# User:Fyre4ce/Retirement plan analysis

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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{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:

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

$\Delta T=-C$ $\Delta R=+C$ $\Delta A=-C\cdot MTR_{n}\cdot (1-MTR_{e})$ 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{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}}\\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{aligned}} 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{aligned}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{aligned}} 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{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 $SS$ 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))