Roth vs. Traditional 401(k): a Mathematical Approach
0. The basic question
Should you contribute to a Roth 401(k) or a traditional 401(k)?
The question sounds simple, but the real‑world answer depends on tax rates now vs. tax rates in retirement. This post walks through the math in several scenarios then draws practical conclusions.
1. Apple‑to‑apple comparison based on max traditional 401(k) contribution, non-maximized case
First define some variables:
- \(C_T\): your traditional 401(k) contribution
- \(C_R\): your Roth 401(k) contribution
- \(\text{MTR}_n\): your marginal tax rate now
- \(\text{MTR}_w\): your marginal tax rate when you withdraw traditional 401(k) money
- \(r\): the annual rate of return
- \(t\): number of years invested
- \(G = (1+r)^t\): growth factor on tax‑advantaged accounts
- \(G_{Tx}\): growth factor on a taxable account (after tax drag)
Here we start with a max traditional contribution \(C_T\) and ask: what Roth contribution gives the same reduction in take‑home pay today?
Traditional 401(k) is pre‑tax. Contributing \(C_T\):
- Tax savings:
- So your paycheck falls by:
Roth 401(k) is after‑tax. If we want the same reduction in take‑home pay as above, the Roth contribution should be:
\[C_R = C_T(1-\text{MTR}_n)\]After \(t\) years, both accounts grow with factor \(G\):
- Traditional 401(k) balance: \(C_T G\)
- After tax at withdrawal:
- Roth 401(k) balance:
Withdrawals are tax‑free, so the Roth after‑tax value is just \(C_T(1-\text{MTR}_n)G\).
Traditional is better if
\[C_T G (1-\text{MTR}_w) > C_T(1-\text{MTR}_n)G\]which simplifies to
\[\text{MTR}_w < \text{MTR}_n\]Intuition: because multiplication commutes, taxing the seed (Roth) or the harvest (traditional) gives the same result if tax rates are equal. Traditional wins if your retirement tax rate is lower than today’s; Roth wins if your retirement tax rate is higher.
2. Apple‑to‑apple comparison based on a Roth 401(k) contribution, non-maximized case
The previous section held the traditional contribution fixed. Now we fix the Roth contribution instead:
- Let \(C_R\) be the amount you put into a Roth 401(k).
To be fair to the traditional side, we must Contribute enough to a traditional 401(k) so that your after‑tax cost today matches a Roth contribution of \(C_R\).
If you contribute \(C_T\) to a traditional 401(k):
- After‑tax cost:
We want this to equal the Roth contribution \(C_R\), so
\[C_T(1-\text{MTR}_n) = C_R \quad\Rightarrow\quad C_T = \dfrac{C_R}{1-\text{MTR}_n}\]After growth and tax at withdrawal, traditional gives
\[\text{Traditional after tax} = \dfrac{C_R}{1-\text{MTR}_n} G (1-\text{MTR}_w)\]Roth 401(k) after tax (with contribution \(C_R\)) is
\[\text{Roth after tax} = C_R G\]Traditional wins when
\[\dfrac{C_R}{1-\text{MTR}_n} G (1-\text{MTR}_w) > C_R G \quad\Rightarrow\quad \text{MTR}_w < \text{MTR}_n\]So even when you include the extra IRA shelter from the 401(k) tax savings, the basic condition is the same: traditional wins if your future marginal rate is lower than today’s.
3. Apple‑to‑apple comparison based on max Roth 401(k) contribution, but not maximized in Roth IRA
Now assume you contribute a max \(C_R\) (after‑tax cost) to Roth 401(k). If we contribute the same max amount to tradtional 401(k), it will generate tax savings of \(C_R \text{MTR}_n\). Clearly we have to invest this tax savings so that we can have the same take home pay forming an apple-to-apple comparison.
We assume further the tax savings \(C_R \text{MTR}_n\) can be contributed to the Roth IRA either by contributing directly if you are eligible or through backdoor Roth IRA conversion. Since this is after tax contribution, the take home pay at withdrawal for the tax savings investment is simply:
\[C_R\cdot\text{MTR}_n \cdot G\]and the after tax traditional 401(k) contribution proceeds are:
\[C_R\cdot G\cdot(1-\text{MTR}_w)\]The sum of these two amounts is \(C_R\cdot G\cdot(1+\text{MTR}_n-\text{MTR}_w)\).
On the other hand, the take home income from the Roth 401(k) investment is:
\[C_R\,G\]Traditional 401(k) plus Roth IRA win when \(\text{MTR}_w \lt \text{MTR}_n\).
4. Apple‑to‑apple comparison with taxable investing
Finally, consider the two cases:
- You contribute max \(C_R\) to a Roth 401(k), and
- You contribute the same max to a traditional 401(k) and invest the tax savings \(C_R \text{MTR}_n\) in a taxable account (because IRA space is full or you choose not to use it).
We already know from previous two sections, that if you invest in the tax savings in either traditional 401(k), deductible IRA, or Roth IRA, the break point is the future tax rate is the same as the current tax rate. Without showing any math, we know that the it requires a lower future tax rate for the traditional 401(k) to win if we invest in a taxable account. It is just a matter to calculate how much.
As before, we find the break even point by evaluating the take home pay at withdrawal time for both cases. Below we derive the same formula in Retirement plan analysis (math). we adopted the same symbols for easy communication with that community.
- Roth 401(k) after tax:
- Traditional 401(k) after tax:
Let after‑tax growth factor is \(G_{Tx}\), so the take home value on the taxable account at withdrawal is
\[C_R \text{MTR}_n G_{Tx}\]The break‑even condition (Roth = traditional + taxable) is
\[C_R G = C_R G (1-\text{MTR}_w) + C_R \text{MTR}_n G_{Tx}\]Cancel \(C_R\):
\[G = G(1-\text{MTR}_w) + \text{MTR}_n G_{Tx} \tag{1}\]Now we model the taxable account.
Let:
- \(r\): total return
- \(y\): dividend yield
- \(\text{MTR}_{div}\): tax rate on dividends
- \(\text{MTR}_{w,cg}\): capital‑gain tax rate at withdrawal
Each year, the after‑tax growth on the taxable account is:
\[1 + r - y\,\text{MTR}_{div}\]So the account value at \(x\) years (per dollar invested) is:
\[v(x) = (1 + r - y\,\text{MTR}_{div})^x\]and the balance factor at the year \(t\) is:
\[v=(1+r-y\cdot\text{MTR}_{div})^t\]Dividends are reinvested after tax, increasing the cost basis of
\[b(x) = (1 + r - y\,\text{MTR}_{div})^x\,y\,(1-\text{MTR}_{div})\]Summing that geometric series plus the original basis gives a total basis factor:
\[\begin{align*} b&=\sum_{x=0}^{t-1} (1+r-y\cdot\text{MTR}_{div})^x\cdot y\cdot(1-\text{MTR}_{div}) + 1 \\ &= \dfrac{(1+r-y\cdot\text{MTR}_{div})^t-1}{r-y\cdot\text{MTR}_{div}}\cdot y\cdot(1-\text{MTR}_{div})+1 \\ &= \dfrac{v-1}{r-y\cdot\text{MTR}_{div}}\cdot y\cdot(1-\text{MTR}_{div})+1 \end{align*}\]The portion taxed as capital gains at withdrawal is:
\[\text{Taxable gain} = v - b = (v-1)\dfrac{r-y}{r - y\,\text{MTR}_{div}}\]After capital‑gains tax, the net growth factor is:
\[G_{Tx} = v - (v-1)\dfrac{r-y}{r - y\,\text{MTR}_{div}}\text{MTR}_{w,cg} \tag{2}\]Substitute (2) into (1) and you get a relationship between today’s marginal rate and the withdrawal‑year rates:
\[G = G\cdot(1-\text{MTR}_w) + \text{MTR}_n\cdot (v - (v-1)\cdot\dfrac{r-y}{r-y\cdot\text{MTR}_{div}}\cdot\text{MTR}_{w,cg}) \tag{3}\]We need to solve the two dependent variables \(\text{MTR}_w\) and \(\text{MTR}_{w,cg}\). Since capital gain rate \(\text{MTR}_{w,cg}\) can only take a few discrete values, we can assume one value and calculate the ordinary tax rate \(\text{MTR}_w\), and re-evaluate if the original assumption makes sense, as was currently done in Traditional versus Roth.
To avoid tracking \(\text{MTR}_w\) and \(\text{MTR}_{w,cg}\) separately, define a marginal effective tax rate at withdrawal, \(\text{MTR}_{we}\), as:
extra tax paid ÷ extra income from both the traditional 401(k) and the taxable account.
Replacing both \(\text{MTR}_w\) and \(\text{MTR}_{w,cg}\) in (3) with \(\text{MTR}_{we}\), we get:
\[G = G(1-\text{MTR}_{we}) + \text{MTR}_n\left[ v - (v-1)\dfrac{r-y}{r - y\,\text{MTR}_{div}}\text{MTR}_{we}\right] \tag{4}\]Solving (4) for \(\text{MTR}_{we}\) yields:
\[\text{MTR}_{we} = \dfrac{\text{MTR}_n} {\dfrac{G}{v} + \dfrac{v-1}{v}\cdot \dfrac{r-y}{r - y\,\text{MTR}_{div}}\cdot \text{MTR}_n} \tag{5}\]This \(\text{MTR}_{we}\) is the break‑even effective tax rate at withdrawal: if your actual effective rate in retirement is lower than this, the traditional + taxable strategy wins; if higher, the Roth 401(k) wins.
5. Example: computing the effective rate
Using the data in Traditional versus Roth, we assume:
- \(\text{MTR}_n = 37.0\%\) (current marginal rate on ordinary income)
- Capital‑gain rate: \(23.8\%\)
- Tax on qualified dividends: \(23.8\%\)
- Tax on nonqualified dividends: \(40.8\%\)
- Total return \(r = 8\%\)
- Dividend yield \(y = 2\%\)
- 90% of dividends are qualified, 10% nonqualified
- Investment horizon \(t = 20\) years
Blended dividend tax rate:
\[\text{MTR}_{div} = 0.9\times 23.8\% + 0.1\times 40.8\% = 25.5\%\]Growth in tax‑advantaged account:
\[G = (1+r)^t = 1.08^{20} \approx 4.6610\]Growth in taxable account (before capital‑gains tax at the end):
\[v = (1 + r - y\,\text{MTR}_{div})^t = (1 + 0.08 - 0.02\times 0.255)^{20} \approx 4.2400\]Plugging \(G\), \(v\), \(r\), \(y\), \(\text{MTR}_n\), and \(\text{MTR}_{div}\) into Equation (5) gives:
\[\text{MTR}_{we} \approx 27.9\%\]So in this example, if your effective tax rate in retirement, considering both ordinary income and capital gains, is 27.9% or lower, the traditional 401(k) plus taxable investing is mathematically favored. If your effective rate is higher than 27.9%, Roth 401(k) is preferred.
Using the same set of parameters except where noted, we calculated 5 different tax rate scenarios and 4 time ranges under each as in the Traditional versus Roth. For each horizon:
- The first number is the break‑even ordinary marginal rate (guess‑and‑verify), that we verified and copied.
- The second number is the break‑even effective rate from Equation (5) we calculated.
| Ordinary income | Qualified dividends | Non‑qualified dividends | Capital gains | 10 years (ordinary / effective) | 20 years (ordinary / effective) | 30 years (ordinary / effective) | 40 years (ordinary / effective) |
|---|---|---|---|---|---|---|---|
| 12.0% | 0.0% | 12.0% | 0.0% | 11.97% / 11.42% | 11.95% / 11.16% | 11.92% / 11.03% | 11.89% / 10.96% |
| 24.0% | 15.0% | 24.0% | 15.0% | 21.87% / 21.28% | 20.58% / 19.91% | 19.67% / 19.05% | 18.96% / 18.42% |
| 32.0% | 18.8% | 35.8% | 18.8% | 28.43% / 27.35% | 26.27% / 25.13% | 24.78% / 23.78% | 23.62% / 22.80% |
| 37.0% | 23.8% | 40.8% | 18.8% | 32.56% / 30.81% | 29.78% / 27.91% | 27.82% / 26.15% | 26.26% / 24.86% |
| 50.3% | 37.1% | 54.1% | 28.1% | 41.32% / 39.16% | 35.90% / 34.23% | 32.23% / 31.29% | 30.56% / 29.13% |
6. Take home conclusion
Talk is cheap; show me the math. However, don’t get lost in the math. Here are practical conclusions you can take home.
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You have to make an apple‑to‑apple comparison. Investing the same dollar amount in a traditional 401(k) is not a fair comparison to investing the same amount in a Roth 401(k), because traditional 401(k) contributions are pre‑tax and Roth 401(k) contributions are after‑tax. It is only a fair comparison if you also invest the tax savings from the traditional contribution, so that both choices leave you with the same after‑tax income today.
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When the additional tax savings are invested in a traditional 401(k), Roth 401(k), or deductible IRA account, the break‑even point occurs when your future marginal tax rate is the same as your current marginal rate.
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However, when you have maximized 401(k) and IRA options and must invest the tax savings in a taxable account, the break‑even point occurs when your future tax rate is lower than your current tax rate. This is mathematically proven by Equation (5). How much lower depends on the parameters; you can either refer to the table in Section 5 or use the simplified formula below.
For the best‑case scenario with no tax drag in the taxable account (i.e., \(y = 0\)), the taxable account growth factor before tax \(v\) equals the 401(k) growth factor \(G\). In that case, Equation (5) simplifies to:
When \(v\) is large enough, \(\dfrac{v-1}{v}\) is close to 1, so the equation can be furether approximated by:
\[\text{MTR}_{we} = \dfrac{\text{MTR}_n} {1 + \text{MTR}_n} \tag{7}\]