**The luckless scientist - prologue**

You are an up-and-coming scientist in the field of microbiology at Harford University, a major research university in the country. You are trying to engineer more rapidly-replicating strains of *E. coli* bacteria for food production. At present, the fastest-growing strain of *E. coli* reproduces once every 20 minutes, and if you put a single cell of that strain into a giant vat of nutrient-filled broth, it will take one day for the bacteria to completely saturate the vat of broth.

After many years of hard work, you have a huge breakthrough! You have engineered a new strain of *E. coli* that divides **10% faster** than the old strain (and still retains its other desirable properties needed for food production). Instead of dividing once every 20 minutes, this new strain divides once every **18.18 minutes**. One cell of this new strain placed into nutrient-filled broth will saturate after one day not one vat, but an incredible **147 vats**! You open a bottle of champagne to celebrate.

A day later, your colleague makes the fortuitous discovery that the addition of an unusual amino acid will increase the growth of your *E. coli* strain by an additional **3%**. This further reduces the dividing time from 18.18 minutes to **17.65 minutes**, a difference of **31.8 seconds**. You are not in the lab at the time because you are still hungover from drinking too much alcohol the night before.

Two weeks later, your colleague is featured in every major scientific news article. He wins the University Medal amongst numerous other awards, and is immediately promoted to the highest professorship at the university.

You complain to the head of your department, "Hey! My discovery increased the growth rate of *E. coli* by 10%, while his finding only boosted it by 3%! Why is he getting all the credit?"

The head replies, "Didn't you read your colleague's article? It was the front cover of the latest issue of *Natural Scientific*! What was the name of the paper again, umâ€¦yes, it was "Addition of unusual amino acid is the main contributor to the increase of *E. coli* growth". While your discovery increased the number of vats per day from 1 to 147, his finding further raised that number to an incredible **763 vats**! In other words, his contribution represented the majority, **80.7%** to be precise, of the total growth of this *E. coli* strain!"

*You go numb with shock. Years later, your engineered strain is used to develop a food production method that solves world hunger. Your colleague goes on to win Nobel Prizes in both Chemistry and Peace for this invention, while you spiral into a life of alcoholism and despair. In the rare times that you are sober, you always harken back to that fateful day, and think to yourself, how can his meager 3% increase be more important than my 10%? Yet, the math was irrefutable...or was it?

(*At the end of this article, an alternative ending to this story is revealed).

**Introduction**

Most of my followers on Seeking Alpha know that I write mostly about my analysis of various exchange-traded products, including ETFs, ETNs and CEFs. However, from time to time I also enjoy using elementary principles to tackle common Wall Street aphorisms that perhaps some investors take for granted, or otherwise assume to be automatically correct.

Previously, basic statistical methods were used to help judge whether the Santa Claus rally, January effect or presidential cycle effect were significant enough to be relied upon. Other analyses, relevant more for the dividend investing community, attempted to investigate whether the reduction in share price on the ex-date could be significantly detected, or whether the choice of dividend yield or dividend growth really mattered.

In this article, I set my sights on another investing aphorism, namely that dividends make up the majority of long-term total returns. In this analysis, I happily found that yes, dividends have made up 93% of total returns over a recent 60-year period. How about capital appreciation, you might ask? Well, it merely made up the other 99%. How can this be possible? Read on to find outâ€¦

**Surveying the literature**

A 2015 Brandes Institute study, referenced in Seeking Alpha expert chowder's recent article entitled "Dividends: A Key Component Of Total Return", indicated that the longer the holding period for equities, the greater the proportion of total return was derived from income (i.e. dividends). For example, Exhibit 2 from the study showed that for 5-year rolling periods from 1926-2014, the income component accounted for 43% of the total return of U.S. equities, whereas for 20-year rolling periods, this percentage increased to 61%.

The authors thus claimed (emphasis mine, and throughout):

For U.S. equities, the income component was

significantfor time horizons as short as five years, anddominantfor horizons of 20 years and longer.

A 2015 study by Dr. Ian Mortimer and Matthew Page from Guinness Atkinson, gives an even more impressive statistic:

If you had invested $100 at the end of 1940, this would have been worth approximately $174,000 at the end of 2011 if you had reinvested dividends, versus $12,000 if dividends were not includedâ€¦ In this period dividends and dividend reinvestments accounted for

over 90% of the total returnfor the index during that time.

Based on this data, Mortimer and Page concluded that:

Over the long term, dividends have been the

main contributorsto total return in equity investments.

**Geometric** **thinking**

Human brains are trained to think linearly rather than geometrically (or exponentially). A recent TED talk by Robert Anderson entitled "Linear Brains in an Exponential World" discusses this very issue. Unless you work in a technical discipline or are crunching the numbers on your investments, how many instances in your daily life do you have to compute numbers geometrically? Not many I'd imagine. This is what makes the tidbits like this so interesting: did you know, that if you folded a piece of paper in half 42 times, you would have a tower that would reach to the moon?

In geometric series, small changes in the base (*b*) can have outsized effects on the output of the function y = *b*^*x* when the exponent (*x*) is very large. As a very simple example, consider a first security that returns 4% p.a., compared to a second security that has a 5% p.a. return. After one year, $100 invested in the first security returns $4, while the same amount invested in the second security returns $5. So far, so good.

Two years later, and assuming compound interest, the $100 invested into the first security has returned $8.16, while the second security has returned $10.25. After 50 years, the first and second securities have returned $611 and $1,047, respectively.

At this point, one could claim, "Well, the moving from a 4% p.a. to a 5% p.a. security boosted the 50-year return from $611 to $1,047, a difference of $436, so that means that this additional 1% p.a. was responsible for $436/$1,047 = 41.6% of the total return of the 5% p.a. security! By inference, the original 4% only contributed 100% - 41.6% = 58.4%!"

The more astute among you may already realize where the problem lies. While the additional 1% p.a. boosted the 50-year total return by $436, which indeed represented 41.6% of the total return over that period, it is misleading to claim that the 1% p.a. was *in itself** responsible* (or "makes up", or "is the major contributor", etc.) for that 41.6% because it only achieved this after being applied "on top" of the existing 4% p.a. multiplier.

On its own, a 1% p.a. return would only give a $64.4 return on $100 invested after 50 years, a far cry from the $1,047 return achieved at 5% p.a. Are we to conclude then that this 1% p.a. contributed only $64.4/$1,047 = 6.2% of the total return of the 5% p.a. security? Meaning that other 4% actually contributed 100% - 6.2% = 93.8%?

Clearly, that "extra" 1% p.a. cannot simultaneously contribute both 41.6% and 6.2% of the 50-year total return of the 5% p.a. security, while the "original" 4% p.a. also cannot simultaneously contribute both 58.4% and 93.8% of the total return. I mean, this is not even math, it's just common sense. In logical terms, this result violates the law of non-contradiction.

Surely, no one would use such misleading analysis and faulty logic to make a point,... or would they?

**Dividends make up 93% of long-term total return, and capital appreciation the other 99%**

Let's go back to the two studies cited at the start of this article, by Brandes Institute and Guinness Atkinson, respectively. In the Brandes paper, they reveal how they calculated the dividend component of total return:

We used two return series: a total return series that included reinvested dividends and capital appreciation and one that was capital appreciation only.

We calculated the income component of returns by subtracting the capital appreciation only series from the total return series.

Hence, we can see that the authors have made the same elementary mistake as described above, and that was also committed by the head of department in my fictitious story. **You cannot simply subtract one component from a compounded total and claim that the remaining component is responsible for the remainder of the returns**.

To put some meat on the bones of this issue, let's work with the numbers given in Guinness Atkinson study. While they did not state their methodology in their article, I have little doubt that it was performed in the same way as in the Brandes study.

If you recall, Guinness Atkinson presented data showing that $100 invested in the S&P500 (NYSEARCA:SPY) at the end of 1940 was worth $174,000 in 2011 had you reinvested dividends, versus $12,000 if dividends were not included. This was used to make the claim that dividends accounted for over 90% of the total return of stocks over this period.

For the sake of simplicity, let's assume that the total return for each year was identical. Simple arithmetic reveals that from 1940 to 2011, a period of 61 years, stocks returned a geometric total return average of 13.0% per year, i.e. $100 x (1.130)^61 = $174,000. Performing a similar calculation for capital appreciation only (no dividends or dividend reinvestment) reveals that capital appreciation provided a average geometric return of 8.2% per year, i.e. $100 x (1.082)^61 = $12,000. Now, $12,000 is only 6.9% of $174,000, meaning that capital appreciation alone provides only 6.9% of the total return of stocks. So far, so good.

Now, can we claim that dividends are responsible for the remaining 100% - 6.9% = 93% of total return as the logic of Brandes and Guinness Atkinson would suggest?

If your answer to the above is yes, then you should also agree with the (faulty) logic of the following.

Since total return = capital appreciation + dividends, we can infer that dividends have provided a geometric average of 4.8% return each year (13.0% - 8.2%) from 1940 to 2011. A "dividend only" (including dividend reinvestment) return value can then be calculated using the formula: $100 x (1.048)^61, which gives a value of $1,795. Since $1,795 is only 1.0% of $174,000, I can hereby claim that capital appreciation accounts for 99% of the total return of stocks over the period!

Can this make sense? Can dividends provide 93% of long-term total return, and capital appreciation the other 99%? Surely not!

The graph below shows the final ending values of dividends only, capital appreciation only and total return on $100 invested for the time period covered by the Guinness Atkinson study. We can see from the below that on their own, each component only provides a small fraction of the total return.

However, stating that dividends account for 1.0% of total return while capital appreciation accounts for 6.9% is similarly unsatisfying, because then where did the remaining 92.1% of total return go?

**The correct way of thinking about dividends vs. capital appreciation**

We can also already see at that this juncture that the claim that dividends provide most of the long-term total returns of stocks is just flat-out wrong. With "capital appreciation only" providing $12,000 after 61 years vs. $1,795 for "dividends only", it is clear that capital appreciation is more important.

How much more important? $12,000 / $1,795 = 6.7 times as important? Wrong again! Hopefully by now you will have realized that it just doesn't make sense to compare the final values of geometric functions when you're trying to assess the relative contributions of each component.

The correct way to think about this, in my opinion, is to go back to the classic equation, total return = capital appreciation + dividends, but to only apply this to discrete return values and not to compounded returns. Using the data from the Guinness Atkinson study, we can calculate that the average 13.0% return of U.S. stocks from 1940 to 2011 is composed of 8.2% capital appreciation and 4.8% dividends. Therefore, capital appreciation provides (on average) 63% of the total return of each year, while dividends provide 37% of the total.

A study on dividends by S&P Dow Jones Indices over a similar time period (1926 to 2012) confirms my analysis. As shown in Exhibit 3 below, dividends have accounted for 34% of the monthly total return of the S&P500 during this time period. The graph also shows that this percentage fluctuates significantly in various decades, which is presumably due to the different capital appreciation returns achieved by stocks during different decades.

**Summary**

Claims that dividends provide the majority of the long-term total returns of stocks range are, depending on your point of view, either misleading or simply incorrect. This is because the logic that is used to arrive at those claims would lead you to believe that dividends can account for 93% of the total return, while simultaneously, capital appreciation can account for the remaining 99%!

Then, what is the best way to interpret the observation that a $174,000 return from $100 over 1940-2011 drops to only $12,000 (-93%) without dividends and dividend reinvestment (i.e. capital appreciation only), but crashes to a measly $1,795 (-99%) without capital appreciation (i.e. dividends + dividend reinvestment only)? I believe that the correct way of thinking about this issue is to not compare compounded values, but to deconstruct the average yearly geometric total return into capital appreciation and dividend components.

Performing this analysis reveals that over 1940 to 2011, dividends accounted for slightly over one-third (37%) of total return during this period, while capital appreciation accounted for slightly under two-thirds (63%) of total return. Similar results were described in the S&P study cited above. Thus, I have to conclude that on a head-to-head basis, capital appreciation is more important than dividends for total return. However, both are absolutely essential for achieving a high total return in the long run.

It is just highly disappointing that even esteemed research institutes like Brandes Institutes or fund houses like Guinness Atkinson can make such an elementary mistake. Hopefully, this article has shed some insight into how geometric compounding works and will help you to think more critically about such claims in the future.

**The luckless scientist - epilogue**

*You go numb with shock. "Sir, I cannot believe that you have made such an elementary error! If I apply your erroneous logic, then I can similarly claim that my colleague's 3% boost on its own would have only increased *E. coli* production from 1 vat to 5 vats. Applying my further 10% enhancement then provides the 763 vats that you describe. So could I similarly claim that my contribution is worth 758 vats, or 99% of the total growth of *E. coli*, while his contribution is only worth 1%?"

Embarrassed, the head of your department mumbles, "I guess you're right. Well, I can't take those awards away from your colleague, because I'd just make a fool of myself. But I'll make sure you get the recognition you deserve".

Years later, your engineered strain is used to develop a food production method that solves world hunger. You win the Nobel Prizes in both Chemistry and Peace for this invention. However, you make sure to mention your colleague's name amply during your Nobel acceptance speech, and even decide to share some of your prize money with your colleague.

How much, you think to yourself. You briefly consider writing him a cheque for only 1% of the total, given that that's what his contribution would have been worth had you applied his faulty logic in reverse. That would serve him right for trying to take credit from you! But you eventually decide to give him 23% of the total amount, because that is what 3% of 13% is. Because you're an upstanding fellow like that.

(Thank you to Seeking Alpha reader Butterfly Seeker's comment for providing the inspiration for this article).

Addendum: Embarrassingly, in my effort to correct the logical mistake of others, I have committed a most elementary mathematical error - that of subtraction. 1940 to 2011 is 71 years, not 61 years as I had stated in my article. Over 71 years, the total return of the S&P 500 has been 11.1% (down from 13.0%), of which 7.0% (down from 8.2%) are from capital appreciation and 4.1% (down from 4.8%) are from dividends and dividend reinvestment. Fortunately, the conclusions of this article remain absolutely unchanged, because with the corrected numbers, capital appreciation still provides 63% of the annual total year of stocks, and dividends and dividend reinvestment 37%. I have decided to provide this addendum here instead of editing my article because of the many comments that have already been posted below.

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

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.