Yield Calculations

With pricing and cost basis adjustments covered, you're ready to tackle the yield calculations that tie everything together. Taxable equivalent yield is the single most tested municipal bond calculation on the Series 7.

Taxable Equivalent Yield (TEY)

TEY allows you to compare a tax-exempt municipal bond yield to a taxable bond yield on an apples-to-apples basis.

Formula:

TEY=Tax-Exempt Yield1−Marginal Tax Rate\text{TEY} = \frac{\text{Tax-Exempt Yield}}{1 - \text{Marginal Tax Rate}}

Example: 4% muni yield, 32% tax bracket:

TEY = 4% / (1 - 0.32) = 4% / 0.68 = 5.88%

This means a taxable bond must yield at least 5.88% to equal the after-tax return of the 4% muni.

Think of it this way: A muni's 4% yield is tax-free, so you keep all of it. A taxable bond has to earn enough extra to still beat 4% after the IRS takes its cut. The higher your tax bracket, the bigger that cut, so the taxable bond needs an even higher yield to compete.

Key insight: Higher tax brackets make munis more attractive because the TEY is higher; the tax savings are worth more.

Exam Tip: Gotchas

  • TEY formula: divide the muni yield by (1 - tax rate), NOT multiply. Multiplying gives a wrong (lower) number.
  • Higher tax brackets make munis MORE attractive (higher TEY). The exam may present two investors in different brackets and ask who benefits more from the muni.

Computing Tax Bracket from TEY

You can also work the formula backwards to find the breakeven tax bracket between two bonds.

Rearranged formula:

Tax Rate=1−Tax-Exempt YieldTaxable Yield\text{Tax Rate} = 1 - \frac{\text{Tax-Exempt Yield}}{\text{Taxable Yield}}

Example: Muni yields 3.5%, taxable bond yields 5%:

Tax rate = 1 - (3.5 / 5) = 1 - 0.70 = 30%

An investor in the 30% bracket would be indifferent between the two bonds. Investors in higher brackets should choose the muni; investors in lower brackets should choose the taxable bond.

Net Yield After Capital Gains Tax

When a bond is purchased at a discount, the capital gain at maturity is taxable. This reduces the effective yield compared to a bond purchased at par.

  • The exam may require calculating the after-tax gain and expressing it as an annual yield adjustment
  • Always account for the tax bite on any capital gain when comparing discount bonds to par bonds

Current Yield

Formula:

Current Yield (CY)=Annual CouponCurrent Market Price\text{Current Yield (CY)} = \frac{\text{Annual Coupon}}{\text{Current Market Price}}

Example: 5% coupon bond priced at 90 ($900):

CY = $50 / $900 = 5.56%

Key points:

  • Current yield ignores the gain or loss at maturity; it only measures income return relative to the current price
  • Current yield > coupon rate when the bond trades at a discount (you paid less, same coupon)
  • Current yield < coupon rate when the bond trades at a premium (you paid more, same coupon)

Exam Tip: Gotchas

  • Current yield ignores capital gains and losses. It measures income return only. The exam may ask which yield measure accounts for gain/loss at maturity (that would be yield to maturity (YTM), not current yield).
  • At par: coupon = CY = YTM. When a bond trades at par, all three yields are equal.

Yield to Call (YTC) on Premium Bonds

For premium callable bonds, yield to call is the most relevant yield because the issuer is likely to call the bond to refund at a lower rate.

  • YTC is lower than YTM for premium bonds because the call price (usually par) is below the purchase price, and the loss is realized over a shorter period
  • Yield to worst = the lowest of YTM, YTC (to each call date), or yield to put. This is the yield that matters most to investors.

Yield Relationships (The "Seesaw")

This table is frequently tested. Worth remembering the order:

Bond Priced AtRelationship
ParCoupon = CY = YTM
DiscountCoupon < CY < YTM
PremiumCoupon > CY > YTM (and YTC is the lowest of all)

For premium callable bonds, add YTC to the chain:

YTC < YTM < CY < Coupon

Memory Aid: For premium bonds, yields shrink as you add more factors: coupon is the biggest number, current yield adjusts for price, YTM adjusts for the loss at maturity, and YTC adjusts for the shorter time to the call. Each step down = one more drag on your return.

For discount bonds, the issuer is unlikely to call (below-market coupon), so YTC is generally not relevant.

Exam Tip: Gotchas

  • For premium callable munis, yield to call is the relevant yield measure (not YTM), because the issuer will likely call the bond.
  • The exam often asks which yield is "lowest" for a premium bond. YTC is always the lowest for a premium callable bond.

Value of a Basis Point

  • A basis point = 0.01% (1/100 of 1%)
  • 100 basis points = 1.00%
  • The dollar value of a basis point (DV01) measures the change in a bond's price for a 1-basis-point change in yield
  • Longer-maturity and lower-coupon bonds have a higher DV01 (more price sensitivity)
  • DV01 is used to compare price sensitivity across different bonds