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| Background concepts for fixed income instruments: zero-coupon bonds, coupon bonds, floating rate bonds, swaps, duration, and convexity — the foundational mechanics underlying all fixed income strategies. |
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Fixed Income Generalities
Section: 5.1 | Asset Class: Fixed Income | Type: Background / Reference
Overview
Fixed income instruments are promises to pay cash flows at future dates, priced today as the present value of those flows. The yield of a bond summarizes its return as a single annualized rate. Duration and convexity characterize how bond prices respond to interest rate changes, and are the primary risk-management tools for fixed income portfolios.
Construction / Mechanics
5.1.1 Zero-Coupon Bonds
A zero-coupon (discount) bond with maturity T pays $1 at time T. Its price at time t is P(t,T), with P(T,T) = 1. The continuously compounded yield is:
R(t,T) = -ln(P(t,T)) / (T - t) (374)
5.1.2 Coupon Bonds
A coupon bond pays principal $1 at maturity T plus n coupon payments of amount kδ at times T_i = T_0 + iδ (i = 1,...,n), where δ is the payment period. Price at time t:
P_c(t,T) = P(t,T) + kδ Σ_{i=I(t)}^n P(t,T_i) (375)
where I(t) = min(i : t < T_i). At issuance (t = T_0), the par coupon rate is:
k = (1 - P(T_0,T)) / (δ Σ_{i=1}^n P(T_0,T_i)) (377)
5.1.3 Floating Rate Bonds
Coupon payments are based on LIBOR. The LIBOR rate at T_{i-1} for period [T_{i-1}, T_i] is:
L(T_{i-1}) = (1/δ) [1/P(T_{i-1},T_i) - 1] (378)
The coupon paid at T_i is X_i = L(T_{i-1})δ = 1/P(T_{i-1},T_i) - 1. The total value at T_0:
V_0 = 1 - [P(T_0,T_n) - P(T_0,T)] (380)
If T = T_n then V_0 = 1 (the bond prices at par).
5.1.4 Swaps
An interest rate swap exchanges fixed rate payments for floating (LIBOR) payments. A long swap = long fixed coupon bond + short floating rate bond. The fixed rate giving zero initial value:
k = (1 - P(T_0,T_n)) / (δ Σ_{i=1}^n P(T_0,T_i)) (383)
5.1.5 Duration and Convexity
Macaulay duration is the present-value-weighted average maturity of cash flows:
MacD(t,T) = (1/P_c(t,T)) [(T-t)P(t,T) + kδ Σ_{i=I(t)}^n (T_i-t)P(t,T_i)] (384)
Modified duration measures relative price sensitivity to parallel yield shifts:
ModD(t,T) = -∂ln(P_c(t,T)) / ∂R(t,T) (385)
For constant yield Y with periodic compounding: ModD = MacD / (1 + Yδ).
Approximate price change: ΔP_c/P_c ≈ -ModD · ΔR
Dollar duration measures absolute price sensitivity:
DD(t,T) = -∂P_c(t,T)/∂R(t,T) = ModD(t,T) · P_c(t,T) (387)
Convexity captures nonlinear (second-order) effects:
C(t,T) = -(1/P_c(t,T)) · ∂²P_c(t,T)/∂R(t,T)² (388)
Full second-order approximation:
ΔP_c/P_c ≈ -ModD·ΔR + (1/2)·C·(ΔR)² (389)
Key Parameters / Signals
- Yield R(t,T): inverse of price; drives all valuation
- Modified duration: primary interest rate risk metric; scales approximately linearly with maturity
- Dollar duration: used for hedging and portfolio construction
- Convexity: scales approximately quadratically with maturity; higher convexity = better protection against parallel yield shifts at the cost of lower yield
Notes
- Duration and convexity formulas assume parallel shifts in the yield curve; non-parallel shifts require more sophisticated treatment.
- Floating rate bonds priced at par (V_0 = 1) when T = T_n because the variable coupons replicate rolling T-bond investments.
- Periodic vs. continuous compounding: MacD and ModD coincide under continuous compounding; differ under periodic compounding by factor (1 + Yδ).