How much do you lose to annual fees after many years?

"" shows how high expense ratios can significantly reduce the amount of money you have in your portfolio. The longer you hold onto funds with high expense ratios, the less money you will have for retirement.

The math works the same way for financial advisors who charge an Asset Under Management (AUM) fee. Whether you have an advisor who charges a 1% AUM fee, or a fund which has a 1% expense ratio, the effect on your portfolio is the same.

Percentages lost to annual fees
Stated more precisely:


 * Assume that you buy an investment and do not touch it for $$n$$ years. In other words, you do a "lump-sum" investment.
 * Assume that the investment has an annual fee $$f$$, expressed as a percentage. For example, a mutual fund might have an annual fee (an expense ratio) of 1%.
 * Assume that the annual fee is taken once per year, at the end of each year. This assumption is a simplification. For example, in reality, a mutual fund calculates its annual fee as a daily fee.
 * Temporarily imagine that the annual fee is zero and that your investment grows for $$n$$ years. The final value of your investment---consider it as "100%."
 * What percentage of your investment do you lose to annual fees? The answer is $$1 - (1 - f)^{n}$$.

Later, we will prove the above formula. First, see the following table and examples.

Example: If you buy an investment with an annual fee of 0.10%, and do not touch the investment for 40 years, you lose 4% to annual fees.


 * The above percentage agrees with a result from user "Lobster," who started the following thread on Bogleheads on 2017-04-20: Visualizing the devastating impact of fees. User "Lobster" linked to a Google Sheets spreadsheet which has a column chart and which has a percentage of 3.92%.
 * Note that $$3.92% = 1 - (1 - 0.10%)^{40}$$.

Example: If you buy an investment with an annual fee of 2%, and do not touch the investment for 10 years, you lose 18% to annual fees.
 * The above percentage agrees with a result from Mr. Richard A. Howard, on his web site buyupside. We boldface the important text:


 * Note that $$18.29% = 1 - (1 - 2%)^{10}$$.

Example: If you buy an investment with an annual fee of 2%, and do not touch the investment for 30 years, you lose 45% to annual fees.
 * The above percentage agrees with a result from Mr. Richard A. Howard, on his web site buyupside. We boldface the important text:


 * Note that $$45.45% = 1 - (1 - 2%)^{30}$$.

Example: If you buy an investment with an annual fee of 2%, and do not touch the investment for 50 years, you lose 64% to annual fees.
 * The above percentage agrees approximately with a result from Mr. John ("Jack") C. Bogle, the founder of Vanguard, in an interview on 2017-11-28. We boldface the important text:


 * Note that $$64% = 1 - (1 - 2%)^{50}$$.

The following examples do not use the above table, but they use the formula which we mentioned earlier.

Example: If you buy an investment with an annual fee of 2%, and do not touch the investment for 25 years, you lose 40% to annual fees.
 * The above percentage agrees with a result from Vanguard on their web page Don't let high costs eat away your returns. We boldface the important text:


 * Note that $$40% = 1 - (1 - 2%)^{25}$$.

Example: If you buy an investment with an annual fee of 2.5%, and do not touch the investment for 65 years, you lose 81% to annual fees.
 * The above percentage agrees approximately with a result from Mr. John ("Jack") C. Bogle, the founder of Vanguard, in an interview on 2006-02-07 with the documentary program Frontline. We boldface the important text:


 * Note that $$81% = 1 - (1 - 2.5%)^{65}$$.

Formula proof
Below, we prove the formula which we mentioned earlier.


 * Assume that you buy an investment and do not touch it for $$n$$ years.
 * Assume that the investment has an annual fee $$f$$, expressed as a percentage.
 * Assume that the annual fee is taken once per year, at the end of each year.
 * Temporarily imagine that the annual fee is zero and that your investment grows for $$n$$ years. The final value of your investment---consider it as "100%."
 * What percentage of your investment do you lose to annual fees? The answer is $$1 - (1 - f)^{n}$$.

Proof.


 * Let $$P$$ be your principal at the start of year 1.


 * Let $$r_{i}$$ be the investment's annual return in year $$i$$.


 * At the end of year 1, you will have $$P(1 + r_{1})(1 - f).$$


 * At the end of year 2, you will have $$P(1 + r_{1})(1 - f) \times (1 + r_{2})(1 - f).$$


 * At the end of year 3, you will have $$P(1 + r_{1})(1 - f) \times (1 + r_{2})(1 - f) \times (1 + r_{3})(1 - f).$$


 * At the end of year $$n$$, you will have $$P(1 + r_{1})(1 - f) \times (1 + r_{2})(1 - f) \times (1 + r_{3})(1 - f) \times \cdots \times (1 + r_{n})(1 - f).$$
 * Let $$I$$ be the last value above. $$I$$ stands for "investment value."


 * Temporarily imagine that the annual fee is zero: $$f = 0$$. Substitute this into the expression for $$I$$, getting $$P(1 - r_{1})(1 - r_{2})(1 - r_{3}) ... (1 - r_{n})$$.
 * Let $$M$$ be the last value above. $$M$$ stands for "max value of investment." Assume that $$M$$ is not zero. Later, we will divide by $$M$$.


 * Note that $$I/M = (1 - f)(1 - f)(1 - f) ... (1 - f) = (1 - f)^{n}.$$ This is the percentage of your investment that you keep.


 * Therefore the percentage of your investment that you lose to annual fees is $$1 - I/M = 1 - (1 - f)^{n}.$$

Generalized formula
We can generalize our above formula. We allow the years to have possibly different annual fees. The formula statement becomes:


 * Assume that the investment has $$n$$ annual fees $$f_{1}$$, $$f_{2}$$, $$f_{3}$$, ..., $$f_{n}$$, where each fee is a percentage.
 * What percentage of your investment do you lose to annual fees? The answer is $$1 - (1 - f_{1})(1 - f_{2})(1 - f_{3}) ... (1 - f_{n})$$.
 * What percentage of your investment do you lose to annual fees? The answer is $$1 - (1 - f_{1})(1 - f_{2})(1 - f_{3}) ... (1 - f_{n})$$.
 * What percentage of your investment do you lose to annual fees? The answer is $$1 - (1 - f_{1})(1 - f_{2})(1 - f_{3}) ... (1 - f_{n})$$.

The proof of the generalized formula is similar to the proof of the original formula.

More related work
On 2011-10-28, in the Vanguard blog post Stopping the silent killer of returns, Dr. John Ameriks posted a table, which we recreate below. Most of his values agree with ours, if we ignore the minus signs. However, he used a different formula: $$1-(1+c)^{-T}$$, where $$c$$ is an annual fee (possibly "c" stands for "cost"), and $$T$$ is the number of years (possibly "T" stands for "time"). This formula is discussed in a thread on Bogleheads which was started by user "Lobster" on 2017-04-20: Visualizing the devastating impact of fees. In that thread, users "Kevin M" and "#Cruncher" ("number cruncher") think that this formula is related to how mutual funds calculate the annual fee as a daily fee.

Cumulative impact of fees on ending wealth at various time horizons

On 2018-02-26, on the blog post How Much Of Your Total Returns Are Lost To Fees?, blogger "Wall Street Physician" posted a table, which we recreate below. The values differ somewhat from ours, possibly because of differences in rounding, or because of other reasons.