What the review changed, and why — every number, with the working

CAS Ratemaking Working Group · Bi-Weekly Call #7 deep dive · Paper v2.0 → v3.0

Thu Aug 27, 2026 · 4:30 PM ET
Companion to the decision poll
Researcher: Pramod Misra
POG: Mondello · Robinson · Paik · Werner · Kozlowski

0. Changed numbers, first

Paper v2.0 was a prose-only revision and said so. v3.0 changes the toolkit, so it changes numbers. If you are holding a v2.0 draft, this is the table you need before anything else.

Quantity	§	v2.0	v3.0	Why
Growth gap per customer	3.2, 6.3, 8.4	$693.40	$376.21	cross-sell credited once, not every term
Mean 5-yr CLV with growth (BG/NBD)	3.2	$3,498.71	$3,181.52	same
Mean 5-yr CLV, base variant	3.2	$2,805.31	$2,805.31	unchanged — no growth term in it
Book CLV, filing basis	3.5	$17,470,511	$16,111,118	flows from the same correction
Credibility, all five quintiles	6.1	0.9614 (identical)	1.0000 / 0.7168 / 0.8419 / 0.7213 / 1.0000	claim counts instead of customer counts
Indicated relativities	6.1	2.8489 … 0.3581	2.8489 … 0.3581	unchanged — credibility does not move the indication
1-yr cohort mean CLV	8.2	$1,870.71	$1,710.82	growth correction
2–3 yr cohort at exactly $0	8.2	not reported	34.59%	new column answering the P25 question

Unchanged in method and result: the six model families and their fits, the tenure loss curve itself, the disparate-impact screen, and the §9 economic-value comparison including its negative result. The cross-sell correction only reaches figures computed on basis C — the growth basis.

1. One formula, and where the correction lands

CLV_i = Σ_t n_i(t) · [ P_i − L_i·m_i(t) − E_i ] + X_i

X_i = V_cs · Σ_t n_i(t)·(1−p)^t−1p + V_up · Σ_t n_i(t)·[1−(1−q)^t]

The bolded term is the whole correction. Through v2.0 the growth value sat inside the per-term margin, which multiplied it by every expected renewal: a $150 cross-sale became $150 × 2.7736 = $416.04 per customer — an annuity of cross-sales. The first-occurrence weights (1−p)^t−1p sum to at most one, so the sale is credited at most once. Upsell keeps a recurring weight, because a coverage upgrade is a permanent step-up in annual margin rather than a one-off event.

Symbol	Name	Plain meaning
S_i(t)	survival	probability customer i is still here at term t
P_i(t)	premium	expected premium in term t, at today's level
L_i(t)	loss	expected loss in term t, including ALAE
E_i(t)	expense	variable expense; acquisition once at inception, renewal thereafter
X_i(t)	growth value	value of a business-development action — a cross-sale or an upgrade
n_i(t)	per-year increment	DERT_i(t) − DERT_i(t−1): discounted survival landing in year t
p	cross-sell hazard	annual probability the FIRST cross-sale happens
q	upsell hazard	annual probability of the FIRST coverage upgrade
V_cs	cross-sell value	the CLV of the secondary product — not a per-term margin
d	discount rate	annual discount rate; toolkit default 0.08

Why this also answers the double-counting question. P_i(t) is carried forward at the customer's current premium, so a limit increase in term n is not already inside it and the upsell term is not a duplicate. The converse is now a numbered limitation (L12): if you project premium trend into P_i(t) directly, you must set upsell_value = 0.

2. Hand-checkable anchor, from published figures only

No toolkit involved — a five-year CLV for an average US auto household using only cited public numbers, so the arithmetic can be checked against the framework independently.

Input	Value	Source
Full-coverage auto premium	$2,300	NerdWallet / Quadrant 2026 average
Industry net loss ratio	0.763	NAIC 2024 report, 2023 accident year
Renewal expense ratio	0.20	toolkit convention, §3.4
Annual retention	0.78	LexisNexis 2025 carrier mono-line auto
Discount rate	0.08	toolkit default, §3.1

Margin per renewal = $2,300 × (1 − 0.763 − 0.20) = $85.10. Discounted expected renewals over five years at 0.78 retention and 0.08 discount = 2.0891. Five-year CLV = $177.78.

Read the sign. At the industry loss ratio of 0.763 and a 0.20 renewal expense load, the renewal margin is $85.10 — barely positive — and a single acquisition charge at 0.40 of premium ($920) exceeds the entire five-year discounted value. That is not an artifact of our synthetic book; it is what the published industry numbers say, and it is the reason this framework treats acquisition cost as the decisive term.

3. Credibility: what the exposure base was doing

1,082 is the classical full-credibility standard for claim counts — the expected claims needed for observed frequency to fall within ±5% of expectation with 90% probability. It has no interpretation as a number of customers.

Segment	Customers	Credibility on customers	Claims	Credibility on claims	Indicated relativity
1	1,000	0.9614	2,433	1.0000	2.8489
2	1,000	0.9614	556	0.7168	0.9322
3	1,000	0.9614	767	0.8419	0.4884
4	1,000	0.9614	563	0.7213	0.4131
5	1,000	0.9614	1,298	1.0000	0.3581

Because the segments are equal-size quintiles, every segment held exactly 1,000 customers, so the customer-count basis returned 0.9614 in all five rows — a column carrying no information, and a credibility weighting that reduced to a uniform 3.9% shrink toward unity. On claim counts the lowest-CLV quintile carries 2,433 claims against 1,000 customers — high frequency is most of what makes it the lowest-CLV quintile — and reaches full credibility, while the thin mid-book segments do not. The segments whose relativities depart furthest from unity are now exactly the ones with the claim volume to support the departure.

4. The exhibit that supports an actual rate decision

The by-tenure exhibit shows that loss ratios improve with tenure. It cannot show whether a segment expected to persist longer may be charged less — that needs loss ratio and retention varying together across the segment.

Segment	Households	Mean retention	Expected lifetime	First-term loss ratio	Lifetime loss ratio	Relativity
1 product	3,205	0.6360	2.72 yrs	0.6515	0.6148	1.1050
2 products	1,545	0.7319	3.57 yrs	0.5939	0.5346	0.9608
3+ products	250	0.9470	7.92 yrs	0.6652	0.5244	0.9425

The 3+ products segment has the worst first-term loss ratio of the three and still indicates a credit. A single-term exhibit ranks it last of three; the lifetime exhibit ranks it second. The entire reversal is persistence, with no willingness-to-pay content anywhere in it — which is what makes it arguable in a prior-approval state.

And the honest caveat. Run the same exhibit on region, where this book's retention spans only 0.872 to 0.894, and the relativities compress into 0.936–1.039. The exhibit is only informative where a segment's retention genuinely differs. The toolkit emits the region view alongside the product view for exactly that reason.

5. How we judge — and why the tenure curve did not survive it

The standard we set for promoting the tenure curve from mechanism to shipped parameter was reproducibility on data other than our own. It failed that standard, and the failure is informative.

Source	What it is	Implied annual credit	Verdict
Seeded synthetic book	5,000 households, 2015–2025	+4.22%	the shipped curve
Wisconsin LGPIF, tenures 0–3	real panel 2006–2010, 1,227 policies	+19.93% (R² 0.949)	~5× the shipped magnitude
Wisconsin LGPIF, all cells	adds one fund-year of catastrophe	-7.93% (R² 0.058)	one year of data flips the sign
Spanish households	10,000 records, explicit seniority	+1.80%, not monotone	no gradient above 5.0 yrs tenure
Schedule P, 144 carriers	cross-carrier loss-ratio dispersion	CV 24.4%	noise floor vs a 35.0% ten-year effect

The designed test cannot be run on public data by anyone. No public source carries carrier identity and policyholder tenure together: Schedule P has 144 carriers but is organised by accident year and exposes no policy-level records; the two public panels that do carry tenure are each a single carrier. Executing it needs a multi-carrier data call or one carrier's internal panel — which is what Q3 option C asks CAS for.

What is not in doubt is the direction. That a revenue-neutral tenure relativity reduces long-tenured CLV follows from the arithmetic of normalisation and holds for a gradient of any magnitude. What is unvalidated is the magnitude — and the magnitude matters, because the margin is thin:

	Flat 0.66 loss ratio	Tenure gradient applied
Short renewer (1 term)	$-392.04	$-354.62
Ten-year renewer	$782.14	$1,959.92
Ten-year renewer's realised loss ratio	0.66	0.52

Crediting the measured gradient more than doubles the ten-year renewer's value and barely moves the short renewer, who has almost no future tenure over which to earn it.

6. Glossary

ALAE / ULAE: Allocated / unallocated loss adjustment expense. ALAE attaches to claims and belongs in the loss projection; ULAE is a claims-operation servicing cost and belongs in the expense ratio.
Base variant (clv_base): P − L − E only. The filing-defensible column. Formerly clv_unconstrained.
BG/NBD: Beta-Geometric / Negative Binomial Distribution — a buy-till-you-die model treating each renewal as a purchase occasion.
Credibility (classical): √(exposure ÷ 1,082) capped at 1, weighting a segment's own indication against the book. Exposure is claim counts.
DERT: Discounted expected residual transactions — the survival-weighted, discounted count of future renewals. The annuity factor of the framework.
Growth value (X): Value from a deliberate business-development action: cross-sell or upsell. Never premium trend, never the value of existing coverage persisting.
Left truncation: A panel that begins mid-relationship, so policies already in force appear as tenure 0. Why the LGPIF tenure-0 cell mixes new with mature business.
Level-effect basis: The tenure curve applied at its own level, producing a level effect. Deliberately not revenue-neutral, and never for a relativity exhibit. Called the planning basis through paper v3.2.
Residual CLV: Value of an in-force customer's remaining renewals, charging renewal expense only — acquisition cost is sunk.
Revenue-neutral basis: The tenure curve rescaled to a loss-weighted mean multiplier of exactly 1.000000, so it redistributes expected loss across tenure without changing the book total. The toolkit default, and the only one of the two that belongs in a relativity exhibit. Called the filing basis through paper v3.2.
Schedule P: NAIC annual-statement loss-reserving exhibit; the CAS Loss Reserving Database is derived from it. Organised by accident year, not tenure.
With-growth variant (clv_with_growth): Base plus growth value. The planning column. Formerly clv_constrained.

Reproduce every figure on this page: python scripts/build_paper_numbers.py · python scripts/validate_tenure_gradient.py · python docs/calls/phase6_review_response_data.py. All under set_seed(42); public figures cite research/data/calibration_benchmarks.md. · This research was funded by the Casualty Actuarial Society.