# Retirement Math: Compound Growing Dividends or Compound Retained Earnings?

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by: David Van Knapp
A recurring topic in dividend-growth investing is the mathematics of compounding growing dividends “versus” the mathematics of a company retaining all of its earnings and compounding them inside the company. The question becomes, which kind of compounding is better for an individual who is saving for or already in retirement.
I put the word “better” in italics, because I believe that reasonable minds can differ on what is better, and the only valid judge for any individual is the individual himself or herself. Once you bring in factors like risk tolerance, SWAN (sleep well at night), and other psychological factors, you realize that the mathematical equations are only part of the story. “The rest of the story” (as Paul Harvey used to say) involves:
1. the assumptions or inputs into the equations,
2. the goals of the investor (i.e., what is “better”), and
3. the impact of Mr. Market on the results.

Here was an example used in a recent comment stream. It concerns two companies, one that pays dividends and one that doesn’t. I have added emphases as italics for discussion later.

Let's say the companies both have 20% ROE, the non dividend paying company [Company A] trades at 2X book value and the dividend paying company [Company B} trades at 3X book value. The retained earnings in the growth company will compound at 20% annually as they are reinvested at book value and the ROE remains constant.

The dividends will get taxed at 15%. So instead of reinvesting \$1 in the business, you reinvest \$0.85. Then instead of investing at book value you're investing at 3X book value. That means instead of investing \$1 back into the business at 20% annually, you're only investing \$0.28 after taxes and the premium to book value.

Over a very short period of time the dividend company may outperform. Over a longer period of time it seems very…unlikely. Consider that each retained dollar compounded at 20% annually will be worth \$38.34 after 20 years. However, the dividend taxed at 15% and reinvested at 3X book will be worth only \$10.73.

If the same 3/2 ratio exists after 20 years then the retained, and compounded dollar, will be worth \$76.68. The taxed dividend reinvested at a premium will only be worth \$32.19.

…Again, the big assumption is that both companies can earn a constant 20% ROE, which is usually unrealistic. But the math is undeniable (in my opinion.)

In fact, I'd argue that based on these numbers, in our hypothetical situation the growth stock should by all rights command a higher multiple than the dividend stock since your stake of the business is growing so much faster.

I don’t question the mathematical formulas in that example. However, I have several problems with the assumptions that the commenter made.
First, note that the example—without ever stating so—is all about amassing the largest capital value possible. Income is considered only to the extent that it helps build the store of capital, and the example therefore becomes a race between two companies in getting their capital values to the highest level possible. The example ignores how the investor accesses his or her money when the race is over and it is time to live off whatever has been accumulated.
Second, it is presumed that “better” equals highest capital value, i.e., that the best result comes from the winner of the capital-maximization race. The idea that an individual might think that it is “better” to build an income stream from his or her portfolio, as opposed to a maximum capital value with no income stream, is not considered.
Third, the example uses internal capital value—book value—as a proxy for the actual worth of the company to the investor. That’s why I italicized “will be worth” a couple of times. The example assumes a linear correspondence between book value and the stock’s price on the market. But we know that is not the way markets work. There are many moments in time that the linear presumption is wildly inaccurate. Those moments might correspond to when you need access to your money.
Fourth, as a simplifying assumption, I agree with the proposition that, “The dividends will get taxed at 15%. So instead of reinvesting \$1 in the business, you reinvest \$0.85.”
However, fifth, I also agree with the comment’s last paragraph about multiples. “Growth” companies usually trade at higher multiples, not lower, than “dividend” companies. So the input of a 2X book value multiple for Company A and a 3X multiple for Company B seems illogical. A more logical example would reverse them and have Company A trade at a 3X multiple and Company B at 2X book. That considerably changes the math of dividend reinvestment. The \$0.28 being reinvested becomes \$0.43—a 50% increase in the compounding power of reinvesting dividends.
Sixth, as indicated by the italicized phrase “in the business,” the example assumes reinvestment of the dividends into the same company. But a dividend-growth investor can accumulate dividends and invest them into any opportunity that exists. He or she does not have to accept Company B’s valuation at the time of reinvestment. They can look for better valuations elsewhere or different investments entirely. That said, many dividend-growth investors do reinvest directly back into the same company, often via DRIP or similar plans.
Seventh, while the example on its face seems to compare internal and external compounding in detail, it actually ignores an important aspect of external compounding: Reinvestment of dividends results in share accumulation. (Internal compounding never produces more shares, just a higher value for the shares owned). So although the reinvestment of dividends suffers from a tax haircut, it does offer the benefit of accumulating more shares. More shares --> more dividends --> more shares etc. So the dividend reinvestor has double compounding: The dividends themselves are rising (that is a true mathematical compounding each year), and reinvesting them creates a second layer of compounding through the purchase of additional shares.
Eighth, when the time comes that the investor is not accumulating assets but rather, is harvesting them (in retirement), taxation on capital gains must be accounted for. While taxation situations vary all over the map, the example used a straight 15% tax on each dividend as it was issued and reinvested, without ever mentioning taxation on capital gains at the time of withdrawal. It is likely that the investor who has maximized capital will redeem shares first that have the least capital gains (highest cost basis). But we are comparing two kinds of gains here (as distinguished from comparing gains to mere return of capital), so I will put a stake in the ground that the taxation on both forms of investment is the same: 15% tax on either the gains from dividends or the gains in capital. I expect disputes about this in the comments.
Please note that I understand that the tax on dividends as they are issued makes compounding within the company more efficient than compounding via reinvestment of dividends, because of the time value of money. That said, the taxation of withdrawals cannot be ignored. And it should be obvious that if an investor has relied solely on capital accumulation to get ready for retirement, he or she must make withdrawals to retrieve the money needed to live on in retirement. That investor has created no income stream. For the dividend investor, there may be zero or reduced withdrawals, as he or she lives off the dividends from all the shares that have been accumulating from the beginning of the dividend strategy.
Ninth, as the example notes, finding a company that can compound earnings at 20% per year for any length of time is rare. I think the example would be improved if we used 15% per year rather than 20%. I don’t believe, however, that would change the validity of the math. It would just scale everything back a bit. So in the theoretical world of the example, a perfect company with perfect management making perfect decisions will compound their retained earnings faster than if some of the earnings were sent out as dividends and then re-invested. Commissions, taxes, and the time value of money make this theoretical internal compounding advantage mathematically inevitable.

But to fulfill its maximum compounding potential, a lot of things have to go right:
• Management has to make consistently excellent decisions about what to do with the retained earnings.
• New projects have to reach the same profitability as the original business.
• Management has to be able to "grow the business to the sky" as the optimal reinvestment of retained earnings causes the company to grow and grow.
• The market has to be perfectly efficient in order for the shareholder to get money out at values that reflect the true value of the business, at the exact times that he or she wants or needs that money.
The real world is not like that:
• Managements often make dumb decisions with retained earnings: wasteful expenditures; overcompensation; ego-driven projects; overpaying for acquisitions; etc.
• As the business grows, formerly plentiful uses of the money that will produce the same rate of return become harder and harder to find and develop.
The final point I would like to make between the two strategies is that market efficiency is a myth. There is no consistent linear relationship between book value and share price. While it is true that “in the long run, the market is a weighing machine,” it is not true over the short run. Market inefficiency hurts retirement strategies based on maximizing capital in at least two ways, and it helps dividend-based strategies in at least one.
• Market inefficiency creates the conditions under which managements so often make share buybacks at inflated valuations. Thus some of the benefits of internal compounding are literally wasted in the repurchase of overvalued shares.
• Market inefficiency allows dividend-growth investors opportunities to purchase their shares at advantageous valuations with higher yields, and to avoid purchasing them at inflated valuations with lower yields.
• Market inefficiency subjects the shareholder to the whims of the market when he or she needs to sell shares to take money out. More shares need to be sold if the sale takes place when they are undervalued. Long-term averages don’t necessarily apply when real life withdrawals are episodic.
It is often under-appreciated (and ignored in the max-cap strategy) how significant is the inflection point when the investor switches from accumulation to retirement mode. For an important part of the total retirement strategy is the exit strategy: How do you get your money out when you need to?
• Pre-retirement, the max-cap investor is depending on the efficiencies of internal compounding exclusively. The higher the rate of compounding, the better off he or she is, because the race is to the highest “Number.” The investor is not accumulating more shares nor any future income rights, just more value in the shares owned.
• Post-retirement, the max-cap investor disposes shares to create retirement income. (“Just sell a few shares when you need money.”) The advantages of efficient internal compounding diminish as the number of shares owned diminishes. High and consistent internal compounding is a necessity, as the only way to keep the account from declining in value is to have the sold shares’ value replaced by increases in value of the remaining shares. The number of shares needing to be sold, of course, depends not only on that continued internal value appreciation, but also on the vagaries of Mr. Market that determine the price at which they can be sold.
• The dividend-growth investor, on the other hand, either sells fewer shares in retirement, or indeed may have reached the point where selling shares is unnecessary. Increase in the value of the shares is a nice-to-have, but it is not a necessity. The income stream will continue to grow whether the share’s market prices advance, decline, or stay flat. Mr. Market plays no role in the value of the dividend stream, having been disintermediated by the strategy from the very beginning.
I want to continue on the impact of Mr. Market, because it becomes way more significant at the inflection point from accumulating to spending. It should be obvious that the market’s lack of continual efficiency often makes it impossible for shareholders to retrieve the company's “fair” value when they want or need that money. Let’s illustrate that with a stock often used (rightfully, in my opinion) to illustrate the power of internal compounding: Berkshire Hathaway (NYSE:BRK.A).
Warren Buffett is one of if not the best capital allocators of our time, and Berkshire stands as a monument to the power of compounding done as well as it can be. Berkshire pays no dividends, preferring to keep all the money inside the company where it can compound most efficiently. The company’s record in this regard is unassailable.
Nevertheless, Berkshire, like any publicly traded company, is subject to the whims of Mr. Market. Buffett is famous for his shareholder letters, and one of the highlights every year appears just inside the front cover. In a table labeled “Berkshire’s Corporate Performance vs. the S&P 500,” Buffett displays a table that compares Bershire’s annual change in book value to the performance of the S&P 500. Needless to say, over many years, he has steamrolled the index.
Years ago, when I first saw that table, I was curious that he chose to compare Berkshire’s record at compounding book value—which a shareholder cannot get at directly—to the S&P 500, which is a measure of attainable shareholder returns. (Those returns can be obtained within a fraction of a percent via a tracking fund or ETF such as SPY that literally mimics the index. You cannot invest directly in the index itself.)
A few weeks ago, I decided to add the “missing columns” to the table: What is Berkshire’s actual stock price—which is accessible to shareholders—compared to its annual changes in book value? And how does Mr. Market treat Berkshire? Remember, since Berkshire does not pay dividends, changes in its stock price equal its total return. I have modified the title of the table slightly. The last two columns are the ones I have added. They illustrate Buffett’s statement, “Even though the business…you own may have economic characteristics that are stable, Mr. Market's quotations will be anything but.”
Berkshire’s Stock Performance vs. the S&P 500
 Year (1) Percentage Change in Book Value of Berkshire (2) Percentage Change in S&P 500 with Dividends Included Relative Results (1) – (2) (3) Percentage Change in BRK.A Stock Impact of Mr. Market on BRK.A (3) – (1) 1968 19.0 11.0 8.0 76.1 57.1 1969 16.2 (8.4) 24.6 19.4 3.2 1970 12.0 3.9 8.1 (4.7) (16.7) 1971 16.4 14.6 1.8 80.5 64.1 1972 21.7 18.9 2.8 8.1 (13.6) 1973 4.7 (14.8) 19.5 (2.5) (7.2) 1974 5.5 (26.4) 31.9 (48.7) (54.2) 1975 21.9 37.2 (15.3) 2.5 (19.4) 1976 59.3 23.6 35.7 129.3 70.0 1977 31.9 (7.4) 39.3 46.8 14.9 1978 24.0 6.4 17.6 14.5 (9.5) 1979 35.7 18.2 17.5 102.5 66.8 1980 19.3 32.3 13.0 32.8 13.5 1981 31.4 (5.0) 36.4 31.8 0.4 1982 40.0 21.4 18.6 38.4 (1.6) 1983 32.3 22.4 9.9 69.0 36.7 1984 13.6 6.1 7.5 (2.8) 20.3 1985 48.2 31.6 16.6 90.6 42.4 1986 26.1 18.6 7.5 16.1 (10.0) 1987 19.5 5.1 14.4 4.7 (14.8) 1988 20.1 16.6 3.5 59.3 39.2 1989 44.4 31.7 12.7 84.6 40.2 1990 7.4 (3.1) 10.5 (23.1) (30.5) 1991 39.6 30.5 9.1 35.6 (4.0) 1992 20.3 7.6 12.7 29.8 9.5 1993 14.3 10.1 4.2 38.9 24.6 1994 13.9 1.3 12.6 25.0 11.1 1995 43.1 37.6 5.5 57.4 14.3 1996 31.8 23.0 8.8 6.2 (25.6) 1997 34.1 33.4 0.7 34.9 0.8 1998 48.3 28.6 19.7 52.2 3.9 1999 0.5 21.0 21.5 (19.9) (20.4) 2000 6.5 (9.1) 15.6 26.7 20.2 2001 (6.2) (11.9) 5.7 6.5 12.7 2002 10.0 (22.1) 32.1 (3.8) (13.8) 2003 21.0 28.7 (7.7) 15.8 (5.2) 2004 10.5 10.9 (0.4) 4.3 (6.2) 2005 6.4 4.9 1.5 0.8 (5.6) 2006 18.4 15.8 2.6 24.1 5.7 2007 11.0 5.5 5.5 28.7 17.7 2008 (9.6) (37.0) 27.4 (32.0) (22.4) 2009 19.8 26.5 (6.7) 2.7 (17.1) 2010 13.0 15.1 2.1 21.4 8.4
Observations:
• In the 43 years covered by the table, Mr. Market has “under-rewarded” shareholders 17 times (40% of the years).
• Interestingly, 6 of those “market underperformance” years have occurred in the past 10 years.
• My own Dividend Growth Portfolio (DGP) has existed since mid-2008. Here are my total return numbers compared to Berkshire’s for those years. (I am including all of 2008 even though my portfolio was not fully constructed until May, 2008.):
• 2008: Berkshire -32.0%; DGP -29%
• 2009: Berkshire +2.7%; DGP +14%
• 2010: Berkshire +21.4%; DGP +17%
• 2011 YTD: Berkshire -4.5%; DGP +7%
• Compounded Annual Gain—2008-present: Berkshire -19%; DGP +1%
• Growth in dividends:
• 2009: Berkshire 0%; DGP +19%
• 2010: Berkshire 0%; DGP +15%
• 2011: Berkshire 0%; DGP on pace for about +16% full-year
• Current Yield on Cost: Berkshire 0%; DGP 5.1%
I realize that a three-year comparison is too small a sample from which to draw any conclusions when you are talking about lifelong strategies that are meant to fund retirement. That said, it is not inconceivable that a dividend-growth strategy may outperform in total return even a compounding machine like Berkshire over longer periods of time. And it is almost assured that the income stream from the Dividend Growth Portfolio will diminish—perhaps even eliminate—the need to sell shares to obtain money for living expenses when that important inflection point is reached. The max-cap strategy, on the other hand, requires that shares be sold to obtain money.

Personally, I willingly give up the book-value maximization powers of even a Berkshire for the predictably of rising income provided by a focused dividend-growth strategy. I consider the likely increase in share price over time a kicker to the income (rather than the other way around), and the diminished need to sell shares is a real bonus. I fully expect my investments (and other sources of) income to exceed my spending needs, so therefore I can look forward to happily selling shares occasionally to finance significant lifestyle enhancements. I can pick my spots for those.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.