User:Fyre4ce/Retirement plan analysis

This page contains a database of analysis and formula derivations for retirement plan-related articles, including Traditional versus Roth and 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

$$MTR_h > MTR_n \cdot (1 - MTR_e)$$

--Fyre4ce 04:44, 10 December 2020 (UTC)

Saver's Credit
$$ \begin{align} MTR_{n, T} &= \text{marginal tax rate now, for the traditional contribution, including Saver's Credit} \\ MTR_{n, R} &= \text{marginal rate now of Saver's Credit for the Roth contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ T &= \text{traditional contribution} \\ R &= \text{Roth contribution} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

For a fair comparison, the two take home pays must be equal: $$A = T \cdot (1 - MTR_{n,T}) = R \cdot (1 - MTR_{n,R})$$

Solving for T and R in terms of A:

$$T = \frac{A}{1 - MTR_{n,T}}$$ $$R = \frac{A}{1 - MTR_{n,R}}$$

The changes in after-tax value of retirement accounts for the two contribution options are:

$$\Delta V_T = \frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w)$$ $$\Delta V_R = \frac{A}{1 - MTR_{n,R}} \cdot G$$

Traditional contributions are preferred when the $$\Delta V_T > \Delta V_R$$

$$\frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w) > \frac{A}{1 - MTR_{n,R}} \cdot G$$

Canceling $$A$$ and $$G$$ (assumed to be the same in both cases), and solving for $$MTR_w$$:

$$\frac{1 - MTR_w}{1 - MTR_{n,T}} > \frac{1}{1 - MTR_{n,R}}$$

$$MTR_w < 1 - \frac{1 - MTR_{n,T}}{1 - MTR_{n,R}}$$

$$MTR_w < \frac{MTR_{n,T} - MTR_{n,R}}{1 - MTR_{n,R}}$$

Employer match
Define variables as follows:

$$ \begin{align} MTR_n &= \text{marginal tax rate now, for traditional contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ m &= \text {employer match rate} \\ T &= \text{traditional balance} \\ R &= \text{Roth balance} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

When making a traditional contribution, the changes in the two types of balances will be:

$$\Delta T_T = \frac{A}{1 - MTR_n} \cdot (1 + m)$$ $$\Delta R_T = 0$$

When making a Roth contribution, the changes in the two types of balances will be:

$$\Delta T_R = A \cdot m$$ $$\Delta R_R = A$$

The after-tax values at withdrawal of the two contribution choices are:

$$\Delta V_T = \frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w)$$ $$\Delta V_R = A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G$$

Traditional contributions are preferred when $$\Delta V_T > \Delta V_R$$:

$$\frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w) > A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G$$

Canceling $$A$$ and $$G$$ (assumed to be the same in both cases):

$$\frac{1 - MTR_w}{1 - MTR_n} \cdot (1 + m) > m \cdot (1 - MTR_w) + 1 $$

Solving for $$MTR_w$$ using a Computer Algebra System (CAS):

$$MTR_w < \frac{(1+m) \cdot MTR_n}{m \cdot MTR_n + 1}$$

Derivation of tax rate boundaries for Social Security taxation
Variables are defined as follows:

$$ \begin{align} SS & = \text{Social Security income} \\ OI & = \text{other income} \\ BT & = \text{bracket threshold} \\ SD & = \text{standard deduction} \\ LB & = \text{lower base} \\ UB & = \text{upper base} \\ RI & = \text{relevant income} = 0.5 \cdot SS + OI \\ \end{align} $$

Point above which 40.7% marginal rate is possible
The point above which 40.7% marginal tax rates is possible is when total taxable income is at the 22% tax bracket threshold and the maximum 85% of Social Security benefits are taxable. It is the combination of $$SS$$ and $$OI$$ that satisfies these two equations:

$$0.85 \cdot SS + OI - SD = BT$$

$$0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = 0.85 \cdot SS$$

Rearranging the first equation to solve for OI gives:

$$OI = BT + SD - 0.85 \cdot SS$$

Save this result for later substitution. Substitute the definition of relevant income into the second equation:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (OI + 0.5 \cdot SS - UB) = 0.85 \cdot SS$$

Substitute in the formula for OI from the rearranged first equation:

$$0.5 \cdot (UB - LB) + 0.85 \cdot \left ([BT + SD - 0.85 \cdot SS] + 0.5 \cdot SS - UB \right) = 0.85 \cdot SS$$

Collecting the SS terms from the left hand side:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (BT + SD - UB) + 0.85 \cdot (0.5 \cdot SS - 0.85 \cdot SS) = 0.85 \cdot SS$$

Simplifying the SS terms on the left hand side:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (BT + SD - UB) - 0.2975 \cdot SS = 0.85 \cdot SS$$

Solving for SS and labeling this value SS* gives:

$$SS^{*} = \dfrac{0.5 \cdot (UB - LB) + 0.85 \cdot (BT + SD - UB)}{1.1475}$$

Recalling the equation above for OI in terms of SS, and labeling this value OI* gives:

$$OI^{*} = BT + SD - 0.85 \cdot SS^{*}$$

22.2% bump begins
For single filers, the 22.2% bump begins in the middle of the 12% bracket when Social Security taxation begins to be taxed at an 85% marginal rate. This occurs when:

$$0.5 \cdot SS + OI = UB$$

Substituting $34,000 for UB gives:

$$OI = $34,000 - 0.5 \cdot SS$$

For married filers, the 22.2% bump begins at the boundary between the 10% and 12% brackets. The line is defined by the solution to these equations:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = p \cdot SS$$

$$OI + p \cdot SS = BT + SD$$

where $$p$$ is the percentage of Social Security income that is taxable. $$p$$ is an unknown variable, but with two equations and three unknowns it should be possible eliminate $$p$$ through substitution. Solving for $$p$$ in the second equation gives:

$$p = \dfrac{BT + SD - OI}{SS}$$

Substituting this value for $$p$$ into the first equation, and also the definition of relevant income, gives:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (OI + 0.5 \cdot SS - UB) = BT + SD - OI$$

Expanding the large term on the left hand side gives:

$$0.5 \cdot (UB - LB) + 0.85 \cdot OI + 0.425 \cdot SS - 0.85 \cdot UB = BT + SD - OI$$

Rearranging to solve for OI:

$$1.85 \cdot OI = BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB - LB) - 0.425 \cdot SS$$

The solution to this set of equations is:

$$OI = \dfrac{BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB-LB)}{1.85} - \dfrac{0.425 \cdot SS}{1.85}$$

$$\dfrac{0.425}{1.85} \approx 0.22973 \approx 0.23$$

22.2% bump ends
The 22.2% bump ends when the maximum of 85% of Social Security benefits becomes taxable. This occurs when:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = 0.85 \cdot SS$$

Substituting the definition for relevant income gives:

$$0.5 \cdot (UB - LB) + 0.85 \cdot (0.5 \cdot SS + OI - UB) = 0.85 \cdot SS$$

Expanding the large term on the left hand side gives:

$$0.5 \cdot (UB - LB) + 0.425 \cdot SS + 0.85 \cdot OI - 0.85 \cdot UB = 0.85 \cdot SS$$

Rearranging to solve for OI gives:

$$0.85 \cdot OI = 0.85 \cdot UB - 0.5 \cdot (UB - LB) + 0.425 \cdot SS$$

Dividing by 0.85 gives:

$$OI = \left ( UB - \frac{0.5}{0.85} \cdot (UB - LB) \right ) + 0.5 \cdot SS$$

40.7% bump begins
For both single and married filers, the 40.7% bump begins at the boundary of the 22% bracket. The formula is the same as for the beginning of the 22.2% bump for married filers, but with a different bracket threshold:

$$OI = \dfrac{BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB-LB)}{1.85} - \dfrac{0.425 \cdot SS}{1.85}$$

40.7% bump ends
The line where the 40.7% bump ends is the same as where the 22.2% bump ends. The only difference is whether the boundary is above or below $$SS^{*}$$.

$$OI = \left ( UB - \frac{0.5}{0.85} \cdot (UB - LB) \right ) + 0.5 \cdot SS$$

-- Section created 02:03, 20 May 2019‎ by Fyre4ce (--LadyGeek 20:33, 20 May 2019 (UTC))