[Paper Review] Generalized semi-Markovian dividend discount model: risk and return

Guglielmo D’Amico|arXiv (Cornell University)|May 9, 2016

Stochastic processes and financial applications20 references3 citations

TL;DR

This paper introduces a generalized discrete-time dividend discount model where the dividend growth rate follows a semi-Markov process on a Borel state space, enabling flexible modeling of stochastic dividend dynamics. It establishes sufficient conditions for price finiteness and transversality, derives exact equations for first- and second-order price-dividend ratios, and provides approximation methods for computing fundamental prices and risk measures.

ABSTRACT

The article presents a general discrete time dividend valuation model when the dividend growth rate is a general continuous variable. The main assumption is that the dividend growth rate follows a discrete time semi-Markov chain with measurable space. The paper furnishes sufficient conditions that assure finiteness of fundamental prices and risks and new equations that describe the first and second order price-dividend ratios. Approximation methods to solve equations are provided and some new results for semi-Markov reward processes with Borel state space are established. The paper generalizes previous contributions dealing with pricing firms on the basis of fundamentals.

Motivation & Objective

• To generalize existing dividend discount models by replacing deterministic or Markovian assumptions on dividend growth with a semi-Markov process on a Borel state space.
• To establish sufficient conditions ensuring the finiteness of fundamental stock prices and the satisfaction of the transversality condition, thus eliminating speculative bubbles.
• To derive explicit equations for the first- and second-order moments of the price-dividend ratio to assess risk and return.
• To develop approximation methods for solving the resulting equations in the context of semi-Markov reward processes with Borel state space.
• To unify and extend prior models, including Gordon-Shapiro, multistage, and Markov-switching models, under a single semi-Markov framework.

Proposed method

• Models the dividend growth rate as a discrete-time semi-Markov chain on a Borel measurable space $(E, \mathcal{E})$, allowing for general continuous dynamics.
• Uses a semi-Markov kernel $Q(x, A, t)$ to describe the transition probabilities and holding times, enabling memory-dependent transitions.
• Derives the fundamental price equation as the discounted expectation of future dividends, with the price process defined recursively via $P(t) = \mathbb{E}[D(t+1) + P(t+1)] / r$.
• Introduces the first- and second-order price-dividend ratios through recursive equations involving the semi-Markov transition kernel and moment-generating functions.
• Applies the Cauchy-Schwarz inequality and limit analysis to verify the transversality condition and ensure price finiteness.
• Proposes numerical approximation schemes based on the structure of the semi-Markov kernel and moment-generating functions for practical computation.

Experimental results

Research questions

• RQ1 Under what conditions is the fundamental stock price finite when the dividend growth rate follows a semi-Markov process with a Borel state space?
• RQ2 How can the transversality condition be satisfied in a semi-Markov dividend model to rule out speculative bubbles?
• RQ3 What are the exact equations governing the first- and second-order price-dividend ratios in this generalized framework?
• RQ4 How can the second-order moment of the price process be computed to quantify stock risk in the semi-Markov setting?
• RQ5 In what way does this model generalize and unify prior models such as the Gordon-Shapiro, multistage, and Markov-switching dividend discount models?

Key findings

• Sufficient conditions for price finiteness and transversality are established using the decay of moment-generating functions and limit analysis of discounted moments.
• The first-order price-dividend ratio satisfies a recursive equation involving the semi-Markov kernel and the expected growth rate.
• The second-order moment of the price process, representing risk, is derived through a system of equations involving $\psi_1(g,v)$ and $\psi_2(g,v)$, with explicit expressions in terms of integrals over the state space.
• The risk measure $p^2(d,g,v)$ is shown to be proportional to $d^2$, with the proportionality factor depending on the semi-Markov kernel and moment-generating functions.
• The model generalizes previous results, including those of D’Amico (2013), by allowing continuous dividend growth rates and more flexible dynamics.
• Approximation methods are provided for solving the moment equations, particularly through recursive computation of $\psi_1$ and $\psi_2$ using the transition kernel and density functions.