Module 9 · Fixed Income

The Term Structure of Interest Rates

EN: Spot, par, and forward rates; bootstrapping; yield-curve shapes and theories.
VN: Lãi suất giao ngay, par, kỳ hạn; bootstrap; hình dạng đường cong và lý thuyết.

1. Spot Rate (Zero-Coupon Yield) Core

About: Yield on a zero-coupon bond for that maturity. Discount factor DF_T = 1/(1+S_T)^T. Building block for valuing any cash-flow stream.Tóm tắt: Yield của trái phiếu zero. DF_T = 1/(1+S_T)^T. Khối xây dựng cho định giá.

\[ \text{Discount factor}: DF_T = \frac{1}{(1 + S_T)^{T}} \]

S_T = spot rate for maturity T years; obtained from prices of zero-coupon government bonds.

Practice problem

1-year spot = 3%. Compute discount factor for $100 received in 1 year.

Show solution

DF = 1/1.03 ≈ 0.9709

PV = 100 × 0.9709

PV = $97.09

2. Forward Rate Core

About: Implied future spot rate locked in today, derived from no-arbitrage between spot rates of different maturities. f_{A,B-A} = forward starting at year A for B−A years.Tóm tắt: Tỷ giá kỳ hạn hiện tại đến tương lai, tính từ no-arbitrage giữa các spot. Khóa vào hôm nay.

\[ (1 + S_B)^{B} = (1 + S_A)^{A} \cdot (1 + f_{A,\,B-A})^{B-A} \]

Notation

f_A,B-A Forward rate starting at year A, lasting (B − A) years.
S_A, S_B Spot rates for A-year and B-year maturities.

Practice problem

2-year spot = 4%, 1-year spot = 3%. Compute the implied 1-year forward rate starting in 1 year.

Show solution

(1.04)² = (1.03)(1 + f) → 1.0816 = 1.03(1 + f)

1 + f = 1.0501

f ≈ 5.01%

3. Par Rate Core

About: Coupon rate that prices a bond at par given current spot rates. Building block of the par yield curve.Tóm tắt: Coupon rate khiến giá = par tại spot hiện tại. Khối xây dựng par curve.

The coupon rate that prices a bond at par given current spot rates. Building block of the par yield curve.

4. Yield-Curve Theories Concept

About: Pure expectations (forwards = expected future spots), liquidity preference (positive term premium → upward bias), segmented markets, preferred habitat.Tóm tắt: Expectations, liquidity preference (term premium dương), segmented, preferred habitat.

Pure expectations Forward rate = expected future spot rate.
Liquidity preference Forwards include a positive term premium → upward bias.
Segmented markets Each maturity has its own supply/demand.
Preferred habitat Like segmented but investors will switch maturities for sufficient yield premium.

5. Yield-Curve Shapes Concept

About: Normal (upward, expansion), flat (transition), inverted (recession signal), humped. Inversion = strongest single signal of upcoming recession.Tóm tắt: Normal, flat, inverted (recession), humped. Inversion = signal recession mạnh nhất.

Four common yield-curve shapes; inverted often signals recession.

Upward (normal) Long yields > short — typical, signals expansion.
Flat All maturities similar — transition.
Inverted Short > long — recession signal.
Humped Mid-maturity highest.