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_{T}&={\text{marginal tax rate for the traditional contribution}}\\mtr_{R}&={\text{marginal tax rate for the Roth contribution}}\\NI&={\text{Net Income (gross pay minus all deductions and tax) before accounting for traditional or Roth contributions}}\\R&={\text{Roth contribution}}\\R_{sp}&={\text{spendable Roth amount in retirement}}\\T&={\text{traditional contribution}}\\T_{sp}&={\text{spendable traditional amount in retirement}}\\mwr_{T=R}&={\text{withdrawal marginal tax rate for the traditional account that makes traditional and Roth results equivalent}}\\\end{aligned}}}

For a fair comparison, the two take home pays must be equal:
${\displaystyle Take\ home\ pay\ traditional\ =NI-T*(1-mtr_{T})}$
${\displaystyle Take\ home\ pay\ Roth\ =NI-R*(1-mtr_{R})}$

Equating them and solving for R or T we get:

${\displaystyle R=T*(1-mtr_{T})/(1-mtr_{R})}$
${\displaystyle T=R*(1-mtr_{R})/(1-mtr_{T})}$

If ${\displaystyle mr_{R}=0}$, then those reduce to the familiar equations for equivalent Roth and traditional contributions.

The equations for spendable amounts are:

${\displaystyle R_{sp}=R\cdot (1+i)^{n}=T*(1-mr_{T})/(1-mr_{R})*(1+i)^{n}}$
${\displaystyle T_{sp}=T\cdot (1+i)^{n}*(1-mtr_{T=R})}$

Equating those and solving for ${\displaystyle mwr_{T=R}}$, we get:

${\displaystyle mwr_{T=R}=(mtr_{T}-mtr_{R})/(1-mtr_{R})}$

For example, take a single filer with $30K gross income. For that person, up to a$2,000 contribution, ${\displaystyle mtr_{T}=22\%}$ and ${\displaystyle mtr_{R}=10\%}$.

For a 2,000 traditional contribution, the equivalent Roth contribution is ${\displaystyle R=\2,000\cdot (1-22\%)/(1-10\%)=\1733}$. The withdrawal marginal tax rate for equivalent results is ${\displaystyle mwr_{T=R}=(22\%-10\%)/(1-10\%)=13.3\%}$. 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. 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: ${\displaystyle 0.5\cdot SS+OI=UB}$ Substituting34,000 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))