Nobody "buys your index fund." People buy and sell stock or stocks on the index, which rises and falls as a result.
So if I understand correctly, the price of an ETF, VTI or VOO, for example (because one of these I'm planning to buy), is determined partly by the success of the individual stocks inside it, in addition to the demand on the ETF itself.
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When you use the word "partly", I am afraid you are not completely appreciative of how strong the dependency is!!
Example: VOO.
Markets are closed in the US now. i.e. the security is *very* illiquid (in the after/before markets) compared to usual periods when your orders by any broker will typically be executed in the regular market.
Last Px: $266.88. Mid Px: $265.775. Bid ask spread right now is: $0.09.
i.e. bid-ask spread = 0.034% of the Mid Px.
Mid Px is a very good approximation of the NPV (i.e. average of the underlying stock prices). This *can* sometimes break down, but is extremely rare in liquid securities like VOO.
Bid Ask spread represents the market inefficiency/illiquidity.
The point of pulling these numbers is that even in the *very illiquid* after/before market, VOO is tracking its intrinsic value with a tracking error < 0.034%/2 = 0.017%.
I don't know about you, to me that is not "partial" dependence - but something closer to a "total and complete" dependence. It is not "mathematical" dependence, but nothing in real world is, and hence that is irrelevant for any real world discussion.
How that almost "total and complete dependence" is maintained is a rather complicated topic involving many actors (specialists and market makers and exchanges and depositories etc. etc. etc.). I don't 100% know how VOO etc work, but if I were to generalize my knowledge writing code for some commodity etfs - then the mechanism is (in very broad strokes):
1. There are underwriters of these etfs, who are on the hook to make up the difference if the intrinsic value and market price diverges AND market participants demand delivery based on the underlying basket. These underwriters have to prove to SEC and other entities they have sufficient financial solvency to take this role, AND maintain sufficient liquidity.
2. Depending on which exchange these ETFs trade on, there will be another set of powerful players called market makers or specialists who will be incentivized to make sure the ETFs are liquid and track closely to the intrinsic value. On each trade, this group make money = Bid-Ask spread. As volume of trade increases, bid-ask spread decreases, netting roughly stable reward for this group to maintain market liquidity.
3. There are horde of traders whose sole job is to arbitrage the ETF market to take advantage of any tracking error. Essentially, if the groups #1 and #2 fail, people in $3 take advantage of it.
Bottomline - for something like VOO/VTI - the price is almost totally dependent on the basket's intrinsic value. Any divergence is almost negligible.
You *are* depending on a lot of actors to do their jobs to maintain this state of affairs when you buy them for retirement!! But this dependency is a lot less (I'd say orders of magnitude less) than if you buy Gibson Guitars, or real estate, or gold - for example!!