**Accounting feature 1 ** The standard derivation of the residual earnings valuation formula from the dividend discount formula formalizes feature 1. Given a constant discount rate, *r*, the value of an asset now (at time *t*) is

- (1)

where *d*_{t+τ} is the expected dividend (cash flow) from the asset in period, *t*+*τ.* (Here and throughout the paper, variables time-subscripted with *τ* > 0 are expected values.) This model is also, of course, a statement of the no-arbitrage price if *r* is the required return for risk borne.^{1} Substituting the clean-surplus relation, *d*_{t}_{+τ}=*Earnings*_{t}_{+τ}− (*B*_{t}_{+τ}−*B*_{t}_{+τ−1}) into equation (1) for all τ > 0,

- (2)

*Earnings*_{t}_{+τ} is earnings on the asset for period *t*+τ and *B*_{t}_{+τ−1} is the book value of the asset on the balance sheet at the end of the prior period, both specified by a particular set of accounting principles. *Earnings*_{t}_{+τ}−*rB*_{t+τ−}_{1}is referred to as residual earnings for year *t*+τ. The model is usually applied to equities but applies to any asset (such as a bond), though for terminal assets (such as a bond) the summation runs only to maturity.^{2}

With no accounting restriction other than the clean-surplus relation, the model holds for all accounting methods. Accordingly, application of the model requires further specification of the accounting, and that accounting is an open issue. For example, one might specify a (mark-to-market) accounting whereby

(as with a liquid, mark-to-market investment fund where investors trade in and out of the fund at book value, ‘net asset value’). This accounting forces an expectation of future residual earnings of zero, so the forecasting task is removed: valuation is satisfied by the accounting for the present. Alternative accounting involves *P*_{t}≠*B*_{t} but, for a given *P*_{t}, means that expected residual earnings is non-zero for some *t*+*τ*. One sees that the accounting determines what is to be forecasted; forecasting is a matter of accounting for the future. The dividend discount model is just a special case where the balance sheet is empty, it reports no book value (except cash). Its unlevered equivalent, the discounted cash flowormula, is just the residual earnings formula stated for an accounting where earnings from operations equals free cash flow and book value equals net debt.^{3}

These observations pose the research question: What is the appropriate accounting for forecasting and valuation? The issue does not arise for infinite-horizon forecasting, for equation (2) is then equivalent to equation (1) for all accounting for earnings and book value; one is indifferent to the accounting. However, practical forecasting must be done over finite horizons, so the question amounts to one of relative forecasting error for a given forecasting horizon.^{4} As with all forecasting, that question might be addressed in terms of assessed error distributions and the standard statistical metrics for evaluating those distributions. But now the accounting also enters in.

For a finite forecasting horizon, *T*, the dividend discount model (1), is stated (consistent with no-arbitrage) as

- ((1a))

By substituting earnings and changes in book value for dividends, it follows that (for all accounting for earnings and book value),

- ((2a))

The last term is the amount of value omitted from the balance sheet at *t*+*T* under the specified accounting; that is, *P*_{t}_{+T}−*B*_{t+T} is the error in the balance sheet in capturing value at the forecast horizon. (It is referred to as the ‘continuing value’ or ‘terminal value’ in textbooks.) Accordingly, a given accounting can be evaluated by the amount of valuation error it produces (in expectation) in the balance sheet for a given forecast horizon. For a particular accounting where *P*_{t}≠*B*_{t} but the accounting is expected to add earnings to book value in the future such that *P*_{t}_{+T}=*B*_{t+T}, the accounting yields zero error for the specified *T* (and correspondingly, residual earnings after *T* are expected to be zero). The case of *P*_{t}=*B*_{t} is a special case, of course, where there is no error at time, *t*.^{5} The claimed dominance of accrual-accounting valuation over discounted cash flow analysis (cash accounting) for equity valuation in based on the observation that *P*_{t}_{+T}−*B*_{t+T} is typically greater under discounted cash flow analysis: Book value under discounted cash flow valuation records only net debt and, as net debt is typically positive (yielding negative book value of equity), *P*_{t}_{+T}−*B*_{t+T} is greater than *P*_{t}_{+T}.

However, in evaluating *ex ante* error for a particular accounting specification, one must recognize that accounting reports an income statement as well as a balance sheet. Under the no-arbitrage condition, successive prices (cum-dividend) are reconciled such that

- (3)

Substituting the accounting relation, *d*_{t}_{+T+1}=*Earnings*_{t}_{+T−1}− (*B*_{t}_{+T+1}−*B*_{t}_{+T}),

- (4)

This substitution recognizes that the stock return in the numerator of equation (3) is always equal to earnings plus the change in the premium over book value in the balance sheet for the earnings period. If the expected change in premium—the error in the balance sheet—is zero, then the expected return equals expected earnings. Thus, just as price equals capitalized expected return, so price is given by capitalized expected earnings:

Accordingly, even though accounting principles produce error in the balance sheet, this is not important if balance sheet errors cancel: *P*_{t}_{+T} is recovered by capitalizing earnings, and a valuation can be implemented by applying the finite-horizon dividend discount model in (1a) with *P*_{t}_{+T}, so determined, as a terminal value.

The idea that error in the balance sheet is unimportant to earnings measurement when that error is a constant was once (in textbooks of old) called the cancelling error principle.^{6} Earnings are just the change in book value (adjusted for net dividends), by the clean-surplus equation, so the effect on earnings from error in the ending balance sheet is cancelled by error in the opening balance sheet. The principle is demonstrated in instruction to first-year accounting students: R&D expense and earnings are the same whether one capitalizes and amortizes R&D expenditures or expenses them immediately provided there is no growth in R&D expenditures. In a valuation context it implies that one is indifferent between two accounting systems that have very different errors in the balance sheet (R&D capitalization versus expensing, for example) if those errors cancel. Even though discounted cash flow analysis has much value missing from the balance sheet (such that typically *P*_{t}_{+T}−*B*_{t+T} > *P*_{t}_{+T}), it survives without error if one expects the premium of price over net debt to be constant.

Penman (1997) adds an accounting feature, *g*, that produces a constant error in expected earnings, in addition to error in the balance sheet, such that *P*_{t+T+}_{1}−*B*_{t+T+}_{1}=*g* (*P*_{t+T}−*B*_{t+T}). This is accounting that depresses earnings (as well as book values). (Feltham and Ohlson, 1995, show that conservative accounting induces this feature as well as balance sheet error.) Correspondingly, residual earnings are expected to grow at the rate, *g*, and this growth rate, induced by the accounting, can be incorporated in the valuation with a capitalization at *r*−*g* rather than *r*:

Accordingly, valuation can tolerate not only error in the balance sheet but also error in the income statement. But note that the growth rate is a property of the accounting for earnings and book values; adding a growth rate to the denominator is a result of accounting with both error in the balance sheet and error in the income statement that results in expected growth in premiums over book value.

Empirical work in Penman and Sougiannis (1998) and Francis *et al*. (2000), compares valuation errors of accrual-based valuation models and cash flow models against observed prices, and broadly affirms that accrual models (based on U.S. GAAP) produce lower valuation error relative to observed prices for a variety of forecast horizons. Consistent with the above, they show, however, that the error with accrual accounting is higher when the premium over book value is higher and when changes (growth) in the premium are expected.

However, little accounting theory has been advanced for evaluating different (accrual) accounting methods for forecasting and valuation. The field is wide open. But it is an important one. Indeed it is at the heart of accounting design and forecasting for valuation. With an eye on the error criterion, one might suggest that the best accounting would be fair value accounting that sets *P*_{t}=*B*_{t}: a perfect balance sheet with *T*= 0 that the removes the need for forecasting. Essentially, accountants do all the forecasting for the investor and analysts disappear. The movement amongst standard setters for fair value accounting and an asset-liability approach (rather than an income statement approach) seems to be inspired by the idea of developing a better balance sheet. So are the prescriptions of those who argue that more ‘intangible’ assets should be recorded on the balance sheet. However, while this accounting may appear to reduce balance sheet error, the question is ultimately that of average *ex post* valuation error using both income statements and balance sheets. Indeed fluffy asset values from Level-3 fair value guesstimates may produce large errors in term of investment outcomes, for imprecise estimates in the balance sheet are compounded in the income statement.^{7} The idea that ‘better’ balance sheet accounting produces a better accounting for valuation is misdirected: It ignores the cancelling error notion. Historical cost accounting leaves value off the balance sheet, but focuses on earnings which, we have seen, has an important role reducing the error from an accounting system.^{8} So there is no problem with omitted intangible assets, for example, if earnings from the assets are flowing through the income statement. For the case where *P*_{t}≠*B*_{t},

if *P*_{t+}_{1}−*B*_{t+}_{1}=*P*_{t}−*B*_{t}. If conservative accounting is applied such as to depress earnings, *P*_{t+}_{1}−*B*_{t+}_{1}=*g*(*P*_{t}−*B*_{t}) and residual earnings are expected to grow at the rate, *g*. The valuation is accordingly modified to accommodate this accounting

- (5)

The Coca-Cola Company has an important brand asset missing from the balance sheet (giving it a price-to-book ratio of about 5), but is easy to value from its earnings on that brand with this simple formula.^{9}

These points aside, clearly much research needs to be done. The main point here is that forecasting must entertain accounting but the evaluation of appropriate accounting (for valuation) must also entertain its use in forecasting. Accordingly, accounting prescriptions might move away from pure accounting concepts (such as ‘measurement attributes’ and definitions of assets and liabilities that absorb much of the current FASB and IASB deliberation documents) to the utilitarian focus on forecasting. Vague accounting concepts such as ‘reliability’ might then take on some bite with a focus on average *ex post* valuation error. Standard metrics for efficient forecasting might be exploited for the task. Fair value accounting and historical cost accounting might be evaluated with the question: How does the accounting help or frustrate the practical task of forecasting and valuation?

**Accounting feature 2 ** It is clear from valuation model (2) that the division of value between current book value and expected future earnings is also a matter of accounting: The difference between price and book value is just the amount of value that the accounting has not yet booked to book value, and that amount will differ for different accounting specifications. Accordingly, it is the accounting for the present that determines the transition from book values and past earnings and dividends to future earnings.

As a statistical model, forecasting might be represented as applying transitional parameters to current and past accounting numbers. For example, with a linear specification,

- (6)

(with *ε*_{t}_{+1} mean zero). The parameters are often estimated from the data. Early research (that conditioned earnings forecasts on past earnings alone) took that approach. Lintner and Glauber (1967) Ball and Watts (1972) estimated a martingale, with drift, for the earnings process and subsequent papers applied Box-Jenkins techniques, popular at the time, to earnings time series. But the process is generated by the accounting and this process should direct the forecasting. This is easily seen in the case where mark-to-market accounting for book value yields *P*_{t}=*B*_{t}. In this case, β_{1}= 0, β_{2}=*r*, and β_{3}= 0, by construction of the accounting that yields a forecast of residual earnings for *t*+ 1 equal to zero. A martingale process in earnings (that sets β_{1}= 1 +*r*, β_{2}= 0, and β_{3}=−*r*, thus accommodating a drift term for retention) implies a valuation model where book value is irrelevant: , that is, the cum-dividend trailing P/E ratio = (1 +*r*)/*r*. (It should be easy to see that this forecasting applies in the case of constant balance-sheet errors earlier.) More generally, the parameters in forecasting equation (6) embed accounting principles, along with the required return. This point is made vividly in Ohlson (1995), which specifies linear dynamics dictated by the accounting, such that the earnings forecast is a weighted average of the book value forecast and the martingale earnings forecast above, with the weights determined by the accounting for earnings and book value. Accordingly, in the general case, the β coefficients in equation (6) involve both the required return and accounting process features.

By depicting forecasting as a process that applies parameters dictated by the accounting, we make the point of linking forecasting to accounting. However, it is unlikely that accounting numbers are generated by a stationary process. For this reason, practical forecasting usually forecasts by modelling pro forma future financial statements with interperiod relations changing period-to-period as indicated by both an analysis of the business and an analysis of the (quality of) accounting. (This is not to exclude parametric approaches to forecasting, however.) Accounting feature 3 talks to the issue of building earnings forecasts from the components of pro forma financial statements.