The formulas
- Macaulay duration = sum of (t x PV of CF_t) / price, in years
- Modified duration = Macaulay / (1 + y/m)
- Convexity = sum of k(k+1) x PV of CF_k / (price x (1 + y/m)^2 x m^2)
- Price change = -modified x dy + 0.5 x convexity x dy^2
How it works
Macaulay duration is the present-value-weighted average time to each payment. Modified duration turns that into price sensitivity: the example bond has a modified duration of 7.67, so its price falls by about 7.67 percent when yield rises one percentage point. Repricing at the higher yield gives a 7.32 percent drop, and the convexity term closes most of that gap.
Duration is a straight-line estimate on a curved price-yield line. Convexity corrects for the curve, and it helps the holder in both directions: prices rise more when yields fall than they drop when yields rise. The last row reprices the bond at the new yield so you see how close the estimate is.
Illustrative figures only. Check the result against your firm's methodology before relying on it.
Common questions
Why is a zero coupon bond duration equal to its maturity?
It has one payment, so the weighted average time is that payment date.
What raises duration?
Longer maturity, a lower coupon and a lower yield.
Is this effective duration?
No. Effective duration reprices bonds with embedded options. For an option-free bond the two match closely.
Formulas on this page are the standard definitions.
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