Abstract
The interaction between the risky growth of investment and future earnings plays a central role in
predicting stock returns, giving rise to an investment puzzle associated with the risk inherent in
growth. Our parsimonious theoretical model provides a connection between stock returns and
uncertainties in investment growth, company profitability, book-to-market ratio, and earnings
systematic risk. This analytic approach to the determinants of stock returns contrasts with
conventional factor models that augment the simple CAPM model with empirically determined
accounting variables. Our analysis sheds light on several empirical anomalies resulting from
interactions between explanatory variables used in empirical analysis and offers insights into
the nature and structure of the book-to-market factor.
Introduction
Empirical studies in finance and accounting frequently document a negative relationship between
growth in investments and subsequent stock returns (e.g. Loughran and Ritter 1995, Ikenberry et al.
1995, Daniel and Titman 2006, Fama and French 2008, Pontiff and Woodgate 2008, Penman and
Zhu 2022). We argue that this investment anomaly can largely be explained by the failure to adequately
incorporate a growth risk factor in empirical investigations. Market returns and investment growth
covary with the economy and each other, introducing a risk term associated with investment growth.
To establish a link between future stock returns and uncertainty in investment growth, we
focus on an all-equity firm and adopt an approach to valuation similar to that carried out by
fundamental analysts, by attributing a market value of equity to the reported accounting
values of equity together with forecasts of future earnings and growth based on firm-specific
and economic circumstances.1 We base this valuation on the estimated present value of
future cash disbursements (or net dividends) to shareholders using a risk-free rate as a dis
count rate and then adjusting for the correlation risk between market returns and fundamental
accounting variables (Rubinstein 1976, Feltham and Ohlson 1999; Pope and Wang 2000). In
the CAPM framework, the difference between the expected (market) return and the risk-free
rate is attributed to the (market) risk premium, where the risk-free rate is an asset’s return in a
risk-neutral economy. This motivates us to introduce the certainty equivalent growth rate,
which is the expected growth rate adjusted by market risk or, equivalently, the growth rate
in a risk-neutral economy. This procedure generates a growth risk premium, defined as the
difference between the expected long-run growth rate and the certainty equivalent growth
rate.2 Given the dynamics of investments and profitability, equity values are then expressed
in terms of accounting fundamentals, including the book value of equity and earnings,
together with adjustments for the risk associated with current earnings and for that associated
with future growth. Based on our equity valuation model, we produce values for stock returns,
where returns in excess of the risk-free rate are expressed as a function of an earnings risk
premium together with a growth risk premium.
Our modelling extends the original valuation models in Ohlson (1995) and Feltham and
Ohlson (1995) based on a constant discount factor, and models in Feltham and Ohlson (1999),
Nekrasov and Shroff (2009) and Lyle et al. (2013) that incorporate the covariance risk
between (abnormal) earnings and market risk in a setting with stochastic discount factors.
Accounting valuation literature often implicitly assumes a time-invariant growth rate, resulting
in models overlooking the property of risky growth and particularly its correlation with other
fundamental determinants of value. Penman and Zhu (2014, 2022) argue that expected return
is indicated by both expected profitability and the growth risk of that profitability. They
provide empirical evidence to show that a variable that forecasts earnings and investment
growth also predicts the expected stock return. More recently, Peng et al. 2024 also show that
information other than bottom-line accounting numbers reflecting investment growth and earn
ings risk is useful for predicting stock returns. However, research has so far had only limited
success in developing an analytic model to separate the two value attributes. In fact, with a
few exceptions, the theoretical literature offers little guidance on the growth risk factor and
the determinants of the growth risk premium. Hou et al. (2015) develop a linear q-factor
model underpinning the factors on profitability and contemporaneous growth of assets on a
static investment framework, and Hou et al. (2021) extend their model to a dynamic setting,
where they choose investment to maximise equity value given a company’s earnings. As a con
sequence, the interaction between the two value attributes, investments and profitability, is neg
lected, and the analytical determinants of growth risk premium are unclear.
The growth risk premium term in our model helps us to explain the investment anomaly. It
suggests that if a variable is positively (negatively) related to future earnings-to-price, then it will
have a negative (positive) growth risk premium associated with future returns. The investment
anomaly can be intuitively explained by investigating the relative magnitudes of the anticipated
accounting rate of return (ROE) and a proxy for the cost of equity capital after controlling for the
growth risk premium. When the expected ROE is larger than the perceived cost of capital or the econ
omic spread is greater than zero, investment growth will increase firm values. As a result, investors
need a smaller growth risk premium and, hence, a lower return for higher-growth firms. Therefore,