# User talk:Fyre4ce/Roth conversion

## Relative value of contributions and conversions

Define variables:

{\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:

$\Delta V_{T}=A{\frac {(1-MTR_{w})}{(1-MTR_{n})}}$ When making a Roth contribution, the change in Roth balance is simply:

$\Delta R=A$ Therefore, the change in value when making a Roth contribution is:

$\Delta V_{R}=A$ When making a Roth conversion, the converted amount is:

$C={\frac {A}{MTR_{n}}}$ Therefore, the change in value when making a Roth conversion is:

$\Delta V_{C}=C-C(1-MTR_{w})={\frac {A}{MTR_{n}}}-{\frac {A}{MTR_{n}}}\cdot (1-MTR_{w})={\frac {A}{MTR_{n}}}(1-(1-MTR_{w}))=A{\frac {MTR_{w}}{MTR_{n}}}$ When $MTR_{n} (current marginal tax rate is less than predicted future marginal tax rate),

$\Delta V_{C}>\Delta V_{R}>\Delta V_{T}$ When $MTR_{n}=MTR_{w}$ (current marginal tax rate equals predicted future marginal tax rate),

$\Delta V_{C}=\Delta V_{R}=\Delta V_{T}$ When $MTR_{n}>MTR_{w}$ (current marginal tax rate is greater than predicted future marginal tax rate),

$\Delta V_{C}<\Delta V_{R}<\Delta V_{T}$ --Fyre4ce 23:10, 10 March 2020 (UTC)

## Conversions on estates subject to estate tax

Define variables:

{\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:

$\Delta V_{h}=\Delta T\cdot (1-MTR_{h})+\Delta R+\Delta A=-C\cdot (1-MTR_{h})+C-C\cdot MTR_{n}\cdot (1-MTR_{e})$ $\Delta V_{h}=C\cdot ((MTR_{h}-1)+1+MTR_{n}\cdot (MTR_{e}-1))=C\cdot (MTR_{h}+MTR_{n}\cdot (MTR_{e}-1))$ It follows that Roth conversions increase the value of the after-tax value of the estate if:

$MTR_{h}+MTR_{n}\cdot (MTR_{e}-1)>0$ or

$MTR_{e}>1-{\frac {MTR_{h}}{MTR_{n}}}$ --Fyre4ce 04:44, 10 December 2020 (UTC)