Three short experiments on invented numbers. In each one you first make a guess, then check the maths, and then read what the result does and does not prove. Every number here is made up to show how the arithmetic works; none of it is data from a real fund or a forecast.
Experiment 1: the average return is zero. Is your money unchanged?
Your guess first. You put €1,000 into an investment. Year one it rises 20%, year two it falls 20%. The average of +20% and −20% is 0%. Do you end with €1,000?
Show the answer
No. €1,000 × 1.20 = €1,200, then €1,200 × 0.80 = €960. You end with €960, a loss of 4%, even though the average of the two returns is zero. The other order gives the same end amount: €1,000 × 0.80 × 1.20 = €960.
| Invented order | After year 1 | After year 2 | With €100 added at each year end |
|---|---|---|---|
| +20%, then −20% | €1,200 | €960 | €1,140 |
| −20%, then +20% | €800 | €960 | €1,180 |
What it shows. A fall needs a bigger rise to make up for it: −20% needs +25% to get back to the start. The yearly growth rate that really matches this example is about −2.02% a year, not 0%. And once you add money along the way, the order of the years matters: the same two returns give €1,140 or €1,180.
What it does not show. It says nothing about which order of years will happen, how large real swings will be, or whether you should wait or invest now.
Experiment 2: this fund is cheaper. What can you conclude?
Your guess first. Two invented funds track the same invented index, which grows 6% a year before costs. Fund A costs 0.10% a year, fund B 0.30%. You put €10,000 into each for 20 years. How big is the difference, and what does it tell you?
Show the answer
After costs fund A grows 5.9% a year and fund B 5.7%. After 20 years: €10,000 × 1.05920 ≈ €31,472 for A and €10,000 × 1.05720 ≈ €30,304 for B. A difference of about €1,168, from 0.20 percentage points a year.
| Invented fund | Yearly costs | Growth after costs | After 20 years |
|---|---|---|---|
| Fund A | 0.10% | 5.9% a year | €31,472 |
| Fund B | 0.30% | 5.7% a year | €30,304 |
What it shows. When everything else is the same, a lower cost leaves you more, and small differences add up over long periods.
What it does not show. Real funds are rarely the same in everything else: they can follow different indexes, lag their index by more or less than their costs, pay out or reinvest dividends, and trade in different currencies. A lower cost on its own does not make a fund the better choice for you. The fund’s own documents show what is different; see our Fact Lab.
Experiment 3: two funds. How different are they really?
Your guess first. Fund “World” holds five invented companies, fund “Big Five” holds four. They have different names. How much of the two funds is the same?
| Invented company | Weight in “World” | Weight in “Big Five” | Shared part |
|---|---|---|---|
| Alpha | 30% | 40% | 30% |
| Beta | 20% | 25% | 20% |
| Gamma | 15% | 20% | 15% |
| Delta | 15% | 15% | 15% |
| Epsilon | 20% | 0% | 0% |
| Total | 100% | 100% | 80% |
Show the answer
80% of the two funds is the same. If you split your money half and half, Alpha becomes 35% of everything you hold (half of 30% plus half of 40%), more than in either fund on its own being spread out; the only company that adds something new is Epsilon, at 10%.
What it shows. A second fund does not automatically spread your money more. Two different names can hold largely the same companies.
What it does not show. Overlap on its own does not say whether a mix is good or bad, how risky it is, or what it will return. Real funds hold hundreds or thousands of companies, and their weights change every day.
These experiments are general explanations with invented numbers, not tax or investment advice. Past results, real or invented, say nothing certain about the future.