Tax-equivalent yield

A tax-exempt yield and a taxable yield cannot be compared as printed, because one of them still has tax to pay. Tax-equivalent yield puts them on the same after-tax footing: it is the taxable quote that, after tax, matches the exempt quote. The identity is $y_{te} = y_{ex}/(1-t)$, where $t$ is the marginal tax rate that would apply to the taxable alternative, not the average rate on all income. A municipal coupon that displaces income at the top of the schedule is saved at the top rate. Using the average rate understates the equivalent yield and makes the municipal look worse than the identity says.

The check is to multiply back. The equivalent times one minus the tax rate must return the exempt yield. At a 20 percent rate, a 4 percent tax exempt yield is equivalent to 5 percent taxable, because 5 percent times 0.80 is 4 percent. Lower brackets shrink the gap, which is why the same municipal quote is worth less, in equivalent-yield terms, to a holder who is not in a high band. The tax-equivalent yield calculator is that conversion.

State tax, credit risk and selling at a gain all sit outside the identity. Exemption is granted by a particular level of government: municipal interest that escapes federal income tax in the United States may still be taxed by a state. A 4 percent municipal is not the same as a 5 percent Treasury just because the tax identity matches. One of them can default. APR against APY is a compounding conversion; this is a tax conversion. They are not interchangeable. Do the compounding step first so both yields are effective annual, then this step so both are after tax.