Deriving the H-Model Dividend Valuation Approximation

# How to approximate a function in the H-model


# How to approximate a function in the H-model












I have been looking to understand the H-model in finance, that is used for stock price valuation. In particular, I wanted to formally derive the final formula:

$$PV=\frac{D}{r-g_2}\left[1+g_2+\frac{H}{2}(g_1-g_2)\right]$$

Here $PV>0$ is the present value (price) of the stock, $D>0$ is the constant dividend payment that is paid forever, $r\in(0;1)$ is the required rate of return on the stock and the growth rate of the stock follows a pattern: it starts with a growth rate of $g_1\in(0;1)$ and at time $H\ge0$ the growth rate switches to $g_2<g_1$, $g_2\in(0;1)$. When $H=0$ the model is equivalent to the Gordon growth model, where we simply evaluate a perpetuity (perpetual stream of discounted and growing dividends). In general the present value should be

$$PV=D\frac{1+g_1}{1+r}+D\left(\frac{1+g_1}{1+r}\right)^2+...+D\left(\frac{1+g_1}{1+r}\right)^{H-1}+D\left(\frac{1+g_1}{1+r}\right)^{H}+D\frac{(1+g_1)^{H}(1+g_2)}{(1+r)^{H+1}}+D\frac{(1+g_1)^{H}(1+g_2)^2}{(1+r)^{H+2}}+...=D\sum_{i=1}^{H}{\left(\frac{1+g_1}{1+r}\right)^i}+{D\left(\frac{1+g_1}{1+g_2}\right)^{H}\sum_{i=H+1}^{\infty}{\left(\frac{1+g_2}{1+r}\right)^i}}$$ I evaluated it to be $$PV=D\frac{1+g_1}{r-g_1}\left(1-\left(\frac{1+g_1}{1+r}\right)^{H-1}\cdot\frac{g_1-g_2}{r-g_2}\right)$$ The paper I found on https://wenku.baidu.com/view/07ef434ae45c3b3567ec8b84.html agrees with me, and it states that the expression I computed can be approximated using the formula of the H-model I showed at the beginning, but does not give a derivation. The paper only gives a strange footnote 'The formal derivation of the H-model ... is available from the authors', but no further elaboration is given. I would be glad to receive any hints as to what technique one should use in order to arrive at the final result

## Answer by William Chong (score 2, accepted)

https://quant.stackexchange.com/a/85785

Sorry that it might be too late for an answer. I got to the same question and finished the derivation. I used Taylor series and expanding centering at $r=g_a$. I claim no originality to the derivation though.

Here is the link where I temporarily store the note containing the derivation (in the title of "A Derivation of an Approximation in H-Model") if you wish to see it in PDF.

## A Derivation of an Approximation in H-Model

#### Abstract

This note is a derivation of a formula of the present value in H-model proposed by Fuller-Hsia.

#### 1. Introduction

Fuller and Hsia [FH84] have developed the famous stock valuation model, H-model, for valuating equity using a simple formula [FH84, Eq 7] as follow:

$$\tag{1} P_0 = \frac{D_0}{r - g_n} \big[ (1 + g_n) + A(g_a - g_n) \big]$$

where $P_0$ is the present value, $D_0$ is the current dividend, $g_a$ is the short-term growth rate, $g_n$ is the long term growth rate, $r$ is the required rate of return and $A$ is the length of the fast-growing period in two-step model.

They mentioned in the text that Equation (1) is an approximation of the following equation of the present value of the ordinary two-step model [FH84, Eq 6]:

$$\tag{2} P_0 = \frac{D_0(1 + g_a)}{r - g_a} \left[ 1 - \left( \frac{1 + g_a}{1 + r} \right)^{A-1} \left( \frac{g_a - g_n}{r - g_n} \right) \right]$$

In this text, we show a derivation of an approximation from Equation (2) to Equation (1) by taking the first Taylor approximation at $r = g_a$ for the function $\left(\frac{1+g_a}{1+r}\right)^{A-1}$. We claim no originality of the proof of the approximation. Fuller and Hsia mentioned that "the formal derivation of the H-model as well as the N-step model is available from the authors" [FH84, Footnote 6]. This note only serves as a reference of a derivation since the derivation seems to be difficult to find online (see e.g. [Iva], [Man]).

#### 2. Taylor Expansion

For this text, we use the first-degree approximation of Taylor expansion.

Theorem 2.1. Let $a, x \in \mathbb{R}$. Let $f$ be a twice differentiable function on the open interval between $a$ and $x$. Then:

$$f(x) = f(a) + f'(a)(x - a) + \frac{1}{2}f''(k)(x - a)^2$$

for some real number $k$ between $a$ and $x$.

Theorem 2.1 gives an approximation as follows:

$$\tag{3} f(x) \approx f(a) + f'(a)(x - a)$$

for $a, x \in \mathbb{R}$ close to each other and $f$ twice differentiable on the open interval between $a$ and $x$.

Example 2.2. Let $f(r) = \left(\frac{1+g_a}{1+r}\right)^{A-1}$, where $g_a > -1$ and $A \ge 1$ are constants. Then $f$ is twice differentiable in $(-1, +\infty)$.

We calculate the derivative of $f$ at $g_a$:

$$f'(r) = (A - 1)\left(\frac{1 + g_a}{1 + r}\right)^{A-2} \cdot (-1) \left(\frac{1 + g_a}{(1 + r)^2}\right) = -\frac{(A - 1)(1 + g_a)^{A-1}}{(1 + r)^A}$$

$$f'(g_a) = -\frac{(A - 1)(1 + g_a)^{A-1}}{(1 + g_a)^A} = -\frac{A - 1}{1 + g_a}$$

We then approximate $f(r)$ by Taylor approximation centered at $r = g_a$ using Equation (3):

$$\tag{4} f(r) \approx f(g_a) + f'(g_a)(r - g_a) = \left(\frac{1 + g_a}{1 + g_a}\right)^{A-1} - \frac{A - 1}{1 + g_a}(r - g_a) = 1 - \frac{(A - 1)(r - g_a)}{1 + g_a}$$

#### 3. Derivation

We use the result in Section 2 to derive the approximation from Equation (2) to Equation (1).

By Example 2.2, we can approximate Equation (2) as:

$$ \begin{aligned} P_0 &= \frac{D_0(1 + g_a)}{r - g_a} \left[ 1 - \left( \frac{1 + g_a}{1 + r} \right)^{A-1} \left( \frac{g_a - g_n}{r - g_n} \right) \right] \\ &\approx \frac{D_0(1 + g_a)}{r - g_a} \left[ 1 - \left( 1 - \frac{(A - 1)(r - g_a)}{1 + g_a} \right) \left( \frac{g_a - g_n}{r - g_n} \right) \right] \quad \text{(By Equation (4))} \end{aligned} $$

We then expand the brackets:

$$ \begin{aligned} P_0 &\approx \frac{D_0(1 + g_a)}{r - g_a} \left[ 1 - \left( 1 - \frac{(A - 1)(r - g_a)}{1 + g_a} \right) \left( \frac{g_a - g_n}{r - g_n} \right) \right] \\ &= \frac{D_0(1 + g_a)}{r - g_a} \left[ 1 - \frac{g_a - g_n}{r - g_n} + \frac{(A - 1)(r - g_a)}{1 + g_a} \left( \frac{g_a - g_n}{r - g_n} \right) \right] \\ &= \frac{D_0(1 + g_a)}{r - g_a} \left[ \left( 1 - \frac{g_a - g_n}{r - g_n} \right) + \frac{(A - 1)(g_a - g_n)}{r - g_n} \left( \frac{r - g_a}{1 + g_a} \right) \right] \\ &= D_0 \cdot \frac{1 + g_a}{r - g_a} \left[ \frac{r - g_a}{r - g_n} + \frac{(A - 1)(g_a - g_n)}{r - g_n} \left( \frac{r - g_a}{1 + g_a} \right) \right] \\ &= D_0 \left[ \left( \frac{1 + g_a}{r - g_a} \cdot \frac{r - g_a}{r - g_n} \right) + \left( \frac{1 + g_a}{r - g_a} \cdot \frac{(A - 1)(g_a - g_n)}{r - g_n} \cdot \frac{r - g_a}{1 + g_a} \right) \right] \\ &= D_0 \left( \frac{1 + g_a}{r - g_n} + \frac{(A - 1)(g_a - g_n)}{r - g_n} \right) \\ &= \frac{D_0}{r - g_n} \big( 1 + g_a + (A - 1)(g_a - g_n) \big) \\ &= \frac{D_0}{r - g_n} \big( 1 + g_n + (g_a - g_n) + (A - 1)(g_a - g_n) \big) \\ &= \frac{D_0}{r - g_n} \big( 1 + g_n + A(g_a - g_n) \big) \end{aligned} $$

which is the same as Equation (1).

#### References

- [FH84] Russell J. Fuller and Chi-Cheng Hsia. A simplified common stock valuation model. Financial Analysts Journal, 40(5):49–56, 1984.

- [Iva] Mr. Ivan. How to approximate a function in the H-model. Retrieved from How to approximate a function in the H-model on August 18, 2026.

- [Man] Management Study Guide. Dividend discount valuation: The H model. Retrieved from https://www.managementstudyguide.com/dividend-discount-valuation-h-model.htm on August 18, 2026.