# User:Fyre4ce/Retirement plan analysis

This page contains a database of analysis and formula derivations for retirement plan-related articles, including Traditional versus Roth and Roth conversion.

## Relative value of contributions and conversions

Define variables:

The overall value of a change to tax-advantaged space is equal to:

Consider a given after-tax investment that can be contributed to a traditional account, a Roth account, or used to pay the taxes on a Roth conversion. When making a traditional contribution, the change in traditional balance is:

Therefore, the change in value when making a traditional contribution is:

When making a Roth *contribution*, the change in Roth balance is simply:

Therefore, the change in value when making a Roth contribution is:

When making a Roth *conversion*, the converted amount is:

Therefore, the change in value when making a Roth conversion is:

When (current marginal tax rate is *less than* predicted future marginal tax rate),

When (current marginal tax rate *equals* predicted future marginal tax rate),

When (current marginal tax rate is *greater than* predicted future marginal tax rate),

--Fyre4ce 23:10, 10 March 2020 (UTC)

## Conversions on estates subject to estate tax

Define variables:

When a Roth conversion is performed on assets, during the owner's life on assets expected to be subject to estate tax, and the taxes can be paid from after-tax assets, the net effect on types of assets are as follows:

The change in after-tax value of the estate to heirs will be as follows:

It follows that Roth conversions increase the value of the after-tax value of the estate if:

or

--Fyre4ce 04:44, 10 December 2020 (UTC)

## Saver's Credit

For a fair comparison, the two take home pays must be equal:

Solving for T and R in terms of A:

The changes in after-tax value of retirement accounts for the two contribution options are:

Traditional contributions are preferred when the

Canceling and (assumed to be the same in both cases), and solving for :

## Maxing out retirement accounts

Define variables as follows:

When contributing a fixed dollar amount to either traditional or Roth accounts, and investing the tax savings in a taxable account, traditional contributions are preferred when:

Canceling and solving for gives:

Rather than plug in the formulas for these factors to create one large equation, it is easier to calculate each factor separately. Assuming annual compounding, the three growth factors can be calculated as follows:

Recall from taxable account performance that:

and

Separate rates of return for traditional, Roth, and taxable accounts allow the comparison between different accounts (eg. IRA or 401(k)) with different investments and fees. Assuming the same investments and fees and , the equations simplifies somewhat to:

with , , and the same as above.

## Employer match

Define variables as follows:

When making a traditional contribution, the changes in the two types of balances will be:

When making a Roth contribution, the changes in the two types of balances will be:

The after-tax values at withdrawal of the two contribution choices are:

Traditional contributions are preferred when :

Canceling and (assumed to be the same in both cases):

Solving for using a Computer Algebra System (CAS):

### Employer match combined with Saver's Credit

The above equations can be modified to also include a Saver's Credit. When making a traditional contribution, the changes in the two types of balances will be:

When making a Roth contribution, the changes in the two types of balances will be:

The after-tax values at withdrawal of the two contribution choices are:

Traditional contributions are preferred when :

Canceling and (assumed to be the same in both cases):

Solving for using a Computer Algebra System (CAS):