This article contains details specific to United States (US) investors. It does not apply to non-US investors.
Retirement plan analysis (math) contains supporting analysis and formula derivations for retirement plan-related articles, including Traditional versus Roth and Roth conversion.
This article is intended for math enthusiasts who want to understand the underlying details. Investors without a math background can safely skip this information.
${\begin{aligned}R&={\text{Roth balance}}\\T&={\text{Traditional balance}}\\C&={\text{Roth-converted amount}}\\V&={\text{Total value of tax-advantaged space}}\\A&={\text{After-tax amount}}\\MTR_{n}&={\text{marginal tax rate now}}\\MTR_{w}&={\text{marginal tax rate at withdrawal}}\\\end{aligned}}$
The overall value of a change to tax-advantaged space is equal to:
$\Delta V=\Delta T(1-MTR_{w})+\Delta R$
Consider a given after-tax investment $A$ that can be contributed to a traditional account, a Roth account, or used to pay the taxes on a Roth conversion. When making a traditional contribution, the change in traditional balance is:
$\Delta T={\frac {A}{(1-MTR_{n})}}$
Therefore, the change in value when making a traditional contribution is:
${\begin{aligned}R&={\text{Roth balance}}\\T&={\text{Traditional balance}}\\A&={\text{After-tax balance}}\\C&={\text{Roth-converted amount}}\\V_{h}&={\text{Total value of estate to heirs after-tax}}\\MTR_{n}&={\text{marginal tax rate now}}\\MTR_{e}&={\text{marginal tax rate on estate}}\\MTR_{h}&={\text{marginal tax rate on heirs}}\\\end{aligned}}$
When a Roth conversion is performed on assets, during the owner's life on assets expected to be subject to estate tax, and the taxes can be paid from after-tax assets, the net effect on types of assets are as follows:
$\Delta T=-C$
$\Delta R=+C$
$\Delta A=-C\cdot MTR_{n}\cdot (1-MTR_{e})$
The change in after-tax value of the estate to heirs will be as follows:
${\begin{aligned}MTR_{n,T}&={\text{marginal tax rate now, for the traditional contribution, including Saver's Credit}}\\MTR_{n,R}&={\text{marginal rate now of Saver's Credit for the Roth contribution}}\\MTR_{w}&={\text{marginal tax rate for traditional contributions at withdrawal}}\\T&={\text{traditional contribution}}\\R&={\text{Roth contribution}}\\A&={\text{after-tax cost of making retirement contributions (traditional or Roth)}}\\G&={\text{growth factor of investments between now and withdrawal}}\\V&={\text{after-tax value of retirement accounts}}\\\end{aligned}}$
For a fair comparison, the two take home pays must be equal: $A=T\cdot (1-MTR_{n,T})=R\cdot (1-MTR_{n,R})$
${\begin{aligned}MTR_{n}&={\text{marginal tax rate now, for traditional contribution}}\\MTR_{w}&={\text{marginal tax rate for traditional contributions at withdrawal}}\\MTR_{div}&={\text{marginal tax rate on dividends}}\\MTR_{cg}&={\text{marginal tax rate on capital gains}}\\C&={\text{contribution (fixed dollar amount for traditional or Roth)}}\\G_{T}&={\text{growth factor on traditional balance, before taxes}}\\G_{R}&={\text{growth factor on Roth balance (tax-free)}}\\G_{Tx}&={\text{growth factor on taxable balance, after taxes}}\\r_{T}&={\text{total rate of return on the traditional balance}}\\r_{R}&={\text{total rate of return on the Roth balance}}\\r_{Tx}&={\text{total rate of return on the taxable balance}}\\y&={\text{yield on the taxable balance}}\\v&={\text{growth factor on the taxable balance}}\\b&={\text{growth factor on the taxable basis}}\\t&={\text{time}}\\\end{aligned}}$
When contributing a fixed dollar amount $C$ to either traditional or Roth accounts, and investing the tax savings $C\cdot MTR_{n}$ in a taxable account, traditional contributions are preferred when:
Rather than plug in the formulas for these factors to create one large equation, it is easier to calculate each factor separately. Assuming annual compounding, the three growth factors can be calculated as follows:
Separate rates of return for traditional, Roth, and taxable accounts allow the comparison between different accounts (eg. IRA or 401(k)) with different investments and fees. Assuming the same investments and fees $(r_{T}=r_{R}=r_{Tx}=r)$ and $G_{T}=G_{R}$, the equations simplifies somewhat to:
${\begin{aligned}MTR_{n}&={\text{marginal tax rate now, for traditional contribution}}\\MTR_{w}&={\text{marginal tax rate for traditional contributions at withdrawal}}\\m&={\text{employer match rate}}\\T&={\text{traditional balance}}\\R&={\text{Roth balance}}\\A&={\text{after-tax cost of making retirement contributions (traditional or Roth)}}\\G&={\text{growth factor of investments between now and withdrawal}}\\V&={\text{after-tax value of retirement accounts}}\\\end{aligned}}$
When making a traditional contribution, the changes in the two types of balances will be:
The above equations can be modified to also include a Saver's Credit. When making a traditional contribution, the changes in the two types of balances will be: