Savings & Capital Gains
You may have noticed that I wrote "mostly" wrong, and not simply wrong. There is a valid point in including shareholder's stake in retained corporate earnings. This is a factor which increases the fundamental value of the stocks that people hold (and thus would enable them to sell some of their shares and still own stocks with the same total value as before) and could thus perhaps just as much as dividends count as income. However, if we do this then we will no longer be talking about household savings as distinct from corporate savings. Instead, that number would measure the level of private sector savings.
A more consistent measure of household's savings as distinct from corporate savings would BTW subtract the money that people invest in newly issued shares from companies. Given the logic that retained earnings shouldn't count as household savings because the money stays within companies, then savings transferred from households to companies shouldn't count as household savings either.
What however should not count as household savings are capital gains that reflect pure asset price inflation. Which is to say, asset price increases not matched by increase in the fundamental value of the asset. Such price increases may benefit sellers, but they only do so at the expense of buyers. From the point of view of the overall economy, it could at best be described as a zero-sum game (it is a zero-sum game in a formal accounting sense. Taking the negative dynamic effects of asset price bubbles into effect it is however worse than that.
The implications of not making such a distinction are absurd if you think about it. If asset price inflation is a form of income and savings, then why not agree to simply double all asset prices each year? With total asset values in the tens of trillions of dollars, Americans could quit working and still enjoy a multifold increase in their current national income of roughly $12 trillion. Who needs hard work and thrift when all you need to do is to send a piece of paper between each other and claim that for each transaction its value has increased! It sounds absurd, and it is indeed absurd, but note that the "logic" in this hypothetical scheme is really no different from the logic of claiming (like for example David Malpass did) that increases in the price of internet stocks or houses unmatched by changes in the fundamental value of these assets constitute income and thus savings. The absurdity in my hypothetical scheme thus reflects the absurdity of claiming that asset price inflation adds to national income and savings.