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I would like to keep it as simple as possible. If I visit Vanguard Benchmark returns page here

https://personal.vanguard.com/us/funds/tools/benchmarkreturns

It shows annualized returns for different funds for up to a decade. Now for example, S&P 500 Index it shows that it has returned 7.31% for last 10 years. Now, I know that average yield on this fund is around 1.90%. Is it factored into the annualized reutun when Vanguard reports it? If not, would it be safe to say that in fact the return was 7.31% + 1.90%?

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No, that link shows returns for an index and not a fund! Note at the bottom:

Index performance is provided as a benchmark. It is not illustrative of any particular investment. An investment cannot be made in an index.

What format would you expect the dividends to actually be handled as some old Unit Investment Trusts would just collect the dividends as cash while some mutual funds may re-invest the proceeds right away? The Handbook of Financial Instruments has this part on page 539:

The dividends must be retained as cash and invested, in effect in money market instruments until the dividend payment is made.

Thus, while you may think your question is simple there are more than a few ways to go. M* has this:

Annual total returns are calculated on a calendar-year and year-to-date basis. Total return includes both capital appreciation and dividends. The year-to-date return is updated daily.

For mutual funds, return includes both income (in the form of dividends or interest payments) and capital gains or losses (the increase or decrease in the value of a security).

Morningstar calculates total return by taking the change in a fund's NAV, assuming the reinvestment of all income and capital gains distributions (on the actual reinvestment date used by the fund) during the period, and then dividing by the initial NAV. Unless marked as load-adjusted total returns, Morningstar does not adjust total return for sales charges or for redemption fees.

Total returns do account for management, administrative, and 12b-1 fees and other costs automatically deducted from fund assets.

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