Which account to save in
Tax now or tax later: multiplication does not care about order, so the decision is a bet on one number, the client's rate today against their rate in retirement.
- Compute after-tax values for a traditional and a Roth contribution
- Show the winner depends only on the two tax rates
- Back out the tax rate paid on a Roth contribution
- Say what changes when the contribution is capped
The intuition
A traditional account takes the tax break now and taxes the withdrawal; a Roth pays the tax now and lets everything out tax-free. Grow $10,000 of pre-tax salary for thirty years at 6% and both accounts multiply the money by the same 5.74. The only difference is when the tax factor is applied, and multiplication does not care about order: if the rate is the same at both ends, the two accounts end exactly equal.
So the decision is a bet on one number. A client at 24% now who expects 12% in retirement keeps 88% of a traditional withdrawal against 76% of a Roth contribution: the traditional wins by the ratio of the two. The return and the horizon do not enter. The one wrinkle is contribution limits: a Roth dollar is an after-tax dollar, so a capped Roth contribution shelters more real money, and the traditional saver can only match it by investing the tax saved outside, where the growth is taxed every year.
Traditional after tax = C × (1 + r)ⁿ × (1 − t_retire). Roth after tax = C × (1 − t_now) × (1 + r)ⁿ. Roth ÷ traditional = (1 − t_now) ÷ (1 − t_retire): the growth cancels. Roth wins when the rate now is lower than the rate in retirement.
Why it works
- The conventions here: one lump contribution of pre-tax salary C, invested at a constant return for n years and withdrawn all at once. Traditional: the full C goes in and the withdrawal is taxed at the retirement rate. Roth: C is taxed now and the remainder goes in tax-free. Flat rates, no penalties or required distributions.
- The growth is sheltered either way. Both accounts multiply by (1 + r)ⁿ; only where the tax factor sits differs, and factors commute. High expected returns are not a reason for either.
- The ratio is the whole answer. Roth over traditional is (1 − t_now) ÷ (1 − t_retire). Below one, the traditional wins; above one, the Roth.
- The classic Roth case is a young client in a low bracket who expects income, and the rate, to rise. The classic traditional case is a peak-earnings client who expects a lower rate in retirement.
- Capped contributions tilt to the Roth. The same dollar amount in a Roth is after tax, so it shelters more. The traditional saver keeps the tax saved today, but investing it in a taxable account taxes the return every year, so it grows at r × (1 − t_inv).
- Run backwards and a Roth balance reveals the rate paid on the way in: divide by the growth factor to see what went in, and compare with the pre-tax amount.
| Growth factor: 1.06³⁰ | 5.7435 |
| Traditional: $10,000 × 5.7435 × (1 − 12%) | $50,543 |
| Roth: $10,000 × (1 − 24%) × 5.7435 | $43,651 |
| Roth ÷ traditional: 0.76 ÷ 0.88 − 1 | −13.64%: the traditional wins |
| Same $10,000 in each: Roth $57,435 against traditional $50,543 + side account $10,673 | Roth $3,781 behind |
Even with the full $10,000 in each account, this client's traditional route wins, because their rate falls by half in retirement. Reverse the rates and the Roth wins by the same ratio.
The formulas
Full contribution in; taxed on the way out.
Taxed on the way in; nothing on the way out.
The growth cancels; only the rates matter.
Run backwards.
The account plus the tax saving invested outside, taxed yearly.
Worked example
Grow the money once, then apply each tax at its own end. The follow-up asks whether the return or the horizon decided the winner, and the answer is neither.
See it move
Same client. Change the contribution, the tax rate now, the expected rate in retirement, the return, the years and the rate the taxable side account pays.
- Raise the return or add years. Both bars rise by the same factor and the percentage gap between them does not move.
- Raise the rate in retirement. The traditional bar falls, the Roth bar does not move, and the curve's crossing shows the rate at which they tie: today's rate.
- Raise the rate now. The Roth bar falls; the traditional bar does not move.
- Raise the tax on the side account. The side account shrinks and the same-dollar comparison tilts toward the Roth.
Run it backwards
A Roth balance after n years is known, and so is the pre-tax salary that funded it. What rate was paid on the way in?
Divide the balance by the growth factor to see what went in after tax; one minus that over the pre-tax amount is the rate.
It is the same identity the whole chapter rests on: the growth factor can be divided out because it applied to both routes alike.
Traps
Say it in the interview
“Roth or traditional for this client?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Traditional = C (1 + r)ⁿ (1 − t_retire); Roth = C (1 − t_now) (1 + r)ⁿ.
- Roth ÷ traditional = (1 − t_now) ÷ (1 − t_retire); the growth cancels.
- Roth wins when the rate now is lower than the rate in retirement.
- A capped Roth shelters more; the traditional's side account is taxed yearly.