# A range, not a number: twenty-five years of possible paths

> Where long-run returns come from, and how to read twenty-five years of simulated paths as a range with a median, a 10th and a 90th percentile.

- Course: [Long-Term Investing: Index Funds, Risk and Your Own Behavior](https://productovi.com/courses/long-term-investing.md)
- Module: What can an investing app honestly promise?
- Lesson 3/32, ~23 min
- Source: https://productovi.com/courses/long-term-investing/range-not-a-number

## Where returns come from

**Grace:** Last week of February at Zapas. Yesterday's screen used a steady 6%. Real markets never move in a straight line, so today we replace that line with a range.

**Ethan:** But a range looks weak. Users want one number. Why can't we just say "about 6%"?

**Grace:** Because nobody knows next year's number, including us. Start with where the return comes from. A stock is a share of a company's profits. A bond is a loan that pays interest. Cash pays a small rate for safety.

**Grace:** Profits grow over decades, but how much the market will pay for them swings with mood, rates and news. That swing is why stock returns are uncertain from year to year, and why we get paid for holding them.

> **Exercise:** Match each holding with what pays its owner over the long run.

## Twenty-five years of paths

**Grace:** So I ran 2,000 plans of 150 dol. a month for 25 years. Each simulated year draws one real return at random from our 40 synthetic Tessaro World years, with repeats allowed. Every path ends up with its own set of 25 years.

**150 dol. a month for 25 years, 2,000 simulated paths**

| Year | 90th percentile | 75th percentile | Median | 25th percentile | 10th percentile | Paid in |
| --- | --- | --- | --- | --- | --- | --- |
| 1 | 1994 | 1925 | 1881 | 1782 | 1705 | 1800 |
| 5 | 13085 | 11944 | 10550 | 9188 | 8017 | 9000 |
| 10 | 34906 | 30024 | 24907 | 20360 | 16877 | 18000 |
| 15 | 69059 | 55963 | 43694 | 34790 | 27560 | 27000 |
| 20 | 120315 | 92234 | 70153 | 52167 | 39490 | 36000 |
| 25 | 189828 | 147273 | 106225 | 76117 | 57544 | 45000 |

*Real dollars; percentiles at the end of each year. Synthetic data, not a forecast.*

**Dataset (CSV):** [Percentiles of 2,000 simulated 25-year plans](https://productovi.com/datasets/long-term-investing/range-not-a-number/paths_25y.csv)

**Sofia:** After 25 years the median is 106,225, on 45,000 paid in.

**Grace:** Yes, and the 10th percentile is 57,544, the 90th is 189,828. One plan in ten ends below 57,544, one in ten above 189,828. That width is the honest answer.

![After 25 years of 150 dol. a month, half the paths end above 106,225, one in ten below 57,544 and one in ten above 189,828, against 45,000 paid in.](https://productovi.com/courses/long-term-investing/diagrams/m1-l3-fan.png)

*Promise the band, not the line.*

> **Exercise:** On the fan chart, what does "10th percentile after 25 years: 57,544" mean?

**Ethan:** Wait. Every path draws from the same 40 years. How can they end so far apart?

**Grace:** Because no two paths get the same 25 years. Our history has three years that lost more than 20% after inflation. I counted them in each path: the bottom tenth drew those years about 3.6 times on average, the top tenth fewer than once.

**Sofia:** So the spread mostly comes from how many bad years a path happens to draw.

> **Exercise:** All 2,000 paths draw their yearly returns from the same 40 synthetic years, so after 25 years they all end within a few thousand dollars of each other.

## The chance of ending below what you paid

**Ethan:** How often does a user end up with less than they put in? That's the scary number.

**Grace:** It depends on the horizon. Below the money paid in: 21.6% of paths after 5 years, 14.1% after 10, 3.8% after 25.

**Share of paths below the money paid in**

| Years invested | Below paid in |
| --- | --- |
| 1 | 30.7 |
| 5 | 21.6 |
| 10 | 14.1 |
| 20 | 6.2 |
| 25 | 3.8 |

*150 dol. a month, real dollars, 2,000 simulated paths. Synthetic data.*

> **Exercise:** In the simulation, what share of paths had a balance below the money paid in after 10 years, in percent?

**Sofia:** So the honest promise is a band plus a horizon, not a line.

> **Exercise:** Which line belongs on Zapas's goal screen for 150 dol. a month over 25 years?

## Testing Ethan's 12%

**Grace:** Now Ethan's "12% a year". At 3% inflation that's 8.7% real. A steady 8.7% for 25 years on 150 dol. a month would give 152,443. Only 22.1% of our paths reach that.

**Ethan:** So my headline describes roughly the luckiest fifth of users. Fine, that's not a promise we can make to everyone.

> **Exercise:** Ethan's "12% a year" is 8.7% real at 3% inflation, which means 152,443 after 25 years of 150 dol. a month. What share of the 2,000 simulated paths reach that balance, in percent?

**Grace:** Not to everyone, no. And notice what the fan does for us: it shows growth and risk in one picture. Users see that most paths end well above what they paid in, and that nobody can say exactly where.

**Grace:** Write the explanation users will see next to the fan chart.

> **Exercise:** Write a 2–3 sentence explanation for users next to the fan chart of 150 dol. a month for 25 years. Use the range and the median, say what the band means, and include a risk line.

> **Key ideas**
>
> - Stocks earn from company profits, bonds from interest, cash from a short-term rate; higher expected return comes with bigger swings.
> - 2,000 simulated plans of 150 dol. a month for 25 years: 10th percentile 57,544, median 106,225, 90th 189,828, on 45,000 paid in.
> - A percentile is a rank, not a floor: one plan in ten ends below the 10th percentile.
> - The spread comes from which years each path draws: the bottom tenth drew the deep-fall years far more often than the top tenth.
> - Below the money paid in: 21.6% of paths after 5 years, 14.1% after 10, 3.8% after 25.
> - "12% a year" (8.7% real) is reached by only 22.1% of paths: promise a band with a horizon, not a line.

> **Educational, not advice**
>
> This course explains how long-term investing works; it never tells you what to buy. Investments can fall, and you can get back less than you put in. Check the current rules where you live or ask a licensed professional.

**Grace:** Tomorrow we put it all together: Ethan's landing page gets rewritten, with a range, a horizon and a risk line.

- Previous lesson: [Compounding with a monthly amount: time beats the rate](https://productovi.com/courses/long-term-investing/compounding-monthly.md)
- Next lesson: [Case: rewrite the "12% a year" landing page](https://productovi.com/courses/long-term-investing/case-landing-page.md)

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