## Summary

- It is commonly stated that one can create his own dividend by selling a few shares.
- Likewise, it is said that receiving a dividend is the same as lowering your investment in the company that sent it.
- Both notions are false.

## What They Say

In the dividend debates, it has been stated unequivocally that one can create his own dividend by just selling a few shares. It has also been stated that receiving a dividend is the same as reducing your investment in the stock that sent it.

Both notions are simplistic and false. They are not true in theory, let alone actual life.

I would like to prove what I just said. Here are the propositions for consideration:

Proposition 1. Selling a few shares is the same as receiving a dividend.

Proposition 2. Receiving a dividend is the same as reducing your investment in a stock by the amount of the dividend received.

The math required to show the actual situation is not hard. Both of the propositions involve these variables.

S = Number of shares

P = Price per share

T = Total dollar market value of shares

D = Dividend per share

I will use the Greek letter "delta" (Δ) to indicate "change in." So ΔS means change in the number of shares, ΔP means change in the price per share, and so on.

The weakness in both propositions is that they do not address ΔS, the change in the number of shares still owned after a shareholder sells a few to "create his own dividend." The change in number of shares exists in real life, and it *cannot* be ignored.

## The Equations Don't Equate

Let's consider two shareholders: One sells shares to create his own dividend, and a second does not. In the beginning, for both shareholders, the total market value of shares equals the number of shares times the price per share. This is a snapshot at a single point in time.

T = S * P

Proposition 1 says that selling a few shares is the same as receiving a dividend. Mathematically, that would be expressed like this:

ΔS * P = D

Here is how to read that: The change in shares (i.e., the number of shares sold) times the price per share equals the dividend received. Let's ignore real-life issues like fractional shares and dollar values that don't line up perfectly. In a theoretical world, that simple equation is Proposition 1.

The *significance* of the proposition is that it says that it does not matter whether you sell shares or receive a dividend to put cash in your pocket, because the two ways are indistinguishable. The two ways are represented by the two sides of the equation. The left side represents selling shares, while the right side represents receiving a dividend. The equal sign says they are equivalent.

Proposition 2 says the same thing in reverse: Receiving a dividend is the same as reducing your investment in a stock by the amount of the dividend received. Let's also represent that with an equation:

D = ΔS * P

Here is how to read that: The dividend received is the same as the change in the number of shares times the price per share. The change in the number of shares represents the reduction in your investment.

As you can see, the equations for Props. 1 and 2 are the same, just rearranged. Mathematically, there is no difference between them.

Here is why the propositions are not valid. Before selling shares, the shareholder owns this:

T = S * P

After selling shares, he owns this:

T = (S - ΔS) * P

ΔS represents the shares sold. After selling them, the shareholder owns fewer shares. That fact cannot be dismissed. Shares are fundamental to the very concept of owning stock. That is how stock is bought and sold: In shares. It is also how dividends are paid: Per share.

Believing that an owner who sold shares is in the same position as one who did not is akin to believing in a free lunch: It is believing that an owner was able to sell some shares *and* retain the same ownership interest in the company. In the real world, there is no such free lunch.

There *is*, however, a condition under which the original value of T can equal the value of T after the sale of a few shares. I believe that this is what is meant to be conveyed by the general propositions: If the price of the shares (NYSE:P) rises sufficiently to offset the effects of the sale, one could correctly say that the same dollar amount (NYSE:T) is still invested in the company even though someone sold some of their shares.

When must this price increase happen? Well, because the propositions are presented as absolutes, it theoretically must happen immediately. But that is unrealistic. Let's give those who believe Props. 1 and 2 some breathing room, and say that it must happen within a year.

At this point, I need to distinguish the original price from the new price and the original number of shares from the new number of shares. Ordinarily, one would use subscripts, but for simplicity I will just say "Orig P" and "New P" for the prices of shares, and "Orig S" and "New S" for the number of shares. (We only need to do this once. Subsequent iterations in later years would look exactly the same.)

So here's what we have. This is the original equation:

Orig T = Orig S * Orig P

After selling a few shares:

New T = New S * New P, where New S = Orig S - ΔS

This means that after the sale, the new (lower) number of shares times the new (higher) price per share equals the new total invested in the company, where both the New T and Old T equal the same number of dollars.

To get Orig T and New T to equal each other, that means that the new number of shares times the new price must equal the original number of shares times the original price:

Orig S * Orig P = New S * New P

How much must P, the price per share, change in order to get that equation to be true? We can solve for that. You can either follow these steps or skip to the last one. I am just using Jr. High algebra here. I will make only one move per step to make it easy to follow. (Each step is based on the associative and commutative properties of multiplication and division: They can be done in any order and grouped or ungrouped without changing the answer.)

Orig S * Orig P = New S * New P

Step 1: New P = (Orig S * Orig P) / New S

Step 2: New P = Orig S * Orig P / New S

Step 3: New P = Orig P * Orig S / New S

Step 4: New P = Orig P * (Orig S / New S)

That's our answer. **In order for the shareholder to have the same dollar amount invested a year later, the original price must have increased by the ratio of the original number of shares divided by the new number of shares.**

This is easy to see if some simple numbers are used. If the owner had 100 shares and sold 10% of them to create his own dividend, then by the end of a year, the price must have increased by 100 / 90 = 11%.

*If it didn't, he is not in the same dollar position, and Propositions 1 and 2 are false. Even if it happens, he still owns fewer shares, and the two propositions are still false, although he has the same dollars invested.*

## Here's the Reality

Those who state that selling a few shares is the same as receiving a dividend are focusing only on T, the dollar value of the investment. Further, they are presuming that the necessary change in price described above will take place - indeed that it always takes place. That presumption is what allows them to ignore the change in the number of shares.

But that does not describe the real world. Prices are determined by the market, and we all know that market prices fluctuate, often irrationally. There is no guarantee that the price increase demanded by the math will actually take place.

Price fluctuations happen independently from two important things:

*Your sales activities*. The price of your stock is obviously disconnected from your ownership of it. The market does not know that you sold a few shares and therefore need the stock's price to go up. It could not care less.*The issuance of dividends*. Dividends are independent of price, because dividends are issued by the company and price is determined by the market. It is true that stock exchanges reduce the price of shares momentarily by the amount of an upcoming dividend, but there is no evidence that this price adjustment is permanent. As many of you know, that subject has been pounded to death recently on this site.

Is this issue important? I believe that it is, at all stages of investing.

But the lack of reality in the theoretical propositions becomes really important when one leaves the accumulation stage and, in retirement, depends on his portfolio to generate income to live on. Ignoring the number of shares owned can lead one to seriously misunderstand what is happening when share are repeatedly sold to create "income." Not only does the number of shares decline because they are being sold, but the rate of decline accelerates if market prices drop.

The retiree cannot simply presume that prices will rise when he needs them to rise. Indeed, sometimes they fall instead of rise. When that happens, the retiree must sell more shares to create the ersatz "dividend" than if the price had risen. That's why Monte Carlo tests are run on withdrawal schemes to estimate the probability that a particular scheme will fail (meaning that the retiree runs out of money while still alive). The Monte Carlo tests attempt to mimic what may actually happen in the market over the course of a 30-year retirement.

In the theoretical world, share prices always go up more than enough, and the shareholder's total wealth rises even though he repeatedly sells some shares to create "income." In the real world, that does not always happen.

That is the danger in the propositions stated at the beginning of this article. *All else equal*, a shareholder is better off receiving a dividend as real income rather than selling shares to create the illusion of income. In the former case, after receipt of each dividend, the shareholder still owns the same number of shares. His ownership stake in the company has not been reduced. In the latter case, the number of shares is relentlessly reduced. They could go to zero.

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