[I don't think this is very original, but it's a fun and instructive metaphor to play with. The most important lesson is the way the metaphor fails.]
Suppose I start a closed-end mutual fund. (Brits call it an "investment trust".) I issue shares, and use the proceeds to buy assets like bonds and stocks. Shares in my mutual fund are traded, but shareholders do not have the right to redeem them for a fixed share of the value of the underlying assets that back those shares. (If they did have that right it would be an open-end mutual fund, what the Brits call a "unit trust".) But I have the right to issue more shares if I want, or to buy back shares, if I want.
Suppose that the shares in my closed-end mutual fund are more liquid than the assets that back those shares. If my shareholders value liquidity at the margin, those shares will have a lower yield in equilibrium than the assets that back them. And if my management expense ratio is smaller than the liquidity yield premium, shares in my mutual fund will trade at a premium to the Net Asset Value. And this means I can earn positive profits from my closed-end mutual fund. I can increase the management expense ratio by paying myself a higher salary, or giving some of the returns of the assets to my favourite charity, until the shares trade at par to Net Asset Value.
Competition from other issuers of closed-end mutual funds could reduce my profits to zero. But maybe I have a legal monopoly. Or maybe I have first-mover advantage in a world of network externalities. Shares in my mutual fund are frequently traded because they are liquid, and they are liquid because they are frequently traded. They have a high velocity of circulation. And if they are the most liquid of all assets, all other goods will be bought and sold for shares in my mutual fund, except for the rare barter deal where there happens to be a coincidence of wants. The liquidity race is a winner-takes-all race. (Carl Menger figured this out in 1892.) So it's a natural monopoly, and it's hard for a competitor to break into my established market.
If shares in my closed-end mutual fund are the most liquid of all assets and so are used as a medium of exchange, they will probably also be used as a unit of account. People will quote prices in terms of the number of shares that are needed to buy the good. So the market value of one of my shares is the reciprocal of the prices of other goods.
Like any monopolist, I face a downward-sloping demand curve for my product. The higher my management expense ratio, the smaller the quantity of my shares that will be demanded, and the fewer shares I can issue while keeping the Net Asset Value at par. Maybe I maximise profits; maybe I am public-spirited; maybe I'm a regulated monopoly; maybe my monopoly gets nationalised. If my monopoly gets nationalised, then my favourite charity becomes the government's favourite charity.
What happens if I issue new shares, and double the total quantity of shares outstanding? That depends.
If I double the number of shares by doing a 2-for-1 stock split, the value of each share will halve. Because the value of the assets that back those shares has not changed.
If I double the number of shares by buying assets, and then give those assets to the shareholders, this is exactly like a 2-for-1 stock split. The shareholders sell me assets in exchange for shares, then I give them back the assets they sold me. the value of each share will halve.
If I double the number of shares by buying assets, then give those new assets to my favourite charity, and if people expect with certainty I will never do this again, it is exactly like a 2-for-1 stock split, plus a forced redistribution of wealth from my shareholders to my favourite charity. The value of each share will approximately halve, but may not exactly halve, because changes in wealth distribution may not have exactly equal and offsetting effects on the demand for my shares.
If I double the number of shares by buying assets, and promised to give all the returns from those new assets to my favourite charity, it's exactly the same as if I gave those assets themselves to my favourite charity.
If I double the number of shares by giving new shares to my favourite charity, and if people expect with certainty I will never do this again, the value of each share will also, approximately, halve. It makes little or no difference whether I give shares instead of assets to my favourite charity, since the charity will sell them to buy what it wants to buy.
I might use a helicopter to deliver the new shares to my favourite charity.
If I double the number of shares by throwing new shares out of a helicopter, and if people expect I will do this again next year, the value of each share will more than halve. Because it is exactly like a big increase in my management expense ratio, used to finance an increased flow of gifts to my favourite charity.
What happens if the assets that back the shares in my closed-end mutual fund are bonds, issued by my favourite charity, that are themselves promises to pay shares in my closed-end mutual fund? If I do a 2-for-1 stock split, and if this halves the value of my shares, I also halve the value of the assets that back my shares. And this redistributes wealth away from those who hold bonds in my favourite charity towards my favourite charity. And this redistribution of wealth may increase or reduce the demand for my shares, so the value of my shares may not exactly halve.
That's probably about as far as we can take this metaphor. But if the issuer of a closed-end mutual fund gets to decide his own management expense ratio and the fund's charitable donations, and can change it whenever he feels like it, and gets to choose the conditions under which he will issue new shares and buy back existing shares, the owners of shares in the closed-end mutual fund can't really be said to "own" the assets that "back" the shares in that mutual fund.
[Update: be sure to read dlr's excellent comment below.]