Lesson 11 of 20 · 6 min
How to read a fan chart
A fan chart shows percentile bands over time, and the bands are not paths.
A fan chart is the standard picture of simulation output: a central line with shaded bands widening to the right. It is informative and it is misread in one specific way that is worth eliminating.
What each element is:
- The central line is the median at each point in time — the value half the paths were above.
- Each band is a percentile range, typically 25th to 75th for the inner band and 5th to 95th for the outer.
- The widening is the square-root-of-time rule from Track 3, visible: uncertainty grows with the square root of the horizon.
- The vertical axis is value; the horizontal axis is time. Every point on the chart is a cross-sectional summary of thousands of paths at that moment.
Here is the misreading. The upper edge of the band is not a path. It is the 95th percentile at each date, and different simulated paths occupy that position at different dates. No single simulation ran along the top edge, and none ran along the bottom edge either. The bands describe the distribution at each instant, not any trajectory through it.
That distinction matters for the question a leveraged holder actually has. Suppose the lower band stays above your trigger value for the whole horizon. It is tempting to conclude the trigger is not reached in 95 percent of scenarios. That conclusion is wrong, because individual paths dip below the band and come back — the band is where the 5th percentile of paths sits at each date, not a floor that paths respect.
In the standard setup the difference is large: 6.8 percent of paths end below the trigger value, while 14.3 percent touch it at some point along the way. A fan chart shows you the first number and you need the second.
Key terms
- Fan chart
- A plot of percentile bands over time from simulation output. Each vertical slice summarises all paths at that date.
- Cross-sectional summary
- A statistic computed across paths at one moment. Percentile bands are cross-sectional and therefore not trajectories.
- Touch probability
- The chance a level is reached at any point along a path. Always at least as large as the chance of ending beyond it, usually much larger.
Worked example
Ending below a level versus touching it
Standard setup: $600,000 collateral, $300,000 balance, 70% threshold, trigger value $428,571, three years.
- Paths ending below the trigger
- 6.8%
- Paths touching the trigger at any point
- 14.3%
- Ratio
- 2.1×
- At 15% volatility, touch probability
- 4.6%
- At 35% volatility, touch probability
- 49.6%
- What changed between those cases
- Only volatility
What a fan chart's endpoint shows.
What actually matters.
The endpoint question and the path question differ by a factor of two here. And at 35 percent volatility, half of all paths reach the trigger.
Check yourself
In the standard setup, 6.8% of paths end below the trigger value but 14.3% touch it. Which number should a borrower care about, and why?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.