Bond duration and convexity formulas explained. Modified duration, Macaulay duration, effective duration, and convexity with calculation examples.
Measure of bond price sensitivity to yield changes. Higher duration = more price volatility. Approximate % price change for 1% yield change.
Weighted average time to receive cash flows. In years. Duration of zero-coupon = maturity.
ModDur = MacDur / (1 + y/m). ΔP/P ≈ -ModDur × Δy. Direct measure of price sensitivity.
Curvature of price-yield relationship. Duration is linear approximation; convexity corrects for non-linearity. Positive convexity: good for investor.
Higher for: longer maturity, lower coupon, lower yield. Portfolio duration = weighted average of bond durations.
Modified Duration
Where: y = YTM, m = compounding periods per year
Price Change (Duration)
Where: First-order approximation
Price Change (Duration + Convexity)
Where: More accurate with convexity adjustment
Effective Duration
Where: Used for bonds with embedded options
Effective Convexity
Where: Curvature correction
Bond Price
PV of coupons + PV of par
Current Yield
Income return only
YTM
Total return if held to maturity
Macaulay Duration
Weighted average time to receive CFs
Modified Duration
Price sensitivity to yield change
Price Change (Duration)
First-order approximation
Price Change (with Convexity)
More accurate for large yield changes
Effective Duration
For bonds with embedded options
Credit Spread
Compensation for credit risk
Expected Loss
PD=prob default, LGD=loss given default
Use when:
Avoid when:
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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