A Dynamic Model of Firm Valuation

• Published date: 01 August 2018
• Author: Natalia Lazzati,Amilcar A. Menichini
• DOI: http://doi.org/10.1111/fire.12164
• Date: 01 August 2018

The Financial Review 53 (2018) 499–531

A Dynamic Model of Firm Valuation

Natalia Lazzati∗

University of California Santa Cruz

Amilcar A. Menichini

Graduate School of Business and Public Policy,Naval Postgraduate School

Abstract

Wepropose a dynamic version of the dividend discount model, solve it in closed form, and

assess its empirical validity. The valuationmethod is tractable and can be easily implemented.

We find that our model produces equity value forecasts that are very close to market prices,

and explains a large proportion of the observed variation in share prices. Moreover, we show

that a simple portfolio strategy based on the difference between market and estimated values

earns considerably positive returns. These returns cannot be simply explained either by the

Fama-French three-factor model (even after adding a momentum factor) or the Fama-French

five-factor model.

Keywords: firm valuation, dividend discount model, Gordon Growth Model, dynamic pro-

gramming

JEL Classifications: G31, G32

∗Corresponding author: Department of Economics, University of California Santa Cruz, 1156 High St.,

Santa Cruz, CA 95064; Phone: (831) 459-5697; Fax: (831) 459-5697; E-mail: nlazzati@ucsc.edu.

We thank the financial support from the Center for Analytical Finance (CAFIN) at UC Santa Cruz, as

well as the research assistance of Luka Kocic. We also thank the helpful comments from Srinivasan

Krishnamurthy,an anonymous referee, Scott Cederburg, Daniel Chi, Joseph Engelberg, Chris Lamoureux,

Charles M. C. Lee, Jun Liu, Timothy McQuade, YuriTserlukevich, Zafer Yuksel,and seminar participants

at U Arizona, UC Santa Cruz, and UMass Boston. Finally, we thank Ken French for making the data

publicly available.

C2018 The Eastern Finance Association 499

500 N. Lazzati and A. A. Menichini/The Financial Review 53 (2018) 499–531

1. Introduction

Wederive a dynamic model of the firm in closed form and show that it can be used

for actual firm valuation.To test its empirical validity,we price firms in COMPUSTAT

in the period 1980–2015 and evaluatethe results from three perspectives. First, we find

that the model produces consistent forecasts of stock prices in the sense that model

predicted values are very close to the actual market values, on average. Second,

we show that the model explains a large fraction (around 92%) of the variation

in current market prices. Third, we find that the temporary or short-run deviations

between market prices and model estimates can be economically exploited. Overall,

we believe these results suggest our model is a valuable pricing tool that may enhance

current approaches to firm valuation.

We use dynamic programming to develop a model of the firm in which the

latter chooses investment, labor, and how to finance its assets in every period. While

this type of model has been used extensively in corporate finance to explain firm

behavior, we introduce three fundamental features that make our model particu-

larly useful for asset pricing purposes. First, we invoke the two-fund separation

principle, which shows that, as long as we discount future cash flows with an appro-

priately risk-adjusted discount rate, we do not need to specify shareholders’ utility

functions in the valuation process beyond the assumptions discussed in Cass and

Stiglitz (1970). Second, we allow the firm to grow in the long run, which could be

interpreted as the firm facing a market size that increases over time, independent

of the short-term fluctuations generated by the business cycle. Third, we intro-

duce risky debt to our model and find an analytic solution, which, to the best of

our knowledge, is novel among existing discrete-time dynamic investment models

of the firm. In particular, debt in our model is protected by a positive net-worth

covenant and, in the event of bankruptcy, the firm pays the bankruptcy costs, is

reorganized under Chapter 11 of the U.S. Bankruptcy Code, and continues its op-

erations. This modeling strategy generates a debt behavior that is in line with the

empirical evidence. For instance, survey results from Graham and Harvey (2001)

suggest that most firms have a target leverage. Consistently, the firm in our model

chooses debt in every period following a target leverage that depends on its own

characteristics.

As mentioned above, an important advantage of our model regarding valuation

is that we solve it analytically. Closed-form equations are strongly preferred to

numerical approximations because the former yield extremely accurate values at very

low computing time. Indeed, a usual problem with the numerical solution of dynamic

programming models is the so-called Bellman’scurse of dimensionality. This problem

arises from the discretization of continuous state and decision variables, since the

computer time and space needed increase exponentially with the number of points

in the discretization (Rust, 1997, 2008). Thus, more accurate firm valuations imply

necessarily exponentially longer periods of computing time. In addition, explicit

solutions allow the user to estimate model parameters with ease.

N. Lazzati and A. A. Menichini/The Financial Review 53 (2018) 499–531 501

After presenting the firm model, we evaluate its actual pricing performance.

We first compute the ratio of the actual market prices to the values predicted by our

model and find that its mean value is around 1. This result means that our model

yields equity value estimates that are, on average, very close to market values. We

regress the market value of equity on the value estimated by our model and find that

we cannot reject the null hypothesis that the intercept is 0 and the slope is 1, which

suggests that our model produces unbiased estimates of stock prices. We also show

that our model predictions can explain a large fraction of the observed variability of

the stock prices. Specifically, the R2coefficient is around 92%. This outcome turns

out to be better than the results reported by related papers (described below), and

implies a strong linkage between model forecasts and market values over time. We

implement this last regression which includes fixed effects at the industry level and

find that, jointly they are not statistically different from zero with a p-value close to

0.67. Moreover, we run an analogous regression at the firm level and find that we

cannot reject the null hypothesis that the intercept is 0 and the slope is 1 for about

71% of the firms at the 5% significance level and for 83% of the firms at the 1%

significance level.

While we show that our estimates are very close to market values, we also

find temporary deviations between stock prices and model estimates. We implement

a simple spread strategy to test whether we can take economic advantage of these

deviations. The spread strategy consists in ranking firms based on their ratios of

market prices to model estimates, then forming quintiles based on those ratios, and

finally buying the firms in the lowest quintile and selling the firms in the highest

quintile. Our results show that this strategy earns, on average, around 20%, 34%,

41%, 44%, and 48% returns after one, two, three, four, and five years of portfolio

formation, respectively.We also study the returns of the spread strategy in the context

of the Fama-French three-factor model (Fama and French, 1993) and find a positive

alpha that is statistically different from zero. This means that the positive returns of

our spread strategy cannot be simply accounted for by these factors. Weobtain similar

results when we add a momentum factor to the Fama-French three-factor model and

when we employ the Fama-French five-factor model (Fama and French, 2015). As

benchmark, we calculate the returns of the spread strategy based on three well-

known financial ratios: market-to-book, price-earnings, and price-dividend. Based

on our model, we find that the spread strategy estimates consistently outperforms the

ones using these ratios.

To do the previous analyses, we estimate the structural parameters at the firm

level for all firms in our data set. In all our estimations, we perform a forward-looking

exercise in the sense that we use data available prior to the valuation period to make

out-of-sample predictions. Doing so is important because this procedure replicates

the situation a user would face when performing actual valuation.

This paper values public firms in COMPUSTAT. However,our valuation method

could also be implemented with firms for which market prices do not exist, as long

as financial statements are available for parameter estimation. These cases include,