Investor expectations and the AIRR model

Abstract

The two-stage growth model (Danielson, 1998) empowers analysts to quantify the growth expectations supporting a firm’s stock price. This article merges that two-stage growth model with the AIRR model developed by Magni (2010, 2013). The new model simplifies the calculation of the firm’s expected competitive advantage period and the expected annual growth rate. The new model can be used to quantify expectations for both dividend and non-dividend paying firms and when the return on new investments or the required return on equity might not be constant in the future.

Acknowledgment

The author thanks two anonymous referees and the associate editor for their constructive suggestions for revising the paper.

Notes

1 For firms that are not currently profitable, Danielson and Lipton (2012) show how to quantify growth expectations from a firm’s price-to-sales ratio.

2 This earnings measure is also used in the finite growth model from Miller and Modigliani (1961).

3 When inflation is neutral, in that it affects $E_1,$ $k$ , and $R_N$ equally (i.e., each term is multiplied by 1 plus the inflation rate), the constant growth model produces identical stock price estimates when stated in either real or nominal terms.

4 One drawback of the Danielson (1998) model is that it uses the assumption that the equity return on new investments drops abruptly from $R_N$ to $k$ after year T. Models that allow this return to decline gradually have been developed by Fuller and Hsia (1984), Leibowitz (1998), and O’Brien (2003). However, for a given stock price, models that allow the equity return on new investments to decline over time also require the initial value of this return to be higher than $R_N.$ Thus, $R_N$ can be interpreted as the average equity return on new investment during the competitive advantage period.

5 A firm’s stock price will be independent of its dividend policy under either of two conditions. First, Brennan (1971) and Rubinstein (1976) argue that a firm’s stock price will be independent of its dividend policy when $R_N=k.$ Second, Miller and Modigliani (1961) claim that dividend policy will be irrelevant in a perfect world with no taxes or bankruptcy costs. However, as pointed out by DeAngelo and DeAngelo (2006), Miller and Modigliani (1961) assume that a firm will distribute a minimum of 100% of its free cash (defined as the firm’s earnings in excess of the amount require to invest in all positive net present value projects) each year. According to DeAngelo and DeAngelo (2006), Miller and Modigliani (1961) simply demonstrate that a firm can pay a dividend in excess of its free cash flow (to existing shareholders) by selling additional shares (to new stockholders). If $R_N=k,$ the DeAngelo and DeAngelo (2006) analysis suggests that a firm should adopt a 100% payout policy (through dividends or repurchases) because of the risk that retained earnings might ultimately earn a return less than $k.$

6 The first step in deriving either Equation (11a) or 11(b) is to multiply each side of Equation (10) by $k.$ Once this is done, Equation (11a) can be derived by taking the natural log of each side of the modified Equation (10) and solving for $T.$ Equation (11b) can be derived by taking the $1/T$ root of each side of the modified Equation (10) and solving for $k+b\left(R_N-k\right).$

7 Brealey, Myers, and Allen (2020) report that the average annual real return on U.S. common stocks over the 1900-2017 period was 8.4%. Within this article it will be assumed, for illustrative purposes, that the risk-adjusted required return on equity (stated in real terms) is $k=8\%,$ approximating the average historical return.

8 The discount rate in the traditional AIRR model is an unspecified, generic cost of capital. In the new application of the AIRR model developed in this article, the discount rate is the required return on equity. More generally, Magni (2020, Chapter 8) uses the AIRR model to analyze in depth all book AIRRs (e.g., average ROA, average ROI, and average ROE) and all associated costs of capital (e.g., required return on assets and required return on equity).

9 Equation (12) is a special case of the unnumbered equation between equations (27) and (28) in Magni (2013) and the unnumbered equation in Proposition 15 in Magni (2016). It is equivalent to Equation (6) from Magni (2010).

10 To obtain the third term in Equation (17), substitute Equation (16) into the numerator of the second term in Equation (17).

11 $0.224=\left(\frac{0.188-0.08}{0.75}\right)+0.08.$

12 https://data.nasdaq.com/data/MULTPL/SP500_PE_RATIO_MONTH-sp-500-pe-ratio-by-month.

13 Different firms can have different risk-adjusted required returns. However, the model can be used to quantify the future growth necessary to generate any given return. For simplicity, the 8% benchmark is used for all six firms in this exercise.

14 The net stock repurchases were calculated as total repurchases minus stock issued pursuant to employee compensation plans.

15 Technically, the P/E ratio as defined by the model is calculated using expected, rather than trailing, earnings. If earnings were expected to grow at the rate $b{R}_N,$ the calculated P/E ratios in this exercise slight overstate the true, but (retroactively) unobservable P/E ratios required by the model.

16 By June of 2023, the expectations embedded in the stock prices of both Lilly and Nvidia had become even more optimistic, as the P/E ratios of the two companies (calculated using trailing earnings) had increased to over 70 and over 200, respectively.

Additional information

Notes on contributors

Morris G. Danielson

Morris G. Danielson is a professor of finance at St. Joseph’s University, Philadelphia, Pennsylvania. He received his Ph.D. in Finance from the University of Washington. Dr. Danielson’s research covers topics in the following areas: valuation ratios and investor expectations, accounting and economic measures of profitability, corporate governance, and small business finance.