Calculating First Moments and Confidence Intervals for Generalized Stochastic Dividend Discount Models

This paper presents models of equity valuation where future dividends are assumed to follow a generalized Bernoulli process consistent with the actual dividend payout behavior of many firms. This uncertain dividend stream induces a probability distribution of present value. We show how to calculate the first moment of this distribution using functional equations. As well, we demonstrate how to calculate a confidence interval using Monte Carlo simulation. This first moment and interval allows an analyst to determine whether a stock is overor under-valued.


Introduction
Dividend discount models are a common feature of most introductory finance textbooks. Normally the Gordon Growth Model [1] is one of the techniques used to illustrate the calculation of the cost of equity capital. Gordon assumes that the dividend stream will increase at a constant geometric growth rate in perpetuity. There have been variations of this model. All of them assume that future dividends will follow a fixed, mechanistic path. These deterministic dividend streams are not really consistent with the payout policies of most firms. Typically a firm will hold a dividend constant until such time as it can see itself being able to increase it and then maintain the increase. Such dividend streams are characterized by uncertainty in that an analyst would never be certain when and by how much a dividend was going to increase. Hurley and Johnson [HJ, [2][3][4] have offered a series of dividend discount models consistent with these two characteristics. A good summary of these models can be found in Hurley and Fabozzi [5].
The contribution of this paper is a complete generalization of these models. I model two components of dividend uncertainty. First, in each period, the dividend is assumed to either increase or stay the same with a given probability. Second, if the dividend does increase, the increase is also modeled as a random variable. In addition, I also consider two functional forms for the dividend increase: in one, the increase is multiplicative; in the other, it is additive. For each valuation model, I show how to compute a first moment and a confidence interval. This allows an analyst to compare the current market price of the stock to this moment and interval and in so doing determine whether or not the share is over-or under-valued.

Stochastic Dividend Discount Models
Going back to J. B. Williams [6], the value, of a common share can be written where t is the dividend in period t, and k is the discount rate. Gordon [1] made the assumption that future dividends would increase at a geometric rate in perpetuity, and, under this assumption, value can be shown to be 1 .
There have been variations of Gordon's assumption. Like Gordon's, all of them posit a future dividend stream that is known and increasing. Table 1 shows a dividend stream for a hypothetical firm, ABC Corp. This dividend payout pattern has all the characteristics of real-world dividend streams. Note that, each period, the dividend either stays the same or increases. I model this stochastic behavior by assuming that the dividend payment in period t is given by 1 1 with probability with probability 1   is a random growth rate, and p is the g E g   I term this dividend process a Geometric Bernoulli Process (GeoBP) and denote its present value by To get its first moment, .
we need to solve the following functional equation: The first term on the right-hand side is the case where the dividend stays the same and the value in one period's time, is discounted to the present by dividing by 1 The second term is the case where the dividend increases by a random amount γ. Hence the value in one period's time is and this has to be discounted one period and integrated over   f  to get an expected present value. Finally these two present values are weighted by 1 -p and p respectively.
In addition, I model an Additive Bernoulli Process (AddBP) where the dividend payment in period t is given by 1 1 with probability with probability 1 where is a random additive increment in the dividend, and p is the probability the dividend increases.
This equation has the same structure as (6) except that the dividend is additive rather than geometric. Both Equations (6) and (8) are functional equations and their solution is discussed in the next section.
Computing first moments provides a point estimate of value and this is certainly beneficial. But the more useful calculation within this modeling structure is a confidence interval for value. Effectively a confidence interval gives a band-width for value and the existing stock price (and indeed the recent history of the stock price movement) can be assessed against this interval to determine whether the share is over-or under-valued. If it were true that the distributions of present value under the assumed stochastic processes were approximately normal, we could calculate the second moments, and using variations of (6) and (8), then get variances and confidence intervals. However these processes tend to give rise to distributions that are skewed and, hence, the normal distribution does not usually apply. For this reason, we resort to computing confidence intervals using a Monte Carlo procedure which we detail herein.

Finding First Moments
To solve the functional Equations in (6) and (8), we need the following lemmas. Lemma 1. The functional equation where     Proof. Substitute the solution in (11) into the righthand side of (9) to get: We then substitute result into the right-hand side of (9) for This gives where y E denotes expectation over Continuing this substitution and evaluation process more times yields . Proof. Substitute the solution in (19) into the righthand side of (17): 1