How the math works
Each month the balance grows by the return and shrinks by one twelfth of the yearly fee. Because the fee is a percentage of the whole balance, it grows as your money grows. The difference between the two results is what the higher fee cost, including the growth that money would have earned.
Examples
Buffett's ten-year bet. In 2007 Warren Buffett bet Protégé Partners that an S&P 500 index fund would beat a group of hedge funds, after fees, over the ten years from 2008 through 2017. In his 2017 letter he reported the result: the index fund gained about 7.1% a year, and the five funds of hedge funds Protégé picked averaged about 2.2%. Buffett put much of the gap down to the layers of fees. The winnings went to Girls Inc. of Omaha.
Cheap and expensive funds today. Large S&P 500 index funds charge a few hundredths of a percent a year. Vanguard's 500 Index Admiral shares, for example, list an expense ratio of 0.04%. Many actively managed stock funds charge 0.5% to 1% or more. With the default settings above, $100,000 left alone for 30 years at 7% ends about $190,000 higher in the 0.04% fund than in the 1% fund.
Explain it like I'm 13
Imagine you keep your savings in a jar, and a helper looks after it. Every year the helper takes one coin out of every hundred as payment. That sounds tiny.
But the helper takes a coin from every hundred every single year, including from the coins your savings earned. After 30 years, the helper has taken a big pile. A helper who takes one coin out of every few thousand leaves almost all of the pile for you.
Questions
What is an expense ratio?
The yearly fee a fund takes, as a percentage of what you have invested. It comes out of the fund automatically, so you never see a bill.
Is a 1% fee a lot?
It sounds small, but it is charged every year on the whole balance, including past growth. Over decades it can take a large share of the final amount.
Are cheaper funds always better?
Not always, but fees are one of the few things an investor can know in advance. Compare funds that hold similar investments.