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

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(Edited Saver's Credit variables to be consistent with other derivations)
(Added analysis with match and Saver's Credit)
 
(7 intermediate revisions by the same user not shown)
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or
 
or
  
<math>MTR_e > 1 - \frac{MTR_h}{MTR_n}</math>
+
<math>MTR_h > MTR_n \cdot (1 - MTR_e)</math>
  
 
--[[User:Fyre4ce|Fyre4ce]] 04:44, 10 December 2020 (UTC)
 
--[[User:Fyre4ce|Fyre4ce]] 04:44, 10 December 2020 (UTC)
Line 106: Line 106:
 
MTR_{n, T} &= \text{marginal tax rate now, for the traditional contribution, including Saver's Credit} \\
 
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_{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} \\
 
T &= \text{traditional contribution} \\
 
R &= \text{Roth contribution} \\
 
R &= \text{Roth contribution} \\
Line 129: Line 130:
 
Traditional contributions are preferred when the <math>\Delta V_T > \Delta V_R</math>
 
Traditional contributions are preferred when the <math>\Delta V_T > \Delta V_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>
+
<math>\frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w) > \frac{A}{1 - MTR_{n,R}} \cdot G</math>
  
 
Canceling <math>A</math> and <math>G</math> (assumed to be the same in both cases), and solving for <math>MTR_w</math>:
 
Canceling <math>A</math> and <math>G</math> (assumed to be the same in both cases), and solving for <math>MTR_w</math>:
Line 139: Line 140:
 
<math>MTR_w < \frac{MTR_{n,T} - MTR_{n,R}}{1 - MTR_{n,R}}</math>
 
<math>MTR_w < \frac{MTR_{n,T} - MTR_{n,R}}{1 - MTR_{n,R}}</math>
  
==Derivation of tax rate boundaries for Social Security taxation==
+
==Maxing out retirement accounts==
  
Variables are defined as follows:
+
Define variables as follows:
  
 
<math>
 
<math>
 
\begin{align}
 
\begin{align}
SS & = \text{Social Security income} \\
+
MTR_n &= \text{marginal tax rate now, for traditional contribution} \\
OI & = \text{other income} \\
+
MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\
BT & = \text{bracket threshold} \\
+
MTR_{div} &= \text{marginal tax rate on dividends} \\
SD & = \text{standard deduction} \\
+
MTR_{cg} &= \text{marginal tax rate on capital gains} \\
LB & = \text{lower base} \\
+
C &= \text {contribution (fixed dollar amount for traditional or Roth)} \\
UB & = \text{upper base} \\
+
G_T &= \text {growth factor on traditional balance, before taxes} \\
RI & = \text{relevant income} = 0.5 \cdot SS + OI \\
+
G_R &= \text {growth factor on Roth balance (tax-free)} \\
 +
G_{Tx} &= \text {growth factor on taxable balance, after taxes} \\
 +
r_T &= \text{total rate of return on the traditional balance} \\
 +
r_R &= \text{total rate of return on the Roth balance} \\
 +
r_{Tx} &= \text{total rate of return on the taxable balance} \\
 +
y &= \text{yield on the taxable balance} \\
 +
v &= \text{growth factor on the taxable balance} \\
 +
b &= \text{growth factor on the taxable basis} \\
 +
t &= \text{time} \\
 
\end{align}
 
\end{align}
 
</math>
 
</math>
  
===Point above which 40.7% marginal rate is possible===
+
When contributing a fixed dollar amount <math>C</math> to either traditional or Roth accounts, and investing the tax savings <math>C \cdot MTR_n</math> in a taxable account, traditional contributions are preferred when:
  
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 <math>SS</math> and <math>OI</math> that satisfies these two equations:
+
<math>C \cdot G_T \cdot (1 - MTR_w) + MTR_n \cdot C \cdot G_{Tx} > C \cdot G_R</math>
  
<math>0.85 \cdot SS + OI - SD = BT</math>
+
Canceling <math>C</math> and solving for <math>MTR_w</math> gives:
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = 0.85 \cdot SS</math>
+
<math>MTR_w < \frac{G_T - G_R + MTR_n \cdot G_{Tx}}{G_T}</math>  
  
Rearranging the first equation to solve for OI gives:
+
Rather than plug in the formulas for these factors to create one large equation, it is easier to calculate each factor separately. Assuming annual compounding, the three growth factors can be calculated as follows:
  
<math>OI = BT + SD - 0.85 \cdot SS</math>
+
<math>G_T = (1 + r_T)^t</math><br>
 +
<math>G_R = (1 + r_R)^t</math><br>
 +
<math>G_{Tx} = (v - (v - b) \cdot MTR_{cg})</math>
  
Save this result for later substitution. Substitute the definition of relevant income into the second equation:
+
Recall from [[Taxable_account#Performance|taxable account performance]] that:
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (OI + 0.5 \cdot SS - UB) = 0.85 \cdot SS</math>
+
<math>v = \frac{V(t)}{V(0)} = (1 + r_{Tx} - y \cdot MTR_{div})^t</math>
  
Substitute in the formula for OI from the rearranged first equation:
+
and
  
<math>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</math>
+
<math>b = \frac{B(t)}{V(0)} = 1 + \left ( \frac{ y \cdot (1-MTR_{div})}{r_{Tx} - y \cdot MTR_{div}} \right ) \left ( (1 + r_{Tx} - y \cdot MTR_{div})^t-1 \right )</math>
  
Collecting the SS terms from the left hand side:
+
Separate rates of return for traditional, Roth, and taxable accounts allow the comparison between different accounts (eg. IRA or 401(k)) with different investments and fees. Assuming the same investments and fees <math>(r_T = r_R = r_{Tx} = r)</math> and <math>G_T = G_R</math>, the equations simplifies somewhat to:
  
<math>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</math>
+
<math>MTR_w < MTR_n \cdot \frac{G_{Tx}}{(1 + r)^t}</math>  
  
Simplifying the SS terms on the left hand side:
+
with <math>G_{Tx}</math>, <math>v</math>, and <math>b</math> the same as above.
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (BT + SD - UB) - 0.2975 \cdot SS = 0.85 \cdot SS</math>
+
==Employer match==
  
Solving for SS and labeling this value SS* gives:
+
Define variables as follows:
  
<math>SS^{*} = \dfrac{0.5 \cdot (UB - LB) + 0.85 \cdot (BT + SD - UB)}{1.1475}</math>
+
<math>
 
+
\begin{align}
Recalling the equation above for OI in terms of SS, and labeling this value OI* gives:
+
MTR_n &= \text{marginal tax rate now, for traditional contribution} \\
 
+
MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\
<math>OI^{*} = BT + SD - 0.85 \cdot SS^{*}</math>
+
m &= \text {employer match rate} \\
 
+
T &= \text{traditional balance} \\
===22.2% bump begins===
+
R &= \text{Roth balance} \\
 
+
A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\
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:
+
G &= \text{growth factor of investments between now and withdrawal} \\
 
+
V &= \text{after-tax value of retirement accounts} \\
<math>0.5 \cdot SS + OI = UB</math>
+
\end{align}
 
+
</math>
Substituting $34,000 for UB gives:
 
 
 
<math>OI = $34,000 - 0.5 \cdot SS</math>
 
 
 
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:
 
 
 
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = p \cdot SS</math>
 
 
 
<math>OI + p \cdot SS = BT + SD</math>
 
 
 
where <math>p</math> is the percentage of Social Security income that is taxable. <math>p</math> is an unknown variable, but with two equations and three unknowns it should be possible eliminate <math>p</math> through substitution. Solving for <math>p</math> in the second equation gives:
 
 
 
<math>p = \dfrac{BT + SD - OI}{SS}</math>
 
 
 
Substituting this value for <math>p</math> into the first equation, and also the definition of relevant income, gives:
 
 
 
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (OI + 0.5 \cdot SS - UB) = BT + SD - OI</math>
 
  
Expanding the large term on the left hand side gives:
+
When making a traditional contribution, the changes in the two types of balances will be:
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot OI + 0.425 \cdot SS - 0.85 \cdot UB = BT + SD - OI</math>
+
<math>\Delta T_T = \frac{A}{1 - MTR_n} \cdot (1 + m)</math><br>
 +
<math>\Delta R_T = 0</math>
  
Rearranging to solve for OI:
+
When making a Roth contribution, the changes in the two types of balances will be:
  
<math>1.85 \cdot OI = BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB - LB) - 0.425 \cdot SS</math>
+
<math>\Delta T_R = A \cdot m</math><br>
 +
<math>\Delta R_R = A</math>
  
The solution to this set of equations is:
+
The after-tax values at withdrawal of the two contribution choices are:
  
<math>OI = \dfrac{BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB-LB)}{1.85} - \dfrac{0.425 \cdot SS}{1.85}</math>
+
<math>\Delta V_T = \frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w)</math><br>
 +
<math>\Delta V_R = A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G</math>
  
<math>\dfrac{0.425}{1.85} \approx 0.22973 \approx 0.23</math>
+
Traditional contributions are preferred when <math>\Delta V_T > \Delta V_R</math>:
  
===22.2% bump ends===
+
<math>\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</math>
  
The 22.2% bump ends when the maximum of 85% of Social Security benefits becomes taxable. This occurs when:
+
Canceling <math>A</math> and <math>G</math> (assumed to be the same in both cases):
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (RI - UB) = 0.85 \cdot SS</math>
+
<math>\frac{1 - MTR_w}{1 - MTR_n} \cdot (1 + m) > m \cdot (1 - MTR_w) + 1 </math>
  
Substituting the definition for relevant income gives:
+
Solving for <math>MTR_w</math> using a Computer Algebra System (CAS):
  
<math>0.5 \cdot (UB - LB) + 0.85 \cdot (0.5 \cdot SS + OI - UB) = 0.85 \cdot SS</math>
+
<math>MTR_w < \frac{(1+m) \cdot MTR_n}{m \cdot MTR_n + 1}</math>
  
Expanding the large term on the left hand side gives:
+
===Employer match combined with Saver's Credit===
  
<math>0.5 \cdot (UB - LB) + 0.425 \cdot SS + 0.85 \cdot OI - 0.85 \cdot UB = 0.85 \cdot SS</math>
+
The above equations can be modified to also include a Saver's Credit. When making a traditional contribution, the changes in the two types of balances will be:
  
Rearranging to solve for OI gives:
+
<math>\Delta T_T = \frac{A}{1 - MTR_{n,T}} \cdot (1 + m)</math><br>
 +
<math>\Delta R_T = 0</math>
  
<math>0.85 \cdot OI = 0.85 \cdot UB - 0.5 \cdot (UB - LB) + 0.425 \cdot SS</math>
+
When making a Roth contribution, the changes in the two types of balances will be:
  
Dividing by 0.85 gives:
+
<math>\Delta T_R = \frac{A}{1 - MTR_{n,R}} \cdot m</math><br>
 +
<math>\Delta R_R = \frac{A}{1 - MTR_{n,R}}</math>
  
<math>OI = \left ( UB - \frac{0.5}{0.85} \cdot (UB - LB) \right ) + 0.5 \cdot SS</math>
+
The after-tax values at withdrawal of the two contribution choices are:
  
===40.7% bump begins===
+
<math>\Delta V_T = \frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w)</math><br>
 +
<math>\Delta V_R = A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G</math>
  
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:
+
Traditional contributions are preferred when <math>\Delta V_T > \Delta V_R</math>:
  
<math>OI = \dfrac{BT + SD + 0.85 \cdot UB - 0.5 \cdot (UB-LB)}{1.85} - \dfrac{0.425 \cdot SS}{1.85}</math>
+
<math>\frac{A}{1 - MTR_{n,T}} \cdot (1 + m) \cdot G \cdot (1 - MTR_w) > \frac{A}{1 - MTR_{n,R}} \cdot m \cdot G \cdot (1 - MTR_w) + \frac{A}{1 - MTR_{n,R}} \cdot G</math>
  
===40.7% bump ends===
+
Canceling <math>A</math> and <math>G</math> (assumed to be the same in both cases):
  
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 <math>SS^{*}</math>.
+
<math>\frac{1 - MTR_w}{1 - MTR_{n,T}} \cdot (1 + m) > \frac{m \cdot (1 - MTR_w) + 1}{1 - MTR_{n,R}} </math>
  
<math>OI = \left ( UB - \frac{0.5}{0.85} \cdot (UB - LB) \right ) + 0.5 \cdot SS</math>
+
Solving for <math>MTR_w</math> using a Computer Algebra System (CAS):
  
-- Section created 02:03, 20 May 2019‎ by Fyre4ce (--[[User:LadyGeek|LadyGeek]] 20:33, 20 May 2019 (UTC))
+
<math>MTR_w < \frac{(1+m) \cdot (MTR_{n,T} - MTR_{n,R})}{m \cdot (MTR_{n,T} - MTR_{n,R}) +1 - MTR_{n,R}}</math>

Latest revision as of 02:18, 1 January 2021

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 :

Maxing out retirement accounts

Define variables as follows:

When contributing a fixed dollar amount to either traditional or Roth accounts, and investing the tax savings in a taxable account, traditional contributions are preferred when:

Canceling and solving for gives:

Rather than plug in the formulas for these factors to create one large equation, it is easier to calculate each factor separately. Assuming annual compounding, the three growth factors can be calculated as follows:



Recall from taxable account performance that:

and

Separate rates of return for traditional, Roth, and taxable accounts allow the comparison between different accounts (eg. IRA or 401(k)) with different investments and fees. Assuming the same investments and fees and , the equations simplifies somewhat to:

with , , and the same as above.

Employer match

Define variables as follows:

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


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


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


Traditional contributions are preferred when :

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

Solving for using a Computer Algebra System (CAS):

Employer match combined with Saver's Credit

The above equations can be modified to also include a Saver's Credit. When making a traditional contribution, the changes in the two types of balances will be:


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


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


Traditional contributions are preferred when :

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

Solving for using a Computer Algebra System (CAS):