A General Affine Earnings Valuation Model

We introduce a methodology, with two applications, that incorporates stochastic interest rates, heteroskedasticity and risk aversion into the residual income model. In the first application, goodwill is an affine (constant plus linear term) function where the constant and linear coefficients are time-varying. Homoskedastic risk gives rise to a constant risk premium, while heteroskedastic risk gives rise to linear state-dependent risk premiums. In the second application, we present a class of models where a non-linear function for the price-to-book ratio can be derived. We show how interest rates, risk, profitability and growth affect the price-to-book ratio. This paper provides a parametric class of models that shows how a firm’s market value relates to accounting data under stochastic interest rates, heteroskedasticity and adjustments for risk aversion. We use the framework of the Residual Income Model (RIM), which expresses the value of a stock as the firm’s book value plus the expected future discounted value of the firm’s abnormal (or residual) earnings. Our methodology builds on the framework of Feltham and Ohlson (1999), who extend the RIM to a no-arbitrage setting to accommodate time-varying interest rates and risk aversion. Feltham and Ohlson give a partial parametric model of stock valuation using accounting information in this setting. In a heteroskedastic environment with stochastic interest rates and risk aversion, we extend this analysis in several ways. First, we apply this methodology to the case where the dynamics of accounting variables are expressed in dollar amounts as in Feltham and Ohlson (1995). Second, we derive a solution for the priceto-book ratio of a firm as a function of stochastic interest rates, accounting rates of return and growth in book. Our first result extends the Linear Information Model (LIM) developed in Ohlson (1995) and Feltham and Ohlson (1995). The LIM presents firm value as a linear function of current observable accounting information and is derived under constant discount rates. This assumption leads to a standard simplification where a single discount factor can be applied to all future periods. The discount factor can incorporate an ad hoc adjustment for risk. There are certain questions, however, which cannot be addressed under this assumption. For instance, is it always possible to incorporate risk aversion as a spread in a constant discount factor? Can a linear solution be found under the addition of time-varying interest rates, heteroskedasticity and risk aversion? Feltham and Ohlson (1999) show how to adjust the RIM for risk but do not provide a complete parametric model to answer these questions directly. They hint, however, that a model as tractable as the LIM might exist under more general conditions. We propose a class of models where an extended Feltham-Ohlson linear form can be preserved under stochastic interest rates and time-varying risk premiums. Under risk neutrality and constant interest rates, our model reduces to the Feltham-Ohlson LIM. Our extension to the LIM expresses firm value as an affine combination (constant plus linear