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)