Taxable equivalent yield (math)

 contains supporting analysis and formula derivations for Tax Equivalent Yield.

=Formula Derivation= We can derive $$TEY$$ for various types of securities by starting with their $$ATY$$, setting them equal, then solving for $$TEY$$.

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

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

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

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

An intuitive way to think of this is that you first reduce the yield on the Treasury or out of state muni to it's 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
 * $$TEY = \frac{ATY}{(1-f) \times (1-s)}$$

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

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

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.

(Here I'll just derive the formulas assuming 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 $10K 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 don't know what se will be for 2019, so I just 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,

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

My estimated compound after-tax yield on Fed MM is $$1.87% \times 0.7124 = 1.33%$$

And:

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

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