Taxable equivalent yield

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Tax-equivalent yield (TEY) is a method to compare the after-tax return of fixed-income investments that have different rates of taxation. For example, US treasuries have lower rates than high yield savings account but are not subject to state and local income tax.

To compare a product with a lower interest rate but subject to lower tax to a product that has a higher interest rate but higher taxes, you can calculate a TEY.

Calculate using either a calculator or the following formulas.

In-state municipal bond
$$TEY = \frac{ATY}{1 - f - s}$$

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

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

Taxable bonds or high yield savings account
TEY for fully taxable fixed income such as HYSA is the stated yield.

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

Formula Derivation when not itemizing deductions
When buying fixed-income securities (Treasuries, in-state municipal bonds, out-of-state munis, corporate bonds, CDs, etc) or bond or money market funds in taxable accounts, you want to compare yields on an taxable-equivalent basis, since each of these may be taxed differently. What matters is the yield you have left to spend after you pay income taxes. You can compare either after tax yield (ATY) or equivalent before tax yield, which is called taxable-equivalent yield (TEY). We can derive TEY for various types of securities by starting with their ATY, setting them equal, then solving for TEY.

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.

Your yield for these is reduced by the factors shown in parentheses in the above equation. In state muni yield = ATY since there are no taxes.

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, then divide by the factor that you would apply to a fully tax-free security, like an in-state muni.

Formula Derivation when itemizing deductions
If you itemize deductions and can fully deduct state income tax on Schedule A (up to 10k), the equations are slightly different.

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 TFYs 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
 * $$TEY = \frac{Yt}{1-s}$$

Formula Derivation 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 $$1.87% \times 1.096 = 2.05%$$

Comments
Please note an individual's top tax bracket is not necessarily the same as their marginal tax rate. You may need to factor in some of the following:


 * Itemized deductions
 * Additional Medicare Tax
 * ACA net investment income tax
 * State Alternative Minimum Tax
 * Federal Alternative Minimum Tax

General rules
Generally:
 * Higher income tax brackets obtain more benefit from tax-advantaged products. While tax-advantaged fixed income products and municipal bonds may advertise TEY, the AFTER tax yield is most important as each individual's marginal tax rate is different.
 * Computing the TEY becomes more complex if you are near the top of a tax bracket and any additional income would be split across two or more tax brackets. This is where "what if" financial spreadsheet modeling for households can be really valuable for evaluating potential investments.
 * US Treasuries are exempt from state and local income taxes but subject to federal income taxes.
 * In-state municipal bonds are exempt from federal, state, or local income taxes.
 * Out-of-state municipal bonds are exempt from federal income tax but subject to state and local income taxes.
 * Taxable bonds or high yield savings account are subject to federal, state, and local income taxes.