What is the correct way to calculate the total amount that mutual fund fees will cost me over a certain time period?

Question

The 3 methods below all have different results, and I'm not sure which is correct.

Principle = $1M
Return: 7%
Fee: 1%
Time period: 3 years

Method 1:
($1000000 * 1.07^3) - ($1000000 * 1.06^3) = $34027

Method 2 (fee taken at end of year):

End of year 1 balance = $1000000 * 1.07 = $1070000
Year 1 fee = $1070000 * .01 = $10700
Year 2 start balance = $1070000 - $10700 = $1059300
End of year 2 balance = $1059300 * 1.07 = $1133451
Year 2 fee = $1133451 * .01 = $11334.51
Year 3 start balance = $1133451 - $11334.51 = $1122116.49
End of year 3 balance = $1122116.49 * 1.07 = $1200664.64
Year 3 fee = $1200664.64 * .01 = $12006.65
Total fee = $10700 + $11334.51 + $12006.65 = $34041.16

Method 3 (fee taken at start of year, this isn't realistic but I wanted to include it to show that it doesn't explain the outcome number in method 1):

Start of year 1 fee = $1000000 * .01 = $10000
Start of year 1 balance = $1000000 - $10000 = $990000
End of year 1 balance = $990000 * 1.07 = $1059300
Start of year 2 fee = $1059300 * .01 = $10593
Start of year 2 balance = $1059300 - $10593 = $1048707
End of year 2 balance = $1048707 * 1.07 = $1122116.49
Start of year 3 fee = $1122116.49 * .01 = $11221.16
Total fee = $10000 + $10593 + $11221.16 = $31814.16

Answer 1

The "return" of a mutual fund is not quite what you think it is, and very few mutual funds would guarantee a fixed rate of return, the way that a bank CD does.

As a more realistic version of your question, suppose that you invested $10K in a mutual fund (most readers would hesitate to invest $1M in a mutual fund in one swell foop, and especially a mutual fund that has an expense ratio of 1%) at the beginning of the year, buying 1000 shares of the fund at the current price of $10 per share. You also told the mutual fund to _reinvest all distributions from the fund. A year later, you own 1024.904 shares because of all the distributions that the mutual fund reinvested for you, and the fund reports that the share price is $10.44. So your investment is now worth $10,700 and your (pre-tax) return is 7%. How much taxes do you owe? Well, that depends on how much the distributions were, and what part of that was Qualified Dividends, what part was ordinary dividends or short-term capital gains, and how much was Long-term Capital Gains, whatever the mutual fund reported to you and the IRS on Form 1099-DIV.

What happened to that 1% annual expense? Well, the expense is included in the share price reported to you (in fact, a fraction of the annual expense is deducted from the mutual fund assets on a daily basis, not at the end of the year), and when the mutual fund reports to you that the share price is $10.44 today, that price is net of the expense which has been silently deducted from the fund assets on a daily basis.

What about the second year? Well, you start off the second year with 1024.904 shares, and distributions may increase the number of shares you own during the second year, but what your account is worth at the end of the second year is just the total number of shares then held times the share price that the mutual fund reports to you, If this number works out to be $11,499, then the fund has "returned" 7% compounded annually to you. Once again, you don't need to take into account the annual 1% expense; that has already been accounted for in the share price reported to you.

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So, what has the 1% annual expense cost you? Each year, it has cost you 1% of the _average value of your account (shares held times the share price reported to you) over the period of one year. If it hadn't been for that constant drain, the mutual fund would have reported larger distributions and larger share prices than it has in the above example. How much larger? Well, the calculations are messy, and the investor does not necessarily have all the information that would be needed to compute the answer down to the last penny. Just think of it as your return would have been 8% instead of 7% each year.

Answer 2

Method 2 is probably the correct calculation of the fees. Method 1 calculates the difference between a 6% return and a 7% return. It's approximating how much you would are losing because of the fees.

Method 3 does not do what you want. You compound the return on top of the fee, which does not match what Method 1 does. To match the Method 1 assumption, we'd have Method 4:

$1,000,000 * .01 = $10,000
$1,000,000 * .07 = $70,000
$1,000,000 + $70,000 - $10,000 = $1,060,000
$1,060,000 * .01 = $10,600
$1,060,000 * .07 = $74,200
$1,060,000 + $74,200 - $10,600 = $1,123,600
$1,144,800 * .01 = $11,236
$1,144,800 * .07 = $78,722
$1,144,800 + $80,136 - $11,448 = $1,213,488
$10,000 + $10,600 + $11,236 = $31,836

But this still doesn't match Method 1. So let's see Method 5:

$1,000,000 * .01 * (1 + 1.06 + 1.06^2) = $31,836

Now we finally get the same results with two methods. Now, how would we write a similar formula for Method 2? Method 6:

$1,000,000 * 1.07 * .01 * (1 + 1.07 * .99 + 1.07^2 * .99^2) = $34,041.16

Note that that's rounded to two decimal places.

The problem with Method 1 is that it doesn't account for the fact that the second and third years are compounded on the results of the first year, both return and fee.

Answer 3

The problem with this type of question is the vague meaning of "will cost me".

For example, you could watch your account and look for the withdrawal of annual fees, write the number down, and add them all up after three years. Method #2 is an application of this meaning.

However, Method #2 represents a fundamental error in evaluating money over time: adding together amounts of money at different times as if they were at the same time.

Another meaning of "will cost me" is: how will my situation at the end of the investment be changed by the application of the annual fee?

This can be calculated very easily, if you concentrate on what you have left at the end of each year, rather than how much you are paying out. The annual fee leaves you with 99% of the end-of-year balance, So, the result after three years leaves you with TF;

TF = 1,000,000 X 1.07 X 0.99 X 1.07 X 0.99 X 1.07 X 0.99 = 1,188,658.00

(this can be massively simplified using exponents)

Without the fee, your balance would have been TNF:

TNF = 1,000,000 x 1.07^3 = 1,225,043.00

So, the fee means that, on the day three years hence when the investment ends, you will have $36,345.00 less in your account.

So this is the cost of the fee, for one definition of cost.

PS: literally as I typed this, the background NFL broadcast was interrupted by a commercial for an investment site, where the fees result in a 30% hit on the final retirement fund (name if considered proper on SE)