# User:Fyre4ce/Tax analysis

## Taxable account performance

For simplicity and clarity, this derivation will assume continuous compounding, but the formula may be modified for period compounding as desired. Continuous compounding formulas use exponentials of base *e* 2.71828. The value of the investment at any future time *t* can be written as follows:

Calculating the basis at any time *t* requires integration. The rate of change of the basis at any time *t* is equal to the value at *t*, multiplied by the yield, multiplied by (1 - ). The product of is the rate of loss to dividend taxes, and the remainder is added to the cost basis:

Integrating this formula (using a Computer Algebra System) from 0 to *t* and setting gives:

Finally, the after-tax value after the investment is sold is given by:

## Derivation of tax rate boundaries for Social Security taxation

Variables are defined as follows:

### Point above which 40.7% marginal rate is possible

The point above which 40.7% marginal tax rates is possible is when total taxable income is at the 22% tax bracket threshold and the maximum 85% of Social Security benefits are taxable. It is the combination of and that satisfies these two equations:

Rearranging the first equation to solve for OI gives:

Save this result for later substitution. Substitute the definition of relevant income into the second equation:

Substitute in the formula for OI from the rearranged first equation:

Collecting the SS terms from the left hand side:

Simplifying the SS terms on the left hand side:

Solving for SS and labeling this value SS* gives:

Recalling the equation above for OI in terms of SS, and labeling this value OI* gives:

### 22.2% bump begins

For single filers, the 22.2% bump begins in the middle of the 12% bracket when Social Security taxation begins to be taxed at an 85% marginal rate. This occurs when:

Substituting $34,000 for UB gives:

For married filers, the 22.2% bump begins at the boundary between the 10% and 12% brackets. The line is defined by the solution to these equations:

where is the percentage of Social Security income that is taxable. is an unknown variable, but with two equations and three unknowns it should be possible eliminate through substitution. Solving for in the second equation gives:

Substituting this value for into the first equation, and also the definition of relevant income, gives:

Expanding the large term on the left hand side gives:

Rearranging to solve for OI:

The solution to this set of equations is:

### 22.2% bump ends

The 22.2% bump ends when the maximum of 85% of Social Security benefits becomes taxable. This occurs when:

Substituting the definition for relevant income gives:

Expanding the large term on the left hand side gives:

Rearranging to solve for OI gives:

Dividing by 0.85 gives:

### 40.7% bump begins

For both single and married filers, the 40.7% bump begins at the boundary of the 22% bracket. The formula is the same as for the beginning of the 22.2% bump for married filers, but with a different bracket threshold:

### 40.7% bump ends

The line where the 40.7% bump ends is the same as where the 22.2% bump ends. The only difference is whether the boundary is above or below .

-- Section created 02:03, 20 May 2019 by Fyre4ce (--LadyGeek 20:33, 20 May 2019 (UTC))

## Variable Marginal Rates with Section 199A Deduction

Define variables:

This analysis will assume a single tax bracket, although because the equation will be differentiated, the results will apply to any tax bracket.

The total tax liability can be written as follows. If total taxable income is below the beginning of the deduction phase-out ($163,300 for single or $326,600 for MFJ, for 2020), total tax is:

In the phase-out range ($163,200-$213,300 for single or $326,600-$426,600 for MFJ), total tax is:

Above the phase-out range ($213,300 for single or $426,600 for MFJ), total tax is:

Above the phase-out, the Section 199A deduction has no effect. Below the phase-out, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

In the phase-out range, the marginal tax rates with respect to QBI and non-QBI are found by taking the partial derivative of T with respect to N and Q:

The second partial derivatives are:

### Example

A MFJ couple has $120,000 of non-QBI income and also earns QBI. They take the standard deduction of $24,800. Below $326,600 of taxable income, they are in the 24% bracket. Their marginal tax rates for non-QBI and QBI income are:

The phase-out begins when their taxable income, after the standard deduction, equals $326,600. This corresponds to a QBI income of **$231,400** ($326,600 + $24,800 - $120,000). Note that although their taxable income is at the 24%/32% threshold, the Section 199A deduction pulls them well down into the 24% bracket. At this income, their marginal tax rates are:

At some point in the phase-out range, the couple will cross into the 32% bracket, and then into the 35% bracket. At the top of the phase-out, when QBI income is **$331,400** ($426,600 + $24,800 - $120,000), marginal tax rates will be:

Above $331,400 QBI, the Section 199A deduction is completely eliminated and the marginal rates become:

### Maximum possible rates

By inspection of the above formulas, the maximum possible rates occur at the very top of the phase-out range, when:

The maximum also occurs when Q is largest, so:

and

Substituting these values into the above equations for the marginal rates give:

and

Note that the two formulas are the same.

For single filers for 2020, UL = $213,300 and R = $50,000, and we will assume the standard deduction D = $12,400. $213,300 taxable income is barely into the 35% bracket, which begins at $207,350 taxable income, so TB = 35%.

The QBI required to achieve this rate is:

For married joint filers for 2020, UL = $426,600 and R = $100,000, and we will assume the standard deduction D = $24,800. $426,600 taxable income is barely into the 35% bracket, which begins at $414,700 taxable income, so TB = 35%.

The QBI required to achieve this rate is:

Note that the single and married joint maximum rates are the same. If itemized deductions are larger than the standard deduction, the maximum rate will be slightly higher.

--Fyre4ce 17:37, 17 February 2020 (UTC)