A stop-loss benchmark is only as good as the number of carriers standing behind each line of it. This page takes one published survey, has your own browser download it from the publisher’s content server, pull every bar label and value out of the file’s own text operators, and then enumerate every whole number of respondents that could have produced those percentages. Eight distributions. Zero of them reachable at 32. Each one reachable at exactly one smaller number.
Observations on the employer stop-loss market — 2024 survey, Milliman white paper, October 2024, by Rob Bachler (FSA, FCAS, MAAA) and Jakob Finney (ASA, MAAA). Five pages, 167,228 bytes. The second sentence sets the base for everything that follows:
“In March 2024, Milliman sent survey participation requests to a wide range of employer stop-loss market participants. Of those receiving a request, 32 provided survey responses.”Introduction, p. 1
That is the only respondent count printed anywhere in the paper. No figure carries its own “n =”.
The introduction breaks the same 32 down twice, and both breakdowns close exactly:
So 32 is not a typo, and the paper’s arithmetic is not generally broken. Several prose claims also reproduce from the charts exactly: “over one-third of respondents reported close ratios of 8% or more” (34%), “nearly half of carriers having a loss ratio estimated to be above target by at least 5%” (23 + 10 + 13 = 46%), “the very largest and the very smallest collectively account for approximately 6%” (4 + 2). The neighbouring arithmetic all reproduces. Only the number of people who answered does not.
Figures 4 to 9 are distributions of respondents, in the paper’s own words: Figure 4 “summarizes each respondent’s persistency”, Figure 5 “the distribution of respondents’ close ratios”, Figure 8 “deviations from target observed in respondents’ aggregate stop-loss loss ratios”. A distribution of respondents is a partition of a whole number of carriers. So each printed percentage pi must equal 100 · ki / N rounded to a whole number, with the ki whole and summing to N. That is a finite search. Run it at N = 32 and it comes back empty — eight times.
Step 1. Download the PDF from Milliman’s content server, hash it, inflate its content streams, and recover the bar labels and values from the text-drawing operators — nothing here is typed in by me.
Step 2. For every recovered distribution, enumerate all whole-number respondent counts from 4 to 64 that could round to those percentages.
Every one of the eight distributions is unreachable at 32, and every one of them has exactly one solution at or below 32 — a different one each time:
| Figure | Printed percentages | At N = 32 | Only N ≤ 32 | The counts |
|---|---|---|---|---|
| 4 — persistency | 33/23/33/10 | impossible | 30 | 10,7,10,3 |
| 5 — close ratio | 34/7/28/31 | impossible | 29 | 10,2,8,9 |
| 6 — decline ratio | 10/45/7/38 | impossible | 29 | 3,13,2,11 |
| 7 — specific loss-ratio deviation | 13/10/23/32/13/3/6 | impossible | 31 | 4,3,7,10,4,1,2 |
| 8 — aggregate loss-ratio deviation | 7/47/10/17/7/13 | impossible | 30 | 2,14,3,5,2,4 |
| 9a — Jan 2024 actual growth | 30/7/13/13/37 | impossible | 30 | 9,2,4,4,11 |
| 9b — 2024 projected growth | 23/3/30/20/23 | impossible | 30 | 7,1,9,6,7 |
| 9c — 2025 projected growth | 21/7/39/14/18 | impossible | 28 | 6,2,11,4,5 |
The larger solutions the search also finds — 39, 48, 58, 60, 64 — are ruled out by the paper itself: only 32 carriers answered at all. So the reading that survives is the ordinary one: between one and four respondents skipped each question, and no chart says which.
It changes what the sentences beside the charts mean. “Nearly half of carriers” having a loss ratio above target by at least 5% is 46% of 31 respondents — about 14 carriers, not 15. “Over one-third” reporting close ratios of 8% or more is 10 of 29. The 2025 growth expectations that a pricing actuary might quote are the view of 28 carriers, of whom 11 sit in a single bucket. A reader who takes 32 as the base overstates every count, and there is no way to tell from the page which questions lost people and which did not.
And it is not fragile. Of the eight distributions, seven cannot be rescued by changing a single printed percentage by one point in either direction — only Figure 5 has such an edit (7% → 6%). The control button runs that test in front of you.
Figures 1, 2 and 3 are also bar charts of percentages in the same paper, and this method would flag them too. They are excluded here because the paper says what they are: “the share of premium attributable to various ranges”. A premium-weighted share is not a count of carriers and has no reason to be a whole-number partition. The claim on this page is only about the charts the paper itself describes as distributions of respondents.
Two more honest notes. In Figure 8 two bars carry no printed value; they are read as zero, and that figure’s percentages sum to 101 rather than 100 — the counts it resolves to sum to 30 exactly. And the tolerance used throughout is the ordinary one for a percentage printed without decimals: a count is accepted when 100 · k / N is within half a point of the printed number.
Nothing here says the survey is wrong. It says the survey does not print, anywhere, the number of carriers behind any individual answer — and that the one number it does print cannot be the base of any of them. For a benchmark that gets quoted into pricing memos and reinsurance submissions, the per-question response count is not a footnote; it is the difference between a result carried by 28 carriers and one carried by 32.
The fix is one line under each chart: n = 29.