Add capital gains distribution planning doc (#818 #817)

changelog.d/818.added.md

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+Documented how capital gains are modelled and imputed (#818, #817) at `docs/book/assumptions/capital-gains-distribution-plan.md`, including the Microcosm UK stages that now fit gains jointly to HMRC Table 3 and what is left to do.

docs/book/assumptions/capital-gains-distribution-plan.md

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+# Capital gains distribution
+
+```{note}
+**Planning page, updated October 2026.** Tracks
+[#818](https://github.com/PolicyEngine/policyengine-uk/issues/818)
+(model gains jointly across income groups) and
+[#817](https://github.com/PolicyEngine/policyengine-uk/issues/817)
+(don't impute gains to households with no wealth). The imputation lives
+in the data build, not in this repo. Most of what this page proposed
+in May 2026 has since been built in
+[Microcosm UK](https://github.com/PolicyEngine/microcosm), which builds
+the replacement for the Enhanced FRS. The certified UK default dataset is
+still the Enhanced FRS from
+[policyengine-uk-data](https://github.com/PolicyEngine/policyengine-uk-data)
+(release 1.58.0). What remains is listed at the end of this page.
+```
+
+## How the model uses capital gains
+
+`capital_gains_before_response` is a person-level input, uprated by OBR
+per-capita GDP growth. `capital_gains_tax` charges it at the main CGT
+rates, with three components charged at their own rates (all added by
+[#1861](https://github.com/PolicyEngine/policyengine-uk/pull/1861)):
+
+- `capital_gains_badr`: gains qualifying for Business Asset Disposal
+  Relief or Investors' Relief, at the relief rate up to the lifetime
+  limit;
+- `capital_gains_residential_property`: gains on UK residential
+  property, at the residential rates;
+- `capital_gains_carried_interest`: carried interest, at the carried
+  interest rates.
+
+Each is a component of the total, not an addition to it.
+
+Behavioural responses: `capital_gains_behavioural_response` changes
+realisations using an elasticity with respect to the retention rate
+(`relative_capital_gains_retention_rate_change`) or the marginal tax
+rate. Gains use `gov.simulation.capital_gains_responses.elasticity`,
+which is 0 by default, so there is no response unless a user sets it.
+When `gov.simulation.capital_gains_responses.separate_badr_elasticity`
+is true (false by default), gains qualifying for BADR respond at
+`badr_elasticity` instead (1.4, the value the OBR used for these gains
+in its January 2025 costing), through `capital_gains_badr_elasticity`.
+
+## How the gains are imputed
+
+### Enhanced FRS: one distribution per income band
+
+The Enhanced FRS (policyengine-uk-data, the certified default) fits a
+spline to the Advani and Summers gains percentiles within each income
+band and draws each household's gains from its band. The spline alone
+cannot produce gains above about £2m, so policyengine-uk-data main (not
+yet in a release) also stacks 30 donor households per HMRC
+size-of-gain band and calibrates to HMRC counts and amounts of gains by
+band. #818 pointed out the problems with the
+band-by-band fit:
+
+- each band's upper tail was fitted on few observations, so the tails
+  were noisy and the joint distribution of income and gains had breaks
+  at band boundaries;
+- two people in the same band drew from the same distribution however
+  different their incomes;
+- gains depended on income only, not on wealth, so they could be
+  imputed to households with no assets (#817).
+
+### Microcosm UK
+
+Microcosm builds gains in six stages (see the UK
+[sources specification](https://github.com/PolicyEngine/microcosm/blob/main/packages/microcosm-build/src/microcosm/build/uk/spec/sources.yaml)):
+
+1. **Support split** (`cgt_support_split`). Within each income band of
+   HMRC Capital Gains Tax statistics Table 3, the households with the
+   most investable wealth (financial wealth, business wealth and
+   property other than the main home) are split into lighter copies, so
+   that there are enough rows to hold the largest gains.
+2. **Who has gains** (`cgt_incidence_clone`). The set and order of
+   gainers come from the incidence distribution in Advani and Summers
+   (2020), *Capital Gains and UK Inequality* (CAGE Working Paper 465).
+3. **How much** (`hmrc_cgt_gains_spine`). Amounts are redrawn so that
+   gains match HMRC Table 3 for 2024-25, which counts taxpayers and
+   gains **jointly by size of gain and taxable income**, conditioned on
+   Tables 1, 2.1a, 5 and 6.
+4. **Weight anchor** (`cgt_incidence_anchor`). Clones below the annual
+   exempt amount carry more weighted people than HMRC publishes, so the
+   excess weight is moved back to each clone's paired original.
+5. **Residential property** (`cgt_residential_split`). Gains on UK
+   residential property are fitted to Table 8 (2024-25 residential
+   property disposals) by splitting households into weighted copies,
+   which fills `capital_gains_residential_property`.
+6. **What kind** (`hmrc_cgt_asset_type_spine`). Of the remaining gains,
+   those qualifying for BADR or Investors' Relief are fitted to Table
+   4.1, which fills `capital_gains_badr`, and the rest are given asset
+   types from Table 7 (gains by asset type, 2023-24).
+
+All the HMRC tables come from the Capital Gains Tax statistics July
+2026 release, through Microcosm's pinned Chronicle feed.
+
+Against #818: gains are now fitted to the published joint distribution
+of gain size and taxable income, not band by band. Against #817:
+the largest gains are placed on the wealthiest households in each
+income band. Microcosm's release checks also include a projection check
+on people just below the annual exempt amount
+(`uk_cgt_projection_entrants` in `gates.json`), so that
+uprating does not push implausible numbers of them into paying CGT in
+later years.
+
+## What is left
+
+- **A zero-wealth test.** Add a test, here or in Microcosm, that no
+  positive gains sit on households with zero investable wealth. Neither
+  repo asserts this directly yet, so #817 should stay open until it
+  does.
+- **Close #818** once a Microcosm UK dataset is the certified default
+  and its diagnostics confirm the Table 3 fit; link them from the issue.
+- **Elasticities by group.** Responses use one main elasticity and one
+  BADR elasticity. With the joint distribution in place, an elasticity
+  that varies by income or size of gain becomes possible, at the cost
+  of more parameters to justify.
+- **Wealth source.** Investable wealth comes from Microcosm's Wealth
+  and Assets Survey imputation. Its quality limits how well #817 can be
+  met.
+
+The quantile regression forest approach proposed here in May is no
+longer needed for this.
+
+## References
+
+- Issues: [#818](https://github.com/PolicyEngine/policyengine-uk/issues/818), [#817](https://github.com/PolicyEngine/policyengine-uk/issues/817).
+- CGT components: [#1861](https://github.com/PolicyEngine/policyengine-uk/pull/1861).
+- HMRC, [Capital Gains Tax statistics](https://www.gov.uk/government/statistics/capital-gains-tax-statistics).
+- Advani, A. and Summers, A. (2020), *Capital Gains and UK Inequality*, CAGE Working Paper 465.
+- Microcosm UK [sources specification](https://github.com/PolicyEngine/microcosm/blob/main/packages/microcosm-build/src/microcosm/build/uk/spec/sources.yaml) (stages `cgt_support_split`, `cgt_incidence_clone`, `hmrc_cgt_gains_spine`, `cgt_incidence_anchor`, `cgt_residential_split`, `hmrc_cgt_asset_type_spine`).
+- OBR (2025), [Costing of changes to the main, BADR and IR rates of capital gains tax](https://obr.uk/docs/dlm_uploads/CGT-supplementary-release-Jan-2025.pdf).