Compound Growth over Decades and How Fees Reduce It

What you will learn

• How to calculate the future value of a lump sum and of yearly savings.
• How to use the rule of 72 for a quick estimate.
• Why a small yearly fee difference becomes a large euro difference over decades.
• How a front-end load compares with an ongoing fee.

Explanation

The individual cost items of a fund (TER, tracking difference, spreads, order costs) were defined in lesson 5. Here we measure their long-term effect.

Compound growth

Compound growth means that returns also earn returns. If a return is reinvested, the next year's return is calculated on a larger base. For a lump sum (one single payment) the future value is:

FV = PV × (1 + r)n

FV is the future value, PV the present value (the amount invested today), r the yearly return as a decimal (6% = 0.06), and n the number of years. The term (1 + r)n is the growth factor.

For a fixed amount saved at the end of every year (PMT, the payment), the future value is:

FV = PMT × ((1 + r)n − 1) ÷ r

Monthly savings compound slightly differently, but the yearly version is easy to check by hand and shows the same pattern.

The rule of 72

To estimate how many years it takes to double money, divide 72 by the yearly return in percent. At 6% this is 72 ÷ 6 = 12 years; at 4% it is 18 years. It is an approximation that works well for returns between about 2% and 12%.

How fees enter the formula

An ongoing fee is taken from the fund's assets every year, so what compounds is the net return: roughly the gross return minus the yearly fee. A fee of 1.6% looks small next to a return of 6%, but it removes more than a quarter of the yearly growth, and this gap compounds every year. Over long periods, the share of the final value lost to fees grows much faster than the fee itself.

A front-end load (German: Ausgabeaufschlag) is a one-off charge when you buy, often charged on actively managed funds sold through banks. A 5% load means only €9,500 of €10,000 is invested. It hurts once; an ongoing fee hurts every year.

Returns are uncertain. The percentages in this lesson are assumptions for calculation, not expected returns. Also remember inflation: a nominal result must be converted into purchasing power, as shown in the macro course (lesson on real returns).

Worked example

Hypothetical example: two investors each invest €10,000 once and also save €2,400 at the end of every year for 30 years. Both funds earn an assumed gross return of 6% per year. Investor A pays total yearly costs of 0.2% (a cheap ETF), so the net return is 5.8%. Investor B pays 1.8% (an active fund plus an account fee), so the net return is 4.2%. We ignore taxes, inflation and any variation in returns.

1. Growth factors: 1.05830 = 5.4271; 1.04230 = 3.4358.
2. Lump sum A: €10,000 × 5.4271 = €54,271. Lump sum B: €10,000 × 3.4358 = €34,358. Difference: €19,913.
3. Savings A: €2,400 × (5.4271 − 1) ÷ 0.058 = €183,191. Savings B: €2,400 × (3.4358 − 1) ÷ 0.042 = €139,190. Difference: €44,001.
4. Totals: A €237,463; B €173,549. Total paid in by each: €10,000 + 30 × €2,400 = €82,000.
5. Difference: €63,914, caused only by a fee gap of 1.6 percentage points per year.
6. Rule of 72 check: at 5.8% money doubles about every 12.4 years; at 4.2% only about every 17.1 years.

If investor B had also paid a 5% front-end load on the lump sum, only €9,500 would have grown at 4.2%: €9,500 × 3.4358 = €32,640, another €1,718 less.

Common mistake

Practice activity

You save €1,800 at the end of each year for 25 years. Compare a net return of 5.0% with a net return of 3.5%. Ignore taxes and inflation.

1. Calculate both growth factors and both final values.
2. How much did you pay in, and how large is the difference between the two results?
3. Use the rule of 72 to estimate the doubling time at 5.0%.

Key takeaways

• Lump sum: FV = PV × (1 + r)n; yearly savings: FV = PMT × ((1 + r)n − 1) ÷ r.
• Fees reduce the net return, and the gap compounds every year.
• A fee gap of 1.6 pp over 30 years can cost far more than the total of the fees paid in any single year.
• All return figures in such calculations are assumptions, not forecasts.

Next step

In the next lesson you will get a general orientation on how funds and ETFs are taxed in Germany.