# The commutative property of dividends

Sometimes it is good to get back to basics. After a recent conversation with a family member, an elementary school teacher, about the school’s decision to rename improper fractions into something else (no, I’m not joking) I ended up thinking about some dividend basics. One of the basic properties in math, the commutative property, also applies to dividends and offers some interesting insights for dividend investors of all types.

Does anyone remember what the commutative property is? It basically says that in a mathematical operation you can move (commute) stuff around without changing the result. So, 2 + 3 = 3+2. Pretty basic. Remember now that the formula for dividends returns, what I called the magic dividend formula, says that Total return = Div yield + Div growth + Change in valuation. Ignoring change in valuation for now, the commutative property says that it does not matter what the composition of total return is. In other words, a stock with a 6% yield and 4% growth is the exact same as a stock with a 4% yield and 6% growth. Seems obvious at face value. I wondered about this when I saw how dividend investor’s opinions strongly fell into one camp or the other. There is a very strong growth over yield camp and an equally a strong yield over growth camp. Why is this considering the commutative property of dividends? Its turns out that it matters what kind of dividend investor you are and where you do your dividend investing from.

First, for dividend investors in non-taxable accounts (IRAs, 401Ks, etc..) and that are re-investing dividends, i.e. are focused on wealth building, then the focus should be an total return. The composition of that return is much less important. The commutative property of dividends applies 100% and a 6% + 4% stock is the same as a 4% + 6% stock. I find that when div growth or div yield advocates disagree on a particular investment the disagreement is often over stocks with different total returns and has nothing to do with yield or growth. For example, a 3% yield, 10% growth stock is obviously a superior investment to a 6% yielding stock with a 4% growth rate. One offers a higher total return than the other irrespective of yield vs growth. Just because there are many more 3% + 10% stocks versus 10% + 3% stocks says more about economics than anything else. The only thing I would say on return composition for these investors is that div yield is more certain than div growth – the company has committed to paying the current dividend but has made no commitments about the future. Also, because of the uncertainty of future growth I think the market mis prices future growth. Why else are there so many dividend stocks that have low to mid dividend yields, 2-4%, yet also offer such strong div growth rates (10%+)? Nonetheless total return should be the primary focus for these investors.

For dividend investors in taxable accounts and also re-investing dividends the composition of total return then matters quite a bit. Why? It comes down to one of the two unquestionable certainties of life – taxes. As a dividend investor in a taxable account, you pay taxes on the div yield component of the stock return but not on the growth component. The growth component is only taxed when you sell the investment. For example, take two 10% total return stocks, a 7% + 3% stock vs a 3% + 7% stock. Pre-tax the returns are the same but after tax the returns are different. It also depends if the dividend are qualified or not. The table below shows these differences. It assumes a qualified dividend tax rate of 15% and an investor in the 25% marginal tax bracket.

As you can see from the table the composition of the return matters. The after tax return of the 7% + 3% stock is 8.95% vs 9.55% for the 3% + 7% stock. Taxes matter a lot. The table also shows the tax dis-advantage of REITs in taxable accounts and the tax advantages of MLPs in taxable accounts. This does not diminish the importance of total return in any way but these taxable considerations could tip the scales in favor of one investment over the other.

In summary, total return is what matters for dividend investors whether the majority of the return comes from the current dividend or dividend growth. The commutative property of dividends holds. The passionate arguments in favor of one component or the other are really just about total return. However for dividend investments in taxable accounts the breakdown of total return into yield and growth can have a differing effect on after tax returns that investors need to be aware of. For these investors, stocks with higher growth components have higher after tax returns. Investors then need to weigh this benefit of higher dividend growth stocks with the accompanying higher uncertainty. In a follow on post I’ll tackle the issue of growth vs yield for investors who are living off their dividends.

Full Disclaimer - Nothing on this site should ever be considered advice, research or the invitation to buy or sell securities. These are my personal opinions only.

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### 3 Responses to The commutative property of dividends

1. David Fleischer says:

Hi Paul,

Something has been nagging me since you first introduced the “Magic Formula”. First let me make sure I understand. The basic formula states:
Total return = Div yield + Div growth

Where Div Yield is the current dividend yield (or the yield based on your initial buy price). And where Div growth is the growth (increase) in the dividend being paid. You arrive at this based on historical dividend growth rates using an average of the past 5 years I believe.

Here is what I don’t understand. There seems to be no time dependency variable in the equation. That is, is the total return based on a 1 year, 5 year or 10 year holding period?

If for example, a stock is yielding 5% and has a 6% dividend growth rate, my total return should be 11%. But this can’t be applied over only one year, can it?

Thanks Paul,

David

• libertatemamo says:

David, great question. I’ve been waiting for someone to ask this. Its one of the major insights into this formula and you’ve hit on it. You are correct that there is no time dependency to the formula. The formula calculates an annual return for as long as you hold the investment and are re-investing the dividends given no change in valuation. Remember that the full formula has an additional term, div yield + div growth + change in valuation.

Lets uses an example. A company prices at \$100, yields 10%(\$10) and has a div growth rate of 10%. In 1 year, the price went from \$100 to \$110. In that year you made the 10% dividend plus the 10% growth in the stock price, you made your 20% just like the formula said. Here is the key – the price of the stock had to go up for the valuation to stay the same. At the new dividend of \$11 the stock price has to be \$110 in order for the dividend yield to stay at 10%. If the stock price stayed at \$100 after year 1, the new dividend yield would be 11% – the valuation changed – the stock got cheaper! So your return in that year the stock price stayed the same was the 10% yield + 10% growth – 10% change in valuation. Remember the full version of the formula us yield + growth + change in valuation.

Imagine if this process continues – the stock price stays the same while the div grows, the stock eventually becomes so cheap that the market would buy it en masse and bid it back up. This process can take a while to happen – sometimes years – but it does happen, over time the market is a weighing machine, not a voting machine. In the post I did on dividends accounting for 80-100% of WW stock returns this is one of the basic conclusions – that over periods of 5yrs dividends account for most of the returns, i.e. the formula works over time.

Of course this process works in reverse as well. The stock price in the example can go to \$120 after 1 year. That year you would have made 30%, 10% yield + 10% div growth + 10% change in valuation.

One of the important points in all this is that price, aka valuation, matters a lot.

Let me know if the clears it up. Its a subtle but awfully important concept to grasp.

Paul