Tax Model Mathematics¶

US Federal Tax Model¶

Ordinary Income Tax¶

Ordinary income (traditional account withdrawals, earned income) is taxed through progressive brackets after applying the standard deduction:

\[\text{taxable income} = \text{gross income} - \text{standard deduction}\]

Tax is computed by applying marginal rates to each bracket:

\[\text{tax} = \sum_{k=1}^{K} r_k \cdot \min(\max(\text{taxable income} - b_{k-1}, 0),\, b_k - b_{k-1})\]

where $r_k$ is the marginal rate for bracket $k$ and $[b_{k-1}, b_k)$ is the bracket range.

2024 Federal Tax Brackets (Single)¶

Bracket	Rate
$0 -- $11,600	10%
$11,601 -- $47,150	12%
$47,151 -- $100,525	22%
$100,526 -- $191,950	24%
$191,951 -- $243,725	32%
$243,726 -- $609,350	35%
Over $609,350	37%

Standard deduction (2024): $14,600 (single), $29,200 (married filing jointly).

Long-Term Capital Gains¶

Taxable account gains are taxed at separate LTCG rates based on total income:

Income Level (Single)	LTCG Rate
Up to $47,025	0%
$47,026 -- $518,900	15%
Over $518,900	20%

Vectorized Implementation¶

Tax calculations are vectorized across all simulation paths. The engine computes taxes for all paths simultaneously using numpy operations rather than per-path Python loops:

compute_annual_tax_vectorized(
    ordinary_income: NDArray,    # (n_paths,)
    ltcg: NDArray,               # (n_paths,)
    filing_status: str,
) -> NDArray                     # (n_paths,)

Required Minimum Distributions (RMDs)¶

Traditional account holders must take minimum distributions starting at age 73 (per current IRS rules). The RMD is calculated as:

\[\text{RMD} = \frac{\text{prior year balance}}{\text{divisor}(\text{age})}\]

IRS Uniform Lifetime Table (Selected Values)¶

Age	Divisor	Implied Rate
73	26.5	3.8%
75	24.6	4.1%
80	20.2	5.0%
85	16.0	6.3%
90	12.2	8.2%
95	8.6	11.6%
100	6.4	15.6%

The divisor table is loaded from taxes/tables/rmd_divisors.yaml.

Flat Tax Model¶

The flat tax model applies a single effective rate to all taxable income:

\[\text{tax} = \text{rate} \times \text{gross income}\]

This is simpler but less realistic. It's useful for:

Quick analysis where tax precision isn't critical
Non-US users who want to model a different tax system
Sensitivity analysis to isolate tax effects

Withdrawal Tax Treatment¶

Account Type	Tax Treatment
Taxable	Capital gains tax on appreciation only
Traditional (401k/IRA)	Ordinary income tax on full withdrawal
Roth	Tax-free

Tax Gross-Up¶

When withdrawing from a traditional account to meet spending needs, the engine applies a tax gross-up:

\[\text{gross withdrawal} = \frac{\text{needed spending}}{1 - \text{effective rate}}\]

This ensures the after-tax amount meets the spending target.