Roth IRA Calculator Formula

See the 2026 Roth IRA contribution, MAGI phaseout, compensation, shared-limit, monthly growth, fee, inflation, and goal formulas.

How do the Roth IRA Calculator formulas work?

This formula page defines the variables used by the Roth IRA Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

The calculator first estimates a direct Roth IRA contribution limit. It caps the editable age-based statutory limit by taxable compensation, applies the 2026 filing-status MAGI phaseout, and subtracts other IRA contributions.

The projection then applies eligible monthly deposits, gross return, fees, inflation, target tracking, and lower and higher return scenarios.

Variable Definitions

Symbol	Meaning	Convention
L_base	Entered annual IRA contribution limit	Currency per tax year
L_catch	Entered age-50 catch-up limit	Currency per tax year
L	Limit after age and compensation	Currency
M	Modified adjusted gross income	Currency
A, Z	Phaseout start and end	Currency
q	Phaseout fraction	0 to 1
C_other	Other traditional or Roth IRA contributions	Currency
B_m	Balance after month m	Currency
r_a, f_a	Annual return and annual fee	Decimal rates
pi	Annual inflation assumption	Decimal rate

Age and Compensation Limit

L_{stat}=L_{base}+L_{catch}\mathbb{1}_{age\ge50}

L=\min(L_{stat},\ taxable\ compensation)

The editable inputs default to the 2026 IRS amounts of $7,500 and $1,100. Taxable compensation can reduce the available amount.

2026 MAGI Phaseout

q=\min\left(1,\max\left(0,\frac{M-A}{Z-A}\right)\right)

L_{income}=\left\lceil L(1-q)\right\rceil_{\$10}

The engine rounds the phaseout fraction to three decimals, rounds the reduced amount upward to the next $10, and applies the IRS $200 minimum when a positive reduced amount would otherwise be below $200.

Shared IRA Limit

L_{shared}=\max(0,L-C_{other})

C_{eligible}=\max(0,\min(L_{income},L_{shared}))

Traditional and Roth IRA contributions share the annual limit. The final direct Roth IRA estimate cannot be negative.

Effective Annual Return and Fee to Monthly Rates

r_m=(1+r_a)^{1/12}-1

f_m=1-(1-f_a)^{1/12}

These conversions preserve the effective annual growth and fee-retention factors.

Monthly Balance Recurrence

B_m=((B_{m-1}+C_b)(1+r_m))(1-f_m)+C_e

Only a beginning or end contribution is used. The monthly amount is capped at one twelfth of that projected year's estimated eligible limit.

Age-50 Catch-Up Modeling

C_{desired,m}=C_{start}+\mathbb{1}_{enabled\land age\ge50}\frac{L_{catch}}{12}

C_m=\min\left(C_{desired,m},\frac{C_{eligible}}{12}\right)

The catch-up increases the desired monthly deposit beginning with the first projected tax year at age 50. Eligibility still caps the actual deposit.

Investment Fees and Fee Drag

Fee\ drag=FV_{fee\ free}-FV_{with\ fees}

r_{net}=(1+r_a)(1-f_a)-1

Fee drag includes deducted fees and the future growth those removed dollars no longer earn.

Inflation-Adjusted Balance

B_{real}=\frac{B_n}{(1+\pi)^t}

The nominal ending balance remains unchanged; this separate output estimates current purchasing power.

Required Monthly Contribution Solver

\min C\quad\text{such that}\quad B_n(C)\ge Goal

The engine checks whether the current balance alone reaches the goal, then uses 80 bisection iterations from $0 to $1 million per month. Every trial remains subject to estimated annual eligibility caps.

Lower, Base, and Higher Return Scenarios

r_{low}=\max(-0.99,r-v),\qquad r_{high}=\min(1.00,r+v)

Only return changes. Eligibility, contributions, fees, inflation, and timing remain constant. The paths are sensitivity comparisons, not probabilities.

Worked Calculation

The default plan uses age 35, single filing status, $100,000 MAGI and compensation, no other IRA contributions, $25,000 invested, and $625 monthly for 30 years.

Income and compensation permit the full $7,500.00 estimated limit.
The first-year contribution is $7,500.00.
New contributions across the projection total $241,500.00.
The monthly recurrence produces $919,717.58.
Projected investment growth is $653,217.58.
Modeled fees total $14,863.45, and fee drag is $30,014.61.
The inflation-adjusted ending value is $438,468.63.

Assumptions and Exclusions

The entered limits and 2026 MAGI phaseout rules remain constant in future projection years.
MAGI, taxable compensation, and other IRA contributions remain constant.
Returns, fees, and inflation remain constant within each scenario.
Catch-up contributions begin in the first modeled year at age 50 when enabled.
Conversions, backdoor Roth transactions, spousal IRAs, withdrawals, taxes, penalties, and distribution rules are excluded.
Displayed currency is rounded to cents; calculations retain full engine precision.

Formula FAQ

Why is the contribution rounded up to $10?

The reduced-contribution process follows the IRS worksheet convention represented in the engine. For a useful Roth IRA Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Why can a large monthly contribution stop increasing the projection?

Actual modeled deposits are capped by annual eligibility. For a useful Roth IRA Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Why can the required contribution be unavailable?

The annual cap may prevent even the solver's upper monthly input from reaching the target. For a useful Roth IRA Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Does this formula determine qualified distributions?

No. Distribution qualification and five-year rules are outside the model. For a useful Roth IRA Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

References

Use the worked Roth IRA examples to reproduce the formulas.

Use the Roth IRA Calculator