User talk:Fyre4ce/Roth conversion

Relative value of contributions and conversions
Define variables:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ C & = \text{Roth-converted amount} \\ V & = \text{Total value of tax-advantaged space} \\ A & = \text{After-tax amount} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_w & = \text{marginal tax rate at withdrawal} \\ \end{align} $$

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 $$MTR_n < MTR_w$$ (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:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ A & = \text{After-tax balance} \\ C & = \text{Roth-converted amount} \\ V_h & = \text{Total value of estate to heirs after-tax} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_e & = \text{marginal tax rate on estate} \\ MTR_h & = \text{marginal tax rate on heirs} \\ \end{align} $$

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)