8.3 Bond Valuation, Clean Price, and Accrued Interest

Price is present value

The value of an option-free fixed-rate bond equals the present value of its remaining coupons and principal. The discount rate reflects benchmark (default-free) rates for each cash flow's timing plus spreads for credit, liquidity, tax, and embedded-option compensation. With one yield per period, price equals the present value of the coupon annuity plus the present value of principal. With spot rates, each cash flow is discounted at its maturity-matched rate; this is the no-arbitrage approach when the curve is not flat.

Worked example: a 4-year, 5% annual-coupon, USD 1,000 par bond priced to a 6% YTM. Coupons are USD 50. PV of the coupon annuity = 50 x [1 - 1.06^-4] / 0.06 = 50 x 3.4651 = 173.26. PV of principal = 1,000 / 1.06^4 = 792.09. Price = 173.26 + 792.09 = 965.35, a discount because the 5% coupon is below the 6% required yield. On the BA II Plus calculator the same problem is keyed as N = 4, I/Y = 6, PMT = 50, FV = 1,000, then CPT PV, returning -965.35 (the sign convention treats the price paid as a cash outflow).

If the same bond paid coupons semiannually, you would instead set N = 8, I/Y = 3, PMT = 25, FV = 1,000, which is the single most common adjustment the exam tests.

Matching the discount rate to the cash flow's risk matters too. The benchmark yield (a government rate of the same maturity) captures the time value of money and inflation expectations, and the spread captures everything else. Two bonds with identical coupons and maturities can price very differently when one carries a wider spread for weaker credit or thinner liquidity. So price reflects both the promised schedule and the market's required compensation, and a candidate should never read a low price as automatically cheap without checking why the required yield is high.

Premium, discount, and par

A bond trades at par when coupon = required yield; at a premium when coupon > required yield; at a discount when coupon < required yield. The logic is mechanical: if investors require 4% and the bond pays 6%, the coupons are rich, so price exceeds par; if investors require 7% and it pays 5%, price falls below par.

As a conventional bond nears maturity with required yield unchanged and no default, price moves toward par. Premium bonds amortize down to par; discount bonds accrete up to par. This is the constant-yield price trajectory.

Flat (clean) and full (dirty) price

Markets quote the flat price, but the buyer pays the full price. Accrued interest exists because the seller held the bond for part of the coupon period; since the buyer collects the entire next coupon, the buyer reimburses the seller for the elapsed portion.

The formula is AI = coupon payment x (days from last coupon to settlement / days in coupon period). Day-count conventions set the exact day count: actual/actual is standard for US Treasuries and many government bonds, while 30/360 is standard for US corporate and municipal bonds. The exam often supplies the day fraction directly.

Pricing timeline

Suppose a bond pays a USD 30 semiannual coupon and settlement is one-third through the period. AI = 30 x (1/3) = USD 10. If the flat price is USD 980, the full price is USD 990; the buyer pays 990 and later receives the full USD 30 coupon. Accrued interest climbs between coupon dates and resets to zero just after each coupon payment, which is why the quoted flat price strips out this predictable saw-tooth and better reflects yield and credit changes.

Day-count choice changes the day fraction. Under 30/360, every month is treated as 30 days and the year as 360 days, so a corporate bond 60 days into a 180-day period accrues 60/180 of the coupon. Under actual/actual, a Treasury uses the real calendar days elapsed over the real days in the coupon period; the two methods can differ by a few cents per USD 100 of par. A bond trades flat (without accrued interest, AI set to zero) only when it is in default or has stopped paying, a detail credit questions occasionally exploit.

The technically exact full price is the flat price compounded forward from the last coupon date, but Level I typically accepts the linear AI approximation above.

Yield changes and price changes

Bond prices move inversely with required yield: yields up, price down; yields down, price up. The relationship is convex. For an equal-size yield move, the price gain from a yield decline exceeds the price loss from a yield increase. Duration gives the first-order (linear) estimate; convexity refines it.

Structured aid: valuation formulas

Exam focus

Read whether the item wants the quoted price (flat) or the amount paid (full/invoice). If it asks for flat price, exclude accrued interest; if it asks for full price or cash at settlement, include it. A flat quote is a percentage of par: 101.25 on USD 1,000 par is USD 1,012.50 before accrued interest.

For calculator work, align coupon frequency with yield periodicity. A semiannual bond with a 6% stated annual YTM uses 3% per period and twice the number of periods; on the BA II Plus set N, I/Y, PMT, and FV on a per-period basis. Many wrong answers come from discounting annual cash flows when the bond actually pays semiannually.