I have not forgotten about this question and promised to post a follow up. So, here it is. I did quite a bit of investigations including getting some answers directly from academics. Thanks to kaneohe for the link to Boglehead article which references a few papers at the end (Principles of Tax-Efficient Fund Placement - Bogleheads
). I ended up studying them to some degree. This also led me to following paper, which had a lot nicely explained details:
I think I have a much deeper appreciation now for what is going on and here it is...
Turned out it's not a simple topic unfortunately, so sorry for the long post, but here is a quick summary
- In most cases it looks like stocks that do not generate much of income are better to be held in taxable account and bonds in tax-deferred. (See below for a summary of how this is derived and therefore the assumptions made.)
- Dollars in tax-deferred accounts are different from dollars in taxable accounts. Correct value of 1 dollar in tax-deferred account is $(1-tn), where tn is the tax you will pay at the end for withdrawing that dollar. This should in part affect how you think about asset allocation.
- Stocks in tax-deferred accounts is a different investment then stocks in taxable accounts because real returns and risk are different for the two. Same applies to bonds.
- Questions I have not resolved (yet?): Accounting for stock dividends in the model will make it more likely that stocks should be in tax-deferred accounts, however to what degree this is true needs to be measured / estimated. On the opposite side, the model had to assume a capital gains tax after each year in taxable account (explained below), and extending this to N years would make stocks in taxable account more preferrable - again degree of this is unknown.
Finally, above conclusion is reached with using standard deviation as the model for risk. There are known problems with modeling risk as standard deviation, and it's unclear to me what the outcome would be if another model were applied (or perhaps there is no better model today, which is why researchers use standard deviation).
Some glory details
for those interested...
(1) I was correct in that dollars should be valued differently in tax-deferred and after-tax accounts. William Reichenstein papers referenced on the Boglehead's page address this point in detail and I posted my explanation of this fact earlier in this thread. Another paper (http://wpweb2.tepper.cmu.edu/spatt/location.pdf
) mentions this in passing and just works with after-tax dollars in tax-deferred accounts assuming you already made the conversion.
Note that while you do not know your future tax rate (tn), thinking of dollars in your tax-deferred 401(k)/TIRA accounts as dollars in your taxable account effectively assumes your tax rate will be 0% (which is true only if you plan to die before the withdrawals).
So, if your future tax rate for 401(k)/TIRA withdrawal is 20%, then to make your asset allocations the same between investments A and B in taxable and tax-deferred accounts, you have to have for every $1000 in tax-deferred account in investment A, $800 in taxable account for investment B.
Note that conversion from nominal pre-tax dollars to after-tax dollars also affects risk and return as explained in (2b) below.
(2) All my calculations are correct in the OP. So in fact, keeping stocks for N years in tax-deferred account would give you higher ending wealth then the opposite scenario. However, I missed 2 important pieces.
(2a) One was actually mentioned by LOL! but I did not give it enough weight at the time with all the other discussion. This relates to rebalancing. If I tried to maintain the same asset-allocation, I would also have to rebalance every year. Otherwise, with each passing year, I am increasing overall risk of the portfolio. I thought it might be non-trivial to put this into the model, but really it's simple - just consider a single year returns. Assuming you rebalance after a year, next year will follow the same "laws" and arrive as same "conclusions". So, just looking at 1 year is what's needed, but remember that minor differnces in 1 year will compund exponentially later one and therefore should not be dismissed even thought they are minor.
Now, an exception to above logic is that if in taxable account you pay taxes only after N years for stocks, how do you account for this when only looking at 1 year? One approach is to assume you pay capital gain tax rate after the 1 year for the earning of that year. However, a more correct variation would make it a lesser tax (since this tax is deferred over multiple years), but by how much?? Seems like this would depend on number of years, the returns on this rebalanced model, etc. For now, I assumed paying the tax after each year on the earning of that year, but some adjustment is needed.
(2b) A much more important part that was missing is my (lack of) accounting for risk. Interestingly, risk and return of stocks in taxable and tax-deferred accounts is different. This is because of taxes again. (See table 3 in the spwfe.fpanet.org paper I mentioned above and explanation for it.) While I had accounted for return differences, I did not account for risk. In short:
- In taxable account with tax t for your investment, your dollar earns R*(1-t) with the risk of SD*(1-t) for an investment with return R and standard deviation SD (measuring risk).
- In tax-deferred account, your after-tax dollar will earn full return R at the full risk SD. (Short explanation: say your nominal pre-tax $1 dollar produces $(1+X), which after taxes tn will give you $(1+X)*(1-tn). Now, recall that nominal $1 is the same as $(1-tn) real after-tax dollars. Thus, your original $(1-tn) after-tax dollars grew to $(1-tn)*(1+X).. In other words, each after-tax dollar grew by a factor of (1+X) giving it the full return of X. Same logic applies to standard deviation.)
What this means is that stocks (and non-zero-risk bonds) in tax-deferred accounts will produce higher returns but with higher risk. In other words, there are no longer just stock and bond investments when looking at taxable and tax-deferred accounts. Instead, there are 4 kinds of investments:
- Real after-tax dollar in taxable accounts can be invested in StocksPreTax and/or BondsPreTax
- Real after-tax dollar in tax-deferred accounts can be invested in StocksAfterTax and/or BondsAfterTax
Each will have its own returns and risk with the (1-t) factor differentiating PreTax and AfterTax investments.
(3) It no longer makes sense to me to say what an asset allocation is in terms of percentage in stocks vs bonds, since both stocks and bonds have to be qualified as indicated above. For example, (after converting to principal to after-tax dollars), $1000-StocksPreTax and $1000-BondsAfterTax allocation is different from $1000-StocksAfterTax and $1000-BondsPreTax. Claiming that both are 50/50 stock-bonds allocation is misleading. They have different risk and return characteristics.
(4) To decide whether it's best to hold StocksPreTax+BondsAfterTax or StocksAfterTax+BondsPreTax, the following technique can be used (it's an extension of a technique in the http://wpweb2.tepper.cmu.edu/spatt/location.pdf
paper where they consider bonds which are risk-free).
(4a) Find out a ratio of tax-deferred and taxable amounts such that standard deviation SD of both combinations is the same. In other words, find ratio between after-tax dollars in taxable and tax-deferred accounts to make risk the same.
For example, say "tc" is tax on capital gains and "to" is ordinary income tax. If bonds are completely risk-free with SDbonds = 0, then for every $1 in tax-deferred account, consider having $1/(1-tc) in taxable account. Then,
- portfolio 1: $1-StocksAfterTax + $1/(1-tc)-BondsPreTax. SDportfolio = SDstocks * [1 / (1+1/(1-tc))] = [(1-tc)/(2-tc)]*SDstocks
- portfolio 2: $1-BondsAfterTax + $1/(1-tc)-StocksPreTax. SDportfolio = [SDstocks * (1-tc)] * [ (1/(1-tc)) / (1+1/(1-tc))] = [(1-tc)/(2-tc)]*SDstocks
(4b) Find out whether portfolio 1 or portfolio 2 has higher expected return (ER). Whichever one has higher return indicates which combination is better since for every portfolio with opposite combination, one could switch stocks and bonds in the proportion found in 4(a) without changing SD and obtaining higher returns. In above example with 0-risk bond, expected returns are:
- portfolio 1: $1-StocksAfterTax + $1/(1-tc)-BondsPreTax. ERportfolio = ERstocks + ERbonds*(1-to)/(1-tc)
- portfolio 2: $1-BondsAfterTax + $1/(1-tc)-StocksPreTax. ERportfolio = ERbonds + ERstocks*(1-tc)/(1-tc) = ERbonds + ERstocks
Since ordinary tax to is expected to be greater than capital gains tax tc, portfolio 2 should be preferred to portfolio 1 for risk-free bonds.
(5) Now, do the same results apply to non-risk-free bonds? I put together a spreadsheet which computes (4a) and (4b) for this more general case. I found that in most cases the answer is the same, but in some combinations it's not. However such combinations are not under "normal" conditions. For example, with 20% cap gains tax, 25% ordinary tax, 8% stock returns with 15% SD, and 4% bond returns with 6% SD (and 0.1 correlation), stocks should be in taxable account. However, moving SD of bonds from 6% to 11% makes them better candidates for taxable account with rest of parameters being the same.