Difference between revisions of "User:Fyre4ce/Retirement plan analysis"

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(Removed equations with plugged-in numbers (which must be updated every year))
(Edited Saver's Credit variables to be consistent with other derivations)
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==Conversions on estates subject to estate tax==
 
==Conversions on estates subject to estate tax==
 
  
 
Define variables:
 
Define variables:
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<math>
 
<math>
 
\begin{align}
 
\begin{align}
mtr_{T} &= \text{marginal tax rate for the traditional contribution} \\
+
MTR_{n, T} &= \text{marginal tax rate now, for the traditional contribution, including Saver's Credit} \\
mtr_{R} &= \text{marginal tax rate for the Roth contribution} \\
+
MTR_{n, R} &= \text{marginal rate now of Saver's Credit for the Roth contribution} \\
NI &= \text{Net Income (gross pay minus all deductions and tax) before accounting for traditional or Roth contributions} \\
+
T &= \text{traditional contribution} \\
 
R &= \text{Roth contribution} \\
 
R &= \text{Roth contribution} \\
R_{sp} &= \text{spendable Roth amount in retirement} \\
+
A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\
T &= \text{traditional contribution} \\
+
G &= \text{growth factor of investments between now and withdrawal} \\
T_{sp} &= \text{spendable traditional amount in retirement} \\
+
V &= \text{after-tax value of retirement accounts} \\
mwr_{T=R} &= \text{withdrawal marginal tax rate for the traditional account that makes traditional and Roth results equivalent} \\
 
 
\end{align}
 
\end{align}
 
</math>
 
</math>
  
 
For a fair comparison, the two take home pays must be equal:<br>
 
For a fair comparison, the two take home pays must be equal:<br>
<math>Take\ home\ pay\ traditional\ = NI - T * (1 - mtr_T)</math><br>
+
<math>A = T \cdot (1 - MTR_{n,T}) = R \cdot (1 - MTR_{n,R})</math>
<math>Take\ home\ pay\ Roth\ = NI - R * (1 - mtr_R)</math><br>
 
 
 
Equating them and solving for R or T we get:
 
  
<math>R = T * (1 - mtr_T) / (1 - mtr_R)</math><br>
+
Solving for T and R in terms of A:
<math>T = R * (1 - mtr_R) / (1 - mtr_T)</math>
 
  
If <math>mr_R=0</math>, then those reduce to the familiar equations for equivalent Roth and traditional contributions.
+
<math>T = \frac{A}{1 - MTR_{n,T}}</math><br>
 +
<math>R = \frac{A}{1 - MTR_{n,R}}</math>
  
The equations for spendable amounts are:
+
The changes in after-tax value of retirement accounts for the two contribution options are:
  
<math>R_{sp} = R \cdot (1+i)^n = T * (1 - mr_T) / (1 - mr_R) * (1+i)^n</math><br>
+
<math>\Delta V_T = \frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w)</math><br>
<math>T_{sp} = T \cdot (1+i)^n * (1 - mtr_{T=R})</math>
+
<math>\Delta V_R = \frac{A}{1 - MTR_{n,R}} \cdot G</math>
  
Equating those and solving for <math>mwr_{T=R}</math>, we get:
+
Traditional contributions are preferred when the <math>\Delta V_T > \Delta V_R</math>
  
<math>mwr_{T=R} = (mtr_T - mtr_R) / (1 - mtr_R)</math>
+
<math>\frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w) > \Delta V_R = \frac{A}{1 - MTR_{n,R}} \cdot G</math>
  
For example, take a single filer with $30K gross income.  For that person, up to a $2,000 contribution, <math>mtr_T = 22%</math> and <math>mtr_R = 10%</math>.
+
Canceling <math>A</math> and <math>G</math> (assumed to be the same in both cases), and solving for <math>MTR_w</math>:
  
For a $2,000 traditional contribution, the equivalent Roth contribution is <math>R = $2,000 \cdot (1 - 22%) / (1 - 10%) = $1733</math>.
+
<math>\frac{1 - MTR_w}{1 - MTR_{n,T}} > \frac{1}{1 - MTR_{n,R}}</math>
  
The withdrawal marginal tax rate for equivalent results is <math>mwr_{T=R} = (22% - 10%) / (1 - 10%) = 13.3%</math>.
+
<math>MTR_w < 1 - \frac{1 - MTR_{n,T}}{1 - MTR_{n,R}}</math>
  
If one expects the actual withdrawal marginal tax rate will be less than 13.3%, traditional is better.  If more than 13.3%, Roth is better.
+
<math>MTR_w < \frac{MTR_{n,T} - MTR_{n,R}}{1 - MTR_{n,R}}</math>
  
 
==Derivation of tax rate boundaries for Social Security taxation==
 
==Derivation of tax rate boundaries for Social Security taxation==

Revision as of 16:43, 20 December 2020

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:

The overall value of a change to tax-advantaged space is equal to:

Consider a given after-tax investment 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:

Therefore, the change in value when making a traditional contribution is:

When making a Roth contribution, the change in Roth balance is simply:

Therefore, the change in value when making a Roth contribution is:

When making a Roth conversion, the converted amount is:

Therefore, the change in value when making a Roth conversion is:

When (current marginal tax rate is less than predicted future marginal tax rate),

When (current marginal tax rate equals predicted future marginal tax rate),

When (current marginal tax rate is greater than predicted future marginal tax rate),

--Fyre4ce 23:10, 10 March 2020 (UTC)

Conversions on estates subject to estate tax

Define variables:

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:

The change in after-tax value of the estate to heirs will be as follows:

It follows that Roth conversions increase the value of the after-tax value of the estate if:

or

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

Saver's Credit

For a fair comparison, the two take home pays must be equal:

Solving for T and R in terms of A:


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


Traditional contributions are preferred when the

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

Derivation of tax rate boundaries for Social Security taxation

Variables are defined as follows:

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 and that satisfies these two equations:

Rearranging the first equation to solve for OI gives:

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

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

Collecting the SS terms from the left hand side:

Simplifying the SS terms on the left hand side:

Solving for SS and labeling this value SS* gives:

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

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:

Substituting $34,000 for UB gives:

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:

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

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

Expanding the large term on the left hand side gives:

Rearranging to solve for OI:

The solution to this set of equations is:

22.2% bump ends

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

Substituting the definition for relevant income gives:

Expanding the large term on the left hand side gives:

Rearranging to solve for OI gives:

Dividing by 0.85 gives:

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:

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 .

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