--------------------------------- EDIT NOTES ---------------------------------

- May 02, 2018. Original post.
- Oct 24, 2019. Added formulas for funds that are partially exempt from state tax, such as Federal Money Market fund.

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. Of course inflation is a factor too, but here I'll just discuss the tax equivalence aspect.

You can compare either after tax yield, I'll call that TFY for Tax Free Yield, or equivalent before tax yield, which is called taxable-equivalent yield, or TEY. The convention seems to be to do the latter, probably because most yields we see quoted are taxable yields.

We can derive TEY for various types of securities by starting with their TFYs, setting them equal, then solving for TEY. For the non-itemizing case, which probably will be vast majority starting in 2018:

TFY = TEY * (1 - f - s) = Yt * (1 - f) = Ym * (1 - s)

Where f and s = federal and state marginal tax rates, Yt is Treasury yield, and Ym is out of state muni yield (in state muni yield = TFY, since there are no taxes). You pay federal and state taxes on fully taxable securities (TEY), federal taxes on Treasuries, and state taxes on out of state munis. So your yield for these is reduced by the factors shown in parentheses in the above equation.

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

In-state-muni: TEY = TFY / (1 - f - s)

Treasury: TEY = Yt * (1-f) / (1 - f - s)

Out-of-state Muni: TEY = Ym * (1 - s) / (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.

If you itemize deductions and fully deduct state income tax on Schedule A, the equations are slightly different. The after-tax value of a fully taxable security in this case is TFY = TEY * (1 - f - s + f*s), which can also be written as TFY = TEY * [ ( 1-f) * (1-s) ]. The latter form is more convenient in solving the equations for TEY. Again, setting the TFYs equal:

TFY = TEY * [ ( 1-f) * (1-s) ] = Yt * (1 - f) = Ym * (1 - s)

If you can do simple algebra in your head, you can see by inspection that the (1 - f) and (1 - s) terms in the numerators are cancelled out by the same term in the denominator when solving for TEY in terms of Yt or Ym, and we get:

In-state-muni: TEY = TFY / [ ( 1-f) * (1-s) ]

Treasury: TEY = Yt / (1-s)

Out-of-state Muni: TEY = Ym / (1-f)

These are the forms of the TEY equations that we see most often. I assume this is because usually investors who are concerned with fixed-income with some sort of tax exemption are likely to have itemized deductions and fully deducted state income taxes on Schedule A. However, with the income tax laws in effect for 2018, this is less likely to be the case, so the slightly more complicated equations are more likely to be applicable.

Thoughts, inputs or questions?

---------------- Oct 24, 2019 EDIT: Funds that are partially exempt from state income tax -------------------------------

I didn't include the formulas for a fund that is partially exempt from state tax when I first posted this. 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.

These formulas also would be applicable if your state allows a state tax exemption for income from US Government Obligations (USGO) even if less than 50% of the assets were in USGO at each quarter-end for the year, which I think is all states except CA, CT, and NY (I am a CA resident, so I tend to forget about this case). For example, many people 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 (but not residents of CA, CT or NY).

(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 * (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 * (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 * se)

(Note that here I'm using

*ATY*for after-tax yield, while in the original post I used

*TFY*for

*tax-free yield*. They mean the same thing in the formulas here, but I think

*after-tax yield*might be more meaningful to people.)

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

ATY = Ypse * (1 - f - s * (1-se))

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

TEY factor = (1 - f - s * (1-se)) / (1 - f - s)

And TEY is:

TEY = Ypse * TEY factor = Ypse * (1 - f - s * (1-se)) / (1 - f - s)

Example.

Using se = 0.78 for Fed MM, current compound yield of 1.87%, and my estimated marginal tax rates of 27% Fed and 8% state,

ATY factor = (1 - f - s * (1-se)) = (1 - 0.27 - 0.08 * (1 - 0.78)) = 0.7124.

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

And:

TEY factor = ATF factor / (1 - f - s) = (1 - f - s * (1-se)) / (1 - f - s) = 0.7124 / (1 - 0.27 - 0.08) = 1.096.

My estimated compound TEY for Fed MM is 1.87% * 1.096 = 2.05%

Kevin