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---
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description: "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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tags: [fixed-income, background, duration, convexity, swaps]
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---
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# Fixed Income Generalities
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**Section**: 5.1 | **Asset Class**: Fixed Income | **Type**: Background / Reference
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## Overview
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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.
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## Construction / Mechanics
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### 5.1.1 Zero-Coupon Bonds
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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:
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```
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R(t,T) = -ln(P(t,T)) / (T - t) (374)
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```
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### 5.1.2 Coupon Bonds
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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:
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```
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P_c(t,T) = P(t,T) + kδ Σ_{i=I(t)}^n P(t,T_i) (375)
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```
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where I(t) = min(i : t < T_i). At issuance (t = T_0), the par coupon rate is:
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```
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k = (1 - P(T_0,T)) / (δ Σ_{i=1}^n P(T_0,T_i)) (377)
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```
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### 5.1.3 Floating Rate Bonds
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Coupon payments are based on LIBOR. The LIBOR rate at T_{i-1} for period [T_{i-1}, T_i] is:
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```
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L(T_{i-1}) = (1/δ) [1/P(T_{i-1},T_i) - 1] (378)
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```
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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:
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```
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V_0 = 1 - [P(T_0,T_n) - P(T_0,T)] (380)
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```
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If T = T_n then V_0 = 1 (the bond prices at par).
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### 5.1.4 Swaps
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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:
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```
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k = (1 - P(T_0,T_n)) / (δ Σ_{i=1}^n P(T_0,T_i)) (383)
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```
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### 5.1.5 Duration and Convexity
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**Macaulay duration** is the present-value-weighted average maturity of cash flows:
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```
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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)
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```
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**Modified duration** measures relative price sensitivity to parallel yield shifts:
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```
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ModD(t,T) = -∂ln(P_c(t,T)) / ∂R(t,T) (385)
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```
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For constant yield Y with periodic compounding: ModD = MacD / (1 + Yδ).
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Approximate price change: ΔP_c/P_c ≈ -ModD · ΔR
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**Dollar duration** measures absolute price sensitivity:
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```
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DD(t,T) = -∂P_c(t,T)/∂R(t,T) = ModD(t,T) · P_c(t,T) (387)
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```
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**Convexity** captures nonlinear (second-order) effects:
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```
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C(t,T) = -(1/P_c(t,T)) · ∂²P_c(t,T)/∂R(t,T)² (388)
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```
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Full second-order approximation:
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```
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ΔP_c/P_c ≈ -ModD·ΔR + (1/2)·C·(ΔR)² (389)
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```
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## Key Parameters / Signals
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- **Yield R(t,T)**: inverse of price; drives all valuation
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- **Modified duration**: primary interest rate risk metric; scales approximately linearly with maturity
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- **Dollar duration**: used for hedging and portfolio construction
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- **Convexity**: scales approximately quadratically with maturity; higher convexity = better protection against parallel yield shifts at the cost of lower yield
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## Notes
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- Duration and convexity formulas assume parallel shifts in the yield curve; non-parallel shifts require more sophisticated treatment.
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- Floating rate bonds priced at par (V_0 = 1) when T = T_n because the variable coupons replicate rolling T-bond investments.
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- Periodic vs. continuous compounding: MacD and ModD coincide under continuous compounding; differ under periodic compounding by factor (1 + Yδ).
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