Cohorts & scenarios: does the loss ratio depend on how long they have renewed?

0. What you asked for, and what this page answers

Both are now built into the toolkit, not just drawn as exhibits. Three findings worth your time, in order of how much they should change your thinking:

• You were right, and the size of it is startling. In our book the loss ratio falls 0.625 → 0.416 across tenure, and mean CLV rises 4.1× from the newest cohort to the 10+ year cohort. CLV is extremely leveraged to this assumption because the margin is thin: crediting the measured gradient more than doubles the value of a ten-year renewer.
• But most of that value is already in the observed loss, not ahead of it. A ten-year customer's advantage has already been realized — it sits in their historical loss ratio. Crediting them a further tenure discount would count it twice. What the curve legitimately adds is the improvement still ahead of the not-yet-mature.
• And on a revenue-neutral filing basis the answer inverts. Recognising that new business improves with tenure means long-tenured business must carry relatively more loss, not less — the 10+ cohort's mean CLV falls 7.0% while every younger cohort rises. This is the opposite of a loyalty discount, and it is the single most important slide here (§5).

Default remains off (loss_trend=0.0, an exact no-op) until you choose a basis. Poll questions Q1–Q3 are exactly these decisions.

1. The unifying formula, and why the old one could not express your question

Through Call #3 every model reported CLV in this form:

A margin that varies by year cannot be factored out of a discounted sum, so that form is structurally incapable of carrying a tenure-varying loss ratio. The identity we now use is:

The decomposition introduces no new estimation: each year's expected renewals is the difference of the cumulative number the model already reported, so it sums back to DERT(H) exactly (measured error 0.0 on all six families), and when mt ≡ 1 it collapses to the old formula to the cent.

2. Where the curve comes from — measured, then smoothed

The gradient is not assumed. retention_adjusted_loss_ratio() measures it directly on the book: loss ratio by tenure year, against a pooled 0.5246 and a survival-weighted lifetime 0.5742.

Raw yearly ratios are noisy — tenure 6 jumps back to 0.5174 on a thin $7.9M of premium. Pricing off that would be indefensible, so the shipped curve is a premium-weighted log-linear fit:

The floor matters: selection wear-off is not unlimited, and an unfloored exponential would eventually price a 25-year customer at an implausibly low loss ratio. This is the actuarial analog of a selected development pattern with a tail cut-off — smooth the observations, then stop extrapolating where credibility runs out.

The indexing subtlety that changes the answer

The multiplier is indexed relative to each customer's own current tenure, mt = curve[τ+t] / curve[τ], not on absolute tenure. This is the crux. A customer's L is their own realized average loss, so it already contains the improvement they have lived through. Multiplying it by an absolute-tenure discount would credit the same improvement twice.

So the honest reading of your point is: tenure value is mostly already in the observed loss; what the curve adds is the expected further improvement for the not-yet-mature.

3. Your cohort, hand-checkable — two renew once, five renew for ten years

Exactly the exhibit you asked for. Persistence is stated, not modelled, so every cell below can be checked with a calculator and no model is involved.

Assumptions (public-benchmark anchored): annual premium $1,450, loss ratio 0.66, renewal expense 0.20, acquisition expense 0.40, discount 8%.

Year by year, one short renewer beside one ten-year renewer

The cohort in terms of profit — flat loss ratio

The profit lens you asked for. Two of seven policyholders destroy value, yet the cohort earns +$3,127. An identical loss ratio of 0.66 on every single policy produces CLVs ranging from −$392 to +$782 — persistence, not loss experience, is doing the work. A book written entirely of one-renewal policyholders at a "profitable" 0.66 loss ratio would lose money.

The same cohort with the tenure credit applied

The ten-year renewer's loss ratio glides 0.6321 → 0.4289 across the ten terms, and their value more than doubles ($782 → $1,960). Cohort value rises +190.7%.

Reproduce: python docs/calls/phase4_cohort_scenario_data.py §B · cas_clv.cohorts.mark_cohort(), micro_cohort_walkthrough()

4. Scaling up — the same question on the whole book

You said: "then you can extrapolate that." Here is the extrapolation on 5,000 synthetic customers, 32,728 policy terms, valued on the hybrid headline model at a 5-year horizon.

The direct answer to your question: mean CLV rises 4.1× (from $1,871 to $7,707) while the loss ratio falls 0.647 → 0.456. The 10+ cohort is 10.4% of customers but 22.9% of book value.

Two honest caveats visible in the table. The 2–3 year cohort's mean is below the 1-year cohort's — early lapses concentrate there, and its P25 and median are both $0.00 (lapsed customers carry zero residual value). And the bands are enormous relative to the means: the 1-year cohort spans $146–$4,650. A single customer's CLV is not a quotable number; a cohort with its band is.

Where each cohort's value comes from — the profit bridge

Net of acquisition cost, the 1-year cohort retains only $477.6K of $1.72M of gross value (28%), while the 10+ cohort keeps $2.92M of $4.01M (73%). The 2–3 year cohort is the thinnest of all at $247K net on $2.12M gross (12%) — it has paid the acquisition cost but not yet earned it back. That trough is where retention spend has the highest return.

Reproduce: §C of the companion script · cohort_clv_summary(), cohort_profit_bridge()

5. The finding that inverts the intuition READ THIS ONE

If the tenure credit is applied on a revenue-neutral filing basis — rescaled so total expected book loss is unchanged, which is what makes it a relativity rather than a rate decrease — then value moves between cohorts and the book total does not move at all ($17,470,511 either way, neutral by construction).

And the direction is the opposite of a loyalty discount:

Why. Relative indexing means the improvement still ahead is what gets credited. Long-tenured customers have almost none left (they are at the curve floor); newer customers have plenty. Rescaling to revenue-neutrality then lifts everyone's multiplier to restore the book total — which, for the cohort with no forward improvement to offset it, means their loss goes up relative to flat.

The two bases and what each is for:

On the planning basis the entire uplift is exactly the modelled loss saving: expected discounted loss falls $27.60M → $24.99M (−$2.62M) and book CLV rises by the same $2.62M. Nothing else moves — that identity is asserted in the test suite.

Reproduce: §D of the companion script · diagnostics: decomposition error 0.0, negative increments clamped 0, loss-weighted mean multiplier 1.000000, mean effective LR year 1 0.6151 → final 0.5307.

6. Scenario planning — the levers, and which ones actually matter

Also from Call #3: "it won't be a static model which will give you one number and that's it. It's more about you have a few levers to change and basis those changes, you can see how this value will evolve." Seven named scenarios now ship, and there is a live planner where you can move the levers yourself.

The result worth flagging: persistence dominates everything else. The soft market — worse margins and worse expenses — comes out marginally ahead of base, because its +3% retention outweighs the margin loss. Meanwhile the hard market's better margins cannot offset an 8% retention decline. CLV is far more sensitive to how long customers stay than to what you earn per year while they do. This is asserted in the test suite so it cannot regress silently.

Is scenario impact real, or model-choice artifact?

Reading across model families at fixed scenario answers the question you would reasonably ask next — whether these movements survive changing the model.

Model choice contributes a stable ~12% spread, and it does not interact with the scenario (1.118 → 1.131 across all five). Scenario effects — up to −54% for recession — are several times larger. Practically: getting the assumptions right matters far more than getting the model right, and all six families agree on the direction of every scenario. That is the strongest argument we have for the transparent, defensible model at filing.

Reproduce: §E of the companion script · cas_clv.scenarios.run_scenarios(), scenario_comparison() · full grid: docs/sensitivity_appendix.md

7. How we judge it — and what would falsify it

8. Open, and honestly not done

• NAIC five-year reproducibility across carriers — the test that decides whether the 4.22%/year credit is a general phenomenon or an artifact of our synthetic generator. Not yet run. Scheduled inside Phase 5 (window closes Aug 19); results at Call #5 (Aug 13) or by email sooner. This is the single biggest caveat on everything above.
• Disparate-impact screen on age bands (not just region) before any filing use of a tenure relativity.
• Curve source is a decision, not a finding — measured / judgmental / credibility-blended is poll Q2. Ours is measured-then-smoothed on one synthetic book, which is the weakest of the three options for a published default.
• The hub is a private HF Space. POG members need Hugging Face accounts added as collaborators, or a PDF export instead. Unresolved from Call #3 action items 1–2 — we need a CAS decision on which route.

9. Glossary — every symbol and abbreviation on this page

10. Reproducibility & provenance

Every figure on this page is generated, not transcribed. Regenerate all of them with:

• Seed: set_seed(42) throughout. Synthetic book: 5,000 customers, 32,728 policy terms, 5,617 claims.
• Assumptions: horizon 5 years, discount 8%, acquisition expense 0.40, renewal expense 0.20, cross-sell $150 + upsell $100 per renewal (constrained variant only).
• Toolkit functions: cas_clv.cohorts — mark_cohort, micro_cohort_walkthrough, micro_cohort_summary, build_cohorts, cohort_clv_summary, cohort_profit_bridge; cas_clv.scenarios — run_scenarios, scenario_comparison, sensitivity_appendix; cas_clv.models — predict_clv (per-year path), tenure_loss_diagnostics; cas_clv.ratemaking — retention_adjusted_loss_ratio.
• Curve provenance: measured by retention_adjusted_loss_ratio(), smoothed by premium-weighted log-linear fit; constants live in models.DEFAULT_TENURE_LOSS_CURVE with the fit recorded in the module docstring. No hand-entered numbers.
• QA gate: pytest 108 passed · ruff clean · mypy clean. New tests: exact no-op per family, per-year sum identity, revenue neutrality, relative-indexing behaviour, and the loss-saving identity on the planning basis.
• Chart colors from the validated data-visualization palette: categorical slots blue #2a78d6 + green #008300 (all-pairs CVD ΔE 26.5, normal-vision 29.0), ordinal blue ramp steps 250–650, diverging blue↔red for the signed exhibit. Validated with the palette checker rather than by eye.
• Not reproducible yet: the NAIC five-year cross-carrier reproducibility test (§8).

Companion materials: call brief · decision poll · interactive scenario planner · Jul-16 case studies