Fixed Income Risk Management: Duration, Convexity, and Key Rate Duration

Fixed income instruments represent the largest asset class in global markets. Understanding their risk characteristics is essential for FRM candidates.

Interest Rate Risk

The primary risk for fixed income portfolios is interest rate risk — the sensitivity of bond prices to changes in interest rates.

Duration

Macaulay Duration

The weighted-average time to receive a bond's cash flows:

D_Mac = Σ [t × PV(CF_t)] / Price
Measured in years

Modified Duration

The percentage price change for a 1% yield change:

D_Mod = D_Mac / (1 + y/k)
ΔP/P ≈ -D_Mod × Δy

Dollar Duration (DV01)

The dollar change for a 1 basis point yield change:

DV01 = D_Mod × Price × 0.0001

Effective Duration

Used for bonds with embedded options (callable, putable):

D_Eff = (P₋ - P₊) / (2 × P₀ × Δy)
Accounts for how cash flows change with rates

Convexity

Why Convexity Matters

Duration is a linear approximation — convexity captures the curvature:

ΔP/P ≈ -D_Mod × Δy + ½ × Convexity × (Δy)²
Positive convexity: bonds gain more from rate drops than they lose from rate rises
Convexity is valuable — investors pay a premium for it

Negative Convexity

Some instruments exhibit negative convexity:

Callable bonds: Price capped at call price when rates fall
MBS: Prepayment increases when rates fall, reducing price upside

Key Rate Duration

Standard duration measures sensitivity to parallel yield curve shifts. Key rate duration measures sensitivity to specific curve points:

2-year KRD, 5-year KRD, 10-year KRD, 30-year KRD
Essential for managing non-parallel yield curve risk
A portfolio can be duration-neutral but have significant key rate exposures

Yield Curve Risk

The yield curve can change in multiple ways:

Parallel shift: All rates move equally
Steepening/Flattening: Long end moves differently from short end
Butterfly: Middle of curve moves differently from ends

Portfolio Duration Management

Cash Flow Matching

Matching asset cash flows exactly with liability cash flows — eliminates interest rate risk but is expensive and inflexible.

Duration Matching (Immunization)

Setting portfolio duration equal to the liability duration — protects against small parallel shifts but not against complex curve movements.

Key Rate Matching

Matching key rate durations at multiple curve points — more precise than simple duration matching.

Practice fixed income risk questions to master these critical concepts!

Frequently Asked Questions

What is the difference between Macaulay and modified duration?▼

Macaulay duration is the weighted-average time to receive cash flows (measured in years). Modified duration = Macaulay / (1 + y/k) and measures the percentage price change for a 1% yield change. Modified duration is more useful for risk management.

Why is convexity important in bond risk management?▼

Duration is a linear approximation that becomes less accurate for large yield changes. Convexity captures the curvature effect: positive convexity means bonds gain more when rates fall than they lose when rates rise. Investors pay a premium for convexity.

What is DV01 and how is it used?▼

DV01 (Dollar Value of a Basis Point) measures the dollar change in a bond's price for a 1 basis point (0.01%) yield change. It is used for precise hedging and comparing interest rate sensitivity across different instruments.

What is key rate duration?▼

Key rate duration measures sensitivity to specific yield curve points (e.g., 2Y, 5Y, 10Y, 30Y) rather than parallel shifts. It enables more precise management of non-parallel yield curve risk than standard duration.