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In a recent Seeking Alpha article (here), I explored how long it takes for dividend income to double when an investor reinvests the dividends. The time to double was explored in terms of initial dividend yield, share price appreciation, and dividend growth rate. That work served a mainly theoretical purpose in that it allowed one to see how each of the different factors (initial yield, share price appreciation, and dividend growth rate) influenced the time for dividend income to double. It lacked, however, for practicality as a tool to serve investors who are looking to make rough predictions for future dividend income when reinvesting dividends.

In this article, I look at the time to double dividend income as a function of the dividend growth rate and the yield the security offers upon reinvestment. After developing the tool, I illustrate with an example from my own portfolio.

## Developing a Rule of 72 for Dividend Reinvestors

In this section, I derive a mathematical equation giving the time to double dividend income as a function of the dividend growth rate and the yield at which reinvestment occurs. This section can safely be skipped to the end for those just interested in the result.

I assume an investor purchases S0 shares of a security that pays D0 dollars in dividend income per share. Such an investor expects to receive S0D0 dollars of dividend income in the first year. Going forwards, we expect the dividend per share to grow at a rate given by DGR. We also reinvest at the prevailing yield at the time of reinvestment, which we denote by Y.

Under the above assumptions, we can model the dividend per share via

dD/dt = DGR,

which gives the equation D(t) = D0e(DGR)t where D(t) represents the dividend per share paid in year t. Similarly, we have

dS/dt = Y,

which gives the equation S(t) = S0eYt where S(t) represents the number of shares owned in year t. Since we are looking to determine when the income provided by the initial investment has doubled, we need to find the time t at which the dividend paid is 2S0D0 = S(t)D(t). This occurs when

t = ln(2) / [DGR + Y].

Approximating ln(2) by .72 yields the familiar "Rule of 72" for doubling time where we divide 72 by the sum of the projected dividend growth rate and the projected yield upon reinvestment. (Note: 72 is chosen instead of a more accurate approximation as it has a lot offactors, which makes it a convenient numerator for mental computation)

An Example: Procter & Gamble (PG)

To illustrate how we might apply this tool, I chose an example from my own portfolio. I purchased shares of PG on July 9, 2012 at a cost basis of \$61.38 / share. At the time, this equated to a dividend yield of 3.66%. Morningstar reports the 5-year dividend growth % of PG as being around 10.8%. In addition, Morningstar reports that PG's dividend yield was 2.9% on average over the last five years.

If we project forwards and assume that PG's dividend will grow annually at a rate of 8%, and that the market will price PG such that reinvestment occurs at the average yield of 2.9%, we would expect the yield on cost of the position to double to 7.32% in between 6 and 7 years, computed via 72/10.9.

In fact, we can go one step further since we can look back to the derivation of this tool and make adjustments as necessary. Suppose we want to know at what point in time we would achieve a yield on cost of 10% given the same assumptions as above. In this case, we need the income produced to grow by a factor of 2.73. Replacing ln(2) by ln(2.73) in the formula given above and taking the anticipated dividend growth and reinvestment yield as above, we would expect to achieve 10% yield on cost in just over 9 years, computed via ln(2.73)/.109.

## Concluding Remarks

Do I expect to open my brokerage statement in 9 years and find that my yield on cost of PG (should I still hold it) is precisely at 10%? Certainly not. There are a number of factors that make this projection very imprecise. We have to predict two numbers pertaining to the future: dividend growth rate and the yield at the time of reinvestment. The second number relies upon Mr. Market behaving as we predict...On the bright side, should the yield upon reinvestment be lower than projected (but dividend growth achieve expectation), we will know it is because of share price appreciation.

Overall, I view this tool as being useful to understand the power of compounding and to help an investor maintain a long-term perspective. Also, it can be used as a yardstick to measure investment success in the future. Should I hold PG long-term and have it achieve a 10% yield on cost in less than 9 years, I will be quite happy. If in the future it looks like this milestone will be very unlikely to occur in year 9, it could prompt me to revisit my investment decision with a more critical eye.

On a final note, those who include the "Chowder Rule" as part of their investment process may be pleased to see that the sum of dividend growth rate and current yield is precisely the key factor that drives this formula.