Interest-Rate Risk

Modified Duration & Price Sensitivity

Modified duration is duration adjusted for the yield level to estimate price sensitivity.

highlighted = computed this step

Duration conversion

Modified duration is Macaulay duration divided by one plus yield. For this bond, that is 3310/1331 periods.

Dmod=DMac1+y=3310/1331≈2.4868D_{\text{mod}}=\frac{D_{\text{Mac}}}{1+y}=3310/1331\approx 2.4868
Modified durationFirst-order price sensitivity to yieldPeriodCouponPrincipalCash flowPV1$100.00$0.00$100.00$90.912$100.00$0.00$100.00$82.643$100.00$1,000.00$1,100.00$826.45Price = Σ PV$1,000.00

Price sensitivity rule

A small yield change moves the price by about minus the modified duration times the yield change, times the price. This is local in y and does not capture curvature for large moves.

ΔP≈−3310/1331⋅Δy⋅P\Delta P \approx -3310/1331\cdot \Delta y \cdot P