Introduction:
Generally, most articles regarding Roth and Traditional taxation structures are titled “Roth vs. Traditional” as if there is an upcoming boxing match between the two. These articles list vague circumstances where one should be used over the other and ignore that an investor can have the best of both worlds.
With the Roth 401k becoming ever popular since its inception in 2006, employees with access to both taxation vehicles have had to choose which taxation structure to pick with little guidance as to which one would be more beneficial. The choice between taxation structures can dramatically impact the lifestyle that an individual receives in retirement. This paper formulates a strategy for maximizing the after-tax payouts of a 401k by using both taxation structures.
The formulation of this strategy should be fiscally rewarding to workers with access to both types of 401ks who are early on in their careers. This paper will also be useful to 401k managers, certified financial planners, employers who are considering the benefits of making a Roth 401k available to their workers, as well as individuals are generally interested in the topic of retirement taxation.
This paper found that the optimal solution for a 401k (in most circumstances) is to choose a Roth taxation structure in the earlier years of employment where income is lower and contributions will grow the longest and have the largest impact on tax rate at withdrawal. Then, at a certain point termed the “Switching Point”, it is beneficial to stop contributions to the Roth 401k, keep the current funds in the Roth 401k, allow them to grow, and then start contributing to a Traditional 401k until retirement. This strategy allows the investor to gain the most money in retirement at no cost other than the time to learn this strategy and open an additional Traditional taxation structure which is generally free and effortless. The switching point times vary based upon a variety of factors such as salary, amount of years contributing, amount of years in retirement, interest rate, amount contributed annually, and additionally taxable income earned during retirement. The tax rate at contribution as well as the tax rate at retirement are not factors as they are determined by the factors previously listed. The tax rate at contribution is determined by the salary earned in the year of contribution while the tax rate at withdrawal is determined by a variety of factors such as choice of taxation structure, contribution amount, interest rate, length of contribution as well as length of withdrawal. Initial Theory required for understanding Pre Tax Contribution Comparison
To begin, there is no difference between the amount gained in a Roth 401k and a Traditional 401k if the tax rates at contribution and withdrawal are equal. Example:
Joe has $500 pretax dollars to invest in either a Roth or Traditional 401k. Joe expects to earn 40% in his investments over a 15 year period which will not differ depending upon his taxation vehicle choice. Joe has a tax rate of 20% currently and will have a tax rate of 20% when he withdraws the funds after 15 years. This example is oversimplified and uses effective tax rates rather than the current progressive scale as well as an already assumed compounded interest multiplier. The simplified example is adequate to prove the general point. Type Roth Traditional
Pre Tax Contribution Amount
$500
$500
Tax Rate at Contribution
20%
20%
Tax
$100
$0
Post-Tax Contribution
$400
$500
Interest Rate
40%
40%
After Interest Amount
$560
$700
Tax Rate at Withdrawal
20%
20%
Tax
$0
$140
After Tax After Interest Amounts
$560
$560
The common mistake is to compare Roth and Traditional vehicles on a post-tax contribution vs. a pre-tax contribution basis. The correct way of comparing a Roth and Traditional structure is used throughout the paper. Effective Tax Rates
Since the U.S. uses a progressive tax bracket, saying that a total amount of income is taxed at 20% is not true if one is using marginal rates. The first $8500 earned is taxed at a 10% rate. The next 26,000 is taxed at a 15% and so on etc. The difference between the marginal tax rates and an effective rate is just a way of showing the actual single rate of tax. Marginal tax rates are used to calculate the total amount of tax on an income. This tax is then divided by the total income being taxed to get the true tax rate. Single effective rates are more useful in this paper because they make tax rates more intuitive for readers to understand.
For example say Joe is single and earns $30,000 annually ignoring any deductions he may receive.
Even though Joe is in a marginal tax bracket of 15%, his actual tax rate is 13.4%. For any income above $8,500, Joe’s effective tax rate will change even if his marginal tax bracket does not change. If his income increases from $5,000 to $8,000, his effective tax bracket would remain at 10%. Interestingly enough, the effective tax rate also increases steadily for those who earn over 379,150, the top marginal tax bracket. The effective rate doesn’t become a steady 35% until an income of 44,712,499,957 is earned in one year which has never been done before. Only if Bill Gates earned his entire fortune in one year would he be subject to the 35% effective rate.
Amortization and even monthly payouts
Typically, individuals prefer to receive even, inflation adjusted payouts in retirement. This ensures an even standard of living and minimizes the risk that their funds will run out before they die. To accomplish this end goal (which also maximizes payouts by decreasing taxes) the amortization method was used. Generally, amortization is commonly used by: businesses to pay debt off or depreciate an asset or banks/credit card organizations to determine monthly payments on a mortgage or credit card bill. A person new to amortization may wonder why the principal wouldn’t be divided by the number of payments and the answer is simple. Interest. While a loan is being paid off, interest is being accumulated every month on the principal. So as someone pays, they are paying the Interest Payment + Principal until the principal reaches 0. Initially, a high % of a person’s debt payment will be the interest payment while a low% will be the principal payment. As more of the principal is paid off, the interest payment will become a smaller % of the payment while the principal payment will increase in % until the initial relationship is reversed and the person is paying mainly principal near the end.
So how does amortization, a method commonly used to pay off debt, related to the payment from a 401k in retirement? The concept of paying off debt is structurally the same to the concept of receiving payments equal payments from a 401k. Think of the amount in the 401k upon retiring as the principal of the loan. The amount in the 401k continues making interest as it is withdrawn. Eventually the individual also wants the full amount to reach 0.
The Equation for calculating monthly payments: i
=
Interest Rate A
=
Amount in Vehicle at Retirement P
=
Number of payments (12 if monthly payments, 52 if weekly payments, 365 if daily payments, etc.) Y
=
Number of Years you will receive monthly/weekly/daily/annual payments
(i/P)*((A*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))) = Amortized Payment in terms of Y Inflation
During the course of this paper the reader may wonder where inflation is factored in. Inflation is factored into this equation by not being factored in. This may initially be confusing, but has a reasonable explanation. All interest rates referred to in the equation are real interest rates, contributions and payouts in reality increase every year with inflation, but on this paper payouts and contribution will appear flat. Tax rates naturally increase with inflation, social security payments increase with inflation with a cost of living increase. Salary is considered to increase with inflation (cost of living raises).
Example:
Our old friend Joe has the following metrics for 2010 and 2011
2009 2010 2011
Income
30000
30690
31395.87
Interest Rate
In terms of the equation the inputs factoring out inflation would be:
2009 2010 2011
Income 30000 30000 30000
Interest Rate 2.7% 3.7%
CPI 0% 0%
Salary Increase 0% 0%
Contribution $5000 $5000 $5000
Effective Tax Rate 0.13636 0.13646
By factoring out inflation throughout the paper, the reader can see figures in terms of the present purchasing power parity vs. an irrelevant future figure. In other words individuals may compare their current expenses to the figures quoted as annual payout figures.
Tax Rate at Withdrawal
While the tax rate at contribution can be calculated from the current income, the tax rate at withdrawal is determined by annual amount of taxable income at retirement. However, the annual amount of taxable income is determined by the amortized amount of taxable income from an individual’s Traditional 401lk. Unfortunately this is a case of circular logic as the choice of taxation structure determines the tax rate at withdrawal and the tax rate at withdrawal determines the choice of taxation structure.
To resolve the above circular reasoning if then scenarios need to be used to determine tax rates for all methods of contribution. This can be done by initially assuming all contributions will go to a Traditional taxation structure.
i = Interest Rate
C = Pretax annual contribution to 401k vehicle
X = Year of First Withdrawal – Year of First Contribution
Tw = Tax rate at withdrawal
Tc = Tax rate at contribution
Y = Number of Years you will receive monthly/weekly/daily/annual payments
P = Number of payments (12 if monthly payments, 52 if weekly payments, 365 if daily payments, etc.)
O = Other Annual Taxable income in Retirement
St = Annual taxable part of social security payout
Z = Year of contribution (if first contribution then 0)
A = Total Annual Amortized pretax payout = ((i/P)*(((C*(1+i)^(X-0))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P + ((i/P)*(((C*(1+i)^(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P + ((i/P)*(((C*(1+i)^(X-(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P
L = Annual Taxable Income = A + O + S
Now that the annual taxable income amount has been determined, it also must be determined for the other possible scenarios (1st contribution Roth and rest Traditional, 1st two contribution Roth and rest Traditional, 1st three contributions Roth and rest Traditional etc.) The reason this is not done in reverse (starting with 1st contribution Traditional and rest Roth) is because this would give preference to contributing to a Roth in later years while contributing to a Traditional in earlier years which would make more sense if expecting a declining income which is not normal for most individuals.
A
A – (((i/P)*(((C0*(1+i)^(X-0))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P)) + O0 + S0,
A - (((i/P)*(((C1*(1+i)^(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P) - ((i/P)*(((C*(1+i)^(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P + O1 + S1 ,
A - (((i/P)*(((C1*(1+i)^(X-2))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P) - ((i/P)*(((C*(1+i)^(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P - ((i/P)*(((C*(1+i)^(X-(X-1))*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P + O2 + S2,
…………………………………..………
A - A
From this list separate tax rates can be determined from all scenarios. Additionally, the total after-tax annual payout can be found by multiplying those tax rates by their respective annual taxable income and adding the amortized Roth payout and annual non-taxable portion of their social security payout. After this is done, one can clearly see the most fiscally beneficial switching point.
The Roth annual payouts can be calculated in a similar way to the Traditional Annual Amortized pretax payout with the exception that these payouts will increase instead of decrease. Therefore if an individual is electing to contribute all of their contributions to their Traditional 401k over their lifetime, they will allocate 0 dollars to their Roth 401k over their lifetime. If an individual elects to contribute all of their contributions to their Roth 401k over their lifetime, they will allocate 0 dollars to their Traditional 401k. The decision is a “this or that” decision. Therefore the Roth scenarios will mirror the Traditional scenarios.
While these equations answer the question of where the switching point exists within any given scenario, they don’t answer the question as for why the switching point changes and what the factors are that cause the change.
Annual Switching Point Factors
There are four main annual switching point factors that add up to be the year to year differences between the different switching point scenarios. As there previously have been no names for these items they have been termed as: “Roth Tax Benefit”, “Roth Natural Benefit”, “Roth Social Natural Benefit”, and “Roth Social Tax Benefit” for the purposes of clarity throughout the paper.
Roth Natural
Roth Natural is probably the most intuitive factor out of the group. It is simply the after-tax amortized future value upon withdrawal of a Roth taxation structure minus the after-tax amortized future value upon withdrawal of Traditional taxation structure.
i = Interest Rate
C = Pretax annual contribution to 401k vehicle
X = Number of Years the contribution will grow from present period until first withdrawal
Tw = Tax rate at withdrawal
Tc = Tax rate at contribution
Y = Number of Years you will receive monthly/weekly/daily/annual payments
Future Value upon withdrawal
(C*(1+i)^X) = Traditional Future Value at withdrawal (pretax)
(C*(1-Tc)*(1+i)^X) = Roth Future Value at withdrawal (posttax)
Amortized Amount of Future Value upon withdrawal
((i/P)*(((C*(1+i)^X)*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P*(1-Tw) = Trad Amort Pmt in terms of Y(posttax)
((i/P)*(((C(1-Tc)*(1+i)^X)*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P= Roth Amort Pmt in terms of Y(posttax)
(Roth Amort Pmt in terms of Y(posttax) – Trad Amort Pmt in terms of Y(posttax))*P = Annual Roth Natural Factor
The above equation can also be simplified into a much easier equation:
(Tw-Tc)*C =Present Value Roth Natural Factor
(((Tw-Tc)*C)*(1+i)^X) = Future Value at withdrawal (posttax) Roth Natural Factor
((i/P)*(( (((Tw-Tc)*C)*(1+i)^X)*((1+(i/P))^(P*Y))/(((1+(i/P))^(P*Y))-1))))*P= Annual Roth Natural Factor
There are two important pieces of information that can be pulled from the above equations. First that the Roth Natural factor is the difference between the after-tax amortized future value of Roth and Traditional taxation structures that have the same pre-tax contribution, interest, rate, years of growth, and periods of amortization. Second, that the present value difference between the two is simply the spread of the two rates multiplied by the pre-tax contribution made in the period.
Roth Tax Benefit
The Roth Tax benefit is the tax benefit from reducing your tax rate by choosing a Roth taxation structure. As shown in the Tax Rate at Withdrawal section, the if then scenario begins with assuming a person chooses Traditional in every year of contribution. If for example an individual was contributing to their respective 401k for 10 years the “if then” scenario would look like this:
0 Contributions Roth, All 10 Contributions Traditional
1st Contribution Roth, 9 Contributions(2nd,3rd…. 10th) Traditional
1st and 2nd Contributions Roth, 8 Contributions(3rd,4th… 10th) Traditional
1st, 2nd, and 3rd Contributions Roth, 7 Contributions(4th, 5th…..10th) Traditional
4 Contributions Roth( 1st, 2nd, …, 4th), 6 Contributions(5th,6th… 10th) Traditional
5 Contributions Roth( 1st, 2nd, …, 5th), 5 Contributions(6th,7th… 10th) Traditional
6 Contributions Roth( 1st, 2nd, …, 6th), 4 Contributions(7th,8th… 10th) Traditional
7 Contributions Roth( 1st, 2nd, …, 7th), 3 Contributions(8th,9th, 10th) Traditional
8 Contributions Roth( 1st, 2nd, …, 8th), 2 Contributions(9th , 10th) Traditional
9 Contributions Roth( 1st, 2nd, …, 9th), 1 Contributions(10th) Traditional
All 10 Contributions Roth, 0 Contributions Traditional
As can be seen, there would be a decreasing taxable income and an increasing amount of nontaxable income. Therefore there would be a decreasing effective tax rate. While the Roth Natural accounts for the benefit of choosing one structure over the other, the Roth Tax Benefit accounts for the fiscal effect a lower tax rate has on other taxable income.
For example:
Imagine that Joe plans to run a pizza shop in retirement that he estimates will earn $10,000 a year. Imagine that Joe is in his 11th year of contribution and already has some funds in his Traditional 401k that will payout 30k of taxable income a year. So regardless of the decision Joe makes regarding his 11th contribution, he will have 40k of taxable income a year. Using the 2011 single tax bracket rates Joe’s taxable rate will be 15.3125%. Let’s say that the Roth natural value is -10 dollars annually, so if Joe chooses a Traditional taxation structure he will gain 10 dollars annually. If Joe chooses a Roth taxation structure he will lose 10 dollars annually.
If Joe decides to stay with a Roth for his 11th contribution this tax rate will remain the same. However if Joe decides to choose a Traditional taxation structure he will gain another 20k in taxable income a year. This will bring his total taxable income a year up to 60k. This would bring up his effective tax rate up to 18.5416667%. This increased tax rate will affect the amount of tax that Joe pays on his pizza shop income. The increased tax rate will also affect the other taxable income that Joe was receiving from his Traditional 401k prior to the decision to go with a Traditional 401k in his 11th contribution which was 30k. Therefore the income being affected by this increased tax rate that is not being counted by the Roth natural factor is 40k. We can measure this factor by simply taking the taxable income that was previously subject to a lower rate before the decision to choose a Traditional 401k structure and multiply it by the difference from the new (increased) tax rate and the old (lower) tax rate.
In Joe’s case, his Annual Roth Tax Benefit Factor would be:
(30k + 10k)*( .185416667-.153125)
40,000*0.032291667 = 1,291.67
As you can see it would be a benefit for Joe even though the Roth Natural Factor shows a 10 dollar a year loss to continue choosing a Traditional structure, it would still be a (1291.67-10) $1281.61 annual benefit to continue contributing to a Roth because the increase in tax rates on the taxable income not involved in the decision would not be worth the minute benefit.
Roth Social Natural Benefit
Social security can be taxable depending upon how much income an individual makes in retirement. There are currently two “bend points” that determine whether a social security payment is even taxable. If an individual makes $34,000 or more in annual taxable income while he is eligible to receive a social security payment (62 and above), then 85% of his social security payment is taxable. If an individual makes between $25,000 and $33,999.99 in taxable income, then 50% of that payment is taxable. If the individual has a taxable income of 24,999.99 or below, then none of his social security payment would be taxable. A person’s taxable income in retirement includes 50% of the raw social security payment before it’s gone through this decision.
For example:
Joe owns a pizza shop in retirement and receives 10k of taxable income. Joe has an annual payment of 20k from his traditional 401k and Joe has a social security payment of 5k. Joe’s total taxable income for purposes of calculating how much of his social security payment is taxable is 22.5k. Since Joe’s total taxable income falls below the 25k bend point, none of his social security payment is taxable and thus his actual taxable income will be 20k.
Joe owns a pizza shop in retirement and receives 20k of taxable income. Joe has an annual payment of 15k from his traditional 401k and Joe has a social security payment of 5k. Joe’s total taxable income for purposes of calculating how much of his social security payment is taxable is 37.5k (20+15+(5/2)). Since Joe’s total taxable income is above the 34k bend point, 85% of his social security payment is taxable and thus his actual taxable income will be 39.25k (15+20+((.85)*5)).
While all other methods of determining an individual’s social security payment are indexed using inflation, the tax thresholds are not and have remained the same since they were enacted in 1984 and 1993. It is for this reason that a Roth Social Benefit will slowly diminish (due to inflation) into only existing for those who do not earn any other taxable income in retirement and go fully with a Roth taxation structure. This law may change, but with the current fiscal austerity measures taking place, it is unlikely.
In terms of calculating the benefit an individual must first calculate their Personal Insurance Amount. In order to calculate their personal insurance amount they must first calculate their average indexed monthly earnings or AIME. To do this involves plotting their highest 35 years of actual salary (including inflation). In the case of modeling, the salary will obviously be estimated for unknown years. There are a six figures that are released annually by the government in order to determine how a both an AIME as well as PIA are calculated: Average National Salary, Maximum, Bend Point 1, Bend Point 2, Tax Threshold 1, and Tax Threshold 2. While the first four figures change annually due to inflation, both tax thresholds have remained the same since inception. When predicting ones PIA, one must predict these four numbers in addition to ones salary to predict the social security payment. Since a history of these figures is available, averages can be calculated for each one
Bend 1 Avg Bend 2 Avg Max Avg Avg Annual Avg
5% 5% 4% 5%
Excluding some outliers for the Max Avg, a clear trend at which to predict future figures emerges. The only other figure left to estimate is salary, which is purely an individual’s best guesswork based upon their current career path and company. From this point an individual can follow backward government instructions to predict a PIA. Once a PIA has been predicted, it can be used in the situational analysis to determine potential taxes between scenarios (full roth, 1 year roth rest traditional, etc.).
Roth Social Tax Benefit
In addition to the reduction in taxes on a social security PIA, the reduction on the taxable PIA also reduces the overall amount of taxable income and thus reduces the overall tax rate. This follows the exact same reasoning as the Roth Tax Benefit, except that this time the drop in taxable income comes from the drop in taxable social security funds which indirectly comes from the choice of taxation structure. This tax benefit is calculated in the following manner:
ST = Annual Taxable social security funds
RT = Annual Taxable funds from Traditional 401k
…….
6 Years Roth Then Rest Trad - ST = 5000 , RT = 28900
7 Years Roth Then Rest Trad - ST = 5000 , RT = 19600
8 Years Roth Then Rest Trad - ST = 2941 , RT = 17400
……
6 to 7 –
Total Taxable Income = (5000 +28900) = 33900 = 0.137463127
Total Taxable Income = (5000+19600) = 24600 = 0.132723577
Tax Benefit = (0.137463127-0.132723577)*24600 = $116.59
7 to 8
Total Taxable Income = (5000+19600) = 24600 = 0.132723577
Total Taxable Income = (2941+17400) = 20341 = 0.129106239
Tax Benefit = (0.132723577-0.129106239)*20341 = $73.58
Roth Benefit = $73.58
It is a relatively simple task to quantify the Tax benefit as a whole. Assigning it to either Roth or Social is also relatively simple in 6 to 7 because the social factor stays constant. In situation such as 7 to 8 where both factors vary, it’s sometimes difficult to assign a share of the benefit to each. The methodology used is to hold one factor the same while allowing the other to vary. The low number is held constant for both periods and a result is calculated, and then the high number is held constant for both periods and the result is calculated. An average is obtained and attributed to the varying factor while the constant is assigned the remainder of the benefit.
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