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# How To Estimate The Average Dividend Growth Rate

Estimating the Average Dividend Growth Rate

By Douglas D. Rose

I often wish to calculate an appropriate average dividend growth figure across a series of years. The figure might help project income from current stocks, help choose among stocks based on expected income, or project the effects of reinvested dividends for retirement.

The right way to do this is to sum up all the dividends across the series, divide by the dividend the year before the series starts, and then model this as a consequence of a single rate of dividend growth. That is, I pretend that a single, average rate of dividend growth has produced the sum of the dividends, then estimate that average from the sum.

It is relatively easy to calculate cumulative dividends when the average dividend growth rate is known. The formula is

((1+g)(n+1) -1)/g -1

times the dividend in the year preceding the series

where g is the dividend growth rate and n is number of years in the series.

Unfortunately, working backward from the cumulative dividend - which can from published reports - to the average growth rate is not easy. A direct solution is impractical for series longer than a couple of years, as the model includes a polynomial of degree equal to the number of years and there is no easy way to estimate the desired rate of growth. Lacking a direct estimate, an approximate figure is used.

What may be the most commonly used approximation is the average of the dividend growth rates in each year in the series. For each year in the series, the dividend growth rate is the dividend for that year divided by the dividend for the prior year, minus one. The simple average of rates is not a good estimate if the rates vary from year to year; if they don't differ, any rate will be the right one. When, for instance, the initial rate and the final rate differ from the average rate by the same amount but in contrasting directions, the simple average of annual dividend growth rates underestimates the cumulative dividend by a multiplicative term that increases as the difference of the initial year from the average grow, that increases as the average dividend growth rate grows, that grows as the length of the series grows, and that grows because all these differences are multiplied.

A better estimate is a weighted average of annual dividend growth rates, where the initial year's dividend growth is weighed by the number of years in the series, the next year by one less, and so on to the final year which is weighed at one. The weighted total is divided by sum of weights, which is the number of years times one larger than the number of years, divided by two. Commonly, this provides the accurate percentage but is often off in the next decimal.

The error in the weighted average is of the opposite sign as the error in the simple average. That is, the weights overcorrect the simple average. So, a good overall estimate is to combine one part simple average to eleven parts weighted average. That usually is accurate to .15%, based on a Monte Carlo simulation with 175 trials of 10 randomly chosen growth rates in the -4% to 20% range, in series of 1 to 10 years for averaging.

The mathematically sophisticated can quickly get an answer to any desired degree of accuracy using tools for converging iterations. For the rest of us, the weighted average or the combined average will probably do.

An even more accurate estimate can be based on treating the first few terms of the cumulative dividend function as a quadratic equation. In this approach, the growth rate is

a + √(a2 - b*(n-c))

where a = -3/(2*(n-1))

b = 6/((n+1)*n*(n-1))

n = number of years

c = cumulative dividends divided by dividend in the year preceding the series

The quadratic provides a good estimate for dividend growth averages across two to five years.

One final method of note is to take the nth root of the ratio of the final dividend to the dividend in the year preceding the series of n years, then subtract one. As long as the rates don't differ much or monotonically decline, this is not much worse than most other methods.

In general, I recommend the combined estimate.

The table below presents an example of how the different estimates behave across a series of one to ten years.

 Year Dividend growth rates Cumulative dividend True average growth rate Simple average Weighted average Combined estimate Quadratic estimate Nth root estimate 1 11% 1.11 11.0% 11.0% 11.0% 11.0% 11.0% 11.0% 2 20% 2.44 14.1% 15.5% 14.0% 14.1% 14.1% 15.4% 3 5% 3.84 12.9% 12.0% 13.0% 12.9% 12.9% 11.8% 4 0% 5.24 11.1% 9.0% 11.4% 11.2% 11.1% 8.7% 5 14% 6.83 10.6% 10.0% 10.9% 10.9% 10.7% 9.8% 6 10% 8.59 10.3% 10.0% 10.4% 10.4% 10.5% 9.8% 7 0% 10.34 9.8% 8.6% 10.1% 10.0% 10.0% 8.4% 8 20% 12.45 9.8% 10.0% 10.1% 10.1% 10.0% 9.7% 9 17% 14.91 10.0% 10.8% 10.2% 10.3% 10.3% 10.5% 10 13% 17.69 10.2% 11.0% 10.4% 10.4% 10.6% 10.8%

 Year Quadratic estimate Nth root estimate 1 11.0% 11.0% 2 14.1% 15.4% 3 12.9% 11.8% 4 11.1% 8.7% 5 10.7% 9.8% 6 10.5% 9.8% 7 10.0% 8.4% 8 10.0% 9.7% 9 10.3% 10.5% 10 10.6% 10.8%