Deriving the Geometric Dividend Growth Valuation Formula

Summary

The document explains why a dividend discount model with constant growth and a constant required return can be reduced from an infinite series to a closed-form expression. The key step is to factor out the initial dividend and recognize that each discounted growing dividend contributes a constant ratio relative to the previous term. The remaining sum is a geometric series, which converges when the growth rate is below the required return.

The answer illustrates the algebra under an additional simplifying assumption that the dividend amount used as the starting value is fixed, then evaluates the series and rewrites the result in terms of the next dividend. It also points out that the original notation leaves an index in the final expression even though the summation has been performed. The exchange is a brief derivation rather than a discussion of how to estimate growth or discount rates, and the closed form depends on the model’s constant-rate assumptions; it should not be read as evidence that actual dividends follow them.

Key ideas

• Constant growth and discount rates make each term of the dividend valuation series part of a geometric sequence.
• The infinite series converges when the growth rate is less than the required return.
• Factoring out the starting dividend leaves a ratio that can be summed in closed form.
• The resulting valuation uses the next period’s dividend and has no remaining time index.
• The formula depends on constant growth and return assumptions that may not hold in practice.

Tags

# Sum disappearing when we assume constant some elements to be constant over time


# Sum disappearing when we assume constant some elements to be constant over time












I have the dividend discount model, which is the following expression:

$$ P_{j,t} = \sum_{\tau=1}^{\infty}D_\tau(1+g)^\tau(1+r)^{-\tau}=\frac{D_{\tau+1}}{r-g} $$

Where $D_t$, is the dividend at time $t$ ,$g$ represents the constant growth over time and $r$ represents the required rate of return which is assumed to be constant over time too.

My questions are: why under constant growth and constant required rate of return can we re-write it this way. Why does the sum sign disappears?

EDIT 1:

@ZRH's solution is correct. I also found a website with more intermediate steps here: http://www.calculatinginvestor.com/2011/05/18/gordon-growth-model/

Thank you

## Answer by ZRH (score 1, accepted)

https://quant.stackexchange.com/a/44752

Using Alex C's link, and further assuming the dividends $D_\tau$ to be constant (else you cannot really come up with a simple formula) you get:

$P=D\sum_{\tau=1}^\infty \left(\frac{1+g}{1+r}\right)^{\tau}=\frac{\frac{1+g}{1+r}}{1-\frac{1+g}{1+r}}=D\frac{1+g}{r-g}$

I presume that in your above formula, you mean $D(1+g)$ when you write $D_{\tau+1}$, as after executing the summation there should not be an index $\tau$ anymore

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.