The fee, every year
A 1% fee is 1% of the money every year, and each year's fee also removes the growth that money would have earned for every year after.
- Compound a return and a fee together over many years
- Show the share of ending wealth a fee removes, whatever the market does
- Find the highest fee a wealth-loss limit allows
- Compute the outperformance an active fund needs just to match an index fund
The intuition
A client says a 1% fee is trivial: it is one percent of their money. It is one percent of their money every year, charged on the whole balance, and each year's fee also takes away the growth that money would have earned for every year after. Over twenty-five years it removes about 22% of the ending wealth, and a far larger share of the gain, because the fee is charged on the capital as well as the return.
The arithmetic is one factor: each year the money grows by (1 + g) and then keeps (1 − fee), so the net factor is the product. Two things fall out. The share of wealth lost to the fee, 1 − (1 − fee)ⁿ, does not depend on the market at all. And an active fund charging more than an index fund must beat the market, gross, every single year, by an amount the same factor gives exactly.
Ending value with a fee = V × ((1 + g)(1 − fee))ⁿ. Share of ending wealth lost = 1 − (1 − fee)ⁿ, whatever the return. Fee that costs a share s = 1 − (1 − s)^(1/n). Alpha to match a cheaper fund = (1 + g)(1 − fee_index) ÷ (1 − fee) − (1 + g).
Why it works
- The conventions here: a constant gross return; the fee is charged on the year-end value, so each year's growth factor is (1 + g)(1 − fee). No contributions, withdrawals or taxes.
- The fee compounds on the whole balance, not on the return. That is why it takes a much larger bite of the profit than its headline rate: a fee of 1% on a 7% return takes a seventh of the return the first year and more than that of the ending gain.
- The market cancels out of the share lost. The fee scales the pot by (1 − fee) each year whatever performance does, so the share of ending wealth lost is 1 − (1 − fee)ⁿ. A good market makes the dollars bigger and leaves the percentage alone.
- Less than n × fee. Thirty years of 1% remove about 26%, not 30%, because each year's fee is charged on a pot earlier fees have already shrunk.
- The fee is the most reliable predictor of net returns an advisor has. Markets are uncertain; costs are not.
- The hurdle for active management is exact. To match an index fund after fees, the active fund must beat the market gross by (1 + g)(1 − fee_index) ÷ (1 − fee) − (1 + g) every year, after its own trading costs, against other professionals trying to do the same.
| No fee: $500,000 × 1.07²⁵ | $2,713,716 |
| With the fee: $500,000 × (1.07 × 0.99)²⁵ | $2,110,787 |
| Cost of the fee | $602,930, 27% of the gain |
| Share of ending wealth lost: 1 − 0.99²⁵ | 22.22%, whatever the market did |
| Highest fee for a 20% limit: 1 − 0.8^(1/25) | 0.889% |
| Alpha to match the index fund: 1.07 × 0.999 ÷ 0.99 − 1.07 | 97 basis points a year, gross |
The fee took 1% of the money each year and 27% of the gain over the period. Switching to the index fund would have ended $535,895 richer.
The formulas
Grow, then keep (1 − fee), every year.
The market cancels out.
The share formula run backwards.
The gross outperformance that pays for the fee gap, every year.
Worked example
One net factor per year, compounded, against the no-fee case. The follow-up gives the fee's share of the gain, which is the honest measure.
See it move
Same client. Change the starting value, the gross return, the fee, the index fund's fee and the number of years.
- Raise the gross return. The dollar cost of the fee rises; the share of ending wealth lost does not move.
- Raise the fee. The cost, the share lost and the outperformance needed all rise.
- Add years. The share lost rises along the curve, but by less each year: the fee is compounding on a shrinking base.
- Raise the index fund's fee. The gap between the two ending values narrows and the outperformance needed falls.
Run it backwards
The two fees and the market's return are known. What must the active fund beat the market by, gross, every year, just to match the index fund after fees?
Set the active fund's net factor equal to the index fund's: (1 + g + α)(1 − fee) = (1 + g)(1 − fee_index). Solve for α.
The follow-up says why that is harder than it sounds: it has to happen every year, after trading costs, against professionals trying to do the same.
Traps
Say it in the interview
“A client says a 1% fee is trivial. What do you tell them?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Net factor per year = (1 + g)(1 − fee); compound it.
- Share of ending wealth lost = 1 − (1 − fee)ⁿ, whatever the market does.
- Judge the fee against the gain, not the portfolio.
- Alpha to justify a fee = the gross outperformance that closes the fee gap, every year.