**Roth conversion outcomes as an n-period game**

Many discussions of Roth conversions get hung up on what I will call “2 period” analyses. Any time you read that “Roth conversions should work out as long as future marginal tax rates are higher than current rates,” a 2-period game will most likely underlie the analysis.

By extension, in that same 2-period game, if future tax rates are unchanged, then a Roth conversion can only breakeven. This breakeven result—zero gain, zero loss from the conversion—follows directly from the commutative property of multiplication. It holds regardless of how many years of returns separate period 1 from period 2 (invariance of outcomes over time), and regardless of fluctuations in the annual returns over those years (invariance across volatility).

But the 2-period analysis turns out to be a special case within the larger class of n-period games, and one that is not very helpful for the retiree who will evaluate conversion outcomes after RMDs begin. If the goal of the conversion was to accumulate greater after-tax wealth at some unspecified future date, by reducing tax exposure consequent to RMDs, then an n-period game must be set up and analyzed.

I will assume throughout that indeed, that was the goal of the conversion: to accumulate greater after-tax wealth at some unknown but distant future date, by reducing tax exposure that would otherwise be incurred by having to take RMDs not needed for living expenses. The claim made in the title of the thread presumes this is the goal of the conversion; if your goal in converting is to achieve something else, this thread may not be relevant to you.

In an n-period game, outcomes are not invariant over time (it matters in which period after-tax wealth is evaluated). As a result, it becomes possible for Roth conversions to accumulate greater after-tax wealth

*even when future tax rates are unchanged--or move lower.*But it will take time; and the more unfavorable the future movement in tax rates, the longer the time it will take for the conversion to produce a pay off.

Put another way, RMDs change the game, and cannot be analyzed within a 2-period framework, since RMDs will continue for n periods, where n = [age at death + 10 – 71]. That means that in most cases n will be a two-digit number >20 and <50. Each time an RMD is taken that would not had to have been taken and taxed, had the conversion occurred, the prospective payoff from the Roth conversion will change.

Once the structure of the “n-period RMD game” is understood, it will become apparent why Roth conversions always pay off, if the time frame is long enough, but almost never produce a substantial pay off prior to age 90, and rarely produce a big pay off at any age < 100 when evaluated in real terms.

**Caveats and Constraints**

1st Caveat: as the tax rate paid on the conversion declines toward zero, the real payoff will expand more and more rapidly, and will begin to explode upward as the tax rate on the conversion drops below 1.0%. Translation: backdoor Roths are great! But this thread focuses on

**Roth conversions**where the tax rate is going to be rather more than one percent. The assertion of only modest outcomes, therefore, mostly applies when the conversion is made at tax rates above 12%, and more especially at marginal rates of 22% and above.

2nd Caveat: The assertion of modest outcomes also mostly applies when future marginal rates are similar to conversion rates. If instead you can convert today at 12%, and your future rate turns out to be 32%--or 39.6%, then a Roth conversion can be quite lucrative indeed. It is when you convert at 22% to save post-TCJA tax at 25%, or any similar increment, that you should expect only modest outcomes.

3rd caveat: these maxims concern only the intrinsic outcome of a Roth conversion that holds both asset allocation and asset location constant. You may be able to turbocharge conversion outcomes by means of clever asset location, i.e., by putting high growth assets in the Roth and leaving low growth assets in the TDA. And if opening the Roth causes you to increase your allocation to high return assets, that too can turbocharge the outcome actually received. Best wishes!

But please don’t attribute your good fortune to the necessary and intrinsic operation of paying a lump sum in tax now to reduce a future stream of tax payments. This thread will be concerned only with the pure effect of reducing future tax exposure by means of a one-time tax debit paid today.

**Analogy**

The most familiar analogue to the n-period RMD game is the homeowner fixed mortgage prepayment game. Here a one-time paydown of the principal can be made at the homeowner’s option at any point. If the amount paid is to any degree substantial, and made in year 2 or at a similarly early point, that paydown will shorten the life of the mortgage, and save a multiple of itself in interest not paid. But it takes a long time for the payoff to be received (typically > year 25), and the dollar amount of the payoff will be far lower if, say, the homeowner refinances in year 10.

Most Roth conversions undertaken to reduce RMDs provide a similar level of payoff as mortgage prepays if evaluated over a mere 15 to 20 years; and with a similar degree of certainty. Conversely, if your planning horizon extends out past age 100, and stocks are part of your asset allocation, a Roth conversion can be rather more lucrative than a mortgage prepayment.

All these points were made in my SSRN paper; but for not a few Bogleheads, the spreadsheet constructed for that paper failed to close the sale. In fact, I started this thread because BH criticism has helped me to devise a better spreadsheet (looking at you cas—your insightful criticism of the old spreadsheet proved extremely helpful). The narrative developed for this thread will in turn provide a foundation for revising the SSRN paper.

Next post introduces the spreadsheet.