Saving 50% of gross or net?

[TLV]

I spent about an hour trying to come up with another way to explain to Bank(rupt$y) why NPV doesn't apply to the 401k tax liability, before realizing that the math came out awfully close for it to be unrelated. Then I spent another hour and figured out out that it actually does work out if you include the compounded tax-free gains.

NPV of tax liability = (P*t-(i/(1-t))*(SUM P*(1-t)*(1+i/(1-t))^q - P*(1-t)*(1+i)^q, q = 0..(n-1))) / (1+i)^n

wolfram alpha reduces it to:


where
P = gross at start
t = tax rate (assumed to be same every taxes apply)
i = interest rate (or rate of return) of the taxable account, after taxes (assuming taxes are annual, e.g. dividends)
n = number of years until 401k is cashed out.

In words, NPV of tax liability = (tax liability if paid now - tax rate * sum of the differences between 401k and taxable account gains over the time horizon) / compounded discount over the time horizon.

The SUM(...) block accounts for the deferred taxes on the compounding gains, but the rest is plain NPV.

If you subtract the tax NPV from the gross, you get the amount that you'd need in a taxable account in order to end up the same as if you put the gross in 401k and then withdrew it all after n years.

So, now that that's all out, caveats: You need to have estimates of ROI, tax rates, AND the number of years before withdrawing from the 401k; it only accounts for gains that are taxed annually (eg dividends); and it assumes you withdraw all the gains and principle at the same time.

Now, returning to the original questions: I use net of tax and tithe (because it's also a % of income) when I calculate my savings rate, without scaling retirement accounts for future taxes. When I'm projecting time until FI I scale the retirement account balances by the expected total tax rate (without discounting) for my current net worth, but I still don't scale the savings rate. And now that I've looked into how complicated it would be to discount properly, I don't think I ever will. Besides, not scaling back the savings rate projection works in the opposite direction of scaling the net worth by too much (because I don't discount the taxes), so it comes out close enough for my tastes.

[grantmeaname]

Trying to parse that equation, written in plain text, makes my head hurt. Is this what you meant? (It's a straight copy and paste into wolfram alpha, removing the brackets. Who knew wolfram alpha was good for something?)

NPV=my image went away. edit: TLV updated his post.

[TLV]

Grant: I had some errors in translating it from spreadsheet to text - I'll update this post when I get it right.

Update: Got it - the reduced one from wolfram alpha produces results matching my spreadsheet.

[SpendyMcSpend]

My opinion is that first, the tax liability will be exactly the same whether you do the 401k or the Roth because with a 401k the amount taxed later is higher.

The reason businesses want to pay their tax liabilities later is because they can use the money saved on taxes NOW for other purposes to increase their wealth.  For us, let's say we saved $200 in taxes by investing in a 401k vs. using a non-tax deferred savings account.  What do we do with $200?  A business would use that to invest in other money-generating activities.  An individual might do that or he might spend it.  It all depends on what you do with that $200.

The savings rate, however, should be the same so you would need to add back .25 or whatever your tax rate is when calculating your savings rate off of net income.

I THINK >:)

[Mr Mark]

Roth: take earnings E, gets taxed to E*T, grows to E*T*G.
401k: take earnings E, grows to E*G, gets taxed to E*G*T.

The only difference is if the taxes are different.

TLV,

Velocistar237's equations above are spot on: there is zero difference between a Roth and a 401k after tax, if the tax rates are same. I tried to see if inflation would have an impact, but it doesn't.

On taxes and delay, the discounting approach is also spot on.

[sol]

there is zero difference between a Roth and a 401k after tax, if the tax rates are same.

While technically true, I think it worthwhile to point out that for anyone who is saving a large portion of their income to retire early, the tax rates will not be the same.

In fact, if you live on half of your current income and save the rest until it can cover your expenses, your tax rate in retirement will be that of someone making half as much money as you were while working, aka significantly lower. 

This is the reason why the 401k is a WAY better deal than a Roth for someone with a high savings rate and an intent to retire early.  The Roth certainly has advantages, but taxes isn't one of them.

[Bank]

In some downtime at work (finally some downtime!!) I was able to track down a source that discusses, in much more artful wording than I was able to muster, the point I was trying to make above.  I don't know if anyone still cares, but if you do... 

http://books.google.com/books?id=XsGdBReOQSgC&pg=PA75&dq=present+value+of+deferring+taxes&hl=en&sa=X&ei=5_rIT9WrFIXc0QGrremwAQ&ved=0CEcQ6AEwAA#v=onepage&q=present%20value%20of%20deferring%20taxes&f=false

If the link doesn't post correctly, it's page 75 of the Encyclopedia of Taxation and Tax Policy.

[TLV]

Good source - what stands out to me is that they list 4 separate ways of calculating the advantage of tax deferral. The previous disagreements probably stemmed from picking different ways of looking at it.

[velocistar237]

We were trying to figure out the scaling factor for different types of savings. Remind me,

1) What type of savings gets a scaling factor of 1?
2) What two types of savings are you comparing when you're discounting?

I would assume a Roth and post-tax principal are both scaled by 1. If the Roth gets a 1, then a 401k gets (1-T), where T is the tax rate upon withdrawal. Long-term capital gains get (1-LTCGT).

If you receive dividends or short-term capital gains, then if you want, you can calculate the effect of compounding tax, or you can just absorb the annual tax into the rate of return, or into your annual expenses.

[Bank]

We were trying to figure out the scaling factor for different types of savings. Remind me,

1) What type of savings gets a scaling factor of 1?
2) What two types of savings are you comparing when you're discounting?

I would assume a Roth and post-tax principal are both scaled by 1. If the Roth gets a 1, then a 401k gets (1-T), where T is the tax rate upon withdrawal. Long-term capital gains get (1-LTCGT).

If you receive dividends or short-term capital gains, then if you want, you can calculate the effect of compounding tax, or you can just absorb the annual tax into the rate of return, or into your annual expenses.

If this is directed at me, I was never comparing two types of savings.  I was only pointing out that the present value of the deferred tax liability for savings made in a tax deferred account was less than the tax if you paid it today.  So multiplying by 1-t actually understates the value of your savings.  Heck, if you plan to never withdraw that principle, or withdraw it in 50 years time, the present value of that tax liability on the principle itself (NOT the earnings) is effectively zero.  In fact, some corporate finance valuation texts make a similar argument, e.g. that deferred tax liabilities which are expected to be carried over from year to year have a present value of $0.  I can try to find some more info on that if you're interested.

But I digress....  in real life, I think this tax driven scaling gets hideously complicated to implement and is dependent on multiple long-term assumptions.  In my line of work (corporate valuation) we joke that anybody forecasting beyond 1-3 years might as well just stare out a window and take a wild guess.

So it's probably easier just to consider tax advantages as differentials in projected portfolio ROI --- as Mr. Mark noted above.  It still has an element of guesswork, but it's simpler guesswork.  But that's just me and my opinion.

[velocistar237]

I was only pointing out that the present value of the deferred tax liability for savings made in a tax deferred account was less than the tax if you paid it today.

I agree that tax deferral strategies work this way, I'm just not sure tax deferred accounts do, or at least I don't see it this way. Maybe it's just differences in the way you and I are accounting for principal and gains and the taxes on both.

If we assume a discount rate of 4%, the present value of your tax liability is $1000/(1.04^15), or $555.  $555/4000 = 13.9%.

The present value of the $4000 principal when you withdraw it is also discounted, to $2220, and $555/$2220 = 25%.

If this is directed at me, I was never comparing two types of savings.

Here, this is how we got started, by trying to find a scaling factor for our savings based on tax liability:

And like Star used to do, I reduce my 401K contributions by a small percentage to reflect the implied tax liability.  It's not by 30%, however, because the present value of that tax liability (realized, say 15 years down the road) is much lower than its current value.  I generally use 10%, just because it's a nice round number that doesn't make my head hurt.

If you use 10% for your 401k, what do you use for your Roth? More than 100%?

[Bank]

If we assume a discount rate of 4%, the present value of your tax liability is $1000/(1.04^15), or $555.  $555/4000 = 13.9%.

The present value of the $4000 principal when you withdraw it is also discounted, to $2220, and $555/$2220 = 25%.

And like Star used to do, I reduce my 401K contributions by a small percentage to reflect the implied tax liability.  It's not by 30%, however, because the present value of that tax liability (realized, say 15 years down the road) is much lower than its current value.  I generally use 10%, just because it's a nice round number that doesn't make my head hurt.

If you use 10% for your 401k, what do you use for your Roth? More than 100%?

$4,000 today is $4,000 today.  That is its present value.  Why would you discount it from the date you withdraw it?  And even if you did want to determine the future value of the $4,000 you wouldn't use your discount rate.  You would use something that reflects its diminished purchasing power, like the CPI.

As for the Roth contributions - I actually can't make them, so I don't have that issue and hadn't thought about it until now.  This may be why I got confused when the conversation went down that road.  However, what I would likely do is count the contribution at its after tax value, and then assume a higher IRR for it vs. my other investments to reflect the fact that I don't pay taxes on the gains, ever.

[velocistar237]

$4,000 today is $4,000 today.  That is its present value.  Why would you discount it from the date you withdraw it?  And even if you did want to determine the future value of the $4,000 you wouldn't use your discount rate.  You would use something that reflects its diminished purchasing power, like the CPI.

Okay, I see this.

And like Star used to do, I reduce my 401K contributions by a small percentage to reflect the implied tax liability.  It's not by 30%, however, because the present value of that tax liability (realized, say 15 years down the road) is much lower than its current value.  I generally use 10%, just because it's a nice round number that doesn't make my head hurt.

I still have a problem with this. On one level, we have two equivalent perspectives: I see a 401k as having a tax on principal and otherwise having tax-free gains, while you see it as tax-deferment on each deposit with further tax on gains. I believe these are mathematically equivalent, which I think TLV showed above (though I never took a close look). However, deferring a tax is about whether it's better to pay tax on $4000 now or later; you're comparing two lumps of pre-tax $4000. That's different from having a pre-tax $4000 in one account and a post-tax $4000 in another account, which is what you're comparing when you're figuring out the scaling factor on a 401k compared to post-tax savings. Discounting will tell you the same 401k advantage that everyone already says much more simply: you're only taxed once. If you try to use discounting to compare an existing 401k balance with an existing taxable account balance, you're double-counting.

For example, assume a long-term capital gains tax rate of 10%, a total income tax burden of 15% when you retire in 15 years, a buy-and-hold strategy, a 4% return, and no inflation. Here is what you would end up withdrawing at the end of 15 years if you started off with $100 in each of these accounts:

Type	Withdrawal	Scaling factor
Roth	$180	1
401k	$153	0.85
Taxable	$172	0.96

The 401k is the best place to put your money, but once you have money in the account, it's the least valuable, because you still have to apply the full income tax rate.

One other note: I only used 30% because that's about what my burden would be if I withdrew it early and incurred the 10% penalty. I expect my total tax burden to be more like 10% with the standard deduction, and the Roth rollover will help avoid the penalty.

[Bank]

First, Star, apologies in advance for the novel.  The force of verbosity is strong in this one…. 

I’m not sure what you’re getting at when you say you see “a 401k as having a tax on principal and otherwise tax-free gains…”  and I don’t fully understand your characterization of my position either.  Also, your example is coming up with results which are non-meaningful because your initial $100 in each investment is on different bases – pre and post-tax.  To run that example fairly, you should start with $100 with pre-tax money for each option.

However, let me explain my perspective on this issue using the helpful three account set-up you use.  First, I think it’s important to separate Principal (savings) from Earnings (returns earned over time).  Different savings options have different advantages, as follows:

401(k) – Tax Deferral on both Principal and Earnings.
Roth – Tax avoidance – on Earnings only.
Taxable Savings – No benefits, deferral or avoidance.

Of these options, only one offers a tax benefit on the PRINCIPAL you invest – the 401(k).  The Roth offers far superior tax benefits on EARNINGS --- but not on Principal – which will make up and match the tax deferral benefits enjoyed by the 401(k) on both Principal and Earnings.  Taxable, of course, offers bupkuss.

So…when I am computing my savings rate --- e.g. how much savings I have accumulated and invested in a given period --- I prefer to recognize the benefit of the 401(k)’s tax deferral up-front, because it accrues to the actual Principal I have saved, whereas a Roth’s benefits come from the tax treatment of its Earnings.  Netting out the PV of taxes from my 401(k) contribution, rather than using the nominal tax liability, reflects the fact that the 401(k) is a better investment than a fully taxable account.  This is partially because the earnings grow tax deferred, but it is ALSO because the present value of the tax liability on the Principal is smaller than if I paid it today.  If I had a Roth (which I don’t) I would likely take into account its badassity by assuming a higher ROI in the spreadsheets I use to project investment performance.  That would seem to me to be the proper way to reflect the advantages of tax free compounding on the Earnings.  But, as I don’t have a Roth, I haven’t really considered the question from every angle and am open to changing my mind about that.

I hope you understand where I’m coming from now.  If you’re comfortable with what you’re doing then keep doing it.  It’s just a savings rate after all, and my way adds more complexity (and yes, rank speculation about withdrawal dates, tax rates, rates of return etc.) that it is perfectly reasonable not to want to deal with for a variety of reasons.  But I do think my way is theoretically sound and not grounded in my ignorance of qualified plans and Roth IRAs, as you stated in one of your earlier comments.  FWIW, I worked for a couple years in the retirement plans department of Wells Fargo (Note:  Not that anyone on this board ever would, but NEVER invest in Wells Fargo retirement products) handling individual accounts and small business retirement plans.  This was back a while back --- SAR/SEPs had just been phased out – but I don’t think the rules have changed that much based on my occasional spin through Pub 590.

Out of curiosity, it seems like you project a lower tax rate in retirement than today.  Given that, have you considered foregoing a Roth investment in favor of all tax deferred accounts?  If so, why did you elect to stay with the Roth despite the higher tax liability today?  Feel free to reply via message, or point me to another thread if it's already been discussed on this board as this is off topic.

[velocistar237]

First, Star, apologies in advance for the novel.  The force of verbosity is strong in this one…. 

I’m not sure what you’re getting at when you say you see “a 401k as having a tax on principal and otherwise tax-free gains…”  and I don’t fully understand your characterization of my position either.

No worries on the novel. I'm still learning things. I hope Meadow doesn't mind. (Are you still reading? :)

The reason I characterize the 401k like a Roth is because the math works out so that they are the same, assuming the same tax rate. The present value of a Roth is earnings deposited minus tax, or E*(1-t). The present value of a 401k, the way you describe it, is earnings deposited minus the discounted tax on earnings minus the tax on future gains, and it turns out that

That is, exactly the same as the present value of a Roth for the same earnings, assuming the same tax rate. I don't understand the need to complicate the 401k when you can treat it exactly like a Roth and simply scale it by the future tax rate.

Also, your example is coming up with results which are non-meaningful because your initial $100 in each investment is on different bases – pre and post-tax.  To run that example fairly, you should start with $100 with pre-tax money for each option.

It looks like our other differences just come down to comparing at the earnings side vs. comparing at the savings side. I usually compare at the earnings side when deciding how to invest, and I compare on the savings side when looking at my current savings. Concerning money I've already saved, I really only care about future tax.

Out of curiosity, it seems like you project a lower tax rate in retirement than today.  Given that, have you considered foregoing a Roth investment in favor of all tax deferred accounts?  If so, why did you elect to stay with the Roth despite the higher tax liability today?  Feel free to reply via message, or point me to another thread if it's already been discussed on this board as this is off topic.

I don't actually contribute to a Roth at the moment, but I have a Roth account from earlier in my career. After I take care of a few things on my financial to-do list that require cash, my plan is to max out my 401k contribution (I already contribute up to the company match, but I can still contribute more). That should push my income down into the 15% bracket, at least for this year, and it probably makes sense to contribute to a Roth before contributing to taxable accounts.

[SpendyMcSpend]

Holy crap with these formulas.  Are you all engineering or finance majors here?

[sol]

Holy crap with these formulas.  Are you all engineering or finance majors here?

Pretty much.

When discussing math, it sometimes help to be able to write it down.  It's not so bad if you take the time to look at it.

[arebelspy]

Holy crap with these formulas.  Are you all engineering or finance majors here?

Don't get discouraged by it.  You can either learn it as a fun challenge, or just let others do it for you and learn from their calculations.  ;)