Market implied future earnings and analysts’ forecasts

Abstract

This study provides evidence on market implied future earnings based on the residual income valuation (RIV) framework and compares these earnings with analyst earnings forecasts for accuracy (absolute forecast error) and bias (signed forecast error). Prior research shows that current stock price reflects future earnings and that analyst forecasts are biased. Thus, how price-based imputed forecasts compare with analyst forecasts is interesting. Using different cost of capital estimates, we use the price-earnings relation and impute firms’ future annual earnings from three residual income (RI) models for up to 5 years. Relative to I/B/E/S analyst forecasts, imputed forecasts from the RI models are less or no more biased when cost of capital is low (equal to a risk-free rate or slightly higher). Analysts slightly outperform these RI models in terms of accuracy for immediate future (1 or 2) years in the forecast horizon but the opposite is true for more distant future years when cost of capital is low. A regression analysis shows that, in explaining future earnings changes, analyst forecasts relative to imputed forecasts do not impound a significant amount of earnings information embedded in current price. In additional tests, we impute future long-term earnings growth rates and find that they are more accurate and less biased than I/B/E/S analyst long-term earnings growth forecasts. Together, the results suggest that the RIV framework can be used to impute a firm’s future earnings that are high in accuracy and low in bias, especially for distant future years.

Appendix: Calculation of earnings forecasts and earnings forecast errors for a sample firm

The following demonstrates the calculation of imputed earnings forecasts, analysts’ earnings forecasts, and forecast errors for American Water Works assuming a 6 % in-horizon growth rate in residual income (RI) and a 2 % risk premium. The table below gives pertinent information:

 

Company name	American water works
Cusip	03041110
Ticker	AWK
SIC	4941
Year (base)	1999
Quarter (base)	3rd
Month (base)	September
Stock price (P 0)	28.938
Book value of common equity (millions $)	1599.986
Shares of common stock outstanding (in millions)	96.831
Dividends per share (d 0)	0.215
Risk-free discount rate	0.0571
Beta	0.477914
Assumed in-horizon annual growth rate (in RI)	0.06
Assumed long-term annual growth rate (in RI) for RIMCVG	0.03
Analysts’ EPS forecasts (Oct. 1999 median consensus)
2000, annual	1.75
1999, annual	1.64
Forecasted long-term growth rate (in EPS)	0.06
Actual EPS
2003, annual	NA
2002, annual	NA
2001, annual	1.69
2000, annual	1.61
1999, annual	1.56
1999, 3rd qtr.	0.58
1999, 2nd qtr.	0.42
1999, 1st qtr.	0.23
Quarterly dividends
1999, 3rd qtr.	0.215
1999, 2nd qtr.	0.215
1999, 1st qtr.	0.215
1998, 4th qtr.	0.205
1998, 3rd qtr.	0.205
1998, 2nd qtr.	0.205
1998, 1st qtr.	0.205
1997, 4th qtr.	0.190

To calculate the first monthly RI per share number, we use Eq. (6) from the manuscript. Equation (6) with T = 60, which is labeled as Eq. (17) in this appendix, is shown below:

$$ {\text{E}}_{ 0} \left[ {x_{1}^{a} } \right] \, = \, {{\left( {P_{0} - B_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{0} - B_{0} } \right)} {\left( {\sum\limits_{t = 1}^{ 6 0} {\left( {1 + g_{m} } \right)^{t - 1} \left( {1 + r_{m} } \right)^{ - t} \, + \, \left( {1 + g_{m} } \right)^{60} \left( {r_{m} - g_{Gm} } \right)^{ - 1} \left( {1 + r_{m} } \right)^{ - 60} } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sum\limits_{t = 1}^{ 6 0} {\left( {1 + g_{m} } \right)^{t - 1} \left( {1 + r_{m} } \right)^{ - t} \, + \, \left( {1 + g_{m} } \right)^{60} \left( {r_{m} - g_{Gm} } \right)^{ - 1} \left( {1 + r_{m} } \right)^{ - 60} } } \right)}} $$

(17)

where g _m is the growth rate of RI for month t in the forecast horizon and g _Gm is the long-term beyond horizon growth rate of RI, which is 0 % for RIMCV_C and is 3 % per year, converted into a monthly rate, for RIMCV_G. Note: The RIM_C model is not applicable in this example since the assumed in-horizon growth rate in RI is different from the assumed long-term growth rate in RI.

Since we are assuming a 2 % risk premium in this case, the discount rate is calculated as follows: Risk-free discount rate + Beta * 0.02 = 0.0571 + 0.477914 * 0.02 = 0.066658.

The discount rate is converted from an annual to a monthly rate as follows:

$$ r_{m} = \root{12} \of {{1 + 0. 0 6 6 6 5 8}} {-1} = 0.005392 $$

The assumed in-horizon and long-term annual growth rates also need to be converted to monthly growth rates as follows:

In-horizon: g _m  = $\root{12} \of {1 + 0. 0 6}{- 1} = 0.004868$

Long-term: g _Gm  = $\root{12} \of {{1 \, + { 0} . 0 3}}{ - 1}= 0.002466$

Since the calculations are done on a per-share basis, the book value of common equity is converted to a per-share basis by taking the book value of common equity (total) and dividing it by the shares of common stock outstanding. Both of those measures are given at the beginning of this appendix. The calculation is as follows:

$$ B_{0} = 1 5 9 9. 9 8 6/ 9 6. 8 3 1= 1 6. 5 2 3 4 8 9 $$

On the right hand side of Eq. (17), the first multiplication in the denominator is as follows:

$$ \sum\limits_{{{\text{t}} = 1}}^{ 6 0} {\left( {1 + g_{m} } \right)^{t - 1} \left( {1 + r_{m} } \right)^{ - t} } $$

Rewriting it and plugging in the necessary numbers, we get:

$$ \sum\limits_{{{\text{t}} = 1}}^{ 6 0} {\frac{{\left( { 1 { } + { 0} . 0 0 4 8 6 8} \right)^{t - 1} }}{{ ( 1 { } + { 0} . 0 0 5 3 9 2 )^{t} }}} $$

which is equal to 58.768992.

The second multiplication in the denominator of Eq. (17) is the following:

$$ \left( {1 + g_{m} } \right)^{60} \left( {r_{m} - g_{Gm} } \right)^{ - 1} \left( {1 + r_{m} } \right)^{ - 60} $$

Rewriting it and plugging in the necessary numbers, for RIMCV_G we get:

$$ \frac{{\left( {1 \, + { 0} . 0 0 4 8 6 8} \right)^{60} }}{{(0.005392{ - 0} . 0 0 2 4 6 6 )\left( { 1 { } + { 0} . 0 0 5 3 9 2} \right)^{60} }} = \frac{1.338261}{0.002926*1.380783} = 331.238698 $$

For RIMCV_C, $g_{Gm}$ = 0. Therefore, the calculation for RIMCV_C is as follows:

$$ \frac{{\left( {1 \, + { 0} . 0 0 4 8 6 8} \right)^{60} }}{{(0.005392{ - 0)}\left( { 1 { } + { 0} . 0 0 5 3 9 2} \right)^{60} }} = \frac{1.338261}{0.005392*1.380783} = 1 7 9. 7 4 8 5 9 6 $$

Plugging the above numbers into Eq. (17), we get the following first month abnormal earnings values for the RIMCV_G and RIMCV_C models (the stock price per share P ₀ is given at the beginning of this appendix):

$$ \begin{gathered} x_{1}^{a} = \, {{\left( { 2 8. 9 3 8 { - 16} . 5 2 3 4 8 9} \right)} \mathord{\left/ {\vphantom {{\left( { 2 8. 9 3 8 { - 16} . 5 2 3 4 8 9} \right)} {\left( { 5 8. 7 6 8 9 9 2 { } + { 331} . 2 3 8 6 9 8} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { 5 8. 7 6 8 9 9 2 { } + { 331} . 2 3 8 6 9 8} \right)}}\quad \left( {{\text{for RIMCV}}_{\text{G}} } \right) \hfill \\ x_{1}^{a} \, = { 0} . 0 3 1 8 3 1\hfill \\ \end{gathered} $$

$$ \begin{gathered} x_{1}^{a} \, = \, {{\left( { 2 8. 9 3 8 { - 16} . 5 2 3 4 8 9} \right)} \mathord{\left/ {\vphantom {{\left( { 2 8. 9 3 8 { - 16} . 5 2 3 4 8 9} \right)} {\left( { 5 8. 7 6 8 9 9 2 { } + { 179} . 7 4 8 5 9 6} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { 5 8. 7 6 8 9 9 2 { } + { 179} . 7 4 8 5 9 6} \right)}}\quad \left( {{\text{for RIMCV}}_{\text{C}} } \right) \hfill \\ x_{1}^{a} \, = { 0} . 0 5 2 0 4 9\hfill \\ \end{gathered} $$

For subsequent months in the forecast horizon (t = 2 to 60), we calculate RI using Eq. (7) from the manuscript, which is as follows:

$$ {\rm E}_{0} \left[ {x_{t}^{a} } \right] = E_{0} \left[ {x_{t - 1}^{a} } \right]\left( {1 + g_{m} } \right). $$

For example, earnings for month t = 2 would be:

$$ x_{2}^{a} \, = { 0} . 0 3 1 8 3 1 { (1 } + { 0} . 0 0 4 8 6 8 ) { } = { 0} . 0 3 1 9 8 6\quad \left( {{\text{for RIMCV}}_{\text{G}} } \right) $$

$$ x_{2}^{a} \, = { 0} . 0 5 2 0 4 9 { (1 } + { 0} . 0 0 4 8 6 8 ) { } = { 0} . 0 5 2 3 0 2\quad \left( {{\text{for RIMCV}}_{\text{C}} } \right) $$

Earnings for month t = 3 would be:

$$ x_{3}^{a} \, = { 0} . 0 3 1 9 8 6 { (1 } + { 0} . 0 0 4 8 6 8 ) { } = { 0} . 0 3 2 1 4 2\quad \left( {{\text{for RIMCV}}_{\text{G}} } \right) $$

$$ x_{3}^{a} \, = { 0} . 0 5 2 3 0 2 { (1 } + { 0} . 0 0 4 8 6 8 ) { } = { 0} . 0 5 2 5 5 7\quad \left( {{\text{for RIMCV}}_{\text{C}} } \right) $$

The same procedure follows for months t = 4 through t = 60.

We rearrange the definition of RI and use Eq. (8) from the manuscript to estimate raw earnings each month. We label it Eq. (18) in this appendix and it is as follows:

$$ {\text{E}}_{0} \left[ {x_{t} } \right] = {\text{E}}_{0} \left[ {x_{t}^{a} } \right] + r_{m} B_{t - 1} . $$

(18)

To calculate future raw earnings, we need future book values (per share). To calculate future book values, we assume the clean surplus relation and label this relation Eq. (19) as shown below:

$$ {\text{E}}_{ 0} \left[ {B_{t} } \right] = B_{t - 1} + {\text{E}}_{ 0} \left[ {x_{t} } \right] - d_{t} $$

(19)

Since future dividends are required to update future book value, we use dividends from the base quarter and calculate future dividends as shown in Sect. 3.6 and below. We label it as Eq. (20).

$$ d_{t} = \left( {d_{0} } \right)\left( {1 + g_{MD} } \right)^{t} , $$

(20)

where d _t is the estimated dividend per share for future month t (t = 1 to 60) in the horizon, d ₀ is the base month dividend estimate, and g _MD is the monthly dividend growth rate. A monthly dividend growth rate is necessary to estimate future dividends per share. To estimate the monthly growth rate, we first estimate an annual dividend growth rate as shown in footnote 17 and below:

$$ g_{AD} = \left( {\sum\limits_{q = 0}^{ - 3} {div_{q} } - \sum\limits_{q = - 4}^{ - 7} {div_{q} } } \right) \left/\sum\limits_{q = - 4}^{ - 7} {div_{q} }\right. . $$

The monthly dividend growth rate is then calculated as follows: $g_{MD} = \left( {1 + g_{AD} } \right)^{{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-\nulldelimiterspace} {12}}}} - 1.$

Using the information given at the beginning of this appendix for American Water Works, the calculation of the annual dividend growth rate is shown below:

$$ g_{AD} \, = \, \frac{{\left[ {\left( { 0. 2 1 5 { } + { 0} . 2 1 5 { } + { 0} . 2 1 5 { } + { 0} . 2 0 5} \right){ - }\left( { 0. 2 0 5 { } + { 0} . 2 0 5 { } + { 0} . 2 0 5 { } + { 0} . 1 9 0} \right)} \right]}}{{ 0. 2 0 5 { } + { 0} . 2 0 5 { } + { 0} . 2 0 5 { } + { 0} . 1 9 0}} = 0.0 5 5 90 1 $$

$$ g_{MD} = \left( { 1 { } + { 0} . 0 5 5 9 0 1} \right)^{{{ 1\mathord{\left/ {\vphantom { 1{ 1 2}}} \right. \kern-\nulldelimiterspace} { 1 2}}}} { - 1} = 0.00 4 5 4 3 $$

Note: If there is a stock split or stock dividend between the first quarter of 1998 (q = −6) and the third quarter of 1999 (q = 0), we need to adjust all quarters prior to the stock split or stock dividend using the Compustat adjustment factor. American Water Works had no stock split or stock dividend during that time period.

To calculate the base month dividend d ₀, we take the current quarter dividend per share of 0.215 and divide it by 3 to convert it to a monthly basis. Therefore, the base month dividend is 0.215/3 = 0.071667. Hence, the estimated future dividends for months t = 1, 2, and 3 within the forecast horizon are calculated using Eq. (20) as follows:

$$ d_{1} = 0.071667*\left( {1 + 0.004543} \right) = 0.071993 $$

$$ d_{2} = 0.071667*\left( {1 + 0.004543} \right)^{2} = 0.072320 $$

$$ d_{3} = 0.071667*\left( {1 + 0.004543} \right)^{3} = 0.072648 $$

The same procedure follows for months t = 4 through t = 60.

The calculations of the raw earnings per share (Eq. 18) for months 1 through 3 are shown below for RIMCV_G and RIMCV_C. The book value per share for t = 0, B ₀, was calculated previously whereas the book values per share (Eq. 19) for months t = 1 and t = 2 are shown below.

RIMCV_G:

• x ₁ = 0.031831 + 0.005392 * 16.523489 = 0.120926

• B ₁ = 16.523489 + 0.120926 − 0.071993 = 16.572422

• x ₂ = 0.031986 + 0.005392 * 16.572422 = 0.121344

• B ₂ = 16.572422 + 0.121344 − 0.072320 = 16.621446

• x ₃ = 0.032142 + 0.005392 * 16.621446 = 0.121765

RIMCV_C:

• x ₁ = 0.052049 + 0.005392 * 16.523489 = 0.141144

• B ₁ = 16.523489 + 0.141144 − 0.071993 = 16.592640

• x ₂ = 0.052302 + 0.005392 * 16.592640 = 0.141770

• B ₂ = 16.592640 + 0.141770 − 0.072320 = 16.662090

• x ₃ = 0.052557 + 0.005392 * 16.662090 = 0.142399

The same procedure follows for months t = 4 through t = 60.

The next step is to compute the annual imputed earnings. Since we are calculating the forecasts as of the end of the third quarter (September base month), the first through third quarter actual earnings are used in calculating the Year +1 earnings. Therefore, the forecasted earnings for Year +1 are calculated as the sum of the actual earnings for the first three quarters and the first three months of imputed earnings, $\sum\nolimits_{t = 1}^{3} {{\text{E}}_{ 0} \left[ {x_{t} } \right]}$. The actual earnings for the first three quarters are given at the beginning of this appendix. As a result, the Year +1 earnings are computed for the RIMCV_G and RIMCV_C models as follows:

Year +1 Earnings:

$$ 0. 2 3 + 0. 4 2 + 0. 5 8 + 0. 1 20 9 2 6 + 0. 1 2 1 3 4 4 + 0. 1 2 1 7 6 5 = 1. 5 9 40 3 5\quad \left( {{\text{RIMCV}}_{{{\text{G}}}} } \right) $$

$$ 0. 2 3 + 0. 4 2 + 0. 5 8 + 0. 1 4 1 1 4 4 + 0. 1 4 1 7 70 + 0. 1 4 2 3 9 9 = 1. 6 5 5 3 1 3\quad \left( {{\text{RIMCV}}_{\text{C}} } \right) $$

The forecasted Year +2 earnings are calculated as the sum of the imputed earnings for months 4 through 15, or $\sum\nolimits_{t = 4}^{15} {{\text{E}}_{ 0} \left[ {x_{t} } \right]}$. Therefore, the Year +2 earnings are calculated for the RIMCV_G and RIMCV_C models as follows (the calculations of the monthly imputed earnings used as inputs for Year +2 are not shown in this appendix):

Year +2 Earnings:

$$ \begin{gathered} 0. 1 2 2 1 9 9 + 0. 1 2 2 6 2 6 + 0. 1 2 30 5 5 + 0. 1 2 3 4 8 5 + 0. 1 2 3 9 1 6 + 0. 1 2 4 3 4 8 + 0. 1 2 4 7 8 2 + 0. 1 2 5 2 1 7 + \hfill \\ 0. 1 2 5 6 5 3 + 0. 1 2 60 9 1 + 0. 1 2 6 5 30 + 0. 1 2 6 9 70 = 1. 4 9 4 8 7 2\quad \left( {{\text{RIMCV}}_{\text{G}} } \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} 0. 1 4 30 4 6 + 0. 1 4 3 6 8 6 + 0. 1 4 4 3 2 8 + 0. 1 4 4 9 7 3 + 0. 1 4 5 6 2 2 + 0. 1 4 6 2 7 3 + 0. 1 4 6 9 2 7 + 0. 1 4 7 5 8 5 { } + \hfill \\ 0. 1 4 8 2 4 5 + 0. 1 4 8 90 8 + 0. 1 4 9 5 7 5 + 0. 1 50 2 4 4 = 1. 7 5 9 4 1 2\quad \left( {{\text{RIMCV}}_{\text{C}} } \right) \hfill \\ \end{gathered} $$

The forecasted Year +3 earnings are computed as the sum of imputed earnings for months 16 through 27, or $\sum\nolimits_{t = 16}^{27} {{\text{E}}_{ 0} \left[ {x_{t} } \right]}$. Also, the forecasted Year +4 earnings are calculated as the sum of imputed earnings for months 28 through 39, or $\sum\nolimits_{t = 28}^{39} {{\text{E}}_{ 0} \left[ {x_{t} } \right]}$. Furthermore, the forecasted Year +5 earnings are the sum of imputed earnings for months 40 through 51, or $\sum\nolimits_{t = 40}^{51} {{\text{E}}_{ 0} \left[ {x_{t} } \right]}$. The Year +3 through Year +5 earnings for the RIMCV_G and RIMCV_C models are shown below:

RIMCV_G:

• Year +3 earnings: 1.558475 Year +4 earnings: 1.624404 Year +5 earnings: 1.692712

RIMCV_C:

• Year +3 earnings: 1.856276 Year +4 earnings: 1.958620 Year +5 earnings: 2.066765

In calculating the forecast error from analysts’ forecasts, the analysts’ forecasts for Year +1 (1999) and Year +2 (2000) are available. However, for American Water Works, explicit analysts’ forecasts are unavailable for Years +3 through +5. Therefore, to calculate the analysts’ forecasts for Years +3 through +5, we compound the Year +2 analyst forecast at the analyst long-term growth rate forecast. The analyst forecasts and calculations, if applicable, are shown below (information is from the table at the beginning of the appendix):

• Analyst forecast (Year +1): $1.64

• Analyst forecast (Year +2): $1.75

• Analyst forecast (Year +3): $1.75 * 1.06 = $1.86

• Analyst forecast (Year +4): $1.75 * 1.06² = $1.97

• Analyst forecast (Year +5): $1.75 * 1.06³ = $2.08

Note: American Water Works had no stock dividends or splits in October. Therefore, the analyst October consensus EPS forecasts do not need to be adjusted to the September base month.

We measure the bias of a forecast in terms of signed forecast error using Eq. (9) from the manuscript. The equation is shown below and labeled as Eq. (21) in this appendix.

$$ FB = {{\left( {F - A} \right)} \mathord{\left/ {\vphantom {{\left( {F - A} \right)} P}} \right. \kern-\nulldelimiterspace} P}, $$

(21)

where F is the forecast (imputed EPS or I/B/E/S median consensus analyst EPS forecast), A is the corresponding actual EPS from I/B/E/S, and P is the firm’s common stock price at the end of the base month. We measure forecast accuracy (FA) in terms of absolute forecast error using Eq. (10) from the manuscript. The equation is shown below and labeled Eq. (22) in this appendix:

$$ FA = {{\left| {F - A} \right|} \mathord{\left/ {\vphantom {{\left| {F - A} \right|} P}} \right. \kern-\nulldelimiterspace} P}. $$

(22)

The calculations of forecast errors in terms of bias and accuracy for RIMCV_G model forecasts, RIMCV_C model forecasts, and analysts’ forecasts are shown below. No bias or accuracy measures can be computed for Year +4 (2002) or Year +5 (2003) because actual EPS numbers are missing for those years. This is due to American Water Works being acquired by German utility company RWE on January 17, 2002. The actual EPS and stock price information are in the table at the beginning of the appendix.

RIMCV_G:

• FB (Year +1) = ($1.594035 − $1.56)/$28.938 = 0.001176

• FA (Year +1) = |$1.594035 − $1.56|/$28.938 = 0.001176

• FB (Year +2) = ($1.494872 − $1.61)/$28.938 = −0.003978

• FA (Year +2) = |$1.494872 − $1.61|/$28.938 = 0.003978

• FB (Year +3) = ($1.558475 − $1.69)/$28.938 = −0.004545

• FA (Year +3) = |$1.558475 − $1.69|/$28.938 = 0.004545

RIMCV_C:

• FB (Year +1) = ($1.655313 − $1.56)/$28.938 = 0.003294

• FA (Year +1) = |$1.655313 − $1.56|/$28.938 = 0.003294

• FB (Year +2) = ($1.759412 − $1.61)/$28.938 = 0.005163

• FA (Year +2) = |$1.759412 − $1.61|/$28.938 = 0.005163

• FB (Year +3) = ($1.856276 − $1.69)/$28.938 = 0.005746

• FA (Year +3) = |$1.856276 − $1.69|/$28.938 = 0.005746

Analysts:

• FB (Year +1) = ($1.64 − $1.56)/$28.938 = 0.002765

• FA (Year +1) = |$1.64 − $1.56|/$28.938 = 0.002765

• FB (Year +2) = ($1.75 − $1.61)/$28.938 = 0.004838

• FA (Year +2) = |$1.75 − $1.61|/$28.938 = 0.004838

• FB (Year +3) = ($1.86 − $1.69)/$28.938 = 0.005875

• FA (Year +3) = |$1.86 − $1.69|/$28.938 = 0.005875