8.3 Bond Valuation, Clean Price, and Accrued Interest
Key Takeaways
- An option-free bond's value is the present value of its promised cash flows discounted at appropriate market rates.
- The full price equals the clean price plus accrued interest.
- Accrued interest compensates the seller for coupon interest earned since the last coupon date.
- Bond prices move inversely with required yield, and the percentage price change is larger when yields fall than when yields rise by the same amount.
- Candidates should be fluent with par, premium, discount, accrued interest, full price, and quoted clean price relationships.
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 should reflect default-free rates for the timing of cash flows plus compensation for credit risk, liquidity risk, taxes, and other required spreads.
For a level-coupon bond valued 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 the rate matching its maturity. The spot-rate method is the no-arbitrage approach when the curve is not flat.
Premium, discount, and par
A bond trades at par when its coupon rate equals the market required yield for comparable risk and maturity. A bond trades at a premium when its coupon rate is above the required yield. It trades at a discount when its coupon rate is below the required yield.
This relationship is mechanical. If investors require 4 percent and the bond pays 6 percent, the promised coupons are attractive, so the price rises above par. If investors require 7 percent and the bond pays 5 percent, the coupons are weak, so the price falls below par.
As a conventional bond approaches maturity, its price tends to move toward par if required yield is unchanged and there is no default. Premium bonds pull down toward par. Discount bonds pull up toward par. Coupon payments and day-count effects can create smaller interim movements.
Clean and full price
Bond markets usually quote a clean price. The clean price excludes accrued interest. The buyer pays the full price, also called the dirty invoice price. The full price equals clean price plus accrued interest.
Accrued interest exists because the bond seller owned the bond for part of the coupon period. The buyer receives the next full coupon, so the buyer compensates the seller for interest earned from the last coupon date through settlement.
The basic formula is: Accrued interest = coupon payment x (days since last coupon / days in coupon period). Day-count conventions define the exact day calculation. Level I questions often provide the fraction or the relevant days.
Pricing timeline
Suppose a bond pays a semiannual coupon of USD 30. If settlement occurs one-third of the way through the coupon period, accrued interest is USD 10. If the clean price is USD 980, the full price is USD 990. The buyer pays USD 990 and later receives the next full USD 30 coupon.
Accrued interest rises between coupon dates and drops to zero after coupon payment. The clean price removes this predictable coupon accrual pattern, so quoted prices better reflect changes in market yield and credit conditions.
Yield changes and price changes
Bond prices move inversely with required yield. If market yields rise, existing fixed cash flows are discounted at higher rates and the price falls. If market yields fall, the price rises. This inverse relation is central to fixed-income risk.
The relation is convex, not a straight line. For the same size yield move, the price gain from a yield decline is larger than the price loss from an equal yield increase, all else equal. Duration gives the first approximation. Convexity refines it.
Structured aid: valuation formulas
| Concept | Formula or rule |
|---|---|
| Full price | Clean price + accrued interest |
| Accrued interest | Coupon payment x elapsed days / coupon-period days |
| One-yield bond value | PV of coupons + PV of principal |
| Spot-rate bond value | Sum of each cash flow discounted by its matching spot rate |
| Price-yield relation | Yield up, price down; yield down, price up |
Exam focus
Read whether the question asks for quoted price or amount paid. If it asks for clean price, exclude accrued interest. If it asks for full price, invoice price, or cash paid at settlement, include accrued interest.
For calculator work, align coupon frequency and yield periodicity. A semiannual bond with an annual stated YTM of 6 percent uses 3 percent per half-year and twice as many periods. Many wrong answers come from using annual cash flows when the bond pays semiannually.
Also be careful with settlement timing. The clean price may be quoted as a percentage of par. A clean quote of 101.25 on USD 1,000 par is USD 1,012.50 before accrued interest. Add accrued interest to determine the full invoice amount.
A bond has a clean price of 98.40 and accrued interest of 1.15, both quoted as a percentage of par. The full price is closest to:
A conventional fixed-rate bond trades at a premium to par when its coupon rate is:
Accrued interest is paid by the bond buyer to the seller because the: