Quick Answer
Municipal bonds price as a percentage of par, with accrued interest calculated on a 30/360 basis. Premium is amortized (not deductible), OID accretes tax-free, and market discount accretes as taxable ordinary income. For callable bonds, always quote the lower of yield to call or yield to maturity, and the taxable equivalent yield formula (divide by one minus the tax rate) is the most heavily tested muni calculation.
This unit covers how municipal bonds are priced, how accrued interest is calculated, and how yields are computed. The taxable equivalent yield (TEY) formula is one of the most frequently tested concepts on the Series 7.
How Is Dollar Price Quoted?
- Quoted as a percentage of par; a price of 102 means the bond trades at $1,020 per $1,000 par
- Premium bond: Dollar price above 100 (above par)
- Discount bond: Dollar price below 100 (below par)
- Inverse relationship: When yields rise, bond prices fall; when yields fall, bond prices rise
How Is Accrued Interest Calculated (30/360 Day Count)?
Municipal bonds use the 30/360 day count basis (the MSRB calculation-of-interest rule):
- Each month is treated as 30 days; each year is 360 days
- Accrued interest is calculated from the last interest payment date up to (but not including) the settlement date
Accrued Interest Formula:
- Buyer pays accrued interest to the seller on settlement
- The buyer will then receive the full next coupon payment (recovering the accrued interest paid)
Odd first coupon:
- If the first coupon period is longer or shorter than 6 months, accrued interest is adjusted accordingly
- A long first coupon (more than 6 months) accrues more interest
- A short first coupon (less than 6 months) accrues less
Example: A $5,000 face amount of a 4% municipal bond with a January 1 and July 1 payment schedule. Trade settles April 1.
- Days accrued: January (30) + February (30) + March (30) = 90 days
- Annual coupon: $5,000 x 4% = $200
- Accrued interest: (90 / 360) x $200 = $50
- Buyer pays the seller $50 in accrued interest at settlement
Exam Tip: Gotchas
- Municipal and corporate bonds both use 30/360 day count (every month = 30 days). U.S. government bonds use actual/actual (actual days in month, 365-day year). This distinction is a common exam trap.
- The buyer pays accrued interest to the seller, then recoups it when the next full coupon payment arrives. The seller does not "keep" interest earned during their holding period.
How Is Premium Amortized?
When a municipal bond is purchased at a premium in the secondary market:
- The premium must be amortized (reduced) over the remaining life of the bond
- Amortization reduces the bondholder's cost basis each year
- For tax-exempt municipal bonds, the amortized premium cannot be deducted as a loss (because the interest income is already tax-free)
- At maturity, the adjusted basis equals par, so there is no capital loss
- Straight-line annual amortization (the Series 7 method): premium / years to maturity
Example: Buy a 5% municipal bond at 108 with 10 years to maturity: cost = $1,080, premium = $80.
- Annual amortization: $80 / 10 = $8 per year
- After 1 year: adjusted basis = $1,080 - $8 = $1,072
- After 5 years: adjusted basis = $1,080 - $40 = $1,040
- At maturity: adjusted basis = $1,000 (no gain or loss)
Exam Tip: Gotchas
- Premium amortization on tax-exempt munis is NOT deductible. Since the interest income is tax-free, you cannot also deduct the premium. The amortization only adjusts your cost basis downward.
How Is Discount Accreted?
Two different types of discount receive different tax treatment:
| Discount Type | Tax Treatment | Cost Basis Effect |
|---|---|---|
| Original issue discount (OID) | Accretion treated as tax-exempt interest income (for tax-exempt munis) | Increases basis annually |
| Market discount (purchased below par in secondary market) | Accreted discount taxed as ordinary income (taxable) at sale or maturity | May be accreted annually or recognized at disposition |
- OID accretion is tax-free for municipal bonds, which is a significant advantage
- Market discount is NOT tax-free; this is a frequent exam trap
- Straight-line annual accretion (the Series 7 method): OID / years to maturity
Example: Buy a new-issue municipal bond at 92 with 20 years to maturity: cost = $920, OID = $80.
- Annual accretion: $80 / 20 = $4 per year
- After 5 years: adjusted basis = $920 + $20 = $940
- If sold at 96 ($960) after 5 years: $960 - $940 = $20 capital gain
- At maturity: adjusted basis = $1,000 (no gain)
Exam Tip: Gotchas
- OID on munis accretes as tax-exempt income; market discount accretes as taxable ordinary income. The difference depends on whether the discount existed at issuance (OID) or was created by market price changes after issuance (market discount).
How Do Maturity and Coupon Affect Price Sensitivity?
| Factor | Price Sensitivity |
|---|---|
| Longer maturity | Greater price sensitivity to interest rate changes (more duration risk) |
| Lower coupon | Greater price sensitivity to interest rate changes |
| Higher coupon | Less price sensitivity (more cash flow returned sooner) |
Think of it this way: A bond with a long maturity and low coupon gives you very little cash flow early on, so most of your return depends on what happens far in the future. That makes it highly sensitive to interest rate changes. Long maturity + low coupon = maximum price volatility.
Exam Tip: Gotchas
- A zero-coupon bond has the highest price volatility of any bond with the same maturity, since it has no coupon at all to cushion the price against rate moves.
- As a bond approaches maturity, its price converges toward par (assuming no default), a pattern called "pull to par."
How Are the Different Yields Calculated?
| Yield Type | Formula/Description |
|---|---|
| Current yield | Annual coupon / Market price |
| Yield to maturity (YTM) | Total return assuming bond held to maturity (accounts for coupon, price, and time) |
| Yield to call (YTC) | Total return assuming bond is called at the first call date |
| Taxable equivalent yield (TEY) | Tax-exempt yield / (1 - Marginal tax rate) |
| Net yield after capital gains tax | Adjusts yield for any capital gains tax owed at maturity on a discount bond |
Which yield to quote for callable bonds:
- For premium callable bonds: YTC < YTM, so quote the lower yield (YTC is the worst case)
- For discount callable bonds: YTM < YTC, so quote the lower yield (YTM is the worst case)
- Always quote the yield that represents the worst case for the investor
- Yield to worst = the lowest of YTM, YTC to every call date, and yield to put; this is the figure that matters most to an investor holding a bond with multiple redemption features
The yield "seesaw" at par, discount, and premium:
| Bond Priced At | Relationship |
|---|---|
| Par | Coupon = Current yield = YTM |
| Discount | Coupon < Current yield < YTM |
| Premium | Coupon > Current yield > YTM |
Basis point: 1/100th of 1% (0.01%). A move from 3.50% to 3.55% is 5 basis points. The dollar value of a basis point (DV01) measures how much a bond's price moves for a 1-basis-point change in yield; longer-maturity, lower-coupon bonds have a higher DV01 because they are more price-sensitive.
Exam Tip: Gotchas
- For callable bonds, always quote the LOWER yield (worst case for the investor). Premium callable bonds: quote yield to call (YTC). Discount callable bonds: quote yield to maturity (YTM).
- TEY formula divides by (1 - tax rate). A common wrong answer multiplies tax-exempt yield by the tax rate instead.
- For a premium callable bond, YTC is always the lowest figure in the seesaw: YTC < YTM < current yield < coupon rate. The exam frequently asks which yield is lowest for a premium bond.
How Do You Calculate Taxable Equivalent Yield (TEY)?
TEY Formula:
This converts a municipal bond's tax-free yield to the equivalent yield a taxable bond would need to offer.
Example: An investor in the 32% federal tax bracket is considering a municipal bond yielding 4%.
- TEY = 4% / (1 - 0.32) = 4% / 0.68 = 5.88%
- The investor would need a taxable bond yielding at least 5.88% to match the 4% muni after taxes
With state tax: If the investor is also in a 6% state bracket and the muni is in-state (triple tax-free):
- Combined rate: 32% + 6% = 38%
- TEY = 4% / (1 - 0.38) = 4% / 0.62 = 6.45%
Working backward: computing the tax bracket from TEY
- Rearranged formula: Tax rate = 1 - (Tax-exempt yield / Taxable yield)
- Example: a muni yields 3.5%, a comparable 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
Exam Tip: Gotchas
- TEY is the most frequently tested muni math concept. Worth memorizing: TEY = Tax-exempt yield / (1 - Tax rate).
- For in-state munis (triple tax-free), combine federal and state tax rates in the denominator. Out-of-state munis only use the federal rate.
- To find the break-even tax bracket, divide the muni yield by the taxable yield and subtract from 1. Do not confuse this with the forward TEY formula.
How Do Bonds in Default Trade?
- A bond in default trades flat (without accrued interest)
- The buyer does not pay accrued interest to the seller
- Income bonds (which pay interest only if earned) also trade flat
What Should You Check on Exam Day?
- Accrued interest uses 30/360: each month = 30 days, each year = 360 days, up to but not including settlement
- Premium amortization on tax-exempt munis is not deductible; at maturity, basis equals par (no gain or loss)
- OID accretes tax-exempt; market discount accretes as taxable ordinary income, a frequent exam trap
- For callable bonds, always quote the lower yield (YTC for premium, YTM for discount); yield to worst covers YTM, every YTC, and yield to put
- TEY = tax-exempt yield / (1 - tax rate); combine federal and state rates only for in-state (triple tax-free) bonds
- Reverse TEY finds the break-even tax bracket: 1 - (muni yield / taxable yield)
- Zero-coupon bonds have the highest price volatility; all bonds pull to par as maturity nears
- Defaulted and income bonds trade flat, with no accrued interest changing hands