What are the investment yield and the tax-equivalent yield?

Investment yield is the bond equivalent yield (BEY) measure of return used by the US Treasury to compare the yield on T-bills with bonds maturing on the same date, calculated as a percent of the T-bill’s purchase price on an actual/actual basis.  The discount yield on a T-bill is a percentage of its face value based on a 360-day year, while investment yield on a T-bill is restated in terms of its purchase price as follows, where F is face value, P the purchase price, and t the number of days to maturity:

Investment Yield_T-Bill = [(F − P)/P] x [365 or 366/t]

The investment yield on a T-bond or T-note held to maturity can be approximated with the following formula, where r is the nominal rate, F is face value, P is purchase price, and y is years to maturity:

Investment Yield_T-Bond = [r + [(F − P)/y]/[(F + P)/2]

To convert the effective annual yield (EAY) on an annual-pay bond (annual-pay YTM) to a bond equivalent yield for comparison with the semiannual-pay YTM of T-notes and T-bonds, the following equation is applied:

BEY_EAY = [(1 + EAY)^0.5 – 1] x 2

Interest income from municipal bonds is not subject to US federal income tax and generally not subject to state and local taxation in the state of issuance.  The tax-equivalent yield (TEY) is the pretax yield of a fully taxable bond used to determine the yield it must offer after taxes in order to yield as much as the municipal bond.  The after-tax yield of a taxable bond is compared to the tax-free yield of a municipal bond, where TEY is tax-equivalent yield, MBY is municipal bond yield, and MTR is marginal tax rate:

TEY = MBY/(1 – MTR)

Alternatively, if the taxable bond yield is known, the required minimum yield on the muni can be determined, where MBY is municipal bond yield, TBY is taxable bond yield, and MTR is marginal tax rate:

MBY = TBY x (1 – MTR)