Level I · Fixed Income

Learning Module 6
Fixed-Income Bond Valuation

Key Outcomes Summary & Practice Problems

Learning Outcomes

What you must be able to do

Curriculum Year: 2026

LOS 1

Calculate a bond's price given a yield‑to‑maturity on or between coupon dates — including full price, flat price, and accrued interest using both 30/360 and actual/actual day counts.

LOS 2

Identify the relationships among a bond's price, coupon rate, maturity, and yield‑to‑maturity — inverse relationship, coupon effect, maturity effect, constant‑yield price trajectory, and convexity.

LOS 3

Describe matrix pricing — estimating the price of illiquid or new bonds using prices of comparable bonds with similar maturity, coupon, and credit quality.

1 · Bond Pricing with a Market Discount Rate

    • Market discount rate (required yield) is the rate of return required by investors given the risk of the bond.

    • Price on a coupon date: PV = Σ PMT/(1+r)ⁿ + FV/(1+r)ᴺ

    • Par bond: Coupon rate = Market discount rate → Price = 100.

    • Discount bond: Coupon rate < Market discount rate → Price < 100.

    • Premium bond: Coupon rate > Market discount rate → Price > 100.

PRICE

PV = PMT/(1+r)¹ + PMT/(1+r)² + … + (PMT + FV)/(1+r)ᴺ
where r = market discount rate per period, N = number of periods

Yield‑to‑Maturity (YTM)

    • The internal rate of return (IRR) that equates the bond's price to the present value of its cash flows.

    • YTM assumptions: (1) hold to maturity, (2) all payments made as scheduled (no default), (3) coupons reinvested at the YTM.

    • If any assumption fails, the realized return will differ from the YTM.

2 · Flat Price, Accrued Interest, and Full Price

    • Full price (dirty price): Price actually paid by the buyer, including accrued interest.

    • Flat price (clean price): Quoted price — full price minus accrued interest.

    • Accrued interest (AI): Portion of the next coupon owed to the seller. AI = (t/T) × PMT

    • Day counts: 30/360 (corporate bonds) and actual/actual (government bonds).

    • Full price between coupons: PVFull = PV × (1 + r)t/T, where PV is the present value as of the last coupon date.

    • PVFull = PVFlat + AI

ACCRUED INTEREST

AI = (t / T) × PMT
t = days from last coupon to settlement; T = days in coupon period

FULL PRICE

PVFull = PV × (1 + r)t/T
where PV = present value as of last coupon date

3 · Relationships Between Bond Prices and Bond Features

Inverse Relationship

    • Bond prices and yields move in opposite directions. Higher discount rate → lower present value.

Coupon Effect

    • For the same maturity, lower‑coupon bonds have greater percentage price changes for a given change in yield.

    • Zero‑coupon bonds have the highest price sensitivity for a given maturity and yield.

Maturity Effect

    • For the same coupon rate, longer‑term bonds have greater percentage price changes for a given change in yield.

    • Exceptions exist only for low‑coupon (but not zero‑coupon), long‑term bonds trading at a discount.

    • The maturity effect always holds for zero‑coupon bonds and bonds priced at par or premium.

Constant‑Yield Price Trajectory

    • Bond prices are "pulled to par" as maturity approaches, assuming the YTM remains constant.

    • Discount bonds → price increases toward par.

    • Premium bonds → price decreases toward par.

Convexity Effect

    • The price‑yield relationship is non‑linear (convex).

    • Percentage price increase for a yield decrease is greater than the percentage price decrease for the same yield increase.

    • This is positive convexity for option‑free bonds.

Feature

Impact on Price Sensitivity

Lower coupon rate

↑ Greater price sensitivity

Longer maturity

↑ Greater price sensitivity (generally)

Yield increase

↓ Price decreases (convex: less than duration estimate)

Yield decrease

↑ Price increases (convex: more than duration estimate)

Time passes (constant yield)

Price pulled toward par

4 · Matrix Pricing

    • Used to estimate the price of illiquid or newly issued bonds when no current market price exists.

    • Compare to frequently traded bonds with similar:

      • • Time‑to‑maturity

      • • Coupon rate

      • • Credit quality

    • Calculate the yields of comparable bonds, then interpolate to estimate the yield for the target bond.

    • Use the estimated yield to calculate the bond's price.

    • Also used in underwriting to estimate the required yield spread over the benchmark rate.

1
Step 1

Identify comparable bonds (similar maturity, coupon, credit quality).

2
STEP 2

Calculate YTM for each comparable bond.

3
STEP 3

Interpolate to estimate YTM for target bond.

4
STEP 4

Calculate price using estimated YTM.