A closed form formula for equity valuation model based on differential equation

The valuation of intrinsic value is the cornerstone of value investing. When shareholders purchase a company’s stock, they are essentially buying claims to its future earnings and shareholder equity. Two main approaches, income-based and asset-based, have been developed to value equities. However, these approaches focus solely on either the company’s earnings or its shareholder equity. To overcome these limitations, the Growth Valuation Model (GVM) was developed, which incorporates both income and asset-based aspects. While the GVM model has the advantage of a closed-form solution, it still relies on a discrete infinite series, a common limitation of equity valuation models. In contrast, continuous differential equations have been successfully employed to develop a variety of option pricing models. However, there is a scarcity of equity valuation models derived using continuous differential equations, which is a gap in the current financial modeling landscape. This paper addresses this gap by developing a theoretical equity valuation model, the Continuous Growth Value Model (CGVM), based on a continuous differential equation, and deriving its closed-form solution. Our study concludes: (1) The GVM and CGVM models are developed under three discrete-typed and three continuous-typed assumptions, respectively. Therefore, they can be considered as the discrete and continuous versions of a valuation approach based on the same core principle: mean reversion of return on equity (ROE). (2) The reciprocal of the exponential decay rate in CGVM plays a similar role to the growth coefficient in GVM. (3) The theoretical curves between ROE and P/B ratio of CGVM and GVM are close to each other, and when ROE equals the required return rate, both models will have a P/B ratio of 1.0. (4) Only when the current ROE is greater than the required return rate, the persistence of ROE does have a positive effect on intrinsic value.