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I am currently reading an investopedia article about expense ratios in funds. This article uses the following numerical example about expense ratios and returns.

A fund has a gross return of 10% and an expense ratio of 1.5%. Therefore the net return of the fund becomes 10% - 1.5% = 8.5%.

I am trying to figure out what mathematical formula, relating gross return and and expense ratio is used to compute this net return. I know the following

Gross return = (Final investment - Initial investment)/Initial investment

And from the investopedia article, expense ratio is defined as

Total fund costs / Total fund assets. Which in this case I would assume is equivalent to

Total fund costs / Final investment.

So setting up the formula used to calculate the net return in the numerical example above, I get.

Net return = (Final investment - Initial investment)/Initial investment - Total fund costs/Final investment

I find this to be a strange result. Simplifying this equation doesn't get me a sensical answer.

So my question is how can you set up the formula which computes the 8.5% net return in the example above which includes the definition of gross return and expense ratio? I cannot seem to find a mathematical answer to this question.

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A fund has a gross return of 10% and an expense ratio of 1.5%. Therefore the net return of the fund becomes 10% - 1.5% = 8.5%.

Isn't that exactly the answer.

The fund based just on the investments had a 10% return. But the expenses for the fund are 1.5% of the value of the fund. Therefore if you had $10,000 in the fund at the start of the year instead of your investment being $11,000 it is lowered by the expenses.

This is how Vanguard handles it:

How expense ratios are calculated at Vanguard

As each fund passes its fiscal year-end, the annual expense ratio is calculated by dividing the fund's operational expenses by its average net assets. If the fund's assets are increasing faster than its costs, you'll enjoy lower expenses as a fund shareholder.

You won't find them on your account statement

The cost of investing is usually associated with trading commissions and account service fees—items you see as "debits" from your accounts.

But expense ratios are less obvious because they're not itemized on your account statements or confirmations. Instead, each fund's expenses are deducted from its total value on a regular basis. And those expenses cut directly into your investment returns.

It looks like the ratio is calculated at the end of the year based on the actual expenses during the year. The money is pulled out during the year.

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