Taxable equivalent yield (math)

 contains supporting analysis and formula derivations for User:Daw007/Tax equivalent yield.

Formula Derivation
We can derive the TEY of a particular type of fixed income starting with the following equation:

$$ATY = TEY \times tt$$

where tt is the total marginal income tax rate applicable to fully taxable fixed income. For the non-itemizing case, which probably will be vast majority under the tax laws as of 2023, tt is the same for all types of fixed income:

$$tt = (1 - f - s)$$

where f = federal marginal tax rate and s = state marginal tax rate. These marginal tax rates are not the same as the top tax bracket; they include anything that affects the true marginal tax rate; e.g., f includes NIIT and AMT if applicable. If you pay local income tax, you'd include that in s, or add another term for it.

Solving for TEY,

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

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

For a fully taxable fixed income, TEY is simply the yield:

$$TEY = Y$$

$$ATY = TEY \times (1-tt)$$

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

$$ATY = Y \times (1-f-s)$$

where Y = yield to maturity. Note that for bank accounts, using APY for yield gives results that generally are close enough for comparisons.

For in-state muni income, which is exempt from federal and state tax, ATY is simply the yield, so:

$$ATY = Y$$

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

$$TEY = \frac{Y}{1 - f - s}$$

For out of state muni income, which is exempt from federal income tax but not from state income tax:

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

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

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

For Treasury income, which is exempt from state and local income tax:

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

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

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

Definitions

 * $$ATY$$ - After Tax Yield
 * $$Ym$$ - out of state muni yield
 * $$Yt$$ - Treasury yield
 * $$f$$ - Total federal marginal tax rate, include NIIT, AMT, and any other items that affect your marginal tax rate. Marginal tax rate is not necessarily the same as an individual's top tax bracket.
 * $$s$$ - Total state marginal tax rate: include state AMT and any other items that affect your actual marginal tax rate.
 * $$TEY$$ - Tax Equivalent Yield
 * $$tt$$ - total marginal income tax rate

When not itemizing deductions
For the non-itemizing case:

$$ATY = TEY \times (1 - f - s) = Yt \times (1 - f) = Ym \times (1 - s)$$


 * Fully Taxable Securities pay both federal and state taxes ($$TEY$$).
 * Treasuries pay only federal taxes and are exempt from state and local taxes.
 * Out of State Munis pay state and local taxes and are exempt from federal taxes.
 * In state muni yield = $$ATY$$ since there are no taxes.

Your yield for these is reduced by the factors shown in parentheses in the above equation.

Solving this equation for $$TEY$$ in terms of $$ATY$$, $$Yt$$ and $$Ym$$:

In-state-muni:
 * $$ATY = Ym$$
 * $$TEY = \frac{ATY}{(1 - f - s)}$$
 * $$TEY = \frac{Ym}{(1 - f - s)}$$

Out-of-state Muni:
 * $$ATY = Ym \times (1 - s)$$
 * $$TEY = \frac{ATY}{(1 - f - s)}$$
 * $$TEY = \frac{Ym \times \left(1 - s\right)}{1 - f - s}$$

Treasury:
 * $$ATY = Yt \times (1-f)$$
 * $$TEY = \frac{ATY}{(1 - f - s)}$$
 * $$TEY = \frac{Yt \times \left(1 - f \right)}{1 - f - s}$$

An intuitive way to think of this is that you first reduce the yield on the Treasury or out of state muni to its after-tax value (or $$ATY$$), then divide by the factor that you would apply to a fully tax-free security, like an in-state muni.

When itemizing deductions
If you itemize deductions and can fully deduct state income tax on Schedule A, the equations are slightly different. Historical, these are the formulas seen most often in TEY calculations. However, the Tax Cuts and Jobs Act of 2017 capped the state and local tax (SALT) deduction at $10,000 and raised the standard deduction for tax years 2018-2025. Because of this, fewer people have incentive to itemize and thus the slightly more complicated equations are more likely to be applicable.

The after-tax value of a fully taxable security in this case is


 * $$ATY = TEY \times (1 - f - s + f \times s)$$


 * which can also be written as


 * $$ATY = TEY \times [(1-f) \times (1-s)]$$

Again, setting the $$ATY$$s equal and then solving for $$TEY$$ for an instate muni, out-of-state muni, and treasury:

$$ATY = TEY \times [(1-f) \times (1-s)] = Yt \times (1 - f) = Ym \times (1 - s)$$

In-state-muni:
 * $$ATY = TEY \times (1-f) \times (1-s)$$
 * simplifies to
 * $$TEY = \frac{ATY}{(1-f) \times (1-s)}$$


 * for in-state-muni: $$ATY = Ym$$


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

Out-of-state Muni:
 * $$ATY = TEY \times (1-f) \times (1-s)$$
 * simplifies to
 * $$TEY = \frac{ATY}{(1-f) \times (1-s)}$$


 * for out-of-state-muni: $$ATY = Ym \times (1 - s)$$
 * $$TEY = \frac{Ym \times (1 - s)}{(1-f) \times (1-s)}$$
 * $$TEY = \frac{Ym}{1-f}$$

Treasury:
 * $$ATY = TEY \times (1-f) \times (1-s)$$
 * simplifies to
 * $$TEY = \frac{ATY}{(1-f) \times (1-s)}$$


 * for Treasuries: $$ATY = Yt \times (1 - f)$$
 * $$TEY = \frac{Yt \times (1 - f)}{(1-f) \times (1-s)}$$
 * $$TEY = \frac{Yt}{1-s}$$

For funds partially exempt from state income tax
Some funds are partially exempt from state tax. This would be the case for anyone who pays state income tax on income from Vanguard Federal Money Market (Fed MM) fund, which had about 78% of income exempt from state income tax for tax year 2018 (U.S. government obligations information: Important tax information for 2018). Since Fed MM is the settlement fund in a Vanguard Brokerage account, this is 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, residents of all states except CA, CT, and NY would get a partial state tax exemption on Prime MM, which had about 28% of income from USGO in 2018, and about 37% of assets in USGO on Dec 31, 2018.

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 Fed MM, so $$1-se = 1 - 0.78 = 0.22$$; i.e., your state would tax about 22% of your Fed MM 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 fed 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.

Using $$se = 0.78$$ for Fed MM, current compound yield of 1.87%, and an estimated marginal tax rates of 27% Fed 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 Fed MM is $$1.87% \times 0.7124 = 1.33%$$

And:

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

The estimated compound TEY for Fed MM is $$1.87% \times 1.096 = 2.05%$$