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:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ C & = \text{Roth-converted amount} \\ V & = \text{Total value of tax-advantaged space} \\ A & = \text{After-tax amount} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_w & = \text{marginal tax rate at withdrawal} \\ \end{align} $$

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 $$MTR_n < MTR_w$$ (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:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ A & = \text{After-tax balance} \\ C & = \text{Roth-converted amount} \\ V_h & = \text{Total value of estate to heirs after-tax} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_e & = \text{marginal tax rate on estate} \\ MTR_h & = \text{marginal tax rate on heirs} \\ \end{align} $$

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
$$ \begin{align} MTR_{n, T} &= \text{marginal tax rate now, for the traditional contribution, including Saver's Credit} \\ MTR_{n, R} &= \text{marginal rate now of Saver's Credit for the Roth contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ T &= \text{traditional contribution} \\ R &= \text{Roth contribution} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

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 $$MTR_w$$:

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

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

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

Maxing out retirement accounts
Define variables as follows:

$$ \begin{align} MTR_n &= \text{marginal tax rate now, for traditional contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ MTR_{div} &= \text{marginal tax rate on dividends} \\ MTR_{cg} &= \text{marginal tax rate on capital gains} \\ C &= \text {contribution (fixed dollar amount for traditional or Roth)} \\ G_T &= \text {growth factor on traditional balance, before taxes} \\ G_R &= \text {growth factor on Roth balance (tax-free)} \\ G_{Tx} &= \text {growth factor on taxable balance, after taxes} \\ r_T &= \text{total rate of return on the traditional balance} \\ r_R &= \text{total rate of return on the Roth balance} \\ r_{Tx} &= \text{total rate of return on the taxable balance} \\ y &= \text{yield on the taxable balance} \\ v &= \text{growth factor on the taxable balance} \\ b &= \text{growth factor on the taxable basis} \\ t &= \text{time} \\ \end{align} $$

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

$$C \cdot G_T \cdot (1 - MTR_w) + MTR_n \cdot C \cdot G_{Tx} > C \cdot G_R$$

Canceling $$C$$ and solving for $$MTR_w$$ gives:

$$MTR_w < \frac{G_T - G_R + MTR_n \cdot G_{Tx}}{G_T}$$

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:

$$G_T = (1 + r_T)^t$$ $$G_R = (1 + r_R)^t$$ $$G_{Tx} = (v - (v - b) \cdot MTR_{cg})$$

Recall from taxable account performance that:

$$v = \frac{V(t)}{V(0)} = (1 + r_{Tx} - y \cdot MTR_{div})^t$$

and

$$b = \frac{B(t)}{V(0)} = 1 + \left ( \frac{ y \cdot (1-MTR_{div})}{r_{Tx} - y \cdot MTR_{div}} \right ) \left ( (1 + r_{Tx} - y \cdot MTR_{div})^t-1 \right )$$

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 $$(r_T = r_R = r_{Tx} = r)$$ and $$G_T = G_R$$, the equations simplifies somewhat to:

$$MTR_w < MTR_n \cdot \frac{G_{Tx}}{(1 + r)^t}$$

with $$G_{Tx}$$, $$v$$, and $$b$$ the same as above.

Employer match
Define variables as follows:

$$ \begin{align} MTR_n &= \text{marginal tax rate now, for traditional contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ m &= \text {employer match rate} \\ T &= \text{traditional balance} \\ R &= \text{Roth balance} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

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:

$$ \begin{align} SS & = \text{Social Security income} \\ OI & = \text{other income} \\ BT & = \text{bracket threshold} \\ SD & = \text{standard deduction} \\ LB & = \text{lower base} \\ UB & = \text{upper base} \\ RI & = \text{relevant income} = 0.5 \cdot SS + OI \\ \end{align} $$

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))

Variable Marginal Rates with Section 199A Deduction
Define variables:

$$ \begin{align} T(N, Q) & = \text{tax liability as a function of income} \\ N & = \text{non-Qualified Business Income} \\ Q & = \text{Qualified Business Income} \\ TB & = \text{tax bracket (eg. 24 percent)} \\ D & = \text{below-the-line deduction (eg. standard deduction)} \\ UL & = \text{upper limit of deduction phase-out (eg. 426,600 for MFJ 2020)} \\ R & = \text{phase-out range (eg. 100,000 for MFJ 2020)} \\ \end{align} $$

This analysis will assume a single tax bracket, although because the equation will be differentiated, the results will apply to any tax bracket.

The total tax liability can be written as follows. If total taxable income is below the beginning of the deduction phase-out ($163,300 for single or $326,600 for MFJ, for 2020), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB - 20% \cdot Q \cdot TB$$

In the phase-out range ($163,200-$213,300 for single or $326,600-$426,600 for MFJ), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB - 20% \cdot Q \cdot \left ( \frac{UL - (Q + N - D)}{R} \right ) \cdot TB$$

Above the phase-out range ($213,300 for single or $426,600 for MFJ), total tax is:

$$T(N, Q) = (N+Q-D) \cdot TB $$

Above the phase-out, the Section 199A deduction has no effect. Below the phase-out, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

$$\frac{\partial T(N, Q)}{\partial N} = TB$$

$$\frac{\partial T(N, Q)}{\partial Q} = (1 - 20%) \cdot TB = 80% \cdot TB$$

In the phase-out range, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

$$\frac{\partial T(N, Q)}{\partial N} = TB - 20% \cdot Q \cdot \left ( \frac{-1}{R} \right ) \cdot TB$$

$$\frac{\partial T(N, Q)}{\partial N} = TB \cdot \left ( 1 + \frac{20% \cdot Q}{R} \right )$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB - \frac{\partial}{\partial Q} \left ( \frac{20% \cdot TB}{R} \cdot (UL \cdot Q - Q^2 - N \cdot Q + D \cdot Q)\right )$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB - \frac{20% \cdot TB}{R} \cdot (UL - 2Q - N + D)$$

$$\frac{\partial T(N, Q)}{\partial Q} = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (2Q + N - UL - D) \right )$$

The second partial derivatives are:

$$\frac{\partial^2 T(N, Q)}{\partial N^2} = 0$$

$$\frac{\partial^2 T(N, Q)}{\partial Q^2} = \frac{2 \cdot TB \cdot 20%}{R}$$

$$\frac{\partial^2 T(N, Q)}{\partial Q \partial N} = \frac{TB \cdot 20%}{R}$$

Example
A MFJ couple has $120,000 of non-QBI income and also earns QBI. They take the standard deduction of $24,800. Below $326,600 of taxable income, they are in the 24% bracket. Their marginal tax rates for non-QBI and QBI income are:

$$\frac{\partial T(N, Q)}{\partial N} = 24%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 80% \cdot 24% = 19.2%$$

The phase-out begins when their taxable income, after the standard deduction, equals $326,600. This corresponds to a QBI income of $231,400 ($326,600 + $24,800 - $120,000). Note that although their taxable income is at the 24%/32% threshold, the Section 199A deduction pulls them well down into the 24% bracket. At this income, their marginal tax rates are:

$$\frac{\partial T(N, Q)}{\partial N} = 24% \cdot \left ( 1 + \frac{20% \cdot $231,400}{$100,000} \right ) = 35.1072%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 24% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot (2 \cdot $231,400 + $120,000 - $426,600 - $24,800) \right ) = 24% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot $131,400 \right ) = 30.3072%$$

At some point in the phase-out range, the couple will cross into the 32% bracket, and then into the 35% bracket. At the top of the phase-out, when QBI income is $331,400 ($426,600 + $24,800 - $120,000), marginal tax rates will be:

$$\frac{\partial T(N, Q)}{\partial N} = 35% \cdot \left ( 1 + \frac{20% \cdot $331,400}{$100,000} \right ) = 58.198%$$

$$\frac{\partial T(N, Q)}{\partial Q} = 35% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot (2 \cdot $339,000 + $100,000 - $415,000 - $24,000) \right ) = 35% \cdot \left ( 1 + \frac{20%}{$100,000} \cdot $331,400 \right ) = 58.198%$$

Above $331,400 QBI, the Section 199A deduction is completely eliminated and the marginal rates become:

$$\frac{\partial T(N, Q)}{\partial N} = \frac{\partial T(N, Q)}{\partial Q} = 35%$$

Maximum possible rates
By inspection of the above formulas, the maximum possible rates occur at the very top of the phase-out range, when:

$$N + Q - D = UL$$

The maximum also occurs when Q is largest, so:

$$N = 0$$

and

$$Q = UL + D$$

Substituting these values into the above equations for the marginal rates give:

$$\frac{\partial T(N, Q)}{\partial N} = TB \cdot \left ( 1 + \frac{20% \cdot (UL + D)}{R} \right )$$

and

$$\frac{\partial T(N, Q)}{\partial Q} = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (2 \cdot (UL+D) + 0 - UL - D) \right ) = TB \cdot \left ( 1 + \frac{20%}{R} \cdot (UL + D) \right ) $$

Note that the two formulas are the same.

For single filers for 2020, UL = $213,300 and R = $50,000, and we will assume the standard deduction D = $12,400. $213,300 taxable income is barely into the 35% bracket, which begins at $207,350 taxable income, so TB = 35%.

$$\left (\frac{\partial T(N, Q)}{\partial N} \right )_{max} = \left (\frac{\partial T(N, Q)}{\partial Q} \right )_{max} = 35% \cdot \left ( 1 + \frac{20% \cdot ($213,300 + $12,400)}{$50,000} \right ) = 66.598%$$

The QBI required to achieve this rate is:

$$Q = $213,700 + $12,400 = $225,700$$

For married joint filers for 2020, UL = $426,600 and R = $100,000, and we will assume the standard deduction D = $24,800. $426,600 taxable income is barely into the 35% bracket, which begins at $414,700 taxable income, so TB = 35%.

$$\left (\frac{\partial T(N, Q)}{\partial N} \right )_{max} = \left (\frac{\partial T(N, Q)}{\partial Q} \right )_{max} = 35% \cdot \left ( 1 + \frac{20% \cdot ($426,600 + $28,400)}{$100,000} \right ) = 66.598%$$

The QBI required to achieve this rate is:

$$Q = $426,600 + $24,800 = $451,400$$

Note that the single and married joint maximum rates are the same. If itemized deductions are larger than the standard deduction, the maximum rate will be slightly higher.

--Fyre4ce 17:37, 17 February 2020 (UTC)