Analysis of Break-Even Tax Rate for Roth Conversions

Introduction

A common rule of thumb is that a Roth conversion is attractive when the expected tax rate in retirement is higher than the tax rate paid on the Roth conversion. This rule is directionally correct. But it is also a weak condition. As will be seen below, the expected tax rate in retirement can afford be lower if the tax on the conversion is paid from a taxable account. (This effectively redirects assets from an inefficient taxable account into a tax efficient Roth account.) 

This post derives, under a simplified set of assumptions, the required tax rate in retirement that makes the Roth conversion beneficial for an after-tax retirement spending scenario. (Vanguard has termed this the Break-Even Tax Rate or BETR.) Below this rate, the modeled Roth conversion produces less after-tax spendable wealth compared to leaving the money in the Traditional IRA.

The comparison is between the following two strategies:

• Default strategy: No Roth conversion. No conversion tax to pay. Hence, no withdrawals from any taxable account to pay conversion tax.
• Roth conversion strategy: An amount $P_0$ is converted from the Traditional IRA (T-IRA) to Roth IRA (R-IRA). Separately, an amount $W_0$ is consumed from a taxable account to pay conversion tax.

Both comparisons are designed to start with the same total wealth ($P_0$ in the T-IRA, $W_0$ in the taxable account). The growth of $W_0$ within the taxable account in the default strategy represents the opportunity cost of paying the conversion tax.

Terminology

Term	Description
$P_0$	Amount converted from the T-IRA to the R-IRA at the start of the investing duration
$W_0$	Amount withdrawn from a taxable account (after selling assets) to cover taxes on Roth conversion
$B_0$	Cost basis of $W_0$
$B_N$	Normalized cost basis $(B_N = B_0/P_0 > 0)$ Caution: This is relative to $P_0$ (not $W_0$)
$r_c$	Capital appreciation rate per year
$r_d$	Dividend yield per year
$m$	Investing duration, in years, for this analysis $(m \ge 0)$
$T_{i0}$	Effective marginal (income) tax rate on $P_0$ when the Roth conversion is effected $(T_{i0} \ge 0)$
$T_{im}$	Effective marginal (income) tax rate on the incremental withdrawal (corresponding to $P_0$) from the T-IRA at the end of $m$ years $(T_{im} \ge 0)$
$T_c$	Capital gains tax rate $(T_c \ge 0)$
$T_d$	Dividend tax rate $(T_d \ge 0)$

Assumptions

• Conversion tax is paid entirely from a taxable account
• No tax interaction with other incomes (like Social Security, or Pensions) is considered
• Investment characteristics are identical across T-IRA, R-IRA, and taxable accounts
• Dividends, net taxes, are reinvested
• $r_c$ and $r_d$ are annual returns measured relative to the beginning-of-year portfolio value
• $T_c$ and $T_d$ are constant over the period of analysis ($m$ years)
• For passive long-term investing, it's reasonable to assume that both $r_c, r_d > 0$ and $0 < B_N \le 1$

Preliminaries

Define$$\begin{align*} G_\text{ira} &= 1 + r_c + r_d\\ G_\text{tax} &= 1 + r_c + r_d(1-T_d) \end{align*}$$where $G_\text{ira}$ and $G_\text{tax}$ are the yearly gain multipliers in the (Traditional/Roth) IRA and taxable accounts respectively. In most cases, $G_\text{tax} < G_\text{ira}$ due to a positive dividend tax drag ($T_d > 0$).

If $W_0$ is withdrawn from a taxable account to cover both (a) taxes due on the Roth conversion of $P_0$, and (b) taxes on capital gains embedded in $W_0$ $(=W_0 - B_0)$, then the amount left over after paying capital gains tax should cover the taxes due on the Roth conversion. To wit,$$ P_0 T_{i0} = W_0 - (W_0 - B_0) T_c = W_0 (1 - T_c) + B_0 T_c $$from which,

$$ \frac{W_0}{P_0} = \frac{T_{i0} - B_N T_c}{1 - T_c} $$

Default strategy

This is defined by:

• No Roth Conversions (i.e., keep $P_0$ in the T-IRA)
• No conversion tax liability
• No withdrawals from the taxable account (i.e., keep $W_0$ in taxable)

At the end of $m$ years, the initial amount $P_0$ becomes$$P_\text{t-ira, pre-tax} = P_0 G_\text{ira}^m$$ and $$P_\text{t-ira, post-tax} = P_0 G_\text{ira}^m (1 - T_{im})$$

Since $W_0$ is not withdrawn from the taxable account it grows with dividend tax-drag. At the end of $m$ years$$ W_\text{taxable, pre-tax} = W_0 G_\text{tax}^m $$

The after-tax portion of dividends is reinvested and becomes additional cost basis in the taxable account. In year $k$ this is$$\Delta B_k = W_0 G_\text{tax}^{k-1}r_d(1-T_d)$$Therefore after $m$ years the basis is$$\begin{align*} B_m &= B_0 + \sum_{k=1}^m W_0 G_\text{tax}^{k-1} r_d (1 - T_d)\\ &= B_0 + W_0 r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1} \end{align*}$$

Post-tax spendable amount in the taxable account is then given by$$ \begin{align*} P_\text{taxable, post-tax} &= W_\text{taxable, pre-tax}(1-T_c) + B_mT_c\\ &= W_0 G_\text{tax}^m(1-T_c)\\ &\quad +\  B_0T_c + W_0 r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1}T_c \end{align*} $$

Normalized to $P_0$, the total after-tax spendable amount in the default strategy is $$ \begin{align*} S_\text{default, post-tax}&= (P_\text{t-ira, post-tax} + P_\text{taxable, post-tax})/P_0\\ \\ &= G_\text{ira}^m (1 - T_{im})\\ &\quad +\ \frac{W_0}{P_0} G_\text{tax}^m(1-T_c)\\ &\quad +\  B_NT_c\\ &\quad +\ \frac{W_0}{P_0} r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1}T_c \end{align*} $$ Substituting for $W_0/P_0$

$$ \begin{align*} S_\text{default, post-tax}&= G_\text{ira}^m (1 - T_{im})\\ &\quad +\ (T_{i0} - B_N T_c) G_\text{tax}^m\\ &\quad +\  B_NT_c\\ &\quad +\ (\frac{T_{i0} - B_N T_c}{1 - T_c}) r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1}T_c \end{align*} $$

Roth conversion strategy

This is defined by:

• Reallocate $P_0$ from the T-IRA to the R-IRA
• Consume $W_0$ from the taxable account to fund the conversion tax

At the end of $m$ years, $P_0$ grows to$$P_\text{r-ira, pre-tax} = P_0 G_\text{ira}^m = P_\text{r-ira, post-tax}$$

Since $W_0$ is withdrawn from the taxable account to pay conversion taxes, there is no growth to be accounted for in the taxable account. (To wit, the opportunity cost of paying the taxes has already been calculated as a 'benefit' under the default strategy and will get incorporated when the relative advantage of the Roth conversion strategy is analyzed.)

Normalized to $P_0$, the after-tax spendable amount in the Roth conversion strategy is $P_\text{r-ira, post-tax}/P_0$, or

$$ S_\text{conversion, post-tax} = G_\text{ira}^m $$

Analysis

The normalized afer-tax benefit of the Roth conversion strategy is$$\begin{align*} \Delta &= S_\text{conversion, post-tax} - S_\text{default, post-tax}\\ &= G_\text{ira}^mT_{im}\\ &\quad -\ (T_{i0} - B_N T_c) G_\text{tax}^m\\ &\quad -\ B_NT_c\\ &\quad -\ (\frac{T_{i0} - B_N T_c}{1 - T_c}) r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1}T_c \end{align*}$$All other things being equal, the higher the $T_{im}$, the larger the $\Delta$. Thus, the Roth conversion strategy certainly benefits from a higher tax rate in retirement.

$\Delta \ge 0$ is the condition for the Roth conversion strategy to not be at a disadvantage compared to the default strategy. When equality applies, neither strategy is better than the other. Imposing $\Delta \ge 0$  gives,

$$ \begin{align*} T_{im} \ge \frac{1}{G_\text{ira}^m} \large[ &(T_{i0} - B_N T_c)G_\text{tax}^m\\ &+ B_NT_c\\ &+(\frac{T_{i0} - B_N T_c}{1 - T_c}) r_d (1 - T_d)\frac{G_\text{tax}^m - 1}{G_\text{tax} - 1}T_c \large] \end{align*} $$

The inequality above gives the floor below which the Roth conversion strategy fails to be beneficial. The value of $T_{im}$ when equality applies is the break-even tax rate or $T_\text{BETR}$  (terminology borrowed from Vanguard's BETR article) for which both strategies yield identical after-tax spendable amounts .

If the expected tax rate in retirement is lower than $T_\text{BETR}$, then the Roth conversion strategy is at a disadvantage and should be avoided.

Whenever $T_{im} > T_\text{BETR}$, the Roth conversion strategy has a net advantage over the default strategy.

Some special cases are instructive. If $T_c = 0$ (no capital gains tax) then$$T_\text{BETR} = T_{i0}\frac{G_\text{tax}^m}{G_\text{ira}^m} < T_{i0}$$as $G_\text{tax} < G_\text{ira}$ in most cases due to dividend tax drag ($T_d > 0$).

Thus, it is not always necessary for $T_{im}$ to be higher than $T_{i0}$ for the Roth conversion to be beneficial.

If $r_d \rightarrow 0$ (low to zero dividend yield), then both $G_\text{tax}, G_\text{ira}$ $\rightarrow  1 + r_c$ and$$T_\text{BETR} = T_{i0} - B_N T_c\left(1 - \frac{1}{(1 + r_c)^m}\right)$$As $m$, $B_N$, and $T_c$ are all $\ge 0$, it follows that $T_\text{BETR} \le T_{i0}$. Again, $T_{im}$ does not need to be higher than $T_{i0}$.

As an illustration, for $B_N = 0.7$, $r_c = 7.177\%/\text{year}$, $r_d = 2.75\%/\text{year}$, $m = 10$ years, $T_{i0} = 24\%$, $T_c = 15\%$, and $T_d = 19\%$ (mix of ordinary and qualified dividends), $T_\text{BETR} = 17.27\%$ which is lower than the $24\%$ tax rate at which the Roth conversion was effected.

Sensitivity of $T_\text{BETR}$ to various parameters can be seen in the following tables where only one parameter, as indicated, is changed and the rest are held constant at the values assumed above.

$m$ (yr)	$T_\text{BETR}$ (%)
$0$	$24.00$
$5$	$19.92$
$10$	$17.27$
$20$	$14.28$
$30$	$12.77$

The longer the investing duration (following the Roth conversion), the lower the break-even tax rate.

$B_N$	$T_\text{BETR}$ (%)
$0$	$23.45$
$0.25$	$21.24$
$0.50$	$19.03$
$0.75$	$16.82$
$1$	$14.62$

The higher the cost basis of the amount withdrawn from the taxable account, the lower the break-even tax rate. A higher $B_N$, often effected by a careful choice of tax lots when selling assets, implies lower capital gains tax and hence a lower $W_0$. This lowers the opportunity cost and, hence, lowers $T_\text{BETR}$.

$r_d$ (%/yr)	$T_\text{BETR}$ (%)
$0$	$18.75$
$2.5$	$17.39$
$5.0$	$16.24$
$7.5$	$15.27$
$10.0$	$14.43$

The higher the dividend yield, the lower the break-even tax rate. This reflects the effective redirection of the taxable assets (used to pay the conversion tax) from the taxable account to the Roth account. Otherwise, the assets used to fund the conversion tax would have remained in the taxable accounts and be subject to a dividend tax drag.

$T_c$ (%)	$T_\text{BETR}$ (%)
$0$	$22.88$
$15$	$17.27$
$20$	$15.30$

The higher the capital gains tax, the lower the break-even tax rate. This again reflects the redirection of taxable assets from the taxable account to the Roth account where they grow tax free. Otherwise, the assets used to fund the conversion tax would continue growing in the taxable account and be subject to the capital gains tax eventually when they are liquidated for after-tax spending.

Conclusion

The break-even tax rate derived here depends on more than the tax rate paid on the conversion. It also depends on how long the converted assets remain invested, the capital-gains tax rate, the dividend yield and dividend tax rate of the investments, and the tax basis of the taxable assets used to pay the conversion tax.

The result is that the tax rate in retirement required to justify a Roth conversion can be substantially lower than the tax rate paid on the conversion today. The mechanisms that drive $T_\text{BETR}$ below the conversion tax rate are:

1. Avoidance of future income tax on the converted assets
2. Avoidance of dividend tax drag inside the taxable account
• Provided the conversion tax is funded by taxable assets that would otherwise have remained in the taxable account
3. The capital-gains tax on the assets used to pay the conversion tax especially when those assets have a substantial basis

This does not mean that Roth conversions are automatically beneficial whenever the expected retirement tax rate is below the conversion rate. The calculation remains sensitive to the assumptions used in the modeling. Nevertheless, this post shows why Roth conversion decisions should be evaluated in the context of the entire portfolio and life circumstances rather than by comparing the current and future income tax brackets in isolation.

Future directions

Aspects that are not covered here:

• Estimation of $T_\text{BETR}$ for a legacy/inheritance goal
• Stochastic returns and uncertainty in tax-rates
• Progressive tax brackets
• Varying asset locations (i.e., T-IRA, R-IRA, and taxable accounts with investments that have different return and tax characteristics)
• In effect, $G_\text{t-ira} \ne G_\text{r-ira} \ne G_\text{tax}$
• Interaction with Social Security, Pensions, Medicare IRMAA, etc.
• Total lifetime after-tax spendable wealth instead of just a single conversion
• Determination of optimal conversion amounts