Step 3 · Scaling
The limit factor across all entry days, why the values are drawn rather than averaged, and what explains the width of the distribution.
The limit factor says up to which risk factor this one history would have kept the account. It is not an estimate but a maximum over a realised series, and therefore has no standard error.
What the factor is, and what it is not
The factor multiplies the daily deltas of this history. It answers the question: what would have happened had the same history been traded at f times the size.
It is not a risk scale. A dial promises that more f means uniformly more risk, and that does not hold everywhere.
The break point is measured. With a soft daily limit the daily loss is
capped at a fixed amount, independent of f, while the upside keeps scaling. A
path that broke at a smaller factor gets capped so often at a larger one that
it reaches the horizon neither passed nor broken. From the audit run of
2026-08-27, EOD Drawdown Evaluation at 100,000, cell statisch/weich, 200
paths:
| Hardness | open paths across six risk factors | monotone? |
|---|---|---|
| soft | 176, 33, 1, 0, 1, 0 | no |
| hard | 176, 33, 1, 0, 0, 0 | yes |
| none | 176, 33, 1, 0, 0, 0 | yes |
That is one violation out of 29,124 cells checked, so it is rare and small. It still shows that "more factor means uniformly more risk" does not hold as a law, which is why the step is called "Scaling": it says what is computed and promises no interpretation on top.
Why a distribution and not one number
The limit factor answers the question for one starting point, namely the beginning of the series. But with a capped floor that is the entry day and not the portfolio: the same series yields very different values depending on entry.
The view therefore computes the limit factor for every possible entry day and shows the distribution. The value "if bought today" sits inside it as its own marker, with its rank.
Drawn, not interpolated
Median and percentile are drawn values, not averaged ones. An interpolated median would be a number that comes out on no entry day at all. Because they are drawn, every one of these values has a real day attached, and that account's path can be opened in step 4.
The same holds for the dots in the chart: every dot sits on a value that actually occurs in the data.
Why there is no confidence interval
The entry days are overlapping windows of the same series. Their results are therefore not independent, and no standard error and no confidence interval follows from their spread.
What is answered is the weaker question: on how many of the possible entry days could it have hit the account at all.
Why the distribution is sometimes wide and sometimes narrow
That depends on the mechanism, and the view says which:
- Entry-dependent (frozen threshold): the threshold freezes and then sits at a level measured from the starting balance. A breach then requires a fall back to exactly that point, and that depends on when someone entered. The distribution spreads.
- Curve property (trailing threshold without freeze): the threshold only measures the distance to the account's own high. The largest pullback sits inside the period regardless of entry, and the distribution is narrow.
Entry days without a breach
An entry day on which no risk factor blows the account is a result, not a missing observation. It counts towards the rank as larger than any finite value. It has no place on the axis and is named as a number beside the chart.
When there is no closed form
For two mechanism combinations the limit factor cannot be given as a formula:
- Soft daily-loss severity. The cut changes the curve itself.
- State-dependent daily amounts (high-water, two-tier, staircase). The threshold amount itself depends on the risk factor.
In both cases the analysis does not silently search for an approximation. A search under the same name would be the worse answer, because the two cases could then no longer be told apart.