Retirement plan analysis (math)

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 contains supporting analysis and formula derivations for retirement plan-related articles, including Traditional versus Roth and Roth conversion.

Relative value of contributions and conversions
Define variables:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ C & = \text{Roth-converted amount} \\ V & = \text{Total value of tax-advantaged space} \\ A & = \text{After-tax amount} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_w & = \text{marginal tax rate at withdrawal} \\ \end{align} $$

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

$$\Delta V = \Delta T (1 - MTR_w) + \Delta R$$

Consider a given after-tax investment $$A$$ 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:

$$\Delta T = \frac{A}{(1 - MTR_n)}$$

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

$$\Delta V_T = A \frac{(1 - MTR_w)}{(1 - MTR_n)}$$

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

$$\Delta R = A$$

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

$$\Delta V_R = A$$

When making a Roth conversion, the converted amount is:

$$C = \frac{A}{MTR_n}$$

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

$$\Delta V_C = C - C(1 - MTR_w) = \frac{A}{MTR_n} - \frac{A}{MTR_n} \cdot (1 - MTR_w) = \frac{A}{MTR_n}(1 - (1 - MTR_w)) = A \frac{MTR_w}{MTR_n}$$

When $$MTR_n < MTR_w$$ (current marginal tax rate is less than predicted future marginal tax rate),

$$ \Delta V_C > \Delta V_R > \Delta V_T $$

When $$MTR_n = MTR_w$$ (current marginal tax rate equals predicted future marginal tax rate),

$$ \Delta V_C = \Delta V_R = \Delta V_T $$

When $$MTR_n > MTR_w$$ (current marginal tax rate is greater than predicted future marginal tax rate),

$$ \Delta V_C < \Delta V_R < \Delta V_T $$

Conversions on estates subject to estate tax
Define variables:

$$ \begin{align} R & = \text{Roth balance} \\ T & = \text{Traditional balance} \\ A & = \text{After-tax balance} \\ C & = \text{Roth-converted amount} \\ V_h & = \text{Total value of estate to heirs after-tax} \\ MTR_n & = \text{marginal tax rate now} \\ MTR_e & = \text{marginal tax rate on estate} \\ MTR_h & = \text{marginal tax rate on heirs} \\ \end{align} $$

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:

$$\Delta T = -C$$

$$\Delta R = +C$$

$$\Delta A = -C \cdot MTR_n \cdot (1 - MTR_e)$$

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

$$\Delta V_h = \Delta T \cdot (1 - MTR_h) + \Delta R + \Delta A = -C \cdot (1 - MTR_h) + C - C \cdot MTR_n \cdot (1 - MTR_e)$$

$$\Delta V_h = C \cdot ((MTR_h - 1) + 1 + MTR_n \cdot (MTR_e - 1)) = C \cdot (MTR_h + MTR_n \cdot (MTR_e - 1))$$

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

$$MTR_h + MTR_n \cdot (MTR_e - 1) > 0$$

or

$$MTR_h > MTR_n \cdot (1 - MTR_e)$$

Saver's Credit
$$ \begin{align} MTR_{n, T} &= \text{marginal tax rate now, for the traditional contribution, including Saver's Credit} \\ MTR_{n, R} &= \text{marginal rate now of Saver's Credit for the Roth contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ T &= \text{traditional contribution} \\ R &= \text{Roth contribution} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

For a fair comparison, the two take home pays must be equal: $$A = T \cdot (1 - MTR_{n,T}) = R \cdot (1 - MTR_{n,R})$$

Solving for T and R in terms of A:

$$T = \frac{A}{1 - MTR_{n,T}}$$ $$R = \frac{A}{1 - MTR_{n,R}}$$

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

$$\Delta V_T = \frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w)$$ $$\Delta V_R = \frac{A}{1 - MTR_{n,R}} \cdot G$$

Traditional contributions are preferred when the $$\Delta V_T > \Delta V_R$$

$$\frac{A}{1 - MTR_{n,T}} \cdot G \cdot (1 - MTR_w) > \frac{A}{1 - MTR_{n,R}} \cdot G$$

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

$$\frac{1 - MTR_w}{1 - MTR_{n,T}} > \frac{1}{1 - MTR_{n,R}}$$

$$MTR_w < 1 - \frac{1 - MTR_{n,T}}{1 - MTR_{n,R}}$$

$$MTR_w < \frac{MTR_{n,T} - MTR_{n,R}}{1 - MTR_{n,R}}$$

Maxing out retirement accounts
Define variables as follows:

$$ \begin{align} MTR_n &= \text{marginal tax rate now, for traditional contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ MTR_{div} &= \text{marginal tax rate on dividends} \\ MTR_{cg} &= \text{marginal tax rate on capital gains} \\ C &= \text {contribution (fixed dollar amount for traditional or Roth)} \\ G_T &= \text {growth factor on traditional balance, before taxes} \\ G_R &= \text {growth factor on Roth balance (tax-free)} \\ G_{Tx} &= \text {growth factor on taxable balance, after taxes} \\ r_T &= \text{total rate of return on the traditional balance} \\ r_R &= \text{total rate of return on the Roth balance} \\ r_{Tx} &= \text{total rate of return on the taxable balance} \\ y &= \text{yield on the taxable balance} \\ v &= \text{growth factor on the taxable balance} \\ b &= \text{growth factor on the taxable basis} \\ t &= \text{time} \\ \end{align} $$

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

$$C \cdot G_T \cdot (1 - MTR_w) + MTR_n \cdot C \cdot G_{Tx} > C \cdot G_R$$

Canceling $$C$$ and solving for $$MTR_w$$ gives:

$$MTR_w < \frac{G_T - G_R + MTR_n \cdot G_{Tx}}{G_T}$$

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:

$$G_T = (1 + r_T)^t$$ $$G_R = (1 + r_R)^t$$ $$G_{Tx} = (v - (v - b) \cdot MTR_{cg})$$

Recall from taxable account performance that:

$$v = \frac{V(t)}{V(0)} = (1 + r_{Tx} - y \cdot MTR_{div})^t$$

and

$$b = \frac{B(t)}{V(0)} = 1 + \left ( \frac{ y \cdot (1-MTR_{div})}{r_{Tx} - y \cdot MTR_{div}} \right ) \left ( (1 + r_{Tx} - y \cdot MTR_{div})^t-1 \right )$$

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 $$(r_T = r_R = r_{Tx} = r)$$ and $$G_T = G_R$$, the equations simplifies somewhat to:

$$MTR_w < MTR_n \cdot \frac{G_{Tx}}{(1 + r)^t}$$

with $$G_{Tx}$$, $$v$$, and $$b$$ the same as above.

Employer match
Define variables as follows:

$$ \begin{align} MTR_n &= \text{marginal tax rate now, for traditional contribution} \\ MTR_w &= \text{marginal tax rate for traditional contributions at withdrawal} \\ m &= \text {employer match rate} \\ T &= \text{traditional balance} \\ R &= \text{Roth balance} \\ A &= \text{after-tax cost of making retirement contributions (traditional or Roth)} \\ G &= \text{growth factor of investments between now and withdrawal} \\ V &= \text{after-tax value of retirement accounts} \\ \end{align} $$

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

$$\Delta T_T = \frac{A}{1 - MTR_n} \cdot (1 + m)$$ $$\Delta R_T = 0$$

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

$$\Delta T_R = A \cdot m$$ $$\Delta R_R = A$$

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

$$\Delta V_T = \frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w)$$ $$\Delta V_R = A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G$$

Traditional contributions are preferred when $$\Delta V_T > \Delta V_R$$:

$$\frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w) > A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G$$

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

$$\frac{1 - MTR_w}{1 - MTR_n} \cdot (1 + m) > m \cdot (1 - MTR_w) + 1 $$

Solving for $$MTR_w$$ using a Computer Algebra System (CAS):

$$MTR_w < \frac{(1+m) \cdot MTR_n}{m \cdot MTR_n + 1}$$

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:

$$\Delta T_T = \frac{A}{1 - MTR_{n,T}} \cdot (1 + m)$$ $$\Delta R_T = 0$$

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

$$\Delta T_R = \frac{A}{1 - MTR_{n,R}} \cdot m$$ $$\Delta R_R = \frac{A}{1 - MTR_{n,R}}$$

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

$$\Delta V_T = \frac{A}{1 - MTR_n} \cdot (1 + m) \cdot G \cdot (1 - MTR_w)$$ $$\Delta V_R = A \cdot m \cdot G \cdot (1 - MTR_w) + A \cdot G$$

Traditional contributions are preferred when $$\Delta V_T > \Delta V_R$$:

$$\frac{A}{1 - MTR_{n,T}} \cdot (1 + m) \cdot G \cdot (1 - MTR_w) > \frac{A}{1 - MTR_{n,R}} \cdot m \cdot G \cdot (1 - MTR_w) + \frac{A}{1 - MTR_{n,R}} \cdot G$$

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

$$\frac{1 - MTR_w}{1 - MTR_{n,T}} \cdot (1 + m) > \frac{m \cdot (1 - MTR_w) + 1}{1 - MTR_{n,R}} $$

Solving for $$MTR_w$$ using a Computer Algebra System (CAS):

$$MTR_w < \frac{(1+m) \cdot (MTR_{n,T} - MTR_{n,R})}{m \cdot (MTR_{n,T} - MTR_{n,R}) +1 - MTR_{n,R}}$$