# Difference between revisions of "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:

{\displaystyle {\begin{aligned}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{aligned}}}

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

${\displaystyle \Delta V=\Delta T(1-MTR_{w})+\Delta R}$

Consider a given after-tax investment ${\displaystyle A}$ 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:

${\displaystyle \Delta T={\frac {A}{(1-MTR_{n})}}}$

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

${\displaystyle \Delta V_{T}=A{\frac {(1-MTR_{w})}{(1-MTR_{n})}}}$

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

${\displaystyle \Delta R=A}$

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

${\displaystyle \Delta V_{R}=A}$

When making a Roth conversion, the converted amount is:

${\displaystyle C={\frac {A}{MTR_{n}}}}$

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

${\displaystyle \Delta V_{C}=C-C(1-MTR_{w})={\frac {A}{MTR_{n}}}-{\frac {A}{MTR_{n}}}\cdot (1-MTR_{w})={\frac {A}{MTR_{n}}}(1-(1-MTR_{w}))=A{\frac {MTR_{w}}{MTR_{n}}}}$

When ${\displaystyle MTR_{n} (current marginal tax rate is less than predicted future marginal tax rate),

${\displaystyle \Delta V_{C}>\Delta V_{R}>\Delta V_{T}}$

When ${\displaystyle MTR_{n}=MTR_{w}}$ (current marginal tax rate equals predicted future marginal tax rate),

${\displaystyle \Delta V_{C}=\Delta V_{R}=\Delta V_{T}}$

When ${\displaystyle MTR_{n}>MTR_{w}}$ (current marginal tax rate is greater than predicted future marginal tax rate),

${\displaystyle \Delta V_{C}<\Delta V_{R}<\Delta V_{T}}$

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

## Conversions on estates subject to estate tax

Define variables:

{\displaystyle {\begin{aligned}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{aligned}}}

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:

${\displaystyle \Delta T=-C}$

${\displaystyle \Delta R=+C}$

${\displaystyle \Delta A=-C\cdot MTR_{n}\cdot (1-MTR_{e})}$

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

${\displaystyle \Delta V_{h}=\Delta T\cdot (1-MTR_{h})+\Delta R+\Delta A=-C\cdot (1-MTR_{h})+C-C\cdot MTR_{n}\cdot (1-MTR_{e})}$

${\displaystyle \Delta V_{h}=C\cdot ((MTR_{h}-1)+1+MTR_{n}\cdot (MTR_{e}-1))=C\cdot (MTR_{h}+MTR_{n}\cdot (MTR_{e}-1))}$

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

${\displaystyle MTR_{h}+MTR_{n}\cdot (MTR_{e}-1)>0}$

or

${\displaystyle MTR_{e}>1-{\frac {MTR_{h}}{MTR_{n}}}}$

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

## Saver's Credit

{\displaystyle {\begin{aligned}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}}\\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{aligned}}}

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

Solving for T and R in terms of A:

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

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

${\displaystyle \Delta V_{T}={\frac {A}{1-MTR_{n,T}}}\cdot G\cdot (1-MTR_{w})}$
${\displaystyle \Delta V_{R}={\frac {A}{1-MTR_{n,R}}}\cdot G}$

Traditional contributions are preferred when the ${\displaystyle \Delta V_{T}>\Delta V_{R}}$

${\displaystyle {\frac {A}{1-MTR_{n,T}}}\cdot G\cdot (1-MTR_{w})>\Delta V_{R}={\frac {A}{1-MTR_{n,R}}}\cdot G}$

Canceling ${\displaystyle A}$ and ${\displaystyle G}$ (assumed to be the same in both cases), and solving for ${\displaystyle MTR_{w}}$:

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

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

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

## Derivation of tax rate boundaries for Social Security taxation

Variables are defined as follows:

{\displaystyle {\begin{aligned}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{aligned}}}

### 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 ${\displaystyle SS}$ and ${\displaystyle OI}$ that satisfies these two equations:

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

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

Rearranging the first equation to solve for OI gives:

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

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

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

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

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

Collecting the SS terms from the left hand side:

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

Simplifying the SS terms on the left hand side:

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

Solving for SS and labeling this value SS* gives:

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

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

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

### 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:

${\displaystyle 0.5\cdot SS+OI=UB}$

Substituting \$34,000 for UB gives:

${\displaystyle OI=\34,000-0.5\cdot SS}$

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:

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

${\displaystyle OI+p\cdot SS=BT+SD}$

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

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

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

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

Expanding the large term on the left hand side gives:

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

Rearranging to solve for OI:

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

The solution to this set of equations is:

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

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

### 22.2% bump ends

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

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

Substituting the definition for relevant income gives:

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

Expanding the large term on the left hand side gives:

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

Rearranging to solve for OI gives:

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

Dividing by 0.85 gives:

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

### 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:

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

### 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 ${\displaystyle SS^{*}}$.

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

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