# The Return-Stages Valuation Model and the Expectations Within a Firm's P/B and P/E Ratios.

Thomas D. Dowedell [*]The return-stages model can quantify the expectations facing a firm from its price-to-book (P/B) and price-to-earnings (P/E) ratios. We illustrate two implications of the model. First, a firm s P/B and P/E ratios can predict the future cash flow pattern earned by a firm. Second, the operating performance consistent with a given stock return differs across four groups of firms. Growth Firms, Mature Firms, Turnaround Firms, and Declining Firms. Our results imply that a firm stock return depends, in part. on how its operating performance compares to the expectations defined by its P/B and P/E ratios.

A firm's stock price (or market value) measures the value of its expected future cash flows. These expected cash flows can follow many different patterns, and these patterns can vary dramatically from firm to firm. For some firms, cash flows might increase in the future. For other firms, cash flows might remain constant, or decline. In either case, these expectations define a performance benchmark for the firm. A firm's stockholders will earn a long-term stock return equal to (or in excess of) the risk-adjusted required return only if the firm's operating performance ultimately meets (or exceeds) these expectations.

Analyst estimates give one measure of the expectations facing a firm, but these estimates usually cover periods of five years or less. For example, the Institutional Brokers Estimate System (IBES) tapes include earnings estimates for periods of up to five years. The number of estimates also decreases with the length of the forecast period. In 1991 (which is one of the base years we use in this paper), IBES lists over 24,000 one-year estimates, but only 543 five-year estimates. Since a firm's stock price depends on both its cash flows over the next five years and its cash flows in later years, analyst estimates do not give a complete description of the expectations facing a firm.

In this paper, we introduce a model that can be used to gain insights about the entire set of expectations facing a firm. We call this model the return-stages model. The return-stages model is concise, writing a firm's stock price as a function of three future return on equity (ROE) levels: the future ROE from the firm's past investments; the ROE from new investments during the firm's growth phase; and the ROE from the firm's total investment base after its growth phase ends. Using this model, we show how a firm's price-to-book (P/B) and price-to-earnings (PIE) ratios can help answer three questions. First, do investors expect the firm's ROE to increase or decrease in the future? Second, is the firm expected to earn a future ROE in excess of the firm's cost of equity (an excess return)? Finally, is the firm expected to earn excess returns from its assets in place, new investments, or both?

To answer these questions, we show how P/B and P/E ratios can be used together to separate firms into four groups: Growth Firms, Mature Firms, Turnaround Firms, and Declining Firms. These four groups define the future ROE patterns that will allow a firm to earn a stock return equal to its required return. For example, a Growth Firm (high P/B and high P/E) is expected to earn excess returns from both new projects and its assets in place. A Growth Firm will earn a stock return that is less than its required return if the firm's performance does not meet these high expectations. At the other extreme, a Declining Firm (low P/B and low P/B) is not required to earn excess returns from either new or existing assets. Because it faces lower expectations, a Declining Firm can earn a higher stock return than a Growth Firm with operating performance that is much worse.

To illustrate the implications of the return-stages model, we use P/B and P/E ratios from a base year(1981, 1986, and 1991) to place three samples of firms into the four groups. We then look at the relations between a firm's P/B and P/E ratios in the base year, and its operating performance and stock returns over the next five years. We find that future operating performance (excess returns) across firms does tend to follow the patterns described by the return-stages model. We also find that a firm's stock return depends on how its operating performance compares to the expectations defined by the return-stages model. Our results show that a firm's performance relative to expectations helps to determine its stock return. More important, our results show how a firm's P/B and P/E ratios can be used to gain a broad understanding of these. expectations.

The paper is organized as follows. Section I derives the return-stages model. Section II develops our analytical framework. In this section, we use P/B and P/E ratios to describe the expectations facing four broad groups of firms: Growth Firms, Mature Firms, Turnaround Firms, and Declining Firms. In Section III, we use empirical data to show how a firm's future operating performance relates to its future stock returns within each of these four groups. Section IV concludes.

I. The Return-Stages Model

Over the life of a firm, it is likely that the returns earned on the firm's investments will change. If the firm can develop a product advantage or a lower cost structure, it could earn excess returns during a portion of its life. The ability of the firm to increase or maintain these excess returns will depend on the competitive forces facing the firm. The excess return growth that is possible will depend on the size of the new markets available to the firm, and on the intensity of the competition in these markets. The ability of a firm to maintain excess returns after it enters a market will depend on the costs it can impose on new entrants. These entry costs could be a function of patents, brand equity, or scale. Since a firm cannot sustain these entry costs forever, the firm's excess returns should disappear over time. See Porter (1980) or Mauboussin and Johnson (1997) for a discussion of these issues.

The standard two-phase model looks at one reason why a firm's ROE may change over time. In particular, the model allows the ROE on new investments during a firm's growth phase to be higher than the ROE on new investments after its growth phase ends. For a discussion of the standard two-phase model, see Miller and Modigliani (1961) and Danielson (1998).

The return-stages model extends the two-phase model by allowing a firm's ROE to change over time for two additional reasons. First, the return-stages model allows the ROE from new investments made during the firm's growth phase to differ from the average ROE of the firm's prior investments. Since the standard two-phase model specifies the level of next year's earnings, but not the ROE produced by those earnings, the two-phase model does allow a firm's marginal and average investment returns to differ. That is, the marginal ROE could be higher than, lower than, or equal to the average ROE from the firm's prior investments. However, the two-phase model does not formally articulate these possibilities.

Second, the return-stages model allows a firm's terminal ROE, which we define as the ROE a firm will earn after its growth phase ends, to differ from the average ROE during its growth phase. If a firm faces increasing competition, then the terminal ROE will be less than the ROE earned in prior periods. It is also possible that the terminal ROE will be greater than the ROE in prior periods. This possibility may describe a firm that can develop a competitive advantage over time, or a firm that can complete a successful restructuring. In either case, the terminal ROE will be higher than the cost of equity only if the firm can build and sustain entry barriers to protect its markets. [1]

A. Assumptions

Like previous models, the return-stages model divides the life of a firm into two phases. The "growth" phase will last for the next [tau] years. During this period, a Growth Firm (or a Mature Firm) might be able to earn excess returns from either new or existing assets. For a Turnaround Firm, [tau] is the length of time before the firm can be expected to complete a successful restructuring. The equilibrium phase will cover all years after t = [tau].

In the basic model, the firm will reinvest 100% of its earnings during its growth phase (i.e., to grow or restructure) and will not pay dividends during these years. We use this assumption to simplify the presentation of the basic model. In reality, the optimal amount of new investment for a firm might be higher or lower than its earnings in any given year. So, a firm might invest more (if it can raise new capital) or less than 100% of its earnings in new projects. Appendix A extends the model to handle these two cases.

On the date the stock is valued, t = 0, the per-share equity investment in the firm (equity book value) is [I.sub.t=0]. This investment base produces a ROE equal to [R.sub.E] during each year from t = 0 through t = [tau]. In each of these years, the firm invests 100% of its earnings in new projects that will earn a ROE equal to [R.sub.N] in each year after the date of the investment through year t = [tau]. Since [R.sub.N] does not have to be equal to [R.sub.E], the ROE from a firm's marginal investments ([R.sub.N]) can be higher or lower than the average ROE from the investment ([R.sub.E]).

During the equilibrium phase, new investments earn a return equal to the required return on equity, k. Since new investments made during these years will not earn excess returns, the firm might choose to pay dividends, rather than to make new investments, during these years. The equity investment at t = [tau], [I.sub.t=[tau]], will earn a terminal ROE of [R.sub.T] in all subsequent years. [2]

A firm's stock price at t = 0, [P.sub.t=0], is the present value of future dividends, discounted using the firm's required return on equity, k. We assume that the required return on equity, k, is constant through time. This assumption requires that new investments have the same risk as the firm's existing assets. In addition, the firm's capital structure must be constant through time. This condition is met if the firm is (and will always be) financed with 100% equity. Or, the firm could be financed with less than 100% equity, as long as each new equity investment is matched with a new debt investment in proportion to the firm's capital structure. [3]

B. Model Derivation

Using these assumptions, we can write the firm's earnings and investment base at t = 1 as Equations 1 and 2, respectively. (Recall that the base case of the model uses the assumption that 100% of the earnings are reinvested in new projects. Appendix A extends the model to allow the firm to invest more or less than 100% of its earnings in new projects.)

[E.sub.t=1] = [R.sub.E] x [I.sub.t=0] (1)

[I.sub.t=1] = [I.sub.t=0] + [E.sub.t=1] = [I.sub.t=0](1 + [R.sub.E]) (2)

During year t = 2, the original investment, [I.sub.t=0], will still earn a ROE equal to [R.sub.E]. The reinvested earnings from t = 1 will earn a ROE equal to [R.sub.N]. So, we can write the firm's earnings at t = 2 as: [E.sub.t=2] = [E.sub.t=1] + [R.sub.N] x [E.sub.t=1]. [I.sub.t=2] can be written as Equation 3:

[I.sub.t=2] = [I.sub.t=1] + [E.sub.t=2] = [I.sub.t=0] + [I.sub.t=0][R.sub.E] + [I.sub.t=0][R.sub.E](1 + [R.sub.N] (3)

Moving forward to t = [tau], the investment base of the firm can be written as Equation 4 or Equation 5. Within Equation 5, PVAF is the present value of an annuity factor for an annuity with a length of [tau] years, using a discount rate of [R.sub.N]. (Appendix B shows how Equation 5 can be derived from Equation 4).

[I.sub.t=[tau]] = [I.sub.t=0] + ([I.sub.t=0] x [R.sub.E])[[[sigma].sup.[tau]].sub.t=1][(1 + [R.sub.N]).sup.t-1] (4)

[I.sub.t=[tau]] = [I.sub.t=0][(1 + [R.sub.N]).sup.[tau]][1 + ([R.sub.E] - [R.sub.N])PVAF([tau],[R.sub.N])] (5)

Any new investments in the year t = [tau] + 1, and in all subsequent years, will earn the required return k. Since the return on new investments is equal to k during these years, the value of the firm does not depend on the firm's investment policies during these years. Therefore, the model uses an assumption that simplifies the calculations: a 100% dividend payout. Using this assumption, the perpetual cash flow (and dividend stream) generated by the firm during these years is the terminal ROE, [R.sub.T] times the investment amount [I.sub.t=[tau]].

[E.sub.t=[tau]+1] = [R.sub.T] x [I.sub.t=[tau]] (6)

The stock price at t = [tau] is the present value of the perpetual annuity in Equation 6: [P.sub.t=[tau]] = ([R.sub.T] x [I.sub.t=[tau]])/k. If the firm does not pay dividends between time t = 0 and t = [tau], then the stock price at t = 0 is the present value of the stock price [P.sub.t=[tau]], evaluated at t = 0: [P.sub.t=0] = [P.sub.t=[tau]]/[( I + k).sup.[tau]]. Using these expressions for [P.sub.t=0] and [P.sub.t=[tau]], along with Equation 5, we write the stock price as Equation 7.

[P.sub.t=0] = [I.sub.t=o] [(1 + [R.sub.N]/1 + k).sup.[tau]] [1 + ([R.sub.E] - [R.sub.N]) PVAF ([tau], [R.sub.N])] ([R.sub.T]/k) (7)

To further simplify Equation 7, we look more closely at the relations between [R.sub.T], [R.sub.E], and [R.sub.N]. The model assumes that the firm's investment base will earn an average ROE of [R.sub.T] during the terminal period. The sustainable level of [R.sub.T] is a function of [R.sub.E], [R.sub.N], and the competitive nature of the firm's industry. We define the weighted average of [R.sub.E] and [R.sub.N] as [WR.sub.E,N]. Within this weighted average, the investment base [I.sub.t=0] earns a return equal to [R.sub.E], and the incremental investment base [I.sub.t=[tau]] minus [I.sub.t=0] earns a return equal to [R.sub.N].

If [WR.sub.E,N] is larger than [R.sub.T], the firm's ROE will be lower in the years after t = [tau] than it was in the years before. This assumption can be reasonable for a firm that faces increasing competition in its markets.

If [WR.sub.E,N] is less than [R.sub.T], the firm's ROE will be higher in years after t = [tau] than before. This assumption may be reasonable for a firm that can develop brand equity or gain advantages of scale overtime, or for a firm that is expected to complete a successful restructuring.

Equation 8 calculates the weighted average of [R.sub.E] and [R.sub.N], [WR.sub.E,N]. Appendix C derives Equation 8.

[WR.sub.E,N] = [R.sub.E]/[1 + ([R.sub.E] - [R.sub.N]) PVAF ([tau], [R.sub.N])] (8)

To derive Equation 9, we multiply the right-hand side of Equation 7 by one ([R.sub.E]/[R.sub.E]). Then, we multiply each side of Equation 8 by [1 + ([R.sub.E] - [R.sub.N])PVAF([tau], [R.sub.N])]. Finally, we substitute the resulting definition of [R.sub.E] into the denominator of the new term ([R.sub.E]/[R.sub.E] in Equation 7 to get Equation 9.

[P.sub.t=0] = [I.sub.t=0] ([R.sub.E]/k)[(1 + [R.sub.N]/1 + k).sup.[tau]] ([R.sub.T]/ [WR.sub.E,N]) (9)

Equation 9 says that a firm's stock price is the product of the equity investment in the firm at t = 0, [I.sub.t=0], and three other components. The second term on the right-hand side of Equation 9 adjusts [I.sub.t=0] for the value of the expected excess returns from the firm's existing assets. The value estimated by this term assumes that these excess returns will continue forever.

The third term on the right-hand side of Equation 9 adjusts [I.sub.t=0] for the expected value of the positive net present value (NPV) investment during the firm's growth phase. The value estimated by this term assumes that the excess returns from these positive NPV investments will continue forever. This term is equal to one if [R.sub.N] is equal to k, or if [tau] is equal to zero.

However, a firm might not be able to maintain the excess returns embedded in [R.sub.E] and [R.sub.N] forever. Therefore, the final term on the right-hand side of Equation 9 adjusts the stock price for any difference between [WR.sub.E,N] and [R.sub.T].

If [WR.sub.E,N] is larger than [R.sub.T], the average profitability of the firm's assets will be lower in the years after t = [tau] than before. In this case, the final term will be less than one.

If [WR.sub.E,N] is smaller than [R.sub.T], the average profitability of the firm's assets will be higher in the years after t = [tau] than before. In this case, the final term will be greater than one.

If [WR.sub.E,N] is equal to [R.sub.T], then [R.sub.T] will maintain the excess returns embedded in [R.sub.E] and [R.sub.N]. In this case, the final term will be equal to one; Equation 9 simplifies into the standard two-phase model. To derive the standard two-phase model from Equation 9, set [WR.sub.E,N] equal to [R.sub.T], and then substitute Equation 1 into Equation 9. These steps yield Equation 10. (Equation 10 can be derived from the standard two-phase model in Miller and Modigliani, 1961, by setting the reinvestment rate equal to one.)

[P.sub.t=0] = [E.sub.t=1]/k [[1 + [R.sub.N]/1 + k].sup.[tau]] (10)

The example in Table I illustrates the model. Panel A shows the growth in the investment base of the firm, as described by Equations 4 and 5. In this example, the firm has an initial investment, [I.sub.t=0], of $100. This investment will earn a return, [R.sub.E], of 2 0% (and a cash flow of $20) in each of the next five years. The firm will reinvest all earnings in new projects, producing a return ([R.sub.N]) of 15% during each of the next five years. With these assumptions, the investment base in the firm will increase from $100 to $234.85 over the five-year period. We can also use either Equation 4 or 5 to calculate the ending investment amount of $234.85.

Panel B lists the stock prices (calculated using Equation 7) at four assumed values of the terminal ROE, [R.sub.T]. When [R.sub.T] is equal to 17.129% (which is [WR.sub.E,N] in this example), we can also use Equation 10 to calculate the stock price. Panel B shows that the assumed size of the terminal ROE can have a large effect on the estimated stock price, which supports the findings in Leibowitz (1998). Within this example, the stock price increases proportionately with the terminal ROE because the firm does not pay dividends until the terminal period starts. For a firm that pays dividends before the terminal period starts, the percentage change in the stock price will be less than the percentage change in [R.sub.T]. However, the key point remains the same. The assumed value of the terminal ROE can have a large effect on the estimated stock price.

II. The P/B and P/E Ratios

P/B and P/E ratios often serve as measures of expected growth. Within the standard two-phase model, the perpetual growth model, and the return-stages model, both ratios increase with the value of a firm's growth opportunities and decrease as the discount rate increases. However, because the return-stages model allows a firm's ROE to change at several key points in a firm's life, the return-stages model gives a more complete description of a firm's P/B and P/E ratios than do previous models.

A. Overview

The return-stages model can be rearranged to measure either the P/B or the PIE ratio. To write the model as the P/B ratio, we divide each side of Equation 9 by [I.sub.t=0].

P/B = [P.sub.t=0]/[I.sub.t=0] = ([R.sub.E]/k])[(1 + [R.sub.N]/1 + k).sup.[tau]]([R.sub.T]/[WR.sub.E,N]) (11)

For a P/B ratio to be greater than one, the firm must be able to earn excess returns in the future. The first term on the right-hand side of Equation 11 will be greater than one if a portion of these excess returns will be generated by the firm's existing assets (i.e., if [R.sub.E] is greater than k). The second term on the right-hand side of Equation 11 will be greater than one if new investments during the firm's growth phase will produce excess returns (i.e., if [R.sub.N] is greater than k). Finally, excess returns can increase (decrease) during the firm's terminal phase if [R.sub.T] is larger than (less than) [WR.sub.E,N]. Equation 11 shows that a firm can have a high P/B ratio (i.e., P/B [greater than] 1) if it is expected to earn excess returns from existing assets, new assets, or both.

To write the return-stages model as the P/E ratio, we divide Equation 9 by [E.sub.t=1]. Using Equation 1 to help simplify, this step yields Equation 12.

P/E = [P.sub.t=0]/[E.sub.t=1] = (1/k)[(1 + [R.sub.N]/1 + k).sup.[tau]]([R.sub.T]/[WR.sub.E,N]) (12)

Equation 12 says that the P/E ratio is the product of three terms. The first term, 1/k, is the P/E ratio of a firm in a perfectly competitive industry (see Leibowitz and Kogelman, 1990, or Danielson, 1998). For this type of firm, k is the maximum sustainable value of [R.sub.E], [R.sub.N], and [R.sub.T]. Assuming that [R.sub.E], [R.sub.N], and [R.sub.T] are equal to k, the P/E ratio in Equation 12 will be equal to 1/k.

The second and third terms on the right-hand side of Equation 12 describe two reasons why a firm might have a P/E ratio that is greater than 1/k. First, investors could expect the firm to invest in new positive NPV projects. In this case, [R.sub.N] is greater than k, and the second term on the right-hand side of Equation 12 will be greater than one. Second, the return from the firm's assets could be expected to improve over time. For example, a firm's operating returns could improve as a result of restructuring activities. In this case, the terminal return, [R.sub.T], could be greater than the weighted average of [R.sub.E] and [R.sub.N]. If so, the final term on the right-hand side of Equation 12 will be greater than one.

Equation 12 shows that a firm can have a P/E ratio greater than 1/k even if investors never expect it to earn returns larger than k. If [R.sub.E] [less than] k, [R.sub.N] = k, and [R.sub.T] = k (i.e., the return on the firm's assets in place improves over time as the firm restructures or divests unproductive assets), the weighted average of [R.sub.E] and [R.sub.N], calculated using Equation 8, will be less than k, and the P/E ratio, calculated using Equation 12, will be greater than 1/k. This example shows that expected improvements to a firm's existing operations, alone, can support a P/E ratio in excess of 1/k.

B. Four Groups of Firms

Fairfield (1994) and Penman (1996) show, using a general dividend discount model, that P/B and P/E ratios can be used to classify firms into four broad groups: Growth Firms, Mature Firms, Turnaround Firms, and Declining Firms. In the following sections, we use the return-stages model to describe the firms in each of the four groups. The return-stages model shows that there can be important differences both between and within the four categories.

1. Growth Firms: P/B [greater than] 1, P/E [greater than] 1/k

The first category, Growth Firms, includes firms for which P/B [greater than] 1 and P/E [greater than] 1/k. These firms are expected to earn excess returns in the future (P/B [greater than] 1), and the dollar amount of the excess returns is expected to increase over time (P/E [greater than] 1/k).

To illustrate these ideas, Table II, Panel A, lists two different future cash flow patterns that will support one combination of P/B and P/E ratios in this category. Firms A.1 and A.2 have the following common inputs: P/B = 3, P/E = 20, [R.sub.E] = 15%, and k = 10%. For Firm A.1, [R.sub.N] = [R.sub.T] = 15%, and [tau] = 15.6 years. For firm A.2, [R.sub.N] = 26.36%, [R.sub.T] = 21.34%, and [tau] = 5 years. In each case, the weighted average of [R.sub.E] and [R.sub.N], calculated using Equation 8, is equal to [R.sub.T]. (Cases in which [WR.sub.E,N] is not equal to [R.sub.T] are most interesting for Mature Firms and Turnaround Firms, and will be discussed with those groups). Firms A.1 and A.2 show that as [tau] decreases, a firm must earn a higher return on new investments, [R.sub.N], to justify a given stock price. If [tau] is short enough (Firm A.2.), the firm must earn a higher return on new investments than it can currently earn in its existing markets.

In a more extreme example of this phenomenon we consider a third firm, one that currently has a P/B ratio of 1.5 and a P/E ratio of 20. Although P/B [greater than] 1 and P/E [greater than] 1/k, the firm is not currently earning excess returns in its existing markets: [R.sub.E] = 7.5% ([R.sub.E] = E/P x P/B = 0.05 x 1.5). To justify its stock price, this firm must earn a significantly higher return in new markets than it does in existing markets, it must improve its return in existing markets, or it must do both. However, the stock of this firm is not necessarily mispriced. For an emerging firm with large new markets, ROE can increase over time as the firm develops brand equity or scale advantages. As a result, [R.sub.N] could be larger than [R.sub.E], or [R.sub.E] could increase over time. Nonetheless, these examples show that there can be important differences within the Group Firm category. Investors expect Growth Firms to earn excess returns in new markets. For some firms, this expected performance is con sistent with the firm's past performance. For others, it is not.

We also note that it is possible for [R.sub.E] to be less than zero, in which case the P/E ratio will be negative. To allow for a continuous ranking of firms when the P/E ratio can be negative, many empirical studies use the E/P ratio. The set of firms for which P/E [greater than] 1/k is similar, but not identical, to the set of firms where E/P [less than] k. The key difference is that the set of firms where E/P [less than] k includes firms with a negative [R.sub.E]. If we use the E/P ratio to define the four groups, the Growth Firm and the Turnaround Firm categories will include firms with a negative [R.sub.E]. From an economic standpoint, these are the two groups in which firms with negative earnings belong. Within the empirical tests in Section III, we place firms with negative earnings into these two groups. Because the model in this paper can be interpreted most directly using the P/E ratio, we use the P/E ratio, rather than the E/P ratio, to explain the analytical framework.

2. Mature Firms: P/B [greater than] 1, P/E [less than] 1/k

The second category, Mature Firms, includes firms for which P/B [greater than] 1 and P/E [less than] 1/k. These firms are expected to earn excess returns in the future (P/B [greater than] 1). However, the dollar amount of the excess returns is expected to decrease over time (P/E [less than] 1/k). The return-stages model shows that a large expected decline in excess returns could cause a firm's P/E ratio to be less than 1/k, even if [R.sub.E], [R.sub.N], and [R.sub.T] all greater than or equal to k. In addition, the return-stages model shows that there is more than one economic reason why excess returns could decline in the future.

Table II, Panel B, shows two future cash flow patterns that will support one combination of P/B and P/E ratios within the Mature Firm category. Firms B. 1 and B.2 have the following common inputs: P/B = 2.5, P/E = 8.33, [R.sub.E] = 30%, k = l0%, and [tau] = l0 years.

Firm B.1 will always earn a ROE that is greater than, or equal to, k: [R.sub.E] = 30%, [R.sub.N] = 12.84%, and [R.sub.T] = 10%. As a result, the firm's P/B ratio is greater than one. Even so, Firm B.1 has a P/B ratio that is less than that of a competitive firm (1/k = 10). Firm B.1 illustrates the idea, from Equation 12, that both expected changes in ROE and expected growth can affect a firm's P/E ratio. For Firm B.1, [R.sub.N] (12.84%) is greater thank. This expectation increases the firm's P/E ratio. At the same time, Firm B.1 is earning a much larger return now (RE = 30%) than it will be able to earn in the future ([R.sub.T] = 10%). For this firm, [R.sub.T] (10%) is less than [WR.sub.E.N] (15.49%). This factor decreases the firm's P/E ratio. In this example, the second factor dominates the first factor, and the P/E ratio is less than 1/k. [4]

Firm B.2 also has a P/B ratio of 2.5, even though its [R.sub.N] is only expected to be 8%. In this case, the high P/B ratio is caused entirely by expected excess returns from the firm's assets in place, rather than by expected excess returns from new projects. The P/E ratio of Firm B.2 is less than 1/k because [R.sub.N] is expected to be less than k. (For this firm, [R.sub.T] is equal to [WR.sub.E.N]. This assumption allows us to focus attention on how the relation between [R.sub.N] and k affects the P/E ratio). Since its low P/E ratio is caused by expected overinvestment (e.g., agency problems), the problems facing Firm B.2 might be more controllable than those facing Firm B.1. For example, the firm could change its management team or its governance (and incentive) structure to better align management and shareholder interests.

Figure 1 looks at why a firm that can earn excess returns in both new and current markets might have a P/E ratio that is less than 1/k. Figure 1 plots changes in the P/E ratio as [tau] changes for three different values of [R.sub.E]: [R.sub.E] = 30%, 15%, and 10%. The firms in this figure share the following common inputs: [R.sub.N] = 30%, [R.sub.T] = 10%, and k = 10%. If [tau] is short, a firm's ROE will drop to [R.sub.T] quickly, and the decreasing returns from the firm's assets in place could exceed the gains from its new investments. In this case, the P/E ratio could be less than 1/k. For example, if [tau] is equal to four, the P/E ratio is less than ten when [R.sub.E] is equal to 15% or 30%. As [tau] increases, it is more likely that the excess returns from the new investment will dominate the decrease in the ROE from the assets in place, and the P/E ratio will be larger than ten.

3. Turnaround Firms: P/B [less than] 1, P/E [greater than] I/k

The third category, Turnaround Firms, includes firms for which P/B [less than] 1 and P/E [greater than] 1/k. These firms are expected to earn, on average, negative excess returns (P/B [less than] 1). We classify these firms as "turnaround" firms because the dollar amount of the negative excess returns is expected to become smaller (in terms of absolute value) in the future (P/E [greater than] 1/k). In this case, the period [tau] is not the period of competitive advantage, but is the length of time it will take the firm to complete its turnaround.

The return-stages model shows that if the expected change in the excess returns is large enough, a firm's P/E ratio could be greater than 1/k even if [R.sub.E], [R.sub.N], and [R.sub.T] are all less than or equal to k. The model also shows that the length of time it will take a firm to complete a turnaround is a critical piece of information when evaluating the firm's current stock price. If the turnaround cannot be completed quickly enough, then the firm might have to earn excess returns in the future (to justify its current stock price), even though it is not currently doing so.

Table II, Panel C, lists two cash flow patterns that will support one combination of P/B and P/E ratios in the Turnaround Firm category. Firms C.1 and C.2 both have the following common inputs: P/B = 0.75, P/E = 015, [R.sub.E] = 5%, [R.sub.N] = 10%, and k = 10%.

Firm C.1 will complete its turnaround in five years. For its stock price to be reasonable, Firm C.1 must be able to earn a terminal ROE of 9.25%. However, the weighted average of [R.sub.E] and [R.sub.N], calculated using Equation 8, is only 6.17%. Firm C.1 shows that it is possible for a firm to have a P/E ratio greater than 1/k even if the firm will not earn excess returns from new or existing assets. The P/E ratio of Firm C.1 is greater than 1/k because the ROE from the firm's investments is expected to increase over time.

Firm C.2 will complete its turnaround in ten years and must earn a terminal ROE of 10.83%. This terminal ROE is larger than 7.22%, which is the weighted average of [R.sub.E] and [R.sub.N]. This terminal ROE is also larger than k. Thus, if a firm cannot complete its turnaround quickly enough, it might have to earn excess returns in the terminal period to justify its stock price.

4. Declining Firms: P/B [less than] 1, P/E [less than] 1/k

The final category, Declining Firms, includes firms for which P/B [less than] 1 and P/E [less than] 1/k. These firms are expected to earn negative excess returns in the future (P/B [less than] 1). In addition, the absolute value of these negative excess returns might become larger in the future (P/E [less than] 1/k). The return-stages model illustrates just how low the market's expectations for these firms are. For these firms, it is likely that [R.sub.E], [R.sub.N], and [R.sub.T], are all less than, or equal to, k.

Table II, Panel D, shows two cash flow patterns that will support one combination of P/B and P/E ratios in the Declining Firm category. Firms D.1 and D.2 have the following common inputs: P/B =0.75, PIE = 8.33, [R.sub.E] = 9%, k = l0%, and [Tau] = 5 years.

For Firm D.1, the expected value of [R.sub.N] is equal to 10%. In this case, the required value of [R.sub.r] is only 7.8%, which is less than the weighted average of [R.sub.E] and [R.sub.N]. Therefore, investors expect the excess returns for Firm D.1 to become more negative over time as the ROE on the firm's assets in place declines further.

For Firm D.2, the expected value of [R.sub.N] is equal to 6.06%. Therefore, the excess returns for Firm D.2 are expected to decline over time as the firm invests in new projects with negative excess returns (i.e., as the firm overinvests).

Although these firms face low expectations, their stock prices are not necessarily too low. A firm could be facing intense competition, or the size of its current markets could be shrinking. For example, the size of a firm's markets could decrease as new technologies create substitute products. Since it can be difficult for some firms to increase their cash flows, the most appropriate stock prices for these firms might fall within the Declining Firm category.

III. Empirical Results

In this section, we use empirical data to illustrate two of the main ideas from this paper. First, a firm's P/B and P/E ratios can predict the future pattern of excess (operating) returns earned by a firm. Second, the operating performance--ROE levels and ROE changes over time--consistent with a given stock market return differs across firms.

Although Fairfield (1994) and Penman (1996) show that a firm's future cash flow patterns can be predicted from its P/B and P/E ratios, neither study links these future cash flow patterns to the future stock returns earned by a firm. Fama and French (1995) analyze the relations between P/B ratios, future ROE, and future stock returns, but do not control for a firm's P/B and P/E ratios together. Therefore, our tests extend the studies of Fairfield (1994), Penman (1996), and Fama and French (1995).

A. Data and Test Design

Within our tests, we first use P/B and P/E ratios from a base year (1981, 1986, or 1991) to classify firms into the four groups (Growth Firms, etc.). We then track the excess returns and stock returns earned by firms over a five-year period after the base year. Our goal is to document the relations between a firm's P/B and P/E ratios, its future excess returns, and its future stock market returns, as predicted by the return-stages model.

Within each of the three non-overlapping samples, we combine data for NYSE, Amex, and Nasdaq firms from the 2000 Compustat databases (Primary-Supplementary-Tertiary, FullCoverage, and Industrial Research) and the Center for Research in Securities Prices (CRSP) database. We use this data to calculate the P/B and P/E ratios, the subsequent stock returns, and the subsequent excess returns for each firm.

1. The P/B and P/E Ratios: Measurement Issues

Within each sample, we calculate the P/B ratio as a firm's stock price at the end of its fiscal year, divided by the book value of equity per share at the end of the year. We calculate the P/E ratio as a firm's stock price at the end of its fiscal year, divided by the firm's earnings per share before extraordinary and discontinued items for that fiscal year. Thus, this P/B ratio does not correspond directly with the P/B ratio defined by the return-stages model, which defines the P/E ratio using expected earnings for the next year. We calculate the P/E ratio using the actual earnings from the past year because the expected earnings for a firm across all investors is not observable. Although we can estimate the expected earnings for many (but not all) firms using analyst forecasts, a number of studies have questioned the accuracy of analyst forecasts for periods even as short as one year (see O'Brien, 1988, and Crichfield, Dyckman, and Lakonishok, 1978). Nevertheless, a P/E ratio that uses last year's earnings is only a rough estimate of the P/E ratio defined by the return-stages model. As a sensitivity test, we repeat our procedures, using an estimate of next year's earnings within the P/E ratio.

Our calculation of the P/B and P/E ratios also differs from the procedures used in some previous studies, such as Penman (1996). To ensure that he measured the stock price after the firm announced its results for the prior fiscal year, Penman calculated these ratios by using the firm's stock price as of 90 days after the end of a firm's fiscal year. For the same reason, many other studies, including Fama and French (1992, 1995), start the calculation of subsequent stock returns 90 to 180 days after year-end.

Measuring the stock price within the P/B and P/E ratios after year-end, or starting the return calculation after year-end, has both advantages and disadvantages. The stock price on the last day of a firm's fiscal year already reflects the actual results from the first three quarters of the year, and is a function of the estimated results for the last quarter of the year. Using a stock price after year-end has the advantage that this stock price will reflect any forecast error relating to the fourth-quarter earnings estimate, whereas the year-end stock price will not. However, a firm's stock price can change for many reasons within a 90- to 180-day period after the end of its fiscal year. The announcement of last year's earnings is certainly an important event that occurs during this period. A firm's stock price can also change as the firm releases information about new products, as it gives guidance about its potential earnings in future periods, as competitors announce news or results, or as macroeconomic e vents cause market-wide price movements.

Thus, for many firms, the events that the return-stages model says the P/E and P/B ratios are supposed to predict start to occur during a 90 to 180 day period after year-end. Therefore, the process of measuring the stock price or starting the return calculation on after year-end both reduces and creates noise. Noise is reduced because this procedure assures that the fourth quarter forecast error will not distort the P/B ratio, the P/E ratio, or the subsequent stock returns. But noise is also created because the portion of the subsequent stock return that is relevant to the periods after year-end is included in the beginning P/B and P/E ratios (this return changes the stock price used in the numerator of these ratios), and is omitted from the subsequent stock returns.

When using the stock price at the end of the year in the two ratios, and as the starting point in the return calculation, we implicitly assume that the fourth-quarter earnings estimates are unbiased estimates of the actual results across the entire sample. If so, the stock returns created by the fourth-quarter forecast errors should cancel out across the sample. To make our results comparable to other studies, we also report results obtained by using the stock price 90 days after the end of the fiscal year to calculate the P/B and P/E ratios, and to start the calculation of the subsequent stock returns. We discuss these results in Section C.

2. The Required Return on Equity, k

The return-stages model uses the required return on equity, k, to separate firms with high P/E ratios from firms with low P/E ratios. As does Penman (1996), we assume that k is equal to 10% for all firms. Although a value of k equal to 10% is admittedly a rough estimate of the cost of equity for many firms, the use of a constant k across all firms lends itself to a second interpretation. When we use a constant k to separate firms into the four groups, the four groups define the excess return patterns required for a firm to earn a stock return equal to (or in excess of) that constant k (i.e., 10%). For example, to earn a stock return of 10%, a Growth Firm must earn a ROE in excess of 10% on both new and existing assets. A Declining Firm can earn a stock return of 10% without earning a ROE of 10% from new or existing assets.

3. Measuring Performance

After separating the firms into four groups using P/B and P/E ratios during the base year, we follow the performance of the four groups of firms over the next five years. We calculate the buy-and-hold stock return, BHR, for each firm using the CRSP daily return files. The BHR is the total (cumulative) return over a given period. For example, if a firm earns an average annual return of 10% across a five-year period, then the BHR for this firm over the five-year period is 61%.

We calculate the excess returns using Equation 13 and data from Compustat. In Equation 13, [B.sub.i], the book value of equity, and [E.sub.i] is the earnings before extraordinary and discontinued items, as of the end of year i. For the 1991 sample, Equation 13 defines the excess return (or residual income), RI, for each firm in year i (where i refers to the six years from 1991 to 1996).

[RI.sub.i] = ([B.sub.i-1]/[B.sub.90])[E.sub.i] - k*[B.sub.i-1]/[B.sub.i-1]=[E.sub.i] - k*[B.sub.i-1]/[B.sub.90] (13)

We note two points of clarification about Equation 13. First, Equation 13 assumes that a firm's asset base at the beginning of a year produces the cash flows earned during the year. As Fisher and McGowen (1983) show, this assumption is useful when one wants to interpret accounting returns in an economic context. Second, the denominator uses the book value from the end of 1990 (which is the beginning balance for the base year, 1991) for all years, i, rather than the book value as of the end of year i - 1. By calculating the excess return in this manner, the excess returns will increase as the scale of the firm increases (even if ROE is constant), as long as the ROE on new investments is greater than k. We use the book value of equity at the end of 1990 in the denominator of Equation 13 so that we can calculate changes in RI from 1991 (the base year) to 1996 (year 5). The calculation of excess returns (residual income) in Penman (1996) is consistent with Equation 13.

We track the median residual income over time for each of the groups because the distribution of the excess returns is highly skewed. [5] For consistency, we focus on the median buy-and-hold stock return for each group, but we also report the corresponding mean returns. For our tests of significance, we use a Wilcoxon rank-sum test to compare median values for each subgroup of firms to the median values across all firms. We use a parametric difference in means test to compare the mean BHR for each subgroup of firms to the mean BHR across all firms.

Finally, to focus more directly on the relations between future ROE and future stock returns, we look at each panel of data individually (rather than pooling data from several panels). Over any period of time, the stock returns earned by a firm will depend on the firm's ROE over that period (as compared to expectations) and on the expectations embedded in the firm's stock price at the end of the period. P/B and P/E ratios can increase or decrease across all firms over time in response to changes in interest rates or other macroeconomic factors. By looking at each panel of data individually, we attempt to hold these general economic factors constant within each sample. We discuss the results of the 1991 sample in detail in Section B. We compare the results from the 1991 sample to the results from the 1986 and 1981 samples in Section C.

B. 1991 Results

Using data from 1991, we place 3,572 firms into the four groups. We then calculate the excess returns and the buy-and-hold stock return for each of the four groups of firms over the 1991 to 1996 period. [6]

Table III reports the median residual income across all firms, and within the four subgroups of firms, for each year from 1991 to 1996. The table also lists the cumulative buy-and-hold stock returns for each of the five-years from 1992 to 1996. The median five-year BHR for all firms over this period is 70.5%, which is equal to an annual return of approximately 11.3%. The mean five-year BHR for all firms over this period was 132.0%, which is equal to an annual return of approximately 18.3%. As a benchmark, Table III also lists the CRSP value-weighted return over this five-year period of 99.2%.

We note that the residual income amounts and the stock returns in this table include only complete year results for those firms that still existed at the end of a given year. That is, the BHRs for 1996 include only complete five-year stock returns for those firms that were still in the sample at the end of 1996. By focusing only on the surviving firms at the end of each year, we can more directly compare the stock returns of these firms to the operating performance (residual income) that helped to create these returns.

Out of the 3,572 firms with available data for 1991, 2,825 (79.1%) survive and have the necessary data for each of the years 1992 through 1996. The survival rate is highest for Growth Firms (81.6%), followed by Mature Firms (80.7%), Declining Firms (74.7%), and Turnaround Firms (70.9%). Since the survival rate is not constant across the four groups, survivorship bias could affect how we interpret the results. For example, the five-year buy-and-hold stock return for the Turnaround Firm group (median BHR = 111.8%) could be high relative to the Growth Firm group (median five-year BHR = 59.2%), because poorly performing Turnaround Firms are more likely to leave the sample than poorly performing Growth Firms. A closer inspection of Table III shows that the superior stock returns in the Turnaround Firm category start in 1992, before many firms had left the sample, and continue through 1996. It does not appear that the process of poorly performing firms leaving the sample is a complete explanation for the high stoc k returns within the Turnaround Firm category. [7]

Nevertheless, survivorship bias could contribute to the differences between the stock returns across the four groups. However, our goal is not just to compare these return levels across the four groups. Instead, we are mainly interested in how the relation between excess returns and buy-and-hold stock returns differs across the four groups. These relations should be present in subsets of the overall sample (such as the subset of firms that survived), as well as in the overall sample. Over the five-year period, Declining Firms that survived tended to earn a higher stock return than Growth Firms that survived, despite operating performance that was much worse. Why? The return-stages model says that this relation can occur because the expected performance of a Declining Firm is much lower than that of a Growth Firm. Survivorship bias does not affect this aspect of the results.

Tables IV through VI take a closer look at the results for the 1991 sample. These tables list three measures of excess returns (RI). RI Change is calculated for each firm as the excess return in 1996, minus the excess return in 1991. RI Average is a firm's average excess return over the five-year period from 1992 to 1996. RI in 1996 is the excess return earned by a firm in 1996. We describe the results of Tables IV through VI in the following sections.

1. Growth Firms

Table IV compares the performance of the 2,020 Growth Firms that survived the five-year period ending in 1996 to the performance of the overall sample of 2,825 surviving firms. The table lists the results for all Growth Firms, and for two subsets of Growth Firms: firms with a ROE less than 10% in 1991, which includes firms with a P/B ratio greater than one and a negative P/E ratio in 1991; and firms with a ROE greater than 10% in 1991. We use a ROE of 10% to separate the firms into subgroups because a firm's ability to earn excess returns in the future (i.e., ROE [greater than] k) could be related to the firm's historical performance.

If it is to earn a stock return equal to k, the return-stages model says that a Growth Firm must earn positive excess returns in the future and increase the dollar amount of its excess returns. Table IV shows that the median five-year BHR for these firms is 59.2%, which approximates an annual return of 10%. Supporting this BHR, the median Growth Firm has a positive RI in 1996 and a positive RI Change from 1991 to 1996.

Table IV also provides some evidence as to why Growth Firms have lower stock returns than other firms for this period: neither the median RI Change, nor the median RI in 1996 for Growth Firms are significantly larger than those across all firms. Since investors expect Growth Firms to experience more growth in excess returns and higher excess return levels than other firms, it is not surprising that Growth Firms had low BHRs relative to the non-growth firms.

The two Growth Firm subgroups show that there can be important performance differences within the growth firm category. The firms that have higher ROE levels in 1991 also have higher measures of future excess returns (RI Average and RI in 1996). The firms with lower ROE levels in 1991 have a larger improvement in excess returns (RI Change) over the next five years. However, the stock returns for the two groups are similar over this five-year period.

2. Non-Growth Firms

Table V summarizes the performance of Mature Firms, Turnaround Firms, and Declining Firms. This table shows that all of the non-growth firm groups have a higher median BHR than the Growth Firms. In addition, all of the median BHRs are higher than 61% (10% per year).

The Mature Firms experience a decrease in excess returns over the five-year period, but the return-stages model predicts this decline. Even after this decline in excess returns, the median RI in 1996 is positive and still significantly larger than the median RI in 1996 across all firms. This performance appears to exceed expectations (if k = 10%), as evidenced by the median BHR of 103.4%. However, the main point is that the return pattern that will produce a high stock return differs between the Mature Firm and the Growth Firm groups. The median RI Change for Mature Firms is negative, while the median RI Change for Growth Firms is positive. Yet, Mature Firms had a much higher median BHR. Why? The return-stages model says that Mature Firms face much lower expectations than do Growth Firms. The evidence suggests that Mature Firms are more likely than Growth Firms to meet or exceed these expectations over this five-year period.

We divide the results for Turnaround Firms into two subgroups: firms with negative P/E ratios in 1991 and those with positive PIE ratios in 1991. Both groups of Turnaround Firms have median BHRs well above 61% (10% per year). Although these firms earn negative excess returns across the entire period, their performance (on average) still appears to meet or exceed the expectations defined by the return-stages model. As predicted by the model, excess returns for these firms increase (become less negative) and converge toward zero over the 1992 to 1996 period. Again, it is the contrast with Growth Firms that is of interest. The median BHR for Turnaround Firms is higher than 61% even though the median RI Average and the median RI in 1996 are both less than zero. Turnaround Firms earn a higher median BHR than do Growth Firms with a lower level of performance.

The highest median BHR is in the Declining Firm category. At the same time, the median RI Change, RI Average, and RI in 1996 for these firms are all very close to, or only slightly larger than, zero. Overall, these results show that most Declining Firms do not grow or "decline" much over this period. It appears that many of the surviving firms in this category beat the low expectations from 1991, and as a result, have high stock returns in this five-year period.

3. High BHR ([greater than] 61%) Firms

Table VI focuses on those firms that had a BHR of 61% (10% per year) or higher between 1992 and 1996. The return-stages model says that the operating performance required for a high stock return (or any given stock return) will differ across the four groups. Table VI illustrates four of these differences.

First, Growth Firms must realize more growth than other firms to earn a high stock return.

The median RI Change for this subset of Growth Firms over the five-year period is 13.1%, while the median across all firms with a BHR of 6l% or higher is 12.6%. This difference is not significant at a 10% level. Although Growth Firms show the same (or slightly higher) amount of growth as the non-growth firms over this period, the median BHR for Growth Firms (135.0%) is significantly less than the median for all firms (156.1%). Consistent with the return-stages model, Growth Firms realized a lower BHR, in part, because they did not experience hgher growth than the non-growth firms.

Second, Mature Firms can realize a high stock return with less growth than other firms. The median RI Change for Mature Firms over the five-year period, 5.7%, is significantly less than the median across all firms, 12.6%, but the median BHR for Mature Firms is slightly higher than the median across all firms. The high BHR for this group of firms is easily explained. The return-stages model says that the residual income for a Mature Firm should decrease over time. However, residual income increases for most of these Mature Firms. Since these firms appear to have beaten expectations, these firms have a high BHR.

Third, Turnaround Firms can earn a high BHR even if they generate a very small amount of residual income. The median values of RI Average and RI in 1996 are both significantly smaller for Turnaround Firms, than they are for the sample of all firms. However, RI in 1996 is positive for this group of Turnaround Firms. These Turnaround Firms earn a high BHR because they not only eliminate their negative residual income overtime, as predicted by the return-stages model, but they also earn positive residual income by 1996.

Finally, Table VI shows that Declining Firms can realize a high BHR with a much lower level of performance than other firms. The median values of RI Change, RI Average, and RI in 1996 are all significantly smaller for Declining Firms, than they are for the sample of all firms. Yet, Declining Firms have a higher median BHR than all groups except for Turnaround Firms. The evidence in this table suggests that the reason is that Declining Firms easily beat expectations. The return-stages model says that the excess returns of a Declining Firm will become more negative overtime. However, for these Declining Firms, all three measures of residual income are positive.

4. Alternative P/E Ratios

The results in Tables III through VI correspond to the case in which we use the year-end stock price in the calculation of the P/B and P/E ratios, and last year's earnings to calculate the P/E ratio. This method has the advantage of simplicity, and it is consistent with the way in which these ratios are frequently calculated in practice. At the same time, we could calculate these two ratios in other reasonable ways. For example, the return-stages model defines the P/E ratio by using an estimate of next year's earnings, while previous empirical studies (such as Penman, 1996) calculate the P/B and P/E ratios by using the stock price 90 days after year-end.

To determine how our method affects the results, we repeat the tests for the 1991 sample, using two alternative methods of calculating the P/B and P/E ratios. First, we use a firm's stock price 90 days after the end of its fiscal year to calculate the P/E and P/B ratios, and then we start the calculation of a firm's BHR 90 days after the end of its fiscal year. Table VII labels this case as [P.sub.+90]/[E.sub.0]. Second, we use an estimate of next year's earnings, which we obtain from IBES, in the calculation of a firm's P/E ratio. Table VII refers to this case as [P.sub.0]/[E.sub.1]. Table VII reports the results of these two additional cases for the 1991 sample and compares these results to the base case ([P.sub.0]/[E.sub.0]). This table lists, for each of the three cases, the number of firms within each group, RI Change, RI Average, RI in 1996, five-year median BHR, and five-year mean BHR.

Table VII shows that the results for the base case, [P.sub.0]/[E.sub.0], and the case in which we measure the stock price after year-end, [P.sub.+90]/[E.sub.0], are similar. The main difference between these two cases is that when we measure the five-year BHR starting at year-end, it is slightly larger than when we measure it starting 90 days later. The similarity of these results is not surprising as the two methods place over 89% of the firms in the same category in the base year, and since the Spearman rank correlation coefficient between the two five-year BHR measures is 0.926.

When we use next year's estimated earnings to measure the P/E ratio, [P.sub.0],/[E.sub.1], the results show more differences relative to the base case. Even so, the same results remain intact. The median Growth Firm appears to realize enough excess return growth during this period to earn a BHR in excess of 6l%. At the same time, the median Growth Firm does not realize any more growth in excess returns than does the median non-growth firm. Not surprisingly, the median non-growth firm had a higher BHR than the median Growth Firm. In this case, the group of firms with the highest median BHR is the Declining Firm group. This BHR makes sense as the excess returns for the median firm in this group does not decline, as the return-stages model predicts. Instead, the median RI Change for this group is 6.1%.

5. Alternative Expectation Measures

The return-stages model shows how to use a firm's P/B and P/E ratios to gain a broad understanding of the expectations facing the firm. Expectations can also be measured using analyst forecasts. To investigate whether the expectations derived from a firm's P/B and P/E ratios are consistent with analyst forecasts, we obtained earnings estimates from IBES for the year ending in 1994, for the firms included in the 1991 sample. Since three-year forecasts are not available for all firms, this requirement reduced the sample size to 840 firms. (Using five-year forecasts reduced the sample size even further, to 249 firms.) For each firm within this new sample, we calculate the expected growth (EG) in earnings per share (EPS) from 1991 to 1994, measured relative to the sales per share (SPS) in 1991, using the following formula:

[EG.sub.91] = [EPS.sub.94] - [EPS.sub.91] / [SPS.sub.91] (14)

The results of this exercise support our assertion that Growth Firms and Turnaround Firms are expected to realize more future growth than Mature Firms and Declining Firms. The 728 Growth Firms in this sample have a median expected growth, calculated using Equation 14, of 4.7%, while the 68 Turnaround Firms have a median expected growth of 6.3%. In contrast, the 31 Mature Firms and the 13 Declining Firms have median expected growth amounts of 1.9% and 1.4%, respectively.

C. Results for 1986 and 1981 Samples

In Tables VIII and IX, we compare selected results from the 1991 sample to those from the 1986 and 1981 samples. For the three samples, each table lists the number of firms within each group, RI Change, RI Average, RI in Year 5, the median five-year BHR, and the mean five-year BHR. In Table VIII, we measure the P/B and P/E ratios by using the stock price at year-end, and start the stock return series immediately after year-end. In Table IX, we measure the P/B and P/E ratios using the stock price 90 days after year-end, and begin the stock return series on that date. The results in these tables show that the ability of the return-stages model to explain the relation between a firm's excess (operating) returns and its stock returns is not limited to the 1991-96 period.

1. Table VIII: No Return Lag

The results in Table VIII show that the 1981, 1986, and 1991 samples differ in three ways. First, the median BHR across all firms is not constant across the samples, ranging from a low of 22.1% in the 1986 sample to a high of 120.7% in the 1981 sample. Second, the relative performance of the four groups does not stay constant across the three samples. For example, the Declining Firm category has the highest median five-year BHR in the 1981 and 1991 samples, but only the third highest median five-year BHR in the 1986 sample. Third, the relative percentages of firms within each of the four groups changes over time. The Declining Firm group is less than 5% of the entire sample in 1986 and 1991, but over 30% of the sample in 1981. The Growth Firm Group is over 70% of the sample in 1986 and 1991, but only 36% of the sample in 1981. Because of these differences, the three sample periods show how the return-stages model can perform over a wide range of economic conditions.

To interpret the results in this table, we construct a simple test of the model. For each of the four groups, the return-stages model defines the future excess returns that a firm must realize to earn an annual stock return in excess of k (i.e., 10%). Our hypothesis is that in those sample periods in which the median excess return performance equals or exceeds the benchmark performance defined by the return-stages model, the median five-year BHR will be equal to or will exceed 61% (10% per year).

According to the return-stages model, investors expect Growth Firms to earn positive excess returns in the future and expect these excess returns to increase over time. The only case in which the median Growth Firm appears to meet these expectations is the 1991 sample. This period is also the only one in which the median Growth Firm shows a five-year BHR close to 61%.

Growth Firms have their worst excess return performance in the 1981 period. Surprisingly, the median five-year BHR (31.1%) during this period is slightly higher than the median five-year BHR in the 1986 sample (21.8%), when the Growth Firms have a slightly better median excess return performance. However, the median five-year BHR for growth firms, relative to the median five-year BHR across all firms, is much lower in the 1981 period (25.8%=31.1/120.7) than in the 1986 period (98.6%=21.8/22.1). The relative BHR of Growth Firms (compared to the BHR of all firms) is consistent with the poor excess return performance of these firms during the 1982 to 1986 period.

The return-stages model implies that Mature Firms will earn positive excess returns in the future, but that these excess returns will decline over time. The median excess returns for Mature Firms fit this broad pattern in all three samples. The median five-year BHR for growth firms exceeds 61% in both the 1991 and 1981 samples, but in the 1986 sample, the median five-year BHR is only 38.7% (a 6.8% annual return). However, the five-year BHR in the 1986 sample does make sense for two reasons. First, all three measures of RI are lower in the 1986 sample for Mature Firms than they are in the 1991 or 1981 samples. Second, Mature Firms have the highest median five-year BHR in the 1986 sample. Mature Firms appear to come the closest to meeting expectations during this period and, therefore, earn the highest median stock return.

According to the return-stages model, Turnaround Firms are expected to earn negative excess returns in the future, but these negative excess returns are expected to converge toward zero over time. In the two sample periods in which the median excess returns for Turnaround Firms fit this description, 1981 and 1991, the median five-year BHR for Turnaround Firms is higher than 61%. In the 1986 sample, the median RI in Year 5 is still -8.1% (compared to -1.1% in 1991 and -2.5% in 1981), indicating that by year 5, the median firm still has not eliminated the majority of its negative excess returns. Not surprisingly, the median five-year BHR for Turnaround Firms in the 1986 sample is only 9.4%.

Finally, the return-stages model predicts that excess returns for Declining Firms will decrease overtime, and that these firms will earn negative excess returns in the future. In the 1991 and 1981 sample periods, the median RI Change for these firms is close to zero, indicating that excess returns did not decrease. Further, the median RI in Year 5 is positive for both of these sample periods. Because the performance of the 1991 and 1981 Declining Firms appears to exceed expectations, the median five-year BHR was much larger than 61% in each of these samples. In 1986, the only sample where the performance of the median Declining Firm actually declined, the median five-year BHR was less than 61%.

2. Table IX: Three-Month Return Lag

The results in Table IX also support the return-stages model. Almost all of the items that were significant in Table VIII are also significant (in the same direction) in Table IX. This outcome is not surprising for two reasons. First, the two methods of calculating the P/B and PIE ratios place most firms into the same groups in each table. The overlap between the two samples is largest in 1991, when 89.4% of the firms are in the same group in each table. In the 1986 and 1981 samples, 88.5% and 85.4%, respectively, of the firms are in the same groups. Second, the two stock return series rank the firms in much the same order. The Spearman rank correlations between the two sets of five-year BHRs were 0.926, 0.939, and 0.951, in the 1991, 1986, and 1981 samples, respectively.

The main difference between the two tables is that the BHRs for the 1981 sample are much higher in Table IX than in Table VIII. The median BHR for all firms in the 1981 sample is over 30% higher in Table IX (159.2%) than it is in Table VIII (120.7%). The median BHR for Growth Firms in the 1981 sample is twice as high in Table IX (62.9%) as it is in Table VIII (31.1%). These differences appear to be caused in part by the strong performance of the market as a whole in the first three months of 1987. The Dow Jones Industrial Average lost almost 6% of its value in the first three months of 1982 (months 1 to 3 for firms with a December 31, 1981 year-end), but gained over 20% in value in the first three months of 1987 (months 61 to 63 for these firms). In Table VIII, we exclude the returns for the first three months of 1987 from the BHRs for the firms in the 1981 sample (for firms with a December 31 year-end), but include these returns in the BHRs in Table IX. [8]

Because of the large market return in early 1987, it is much harder to interpret the results for the Growth Firms in Table IX. The 1981 Growth Firms have a higher five-year BHR than either the 1986 or 1991 Growth Firms, but their operating performance is much worse. This result appears to conflict with the return-stages model. However, we note that the median BHR of Growth Firms relative to the median BHR across all firms was much lower in 1981 (39.5%=62.9/159.2), than in 1986 (92.3%=13.1/14.2) or 1991 (84.3%=55.5/65.8). We also note that the Growth Firms appear to perform the worst, relative to the expectations, of all groups of firms within the 1981 sample. Thus, Growth Firms had the lowest median (and mean) BHR in the 1981 sample. After controlling for the market-wide price movements in this way, the results for Growth Firms in Table IX support the return-stages model.

The differences between the results in Tables VIII and IX do not suggest that either method of calculating returns creates systematic biases in this application. Keep in mind, the main differences between the two tables--the BHRs for the 1981 sample--are caused to a greater extent by events in months 61 to 63 after year-end, than by events in months 1 to 3 (the period in which the fourth quarter results from the base year are announced). The differences between these two tables do illustrate when we would expect the return-stages model to be less effective in explaining the link between stock returns and operating performance. In periods where market-wide price movements are large relative to changes in operating performance (i.e., April 1982 to March 1987), the return-stages model will be less effective in explaining this link.

D. Empirical Discussion

The three samples used in this empirical example show that future excess return patterns do differ across the four groups. This finding is similar to those in Fairfield (1994) and Penman (1996). These samples also show that the type of future operating performance required to earn a stock return in excess of 10% per year differs across the four groups, as predicted by the return-stages model. By linking a firm's stock returns to its operating performance, our results extend the work in Fairfield (1994) and Penman (1996). Because we control for both P/B and P/E ratios, our results also extend those from Fama and French (1995).

Some of our results are consistent with Fama and French (1995), but the results of their study and ours differ in important ways. Like our study, Fama and French (1995) show that a firm's current P/B ratio is related to its future operating performance. For example, they show (in their Figure 1) that the ratio of earnings to book equity (ROE) decreases over a five-year period (after measuring the P/B ratio) for firms with high P/B ratios, and increases over a five-year period for firms with low P/B ratios. But after showing this link between P/B ratios and earnings, Fama and French (1995) cannot document how P/B ratios and subsequent operating performance (ROE) combine to determine a firm's stock return. In contrast, we find that stock returns do respond to how a firm's future excess returns compare to the expectations defined by a firm's P/B and P/E ratios.

There are three key differences between our study and that of Fama and French (1995) that could explain why the two studies reach these different conclusions. First, we use P/B and P/E ratios to separate firms into four groups, while Fama and French (1995) use P/B ratios to separate firms into two groups. This difference is important, because groups of high and low P/B firms both combine two distinct types of firms. High P/B firms include both Growth Firms and Mature Firms, while low P/B firms include both Turnaround Firms and Declining Firms.

Therefore, the relation between future excess returns and future stock returns is not constant within either the high P/B or the low P/B group of firms. For example, a Growth Firm must realize a substantial amount of growth in excess returns to realize a "high" stock return (i.e., in excess of 10%), but a Mature Firm can realize a high stock return even if its excess returns decrease overtime. Since Fama and French (1995) use only the P/B ratio to classify firms, it is not surprising that their study cannot identify these more subtle, but important, relations within the P/B groups.

A second difference between the two studies is that we look at three panels of data individually, while Fama and French (1995) pool data over a twenty-five year period. In our study, looking at each panel of data individually has three advantages. First, during any given period, a firm's stock return will depend on the firm's operating performance during that period and on any changes in future expectations that occur during that period. During some periods, P/E and P/B ratios can increase (or decrease) across all stocks, creating higher (or lower) stock returns. By looking at each panel of data individually, we can hold these market wide movements constant across each sample. Second, as Table VIII shows, many sample characteristics change over time: the total number of firms, the percentage of firms in each category, and the relative performance (excess returns and stock returns) of each group. If we combine many panels, we give more weight to the experience of those panels with the most observations. Final ly, within some panels, a specific group of firms might do better than expected, and within other panels the same group of firms might do worse than expected. When panels are combined, these periods of over- and under-performance tend to cancel out. By looking at the panels of data separately, we can more readily identify the periods of unusual performance. The results of our tests show that stock returns do respond to unusual periods of excess return performance in ways that are predicted by the return stages model.

A third difference between our study and that of Fama and French (1995) is that they measure future performance using ROE, but we use a measure of excess returns (ROE less k, adjusted for changes in firm scale). To illustrate the difference between these two measures, we consider a firm with a ROE on new and prior investments of 15% and a cost of equity of 10%. As this firm grows by investing in new projects, ROE will remain constant while excess returns, in dollars, will grow. In the 1991 sample, the two measures do give different pictures of how much the Growth Firms actually grow. The median five-year growth in ROE was 0.7%, but the median five-year growth in excess returns (Table IV) was 3.7%.

Since we analyze only three panels of data, we do not present our results as a rigorous test of the model. Nevertheless, our results show that the return-stages model can explain the relations between operating performance and stock returns over three distinct periods that cover a total of 16 calendar years (1981 through 1996). To further validate the model, future studies could analyze additional periods, or attempt to measure excess returns in a more precise manner. Furthermore, by limiting our tests to four groups, we might be masking important differences within the four groups. Future research could attempt to identify these differences, and relate these differences to the model.

IV. Conclusions

The return-stages model shows how to gain a broad understanding of the expectations facing a firm from its P/B and P/E ratios. Because the model allows the returns earned by a firm to change over the course of its life, the model can be applied to both growth firms and to a wide range of non-growth firms. The insights offered by the return-stages model can be useful in a number of corporate finance applications and within many research projects.

For example, when a firm is evaluating a potential merger or acquisition, the firm must decide the appropriate price to pay, or to accept. The return-stages model can help managers gain a first-cut estimate of the type of future performance an acquired unit must produce (or the level of synergies that must be realized) to justify the purchase price. By doing so, the return-stages model can help guide a more detailed discounted cash flow analysis of a transaction.

As another practical application, many firms use stock options to motivate the future performance of employees and managers. The insights managers can gain from using the return-stages model can help them decide when to issue options and can help employees decide when to accept options. If the expectations implied by the firm's P/B and P/E ratios are too high (relative to the market opportunities available to the firm), then the employees might be better off with more compensation in cash. But if the expectations facing a firm are too low, the firm might want to delay issuing stock options. Shareholders of a firm might not benefit if employee stock options are likely to pay off, regardless of how well the employees perform.

The return-stages model also suggests that P/B and P/E ratios can play expanded roles within research studies. The P/B and P/E ratios have been used individually to classify firms into groups within empirical capital structure studies. McConnell and Servaes (1995) use P/E ratios to group firms within a study of the relation between growth and leverage. Jegadeesh (2000) uses P/B ratios to group firms within a study of the long-term performance of firms that issue seasoned equity. Our analysis in this paper suggests that further insights may be gained by using the two ratios together to sort firms.

In summary, the return-stages model can be used in both practical and research applications. In each case, the economic intuition offered by the model can help guide more detailed tests and procedures. We thank Rocky Higgins, Ken Kopecky Eric Press, Jack Ritchie, Jon Scott, and two referees far comments on previous drafts of this paper. We also gratefully acknowledge the contribution of IBES International Inc. for providing earnings per share forecast data, available through the Institutional Brokers Estimate System. This data has been provided as part of a broad academic program to encourage earnings expectations research.

(*.) Morris G. Danielson is an Assistant Professor of Finance at Temple University Thomas D. Dowdell is a Ph.D. student in Accounting at Temple University

(1.) In a related model, Leibowitz (1998) modifies the standard two-phase model to allow a firm's ROE to decline to its cost of equity after the end of the firm's growth phase. The return-stages model differs from the model in Leibowitz (1998) in that it allows a firm's terminal ROE to be either higher or lower than the ROE in prior periods. For this reason, the return-stages model can be applied more directly to non-growth firms.

(2.) We note that the variables [R.sub.E], [R.sub.N], [R.sub.T], and I are all economic, rather than accounting, measures. The model assumes that the firm's investment base does not decline (depreciate) over time because each year the firm reinvests an amount equal to the economic depreciation of its assets to maintain the productive capacity of those assets. Because of this assumption, the economic earnings are equal to the cash flows available for growth-related investments (i.e., the economic depreciation and the maintenance portion of the capital expenditures offset each other). The standard two-phase model also uses this assumption.

(3.) We write the model in terms of equity returns and equity investment so that it mirrors the standard two-phase model. The model can also be written in terms of asset returns and total investment. In this ease, [R.sub.E], [R.sub.N], and [R.sub.T] measure return on assets, not return on equity; the investment, I, will measure the total investment, rather than the equity investment, in the firm.

(4.) Using Equation 12, we can recalculate Firm B.1's P/E ratio as follows: The second term in Equation 12, the value expected to be created by new investment, is 1.29 [= (1.1284/1.1) [10]]. The third term in Equation 12, the ratio [R.sub.[tau]]/[WR.sub.E.N], is 0.646 (= 10%/15.49%). Therefore, the P/E ratio is: 8.33 = 10 x 1.29 x 0.646. We note that a P/E ratio less than 1/k implies that ROE is expected to decline. This P/E ratio does not imply that the dollar amount of the firm's earnings will necessarily decline. If we assume that [I.sub.1=0] for Firm B.1 is $10, then the earnings during year t = 1 will be $3. Using Equation 5, we can estimate [I.sub.t=10] for firm B.1 as $64.80. Assuming that the terminal ROE will be 10%, we can use Equation 6 to calculate the terminal earnings of $6.48.

(5.) Fairfield (1994) and Penman (1996) also report median values for this reason.

(6.) We place firms with a negative P/E ratio into the Growth Firm category if the firm's P/B ratio is greater than 1 (461 firms in 1991), and into the Turnaround Firm category if the firm's P/B ratio is less than 1 (310 firms in 1991). As do Fama and French (1995), we exclude from our tests all firms with a negative book value of equity in the base year (495 firms in 1991).

(7.) Our review of CRSP deletion codes shows little difference between the four groups of firms as to why the firms left the sample. For example, within the Growth Firm category, 56% of the firms leaving the sample either merged with, or were taken over by, other firms. Within the Mature Firm, Turnaround Firm, and Declining Firm categories, 63%, 54%, and 57%, respectively, of the firms left the sample for this reason.

(8.) The strong performance of the market in early 1987 also affects the comparison of the 1986 sample results between Tables VIII and IX, but to a lesser extent. For the 1986 sample, the Dow Jones Industrial Average gained over 20% in value in the first three months of 1987 (months 1 to 3 for firms with a December 31, 1986 yearend), but gained only 2.1% in the first three months of 1992 (months 61 to 63 for these firms).

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Table I. Stock Price Calculation Example This table provides a detailed calculation of a stock price using the return-stages model. In this example, we assume that [R.sub.E], the ROE from the firm's investment base at t = 0, is 20%; [R.sub.N], the ROE from new investments, is 15%; k, the required return on equity, is 10%; [tau], the length of the firm's growth phase, is five years; and [I.sub.t=0], the investment base at t = 0, is $100. We calculate the stock price for four assumed values of the terminal ROE, [R.sub.T] Panel A shows how the investment base in the firm increases from t = 0 to t = 5. In Panel A, each number in the Earnings From Growth column can be calculated as: (Beginning Investment - 100) x [R.sub.N]; where [R.sub.N] is equal to 15%. Panel B shows the stock price calculation. In Panel B, [E.sub.t=6] is defined as [R.sub.T] x [I.sub.t=5], see Equation 6. Each stock price, [P.sub.t=0], can be calculated using Equation 7. Each stock price is also the present value of a perpetual annuity in the amount [E.sub.t=6], valued as of t = 0. Panel A. Calculation of Investment Base at t = 6 Beginning Earnings Total Time Investment [R.sub.E] X [I.sub.t=0] From Growth Earnings 1 $100.00 $20.00 $ 0.00 $20.00 2 120.00 20.00 3.00 23.00 3 143.00 20.00 6.45 26.45 4 169.45 20.00 10.42 30.42 5 199.87 20.00 14.98 34.98 Ending Time Investment 1 $120.00 2 143.00 3 169.45 4 199.87 5 234.85 Panel B. Stock Price Calculation Variable [R.sub.T] = 10% [R.sub.T] = 15% [R.sub.T] = 17.129% [I.sub.t=5] $234.85 $234.85 $234.85 [E.sub.t=6] 23.49 35.23 40.23 [P.sub.t=0] 145.82 218.73 249.78 Variable [R.sub.T] = 20% [I.sub.t=5] $234.85 [E.sub.t=6] 46.97 [P.sub.t=0] 291.64 Table II. ROE Patterns Within Four Groups of Firms This table illustrates some possible ROE patterns within four groups of firms: Growth Firms, Mature Firms, Turnaround Firms, and Declining Firms. For all cases, k = 10%. These return patterns are described by [R.sub.E], the ROE from the firm's investment base at t = 0; [R.sub.N], the ROE from new investments during the firm's growth phase; and [R.sub.T], the ROE during the firm's terminal period, which will begin in [tau] years. The table also ists the terminal ROE value that will allow the returns [R.sub.E] and [R.sub.N] to continue forever: [WR.sub.E,N]. [WR.sub.E,N] can be recalculated using Equation 8. Each P/B and P/E ratio can be recalculated using Equations II and 12, respectively. Firm P/B P/E [tau] Panel A. Growth Firms (P/B [greater than] 1; P/E [greater than] 1/k) A.1 3 20 15.6 A.2 3 20 5.0 Panel B. Mature Firms (P/B [greater than] 1; P/E [less than] 1/k) B.1 2.5 8.33 10 B.2 2.5 8.33 10 Panel C. Turnaround Firms (P/B [less than] 1; P/E [greater than] 1/k) C.1 0.75 15 5 C.2 0.75 15 10 Panel D. Declining Firms (P/B [less than] 1; P/E [less than] 1/k) D.1 0.75 8.33 5 D.2 0.75 8.33 5 Firm [R.sub.E] [R.sub.N] Panel A. Growth Firms (P/B [greater than] 1; P/E [greater than] 1/k) A.1 15% 15.00% A.2 15% 26.36% Panel B. Mature Firms (P/B [greater than] 1; P/E [less than] 1/k) B.1 30% 12.84% B.2 30% 8.01% Panel C. Turnaround Firms (P/B [less than] 1; P/E [greater than] 1/k) C.1 5% 10.00% C.2 5% 10.00% Panel D. Declining Firms (P/B [less than] 1; P/E [less than] 1/k) D.1 9% 10.00% D.2 9% 6.06% Firm [R.sub.T] [WR.sub.E,N] Panel A. Growth Firms (P/B [greater than] 1; P/E [greater than] 1/k) A.1 15.00% 15.00% A.2 21.34% 21.34% Panel B. Mature Firms (P/B [greater than] 1; P/E [less than] 1/k) B.1 10.00% 15.49% B.2 12.12% 12.12% Panel C. Turnaround Firms (P/B [less than] 1; P/E [greater than] 1/k) C.1 9.25% 6.17% C.2 10.83% 7.22% Panel D. Declining Firms (P/B [less than] 1; P/E [less than] 1/k) D.1 7.80% 9.35% D.2 8.01% 8.01% Firm [R.sub.T]-[WR.sub.E,N] Panel A. Growth Firms (P/B [greater than] 1; P/E [greater than] 1/k) A.1 0 A.2 0 Panel B. Mature Firms (P/B [greater than] 1; P/E [less than] 1/k) B.1 -5.49% B.2 0 Panel C. Turnaround Firms (P/B [less than] 1; P/E [greater than] 1/k) C.1 3.08% C.2 3.61% Panel D. Declining Firms (P/B [less than] 1; P/E [less than] 1/k) D.1 -1.55% D.2 0 Table III. Descriptive Statistics by Year (1991 Base Year) This table lists the number of firms, the median residual income (RI), and the median and mean buy-and-hold return (BHR) within each of the four groups of firms during the 1991 to 1996 period. We define a firm's RI during each year as a firm's return on equity in excess of 10%, adjusted by the change in the scale of the firm since 1991, as shown in Equation 13. As a benchmark, we also list the CRSP value-weighted BHR over this period. Group 1991 1992 1993 1994 1995 All Firms Number of Firms 3,572 3,464 3,365 3,151 2,986 Median RI -0.021 -0.011 -0.004 0.013 0.025 Median BHR (%) 9.2 25.4 23.8 51.1 Mean BHR(%) 19.4 53.5 52.7 100.7 Growth Firms Number of Firms 2,475 2,404 2,336 2,220 2,110 Median RI 0.007 0.017 0.013 0.031 0.035 Median BHR (%) 6.6 18.5 15.0 41.8 Mean BHR (%) 12.8 37.5 35.7 77.6 Mature Firms Number of Firms 218 213 211 197 190 Median RI 0.118 0.071 0.078 0.048 0.076 Median BHR (%) 20.3 37.0 35.9 68.1 Mean BHR (%) 23.4 54.7 53.5 95.5 Turnaround Firms Number of Firms 721 692 644 598 557 Median RI -0.124 -0.086 -0.059 -0.033 -0.019 Median BHR (%) 16.5 47.2 52.7 87.8 Mean BHR (%) 35.9 95.7 101.4 175.5 Declining Firms Number of Firms 158 155 144 136 129 Median RI 0.008 -0.015 -0.018 -0.007 0.003 Median BHR (%) 20.5 58.0 61.2 95.6 Mean BHR (%) 42.9 123.2 115.1 162.8 CRSP Value-Weighted BHR 9.0 21.5 20.6 65.2 Group 1996 All Firms Number of Firms 2,825 Median RI 0.035 Median BHR (%) 70.5 Mean BHR(%) 132.0 Growth Firms Number of Firms 2,020 Median RI 0.045 Median BHR (%) 59.2 Mean BHR (%) 100.6 Mature Firms Number of Firms 176 Median RI 0.089 Median BHR (%) 103.4 Mean BHR (%) 138.9 Turnaround Firms Number of Firms 511 Median RI -0.011 Median BHR (%) 111.8 Mean BHR (%) 224.5 Declining Firms Number of Firms 118 Median RI 0.016 Median BHR (%) 138.3 Mean BHR (%) 257.6 CRSP Value-Weighted BHR 99.2 Table IV. The Performance of Growth Firms (1991 Base Year) This table analyzes the performance of two sets of Growth Firms from the 1991 (base year) sample. The first set of Growth Firms includes firms that had a ROE less than 10% in 1991. The second set of Growth Firms includes firms that had a ROE greater than 10% in 1991. For each set of firms, the table lists the median change in residual income (RI) from 1991 to 1996 (RI Change), the median RI over the 1992-1996 period (RI Average), the median RI in 1996 (RI in 1996), and the median and mean buy-and-hold return (BHR) over the 1992-1996 period. We calculate RI using Equation 13. We compare each median value to the median value across all firms using a Wilcoxon rank-sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test, p-values (two-tailed tests) are listed in parentheses. RI Change: RI Average: RI in # of Median Median 1996: Group Firms (1991-96) (1992-96) Median All Firms 2,825 0.046 0.006 0.035 All Growth Firms 2,020 0.037 0.023 0.045 (0.179) (0.006) (0.239) Growth Firms 874 0.089 -0.041 -0.007 ROE (91) [less than]10% (0.000) (0.000) (0.000) Growth Firms 1,146 0.011 0.060 0.077 ROE (91) [greater than] 10% (0.000) (0.000) (0.000) BHR BHR Group (Median, in %) (Mean, in %) All Firms 70.5 132.0 All Growth Firms 59.2 100.6 (0.000) (0.000) Growth Firms 59.7 107.1 ROE (91) [less than]10% (0.000) (0.008) Growth Firms 59.0 95.6 ROE (91) [greater than] 10% (0.001) (0.000) Table V. The Performance of Non-Growth Firms (1991 Base Year) This table analyzes the performance of Mature Firms, Turnaround Firms, and Declining Firms from the 1991 (base year) sample. For each set of firms, the table shows the median change in residual income (RI) from 1991 to 1996 (RI Change), the median RI over the 1992-1996 period (RI Average), the median RI in 1996 (RI in 1996), and the median and mean buy-and-hold return (BHR) over the 1992-1996 period. We calculate RI using Equation 13. We compare each median value to the median value across all firms using a Wilcoxon rank-sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test. p-values (two-tailed tests) are listed in parentheses. RI Change: RI Average: RI in # of Median Median 1996: Group Firms (1991-96) (1992-96) Median All Firms 2,825 0.046 0.006 0.035 Mature Firms 176 -0.046 0.069 0.089 (0.000) (0.000) (0.000) All Turnaround Firms 511 0.102 -0.055 -0.011 (0.000) (0.000) (0.000) P/E (91) [less than] 0 256 0.199 -0.066 -0.007 (0.000) (0.000) (0.001) P/E (91) [greater than] 0 255 0.046 -0.044 -0.013 (0.937) (0.000) (0.001) Declining Firms 118 0.004 -0.009 0.016 (0.011) (0.282) (0.377) BHR BHR Group (Median, in %) (Mean, in %) All Firms 70.5 132.0 Mature Firms 103.4 138.9 (0.004) (0.643) All Turnaround Firms 111.8 224.5 (0.000) (0.000) P/E (91) [less than] 0 126.1 267.0 (0.000) (0.000) P/E (91) [greater than] 0 103.8 181.8 (0.000) (0.010) Declining Firms 138.3 257.6 (0.000) (0.000) Table VI. Buy-and-Hold Returns in Excess of 10% (1991 Base Year) This table analyzes the performance of those firms that earn an annual buy-and-hold return (BHR) in excess of 10% (cumulative BHR of 0.61) over the 1992 to 1996 period. For each set of firms, the table lists the median change in residual income (RI) from 1991 to 1996 (RI Change), the median RI over the 1992-1996 period (RI Average), the median RI in 1996 (RI in 1996), and the median and mean BHR over the 1992-1996 period. We calculate RI using Equation 13. We compare each median value to the median value across all firms in a Wilcoxon rank- sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test, p-values (two-tailed tests) are listed in parentheses. RI Average: RI Average: RI in # of Median Median 1996: Group Firms (1991-96) (1992-96) Median All Firms 1,512 0.126 0.054 0.107 Growth Firms 991 0.131 0.079 0.125 (0.285) (0.000) (0.001) Mature Firms 113 0.057 0.143 0.164 (0.000) (0.000) (0.001) Turnaround Firms 323 0.167 -0.021 0.035 (0.015) (0.000) (0.000) Declining Firms 85 0.029 0.016 0.037 (0.000) (0.005) (0.000) BHR BHR Group (Median, in %) (Mean, in %) All Firms 156.1 252.6 Growth Firms 135.0 214.2 (0.000) (0.001) Mature Firms 180.8 218.1 (0.493) (0.083) Turnaround Firms 226.8 355.1 (0.000) (0.000) Declining Firms 190.1 357.0 (0.003) (0.084) Table VII. Results Using Alternative P/E Ratio Definitions This table compares the 1991 results using three alternative P/E ratios. [P.sub.0]/[E.sub.0] measures the stock price and starts the buy-and-hold return (BHR) calculation at the end of the base year. [P.sub.+90]/[E.sub.0] measures the stock price and starts the BHR calculation 90 days after the end of the base year. [P.sub.0]/[E.sub.1] uses an estimate of next year's earnings in the P/E ratio. For each sample, the table lists the median change in residual income (RI) from 1991 to 1996 (RI Change), the median RI over the 1992-1996 period (RI Average), the median RI in 1996 (RI in 1996), and the median and mean buy-and-hold return (BHR) over the 1992-1996 period. We calculate RI using Equation 13. We compare each median value to the median value across all firms using a Wilcoxon rank-sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test. [P.sub.0]/[E.sub.0] [P.sub.+90]/[E.sub.0] All Firms Number of Firms 2,825 2,838 RI Change: Median 0.046 0.046 RI Average: Median 0.006 0.006 RI in 1996: Median 0.035 0.035 Median BHR (%) 70.5 65.8 Mean BHR (%) 132.0 119.7 Growth Firms Number of Firms 2,020 2,101 RI Change: Median 0.037 0.039 RI Average: Median 0.023 *** 0.023 *** RI in 1996: Median 0.045 0.045 Median BHR (%) 59.2 *** 55.5 *** Mean BHR (%) 100.6 *** 96.4 *** Mature Firms Number of Firms 176 160 RI Change: Median -0.046 *** -0.037 *** RI Average: Median 0.069 *** 0.062 *** RI in 1996: Median 0.089 *** 0.093 *** Median BHR (%) 103.4 *** 106.3 *** Mean BHR (%) 138.9 128.7 Turnaround Firms Number of Firms 511 473 RI Change: Median 0.102 *** 0.101 *** RI Average: Median -0.055 *** -0.059 *** RI in 1996: Median -0.011 *** -0.013 *** Median BHR (%) 111.8 *** 95.9 *** Mean BHR (%) 224.5 *** 202.1 *** Declining Firms Number of Firms 118 104 RI Change: Median 0.004 ** -0.019 *** RI Average: Median -0.009 -0.031 *** RI in 1996: Median 0.016 -0.001 Median BHR (%) 138.3 *** 115.7 *** Mean BHR (%) 257.6 *** 200.2 ** [P.sub.0]/[E.sub.1] All Firms Number of Firms 1,579 RI Change: Median 0.052 RI Average: Median 0.028 RI in 1996: Median 0.060 Median BHR (%) 78.4 Mean BHR (%) 125.8 Growth Firms Number of Firms 1,125 RI Change: Median 0.050 RI Average: Median 0.041 *** RI in 1996: Median 0.072 Median BHR (%) 68.3 ** Mean BHR (%) 112.1 Mature Firms Number of Firms 220 RI Change: Median 0.027 *** RI Average: Median 0.024 RI in 1996: Median 0.060 Median BHR (%) 96.6 Mean BHR (%) 124.1 Turnaround Firms Number of Firms 123 RI Change: Median 0.122 *** RI Average: Median -0.046 *** RI in 1996: Median -0.001 *** Median BHR (%) 109.2 Mean BHR (%) 187.2 *** Declining Firms Number of Firms 111 RI Change: Median 0.061 RI Average: Median -0.021 *** RI in 1996: Median 0.031 ** Median BHR (%) 136.8 *** Mean BHR (%) 201.1 *** (***)Significant at the 0.01 level. (**)Significant at the 0.05 level. Table VIII. Comparison of 1991, 1986, and 1981 Samples This table compares the performance of the 1991 sample to the performance of samples with base years in 1986 and 1981. Within each sample, we measure the P/E ratio using the year-end stock price, and we start the buy-and-hold return (BHR) calculation at year-end. For each sample, the table lists the median change in residual income (RI) from year 0 to year 5 (RI Change), the median RI over years 1 to 5 (RI Average), the median RI in year 5 (RI in Year 5), and the median and mean BHR over a five-year period following the base year. We calculate RI using Equation 13. We compare each median value to the median value across all firms using a Wilcoxon rank-sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test. 1991 1986 1981 All Firms Number of Firms 2,825 2,946 2,080 RI Change: Median 0.046 -0.014 -0.020 RI Average: Median 0.006 -0.015 0.013 RI in Year 5: Median 0.035 -0.021 0.010 Median BHR (%) 70.5 22.1 120.7 Mean BHR (%) 132.0 62.2 177.3 Growth Firms Number of Firms 2,020 2,199 745 RI Change: Median 0.037 -0.007 -0.079 *** RI Average: Median 0.023 *** -0.001 -0.014 *** RI in Year 5: Median 0.045 -0.011 -0.048 *** Median BHR (%) 59.2 *** 21.8 31.1 *** Mean BHR (%) 100.6 *** 61.2 88.0 *** Mature Firms Number of Firms 176 283 442 RI Change: Median -0.046 *** -0.105 *** -0.075 *** RI Average: Median 0.069 *** 0.035 *** 0.052 *** RI in Year 5: Median 0.089 *** 0.024 ** 0.028 Median BHR (%) 103.4 *** 38.7 114.1 Mean BHR (%) 138.9 59.5 164.0 Turnaround Firms Number of Firms 511 348 228 RI Change: Median 0.102 *** 0.035 *** 0.082 *** RI Average: Median -0.055 *** -0.081 *** -0.065 *** RI in Year 5: Median -0.011 *** -0.081 *** -0.025 *** Median BHR (%) 111.8 *** 9.4 103.3 Mean BHR (%) 224.5 *** 72.8 200.2 Declining Firms Number of Firms 118 116 665 RI Change: Median 0.004 ** -0.090 *** 0.002 *** RI Average: Median -0.009 -0.056 ** 0.030 *** RI in Year 5: Median 0.016 -0.053 ** 0.043 *** Median BHR (%) 138.3 *** 16.9 222.2 *** Mean BHR (%) 257.6 *** 54.0 278.2 *** (***)Significant at the 0.01 level. (**)Significant at the 0.05 level. Table IX. Comparison of 1991, 1986, and 1981 Samples Using [P.sub.+90]/[E.sub.0] This table compares the performance of the 1991 sample to the performance of samples with base years in 1986 and 1981. Within each sample, we measure the P/E ratio 90 days after the end of the base year, and we start the buy-and-hold return (BHR) calculation at that date. For each sample, the table lists the median change in residual income (RI) from year 0 to year 5 (RI Change), the median RI over years 1 to 5 (RI Average), the median RI in year 5 (RI in Year 5), and the median and mean BHR over a five-year period following the base year. We calculate RI using Equation 13. We compare each median value to the median value across all firms using a Wilcoxon rank-sum test. We compare the mean BHRs to the mean BHR across all firms in a parametric difference in means test. 1991 1986 1981 All Firms Number of Firms 2,838 2,955 2,119 RI Change: Median 0.046 -0.015 -0.019 RI Average: Median 0.006 -0.014 0.014 RI in Year 5: Median 0.035 -0.021 0.011 Median BHR (%) 65.8 14.2 159.2 Mean BHR(%) 119.7 52.0 214.1 Growth Firms Number of Firms 2,101 2,348 673 RI Change: Median 0.039 -0.009 -0.069 *** RI Average: Median 0.023 *** -0.007 ** -0.003 RI in Year 5: Median 0.045 -0.013 -0.030 *** Median BHR (%) 55.5 *** 13.1 62.9 *** Mean BHR (%) 96.4 *** 52.8 137.2 *** Mature Firms Number of Firms 160 242 448 RI Change: Median -0.037 *** -0.100 *** -0.091 *** RI Average: Median 0.062 *** 0.026 ** 0.057 *** RI in Year 5: Median 0.093 *** 0.022 ** 0.029 Median BHR (%) 106.3 *** 44.1 143.1 Mean BHR (%) 128.7 43.6 192.0 Turnaround Firms Number of Firms 473 290 252 RI Change: Median 0.101 *** 0.032 *** 0.087 *** RI Average: Median -0.059 *** -0.088 *** -0.065 *** RI in Year 5: Median -0.013 *** -0.075 *** -0.027 *** Median BHR (%) 95.9 *** 9.0 164.4 Mean BHR (%) 202.1 *** 55.1 247.8 Declining Firms Number of Firms 104 75 746 RI Change: Median -0.019 *** -0.096 *** -0.004 ** RI Average: Median -0.031 *** -0.066 ** 0.025 RI in Year 5: Median -0.001 -0.070 ** 0.035 *** Median BHR (%) 115.7 *** 17.5 238.9 *** Mean BHR (%) 200.2 ** 42.8 285.4 *** (***)Significant at the 0.01 level. (**)Significant at the 0.05 level.

Appendix A. Dividend Paying Stocks

The model in this paper uses the assumption that from an investment standpoint, the life of a firm has two distinct phases. During the growth phase, the firm will invest 100% of its earnings in new (positive NPV) projects and will not pay dividends. After the growth phase ends, the firm will not be able to invest in positive NPV projects. For many firms, the optimal investment amount could be more or less than the firm's current earnings, and many firms do pay at least some dividends during their growth phase. Therefore, the basic model derived in this paper might not apply to all firms.

Danielson (1998) shows how to derive a model to approximate the stock price of a firm that invests either more or less than the amount of its earnings in new projects. This approach uses the following assumptions.

During each year of its growth phase, year 1 to year [tau], the firm will invest a portion of its earnings in positive NPV projects. The dollar amount of this investment is the total earnings times the reinvestment rate, [rho]. These investments will earn an equity return equal to [R.sub.N].

The value [rho] can be greater than, equal to, or less than, one. If [rho] is greater than one, then the firm raises new financing and invests an amount greater than its earnings in positive NPV projects. If the new financing does not change the firm's capital structure or risk, then the cost of the equity portion of this financing will be equal to k. If p is less than one, then the firm will invest an amount less than its earnings in positive net present value projects.

The remainder of the earnings can either be paid as dividends, or invested in projects with an equity return equal to the firm's required return on equity, k. The model assumes that any earnings in excess of the amount [rho][E.sub.t=1] will be in vested in projects with an equity return equal to k. When the return on new investments is equal to k, firm value will be same, whether the firm invests in the new projects or pays dividends. Thus, this assumption should not affect the estimated firm value. However, this assumption does allow us to simplify the model to a more concise form. [a]

With the exception of these additional assumptions, the model in this appendix uses the same assumptions as the basic model. The firm's investment base at t = [tau] can be written as Equation A. 1:

[I.sub.t=[tau]] = [I.sub.t=0] + ([I.sub.t=0] X [R.sub.E])[[[sigma].sup.[tau]].sub.t=1][[[rho](1 + [R.sub.N]) + (1 - [rho])(1 + k)].sup.t-I] (A.1)

Following the same steps used to derive Equation 7, the model for a dividend paying firm can be written as Equation A.2:

[P.sub.t=0] = [I.sub.t=0]([R.sup.T]/k)[(1 + [[R.sup.D].sub.N]/1 + k).sup.[tau]] [1 + ([R.sub.E] - [[R.sup.D].sub.N])PVAF([tau], [[R.sup.D].sub.N])] (A.2)

where:

[[R.sup.D].sub.N] = k + [rho]([R.sub.N] - k) (A.3)

Appendix B. Derivation of Equation 5

This appendix shows how Equation 5 can be derived from Equation 4. First, we note that the summation term (S) in Equation 4 can be written as Equation B.1.

[[[sigma].sup.[tau]].sub.t=1][(1 + [R.sub.N]).sup.t-1] = S = 1 + U + [U.sup.2] +...+[U.sup.[tau]-1] (B.1)

where:

U = (1 + [R.sub.N]) (B.2)

To eliminate the summation sign from Equation 4, we multiply the sum S by U, and subtract this product from S. This step yields Equation B.3.

S-US = 1 - [U.sup.[tau]] (B.3)

We solve Equation B.3 for S, substitute back for U, and plug the result into Equation 4. This step yields Equation B.4.

[I.sub.t=[tau]] = [I.sub.t=0] + ([I.sub.t=0] X [R.sub.E])[[(1 + [R.sub.N]).sup.[tau]] - 1/[R.sub.N]] = [I.sub.t=0][1 + [R.sub.E][[(1 + [R.sub.N]).sup.[tau]] - 1/[R.sub.N]]] (B.4)

We add and subtract [(1+[R.sub.N]).sup.[tau]] within the brackets on the right-hand side of Equation B.4, and rearrange terms to get Equation B.5.

[I.sub.t=[tau]] = [I.sub.t=0][[(1 + [R.sub.N]).sup.[tau]] + [1 - [(1 + [R.sub.N]).sup.[tau]]] + [R.sub.E][[(1 + [R.sub.N]).sup.[tau]] - 1/[R.sub.N]]] (B.5)

After multiplying the second term within the large brackets on the right-hand side of Equation B.5 by [R.sub.N]/[R.sub.N], we can combine this term with the final term within the brackets.

[I.sub.t=[tau]] = [I.sub.t=0][[(1 + [R.sub.N]).sup.[tau]] + ([R.sub.E] - [R.sub.N])[[(1 + [R.sub.N]).sup.[tau]] - 1/[R.sub.N]]] (B.6)

Next, we move the term [(1+[R.sub.N]).sup.[tau]] outside the brackets on the right-hand side of Equation B.6.

[I.sub.t=[tau]] = [I.sub.t=0][(1 + [R.sub.N]).sup.[tau]][1 + ([R.sub.E] - [R.sub.N])[1 - 1/[(1 + [R.sub.N]).sup.[tau]]/[R.sub.N]]] (B.7)

Finally, we note that the last term within the brackets on the right-hand side of Equation B.7 is the formula for the present value of an annuity of length [tau], using a discount rate, [R.sub.N]. So, Equation B.7 can be rewritten as Equation B.8, where the notation PVAF is the present value of an annuity factor for an annuity with a length [tau], using a discount rate, [R.sub.N]. Equation B.8 is equal to Equation 5.

[I.sub.t=[tau]] = [I.sub.t=0][(1 + [R.sub.N]).sup.[tau]][1 + ([R.sub.E] - [R.sub.N])PVAF([tau],[R.sub.N])] (B.8)

Appendix C. Derivation of Equation 8

This appendix derives Equation 8, the weighted average of [R.sub.E] and [R.sub.N] as of t = [tau]. The weighted average assumes that the firm will continue to earn a ROE equal to [R.sub.E] from the investment amount [I.sub.t=0], and that investments made between year 1 and year [tau] will continue to earn a ROE equal to [R.sub.N] after t = [tau]. Equation C.1 writes this weighted average in its most direct form:

[WR.sub.E,N] = [R.sub.E]([I.sub.t=0]/[I.sub.t=[tau]])+ [R.sub.N]([I.sub.t=[tau]] - [I.sub.t=0]/[I.sub.t=[tau]]) (C.1)

To derive Equation 8, we first add and subtract [I.sub.t=0], [(l+[R.sub.N]).sup.[tau]] to the numerator within the first set of brackets on the right-hand side of Equation C.1. This step yields Equation C.2.

[WR.sub.E,N] = [R.sub.E]([(1 + [R.sub.N].sup.[tau]] [I.sub.t=0]/[I.sub.t=[tau]])+Z (C.2)

where:

Z = [R.sub.E]((1-[(1 + [R.sub.N]).sup.[tau]]) [I.sub.t=0]/[I.sub.t=[tau]])+ [R.sub.N]([I.sub.t=[tau]] - [I.sub.t=0]/[I.sub.t=[tau]]) (C.3)

By substituting Equation 5 into the denominator of the first term in Equation C.2, Equation C.2 can be simplified to Equation C.4.

[WR.sub.E,N] = [R.sub.E]/[1+([R.sub.E] - [R.sub.N])PVAF([tau], [R.sub.N])]+Z (C.4)

Thus, Equation C.1 can be rewritten as Equation 8 if Z = 0. To show that Z is equal to zero, we substitute Equation B .4 for [I.sub.t=[tau]] in the numerator of the second bracketed term on the right-hand side of Equation C.3. After rearranging terms, this step yields Equation C.5.

Z = [I.sub.t=0]/[I.sub.t=[tau]] [[R.sub.E](1-[(1 + [R.sub.N].sup.[tau]])+ [R.sub.N]([R.sub.E][[(1+[R.sub.N]).sup.[tau]] -1/[R.sub.N]])] (C.5)

All of the terms within the largest set of brackets on the right-hand side of Equation C.5 cancel each other out. Therefore, Z is equal to zero, and Equation C.1 is equal to Equation 8.

(a.) When the equity return is equal to k, dividend policy should not affect firm value. Within the model in this appendix, this assumption does have a slight effect on the estimated firm value because the assumption increases the amount assumed to be invested in positive NPV projects in future years. As Danielson (1998) shows, this distortion should not have a large effect on the estimated stock price of most firms.

[Graph omitted]

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Author: | Danielson, Morris G.; Dowedell, Thomas D. |
---|---|

Publication: | Financial Management |

Geographic Code: | 1USA |

Date: | Jun 22, 2001 |

Words: | 20301 |

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