Module 6 · Fixed Income

Fixed-Income Bond Valuation: Prices and Yields

EN: PV pricing, price–yield relationship, and accrued interest.
VN: Định giá theo PV, quan hệ giá–lợi suất, lãi tích lũy.

1. Bond Price as PV of Cash Flows Core

About: Bond price = PV of all coupon payments + PV of face value at YTM. The single most fundamental fixed-income calculation.Tóm tắt: Giá trái phiếu = PV coupon + PV face. Tính toán FI nền tảng nhất.

\[ P_0 = \sum_{t=1}^{N} \frac{C}{(1 + r)^{t}} + \frac{F}{(1 + r)^{N}} \]

Components

C Periodic coupon = (coupon rate × face) / periods per year.
F Face / par value.
r Periodic discount rate (YTM / periods per year).
N Total number of periods to maturity.

Practice problem

5-year, 4% annual coupon, face $1,000, YTM 5%. Compute price.

Show solution

Annuity factor (5y, 5%) ≈ 4.3295

PV(coupons) = 40 × 4.3295 ≈ 173.18

PV(face) = 1000/1.05^5 ≈ 783.53

Price ≈ $956.71 (discount)

2. Price–Yield Relationship Core

About: Inversely related and convex. Coupon < YTM → price < par (discount). Coupon > YTM → price > par (premium). Exam staple.Tóm tắt: Quan hệ ngược, lồi. Coupon < YTM → discount. Coupon > YTM → premium.

Duration linearizes a convex relationship — convexity adjusts the gap.

Three states

Premium Coupon > YTM → P > Par.
Par Coupon = YTM → P = Par.
Discount Coupon < YTM → P < Par.

Price ↔ yield are inversely related and convex (price rises more for a given yield drop than it falls for the same yield rise).

3. Pricing Between Coupon Dates Core

About: Full (dirty) price = quoted clean price + accrued interest. Accrued = pro-rata coupon since last payment. Buyer pays full; quote is clean.Tóm tắt: Full price = clean + accrued. Accrued = coupon × (days since last/days in period).

\[ \text{Full (dirty) price} = PV_{\text{flat}} \cdot (1 + r)^{t/T} \] \[ \text{Accrued interest} = C \cdot \frac{t}{T} \] \[ \text{Flat (clean) price} = \text{Full price} - \text{Accrued interest} \]

Components

t Days since last coupon.
T Days in the coupon period.

Practice problem

Annual-coupon bond, 30 days into a 365-day period; flat (clean) price = $980, coupon = $50. Compute accrued interest and full price.

Show solution

AI = 50 × 30/365 ≈ $4.11

Full = 980 + 4.11

AI ≈ $4.11; Full price ≈ $984.11

4. Matrix Pricing Concept

About: Estimate yield/price of an illiquid bond using yields of comparable bonds (same credit, similar maturity). Standard for off-the-run issues.Tóm tắt: Ước lượng yield trái phiếu ít giao dịch dùng comparable cùng credit và maturity.

Estimate the YTM/price of an infrequently-traded bond using yields of comparable bonds with similar credit and maturity. Common for off-the-run, illiquid issues.

Practice problem Practice

Practice problem

A 2-year bond has face $1,000, 6% annual coupon, and YTM 4%. Compute the price.

Show solution

Year 1 CF = 60; Year 2 CF = 1,060

PV = 60/1.04 + 1060/1.04$^{2}$

= 57.69 + 980.03

≈ $1,037.72 (premium)