Taxable equivalent yield (math)

 contains supporting analysis and formula derivations for Taxable equivalent yield.

Definitions

 * $$ATY$$ - After Tax Yield
 * $$f$$ - Total federal marginal tax rate, include NIIT, AMT, and any other items that affect your marginal tax rate. Your marginal tax rate is not necessarily the same as your top tax bracket.
 * $$s$$ - Total state marginal tax rate: include state AMT and any other items that affect your actual marginal tax rate.
 * $$TEY$$ - Taxable Equivalent Yield
 * $$tt$$ - Total marginal income tax rate applicable to fully taxable fixed income.
 * $$Y$$ - Yield

When not itemizing deductions
The Tax Cuts and Jobs Act of 2017 capped the state and local tax (SALT) deduction at $10,000 and raised the standard deduction to $12,000 for single filers and $24,000 for married couples filing jointly for tax years 2018-2025. Most people will not derive any income tax benefit with respect to investment income by itemizing deductions, and the derivations in this section are based on this assumption.

We start by considering fully taxable income, for which

$$ATY = Y \times (1-tt) \implies Y = \frac{ATY}{1-tt}$$

where $$Y$$ is the yield and $$tt$$ is the total marginal income tax rate.

For fully taxable income, $$TEY = Y$$, and we can rewrite the equation as:

$$TEY = \frac{ATY}{1-tt}$$

This equation only applies to fully taxable income (for bank accounts, using annual percentage yield (APY) for $$Y$$ gives results that generally are close enough for comparisons).

For example, for marginal income tax rates of 22% federal and 9.3% state, with no NIIT, AMT or other credit phase-ins or phase-outs,

$$tt = f + s$$

$$tt = 22% + 9.3% = 31.3%$$

These marginal tax rates are not the same as the top tax brackets; they include anything that affects the true marginal tax rate; e.g., ACA net investment income tax (NIIT) and Federal Alternative Minimum Tax (AMT), and local income tax. For example, for marginal income tax rates of 35% federal, 3.8% NIIT, and 5% state,

$$tt = (35% + 3.8% + 5%) = 43.8%$$

Next, we determine $$ATY$$ for different types of income, and use that to compute the TEY for those incomes.

In-state municipal bond
In-state municipal bond income is exempt from both federal and state tax.

First we solve for the ATY of an in-state municipal bond. (Note: the after-tax yield is simply the yield in this situation.)

$$ATY = Y \times (1-tt)$$
 * $$ATY = Y$$

where $$Y$$ is the yield of an in-state municipal bond. Then we calculate the TEY for an in-state municipal bond:

$$TEY = \frac{ATY}{1-tt}$$
 * $$TEY = \frac{Y}{1-tt}$$

Note that $$tt$$ is the same in all the TEY equations, so we would use the same $$tt$$ as in the previous examples; e.g., 31.3% or 43.8%.

Out-of-state municipal bond
Out of state municipal bond income is exempt from federal income tax but not state income tax.

First we solve for the ATY of a out-of-state municipal bond:

$$ATY = Y \times (1-tt)$$
 * $$tt = s$$
 * $$ATY = Y \times (1-s)$$

where $$Y$$ is the yield of an out-of-state municipal bond. Then we calculate the TEY for an out-of-state municipal bond:

$$TEY = \frac{ATY}{1-tt}$$
 * $$ATY = Y \times (1-s)$$
 * $$TEY = \frac{Y \times (1-s)}{1-tt}$$

Treasuries
Treasury bond income is exempt from state and local income tax but not federal income tax.

First we solve for the ATY of a Treasury bond:

$$ATY = Y \times (1-tt)$$
 * $$tt = f$$
 * $$ATY = Y \times (1-f)$$

where $$Y$$ is the yield of a Treasury bond. Then we calculate the TEY for a Treasury bond:

$$TEY = \frac{ATY}{1-tt}$$
 * $$ATY = Y \times (1-f)$$
 * $$TEY = \frac{Y \times (1-f)}{1-tt}$$

When itemizing deductions
If you itemize deductions and can fully deduct state income tax on Schedule A, the equations are slightly different. Historically, these are the formulas seen most often in TEY calculations. Thus the slightly more complicated equations in the above section are more likely to be applicable.

The after-tax value of a fully taxable security as derived above:

$$TEY = \frac{ATY}{1-tt} \implies ATY = TEY \times (1-tt)$$
 * $$tt = f + s - f \times s$$ where $$f \times s$$ is deducting state income tax from your federal tax
 * $$ATY = TEY \times (1-(f + s - f \times s))$$
 * $$ATY = TEY \times (1-f-s+f \times s)$$
 * $$\implies ATY = TEY \times [(1-f) \times (1-s)]$$
 * $$\implies TEY = \frac{ATY}{(1-f)\times (1-s)}$$

In-state municipal bond
In-state municipal bond income is exempt from both federal and state tax.

First we solve for the ATY of an in-state municipal bond. (Note: the after-tax yield is simply the yield in this situation.)

$$ATY = Y \times (1-tt)$$
 * $$ATY = Y$$

where $$Y$$ is the yield of an in-state municipal bond. Then we calculate the TEY for an in-state municipal bond using the derived formula above:

$$TEY = \frac{ATY}{(1-f)\times (1-s)}$$
 * $$TEY = \frac{Y}{(1-f)\times (1-s)}$$

Out-of-state municipal bond
Out of state municipal bond income is exempt from federal income tax but not state income tax.

First we solve for the ATY of a out-of-state municipal bond:

$$ATY = Y \times (1-tt)$$
 * $$tt = s$$
 * $$ATY = Y \times (1-s)$$

where $$Y$$ is the yield of an out-of-state municipal bond. Then we calculate the TEY for an out-of-state municipal bond:

$$TEY = \frac{ATY}{(1-f)\times (1-s)}$$
 * $$ATY = Y \times (1-s)$$
 * $$TEY = \frac{Y \times (1 - s)}{(1-f) \times (1-s)}$$
 * $$TEY = \frac{Y}{1-f}$$

Treasuries
Treasury bond income is exempt from state and local income tax but not federal income tax.

First we solve for the ATY of a Treasury bond:

$$ATY = Y \times (1-tt)$$
 * $$tt = f$$
 * $$ATY = Y \times (1-f)$$

where $$Y$$ is the yield of a Treasury. Then we calculate the TEY for a Treasury:

$$TEY = \frac{ATY}{(1-f)\times (1-s)}$$
 * $$ATY = Y \times (1-f)$$
 * $$TEY = \frac{Y \times (1 - f)}{(1-f) \times (1-s)}$$
 * $$TEY = \frac{Y}{1-s}$$

For funds partially exempt from state income tax
Some funds are partially exempt from state tax. For the tax year 2018, Vanguard Federal Money Market Fund (VMFXX) had about 78% of income exempt from state income. (U.S. government obligations information: Important tax information for 2018). Since the Federal Money Market fund is the settlement fund in a Vanguard Brokerage account, this is a fund that many people might want to be able to compare to other funds on a taxable-equivalent basis.

Notably, California, Connecticut, and New York require that 50% of the fund’s assets at each quarter-end within the tax year consist of U.S. government obligations (USGO) for a state tax exemption. For example, in 2018, residents of all states except California, Connecticut, and New York would get a partial state tax exemption on a Prime Money Market fund holding, which had about 28% of income from USGO in 2018, and about 37% of assets in USGO on Dec 31, 2018.

For 2022, however, unlike 2018, VMFXX did not meet the 50% requirement for California, Connecticut, and New York, so fund holders in these states receive no state tax exemption. This highlights the risk in assuming the state tax exemption percentage for the current year.

From here, the formulas assume no federal deduction for state income tax on marginal income, which is the most common case, either because of the high standard deduction or hitting the $10,000 deduction limit for SALT.

For such a fund, the state tax rate on partially state exempt income is $$s \times (1 - se)$$, where $$s$$ is the marginal state tax rate and $$se$$ is the state-tax-exempt portion of fund income. For example, for 2018, $$se$$ was about 78% for the Federal Money Market fund, so $$1-se = 1 - 0.78 = 0.22$$; i.e., your state would tax about 22% of your Federal Money Market fund income. We do not know what $$se$$ will be for 2019, so we assume it will be the same as for 2018 in estimating TEY for 2019.

So the after-tax yield (ATY) factor for income that is partially exempt from state tax is:

$$ATY factor = (1 - f - s \times (1-se))$$

where $$f$$ = marginal federal tax rate, $$s$$ = marginal state tax rate, and $$se$$ = state-exempt percentage of income.

This can also be written as:

$$ATY factor = (1 - f - s + s \times se)$$

You multiply the partially-state-tax-exempt fund yield ($$Ypse$$) by the ATY factor to get the after tax yield:

$$ATY = Ypse \times (1 - f - s \times (1-se))$$

As derived in the original post, the $$TEY factor$$ is just the $$ATY factor$$ divided by $$(1 - f - s)$$:

$$TEY factor = \frac{(1 - f - s \times (1-se))}{(1 - f - s)}$$

And $$TEY$$ is:

$$TEY = Ypse \times TEY factor = Ypse \times \frac{(1 - f - s \times (1-se))}{(1 - f - s)}$$

Example calculation
Using $$se = 0.78$$ for the Federal Money Market fund, current compound yield of 1.87%, and estimated marginal tax rates of 27% Federal and 8% state,

$$ATY factor = (1 - f - s \times (1-se)) = (1 - 0.27 - 0.08 \times (1 - 0.78)) = 0.7124$$

The estimated compound after-tax yield on Federal Money Market fund is $$1.87% \times 0.7124 = 1.33%$$.

And:

$$TEY factor = \frac{ATY factor}{(1 - f - s)} = \frac{(1 - f - s \times (1-se))}{(1 - f - s)} = \frac{0.7124}{(1 - 0.27 - 0.08)} = 1.096$$

The estimated compound TEY for the Federal Money Market fund is $$1.87% \times 1.096 = 2.05%$$.