow.

The dynamic (ID2) deals with "superior" growth, which goes beyond the growth due to retained earnings in the dynamic (ID2.1).

O95 suggest that conservatism arises from two different sources. First, there exist positive net present value (NPV) projects. Second, the accounting rules can be basically conservative.

As to the first point, a firm may be observed to have the opportunity to undertake positive NPV projects. Such projects do not affect the accounting today, however the same is not true for today's value. Thus it goes without saying that the market value today can exceed both book value and next period's expected net income capitalized. (A general analysis exploits the (I) model and the (II) model).

Even so, equation (10) and (11) make it clear that this source of conservatism can happen only in the context of growth. will not exist with a positive NPV setting.

It is seen that the import of growth asserts any introduction of positive NPV projects. It may seem that positive NPV is not only sufficient but also necessary for the conservatism. Such is not the case, however. O95 provide a second point: the accounting rules themselves can be conservative besides not recognizing positive NPV opportunities. The accelerated depreciation, R&D and excess write-offs cases. Hence one can easily obtain an inequity where price exceeds book value only as a result of a downward bias in the balance-sheet valuation model. If in addition there is firm growth, then net income will also be relatively discouraged because they will be effectively "deferred".

For reference purpose, we put Yeh (2001) Residual Income valuation mode (RIVM) in endnote 7.

The paper proceeds in three steps. First it presents and critiques the extant valuation approaches namely the present value of expected dividends (PVED) and the residual income valuation model (RIVM). Second, it presents a framework to unify these extant models and to derive a model based on residual income and growth on residual income, which are the two most heavily watched metrics in the real world. Third, it presents a parsimonious parameterization of the net income-based model that is easy to implement and yet gives powerful insights into a firm's value and its perceived risk.

This paper provides a conceptually useful foundation for the study of net income, book values, and dividends as to how these variables relate to equity value. The discussion makes the case that the analysis is also of empirical interest.

Accounting, or the financial reporting model, has its own rules, and these make their presence felt all the time. The CSR has a role to interlock the book values and net income. Book values differ from net income because the latter need a capitalization factor to be of the same order of scale as book values. With respect to dividend, this paper discriminates the creation of wealth from its distribution and its accumulation: the dividend does indeed differ from the net income and book values, and the shift from PVED to various expressions that incorporate net income and book values implants the simple but important idea that the distribution of value must bring together and be in agreement with its creation and its accumulation. Wealth creation takes on fame insofar that the dividend policy itself cannot create any value — that is, DPI applies. Performance of this concept depends critically — and attractively — on the five accounting concepts: (i) dividends do not influence same-date net income (), (ii) dividends decrease book value () (iii) dividends decrease subsequent net income () since the decreased book value represents fewer resources essential to generate future net income, An increase in dividends at any given date decrease the subsequent period's expected earnings. Because risk neutrality obtains, the marginal effect of a dollar of dividends on next period's foregone expected earnings equals the risk-free rate. (iv) Dividends do not influence residual net income (), and (v) dividends do not influence increment () in residual net income ().

Does the analysis result in useful empirical implications? I think so, for the simple reason that investor starts from the principle that the growth of expected net income should justify the P/E ratio. And this is the principle that the analysis elaborates on, including "why net income" and the nature of the growth shapes. It certainly improves on the traditional, textbook, so-called constant growth model. The popularity of this model has been more dependent on the importance of the empirical issues that it can manage than on its intrinsic attraction.

• The market-to-book (V/B) also called price-to-book (P/B), a ratio used to find the value of a company by comparing market value of a firm to its book value. Market value is determined in the stock market through its market capitalization. Book value is calculated by looking at the firm's historical cost, or accounting value. The market-tobook ratio attempts to identify undervalued or overvalued securities by taking the market value and dividing it by book value. In basic terms, if the ratio is above 1 then the stock is overvalued (growth stocks); if it is less than 1, the stock is undervalued (value stocks). Growth stocks returned 6.41% per year compared with value stocks which returned 23.28%.
• A measure of the price-to-earnings ratio (P/E) using forecasted earnings for the P/E calculation. While the earnings used are just an estimate and are not as reliable as current earnings data, there is still benefit in estimated P/E analysis. The forecasted earnings used in the formula can either be for the next 12 months or for the next full-year fiscal period. Long-term investors—investors who commit their money to an investment for ten full years—did do well when prices were low relative to earnings at the beginning of the ten years. The lower P/E ratio stock you buy the higher return you earn. Note that at the height of the Dotcom bubble P/E had raised to 32. The collapse in earnings caused P/E to rise to 46.50 in 2001. It has declined to a more sustainable region of 17. Its decline in recent years has been due to higher earnings growth.
• Return on Equity (ROE) between 15% and 20% are considered desirable. High ROE yields no immediate benefit. Since stock prices are most strongly determined by earnings per share (EPS), you will be paying twice as much (in Price/Book terms) for a 20% ROE company as for a 10% ROE company. The benefit comes from the earnings reinvested in the company at a high ROE rate, which in turn gives the company a high growth rate. ROE is presumably irrelevant if the earnings are not reinvested. The sustainable growth model shows us that when firms pay dividends, earnings growth lowers. If the dividend payout is 20%, the growth expected will be only 80% of the ROE rate.
  What Does ROE Mean? The amount of net income returned as a percentage of shareholders equity. Return on equity measures a corporation's profitability by revealing how much profit a company generates with the money shareholders have invested. ROE is expressed as a percentage and calculated as: Return on Equity = Net Income/Shareholder's Equity.
• (I) Setting x_t = B_t, The firm in our example is Taiwan Semiconductor Manufacturing Company (TSMC). TSMC shares were trading at $50 at the end of 2010 and the book value per share was $25. Market-to-book ratio = 2. Bloomberg's estimate of the cost of capital on TSMC was 15 percent. Value Line forecasted earnings for 2011 of $5. The implied long-term growth in residual income g, is 10 percent.
  
  
  The relations between prices, book value, and growth in book value are illustrated in equation (7.1)
  
  (7)

  
  (7.1)

  
  
  Notice that the price-to-book ratio of 2 is greater than one — That is, the balance sheet is conservative. This conservative accounting is associated with short-term growth in (dividend-adjusted) book value (20 percent) and long-term growth in residual income of 10 percent; The observation that the expected short- term rate of growth in dividend-adjusted book value (i.e., the expected return on equity) of 20 percent is greater than the cost of capital (15 percent) shows that the 2011 accounting is beginning to "correct" for the conservatism in the 2011 balance sheet. The long-term growth rate in residual income of 10 percent shows that more growth is expected in future accounting earnings to complete the "correction" for conservatism.
• (II) If , is capitalized earnings, y is capitalized t abnormal earnings growth, g is (one plus) change in abnormal earnings growth, and the abnormal earnings growth model may be written as
  
  (9)

  
  Where,
  
  (9.1)

  
  (9.2)

  
  The adjustment to capitalized earnings in this form of the model is additive in (II); the alternate form differs in that the adjustment is multiplicative in (I). This form highlights growth in earnings — both short-term and long-term. The relations between TSMC's prices, earnings, and growth in earnings are illustrated in equation (9.1) and (9.2). (In addition to data seen before we now note that Value Line's forecast of TSMC's 2012 earnings will be $5.5 and TSMC's 2011dividends will be $2.50). Notice that the forward-earnings-to-price ratio of is less than r (the cost of capital on TSMC was 15 percent) — that is, the income statement is conservative. This conservative accounting is associated with short-term growth in (dividend-adjusted) earnings of 20 percent and long-term growth/decay from this high base-level of abnormal earnings growth, 10 percent (γ = 1.10).
• The firm in our example is TSMC. Data same as above, other information is close to zero (0₁ = 0.1).
  
  (11)

  
• Residual Income (Residual-Income, I R).
  
  (A1)

  
  I_t : Net Income=NI, Earnings After Tax=EAT=EPS
  r = denotes the nominal cost of capital and
  Discounted Dividend Valuation Model (DDVM)
  (A2)

  
  Under the clean surplus rule (CSR), dividend-adjusted incomes are added to shareholder's book value.
  
  (A3)

  
  Combine (A1) 、 (A2) 、 (A3), we get (A4) ：
  
  (A4)

  
  Linear Information Models, LIM
  
  (A5)

  
  (A6)

  
  Combine (A4) 、 (A5) 、 (A6), we get (A7) :
  
  (A7)

  
  Where
  
  
  
  A restatement of the model (A7)
  Using the definition of I_t^R, the model (1.7) also equals . If one further replace B_t-1 with CSR, then, we get
  
  
  Where
  
  
  
  
  These valuation weights are purely functions of the cost of equity and the persistence of residual income.