💵 Fixed Income

Bond Duration & Convexity — Modified, Macaulay & Effective Duration

Bond duration and convexity formulas explained. Modified duration, Macaulay duration, effective duration, and convexity with calculation examples.

Key Concepts

Duration

Measure of bond price sensitivity to yield changes. Higher duration = more price volatility. Approximate % price change for 1% yield change.

Macaulay Duration

Weighted average time to receive cash flows. In years. Duration of zero-coupon = maturity.

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Formulas

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Modified Duration

ModDur = MacDur / (1 + y/m)

Where: y = YTM, m = compounding periods per year

Price Change (Duration)

ΔP/P ≈ -ModDur × Δy

Where: First-order approximation

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Master Formula Sheet -- Fixed Income

Bond Price

P = Σ[C/(1+r)ᵗ] + FV/(1+r)ⁿ

PV of coupons + PV of par

Current Yield

CY = Annual Coupon / Price

Income return only

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Decision Frameworks

How to manage interest rate risk?

Use when:

  • Match duration of assets and liabilities (immunization)
  • Extend duration if expecting rates to fall
  • Shorten duration if expecting rates to rise

Avoid when:

  • Using Macaulay duration for callable bonds (use effective duration)
  • Ignoring convexity for large yield changes

Test Your Understanding

If a bond has a modified duration of 6 and yields increase by 100 basis points, the approximate price change is:

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