*By Elli Malki*

Towards the end of the last century, academic finance economists came to take seriously the view that aggregate stock returns are predictable.

This somewhat surprising statement comes from John Y. Campbell, an economics professor at Harvard University, and Samuel B. Thompson, manager of research in Arrowstreet Capital, L.P., in an article that they published recently.[^{1}]

Campbell and Thompson investigated whether value ratios, such as P/E, P/B and D/P, were helpful in predicting the future returns of the overall stock market during the period of 1926-2005. Their first conclusion was that value ratios showed an ability to predict future returns, e.g., periods with low P/E or conversely high D/P ratios were usually followed by periods with higher-than-normal returns in the stock market.

This conclusion should not come as a surprise to value investors, who follow the footsteps of Benjamin Graham, but it may raise some eyebrows in the academy. The second, and more interesting conclusion, was that the best predictions of future returns were obtained once the parameters of the forecast model were computed using the Dividend Growth Model, known also as the Gordon Model.

The Gordon Model, in its simplest form, determines the relation between the dividend yield (D/P), the expected growth in dividends (g) and the expected rate of return (r).

(1) D/P = r - g

In a previous article ("The Risk Premium Puzzle"), I showed that this simple relation can be used to detect abnormally large or small risk premiums. Another extension of the Gordon Model is the Abnormal Earnings Model (AEM). The AEM, which is also known as the Residual Income Model, can be derived from the Gordon Model with the addition of the long-term relation between the dividend growth rate (g), the return on equity (R) and the earnings retention rate (b):

(2) g = R * b

The retention rate (b) is the percentage of earnings (E) that is not paid out as dividends (D), but rather, is retained to support future growth:

(3) b = 1 - D/E = 1 - (D/P) * (P/E)

Using equations (1) and (2), the derivation of the AEM is fairly straightforward (for the interested reader, I have added an appendix that presents the mathematical proof of (2) and (4) and some other methodological issues that are related to the calculations hereinafter):

(4) P/B = 1 + (R - r) / (r - R*b)

Campbell and Thompson used equations (1) and (4) to predict the expected rate of return on the stock market. I propose a slightly different approach: From equation (4), it is possible to extract the expected return on equity (R):

(5) R = (P/B) * r / (1 + b * ((P/B) - 1))

The expected return on equity (R) is the level of profitability that conforms to the current level of price and the expected rate of return. However, it is not necessarily equal to the actual return on equity (hereinafter ROE) or to the historical level of the return on equity. Comparing the level of R to either the actual ROE or to the historical level of the return on equity can provide a useful insight on the current level of prices. High or low levels of R, in comparison to these benchmarks, may serve as a sign that stocks are over- or undervalued.

Based on data from Morningstar, I performed such comparison for three ETFs that follow stock indexes of large American corporations: SPY—which follows the S&P 500; IVW—which follows the S&P 500 Growth Index; and PRF—which is a fundamental index that follows the RAFI 1000, the fundamental equivalent of a large-cap U.S. index. These indexes represent the three investment styles defined by Morningstar: Large Blend (NYSEARCA:SPY), Large Growth (NYSEARCA:IVW) and Large Value (NYSE:PRF).

The results of the comparison are presented in Table 1.

**Table 1**

To calculate R, it is necessary to determine the expected rate of return, which is the sum of the risk-free rate (rf) and the equity risk premium. There are several estimates of the risk premium; I chose the highest of them—6.5%—which was calculated by Goetzmann & Ibbotson.[^{2}]

In the first part of Table 1, I used the current yield on 10-year U.S. government bonds (currently 2.2%) as the risk-free rate. The respective expected rate of return is 8.7%. The calculated R, based on this rate of return, varies between 9%-10%. In comparison, the historical level of the return on equity in the last decade was much higher: 12%-18%. However, such high level of return on equity is only typical to the last decade. Data presented by Campbell show that during the '80s and the early '90s, the return on equity was in the range of 8%-10%, not far from the calculated levels of R.[^{3}]

In the second part of Table 1, I used a higher risk-free rate (5%), based on the notion that investors relate to the current low yields of government bonds as temporary, and their expected rate of return is based on the long-term return of government bonds. Based on the higher expected rate of return (11.5%), the calculated R is in the range of 12%-13%. This level of R is at the low end of the historical return on equity in the last decade, but somewhat higher if the comparison is made with earlier periods.

The results of the comparison of R to the historical return on equity are not conclusive. On one hand, the levels of R seem quite low compared with the return on equity that we used to see since the mid-90s; on the other hand, we cannot exclude the possibility that long-term return on equity will be much lower in the next decade. In other words, the U.S. market looks cheap now only if we believe that the shift in the long-term return on equity, that occurred in the mid-90s, is permanent.

The cross-section comparison of the figures in Table 1 is much more revealing. Here I compared the computed level of R to the actual ROE of each index. The calculation of the actual ROE is based on its definition as the ratio between earnings (E) and book value (B).

(6) ROE = E / B = (P/B) / (P/E)

The actual ROE varies in a wide range: 19.5% for the growth index (IVW) and 10.8% for the value index (PRF). This should not come as a surprise since growth companies are expected to generate higher return on equity in order to maintain their growth prospects. The comparison of the calculated R to the actual ROE reveals large disparity between the indexes (the yellow column in Table 1). Regardless of the assumption about the correct expected rate of return, the growth index (IVW) stands out with a very high difference between ROE and R.

What is the interpretation of the difference between ROE and R? It is the extent to which profitability can decrease, but remain in accordance with the current level of prices. If we use the current yield on government bonds as the risk-free rate, the current price of the growth index (IVW) will be in accordance with the index's fundamental values, even if the return on equity will decrease from 19.5% to 10% (which implies a decrease of 50% in earnings).

In the same manner, if we use the long-term risk-free rate, then the current price of IVW will be in accordance with the index's fundamental values, even if the return on equity decreases from 19.5% to 13% (which implies a decrease of 35% in earnings). Thus, we can think of the difference between ROE and R as the "margins of safety" (using Benjamin Graham's terminology) between the price of the index and its fair value. The higher the value of ROE - R the cheaper the price of the index in relation to its fair value.

The analysis in Table 1 applies the concept of Value Investing to indexes and consequently to ETFs. The somewhat surprising conclusion for the U.S. stock market is that the growth index seems to be cheaper nowadays than the value index.

**Appendix**

*A) Derivation of equation (2)*

The relation between dividends (D), earnings (E) and the retention rate (b) is defined as follows (the index numbers relate to years 1 and 2):

(I) D_{1} = E_{1} * (1 - b)

If the company has a constant return on equity (R) and its book value is (B), its second-year earnings will be:

(II) E_{2} = R * (B_{1} + E_{1 }* b) = E_{1} + R * b * E_{1} = E_{1} * (1 + R * b)

The second year's dividend will be:

(III) D_{2} = (1 - b) * E_{1} * (1 + R * b)

Thus, the dividend growth rate will be:

(IV) D2 / D1 - 1 = R * b

*B) Derivation of AEM - equation (4)*

Placing relation (2) into equation (1) and substituting D by (1 - b) * E, results in the definition of P/E:

(I) P/E = (1 - b) / (r - R * b)

Using the definition in (6) we can obtain the relation between P/E and P/B:

(II) (P/B) = R * (P/E)

Placing (I) into (II) results in the AEM:

(III) (P/B) = R * (1 - b) / (r - R * b) = (R - r + r - R * b) / (r - R * b) =

= 1 + (R - r) / (r - R * b)

*C) Moving from company's level to index level*

The relations between accounting data and market data, that are presented by equations (1) to (6), are theoretically correct at the company's level. When we move from the company's level to the index level, we should define the meaning of these relations. Ideally, the index level figures should reflect average values; a kind of a representative company, that can be compared between the indexes. Such methodology is employed by Morningstar in the calculation of the P/B, P/E and D/P ratios for ETFs (I have discussed some conceptual issues emanating from this methodology in a previous article: "Has the U.S. Stock Market Become Cheaper?"). However, when these average value ratios are used to derive other accounting ratios, the results of these indirect calculations may differ from their actual (unobservable) average values. This concern holds for the calculation of the retention rate (b) in equation (3) and the actual ROE in equation (6).

In order to check the implications of this issue, I have constructed a simulation using three synthetic portfolios of 20 stocks each. The portfolios were constructed according to the three investment styles: blend, growth and value. The value ratios for each stock were selected randomly, but within predetermined boundaries according to the portfolio's style. For the simulation of the retention rate (b) I have used the following boundaries:

- P/E for growth stocks - 15 to 40; D/P for growth stocks - 0% to 1%
- P/E for blend stocks - 10 to 20; D/P for blend stocks - 1% to 3%
- P/E for value stocks - 5 to 15; D/P for value stocks - 3% to 7%

For each portfolio, I generated 20 iterations of random P/E and D/P, and for each iteration, I calculated the average retention rate (marked as "b - Average") and the index retention rate from equation (3) (marked as "b - Fund"). The results are presented in Table 2.

**Table 2**

The blue columns named "Difference" present the deviations of the calculated values of the retention rate (b - Fund) from the actual averages (b - Average). These deviations do not seem to be systematic; their magnitude is very small and their average is very close to zero.

A similar simulation was performed for the calculation of the ROE with the following boundaries for the relevant value ratios:

P/E for growth stocks - 15 to 40; P/B for growth stocks - 2.0 to 5.0

P/E for blend stocks - 10 to 20; P/B for blend stocks - 1.0 to 2.5

P/E for value stocks - 5 to 15; P/B for value stocks - 0.5 to 1.5

In the same manner, I generated 20 iterations of random P/E and P/B and calculated the average ROE (marked as "ROE - Average") and the index ROE from equation (6) (marked as "ROE - Fund"). The results are presented in Table 3.

**Table 3.**

As in Table 2, the blue columns named "Difference" present the deviations of the calculated ROE from the actual (unobservable) average ROE. Unlike the previous case, here the simulation reveals a positive systematic deviation, with an average value of 0.5%-1%. Nevertheless, this systematic deviation does not change the conclusions of the previous analysis. The "margins of safety" that were used to measure which index is relatively cheaper were computed as the difference between the calculated ROE and R. Thus, if the average ROE is systematically larger than the calculated R, the "margins of safety" are even larger than the previous model predicts.

**Endnotes**

1. "Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?" John Y. Campbell and Samuel B. Thompson, *The Review of Financial Studies*, vol. 21, Issue 4, pp. 1509-1531, 2008

2. "History and the Equity Risk Premium," W. N. Goetzmann & R. G. Ibbotson, Yale ICF Working Paper No. 05-04, April 2005.

3. "What Is the Equity Premium Today?" John Y. Campbell, Arrowstreet Capital, LP, July 2007

*Elli Malki is an economic consultant and the editor of **http://www.inbest.co.il/**, an Israeli Web site that provides a decision support service for savers and investors. He can be reached at **malki@netmedia.net.il**This e-mail address is being protected from spambots. You need JavaScript enabled to view it **. *