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101 Concepts for the Level I Exam

Concept 85: Macaulay, Modified, and Effective Ddurations


Bond duration measures the sensitivity of the bond’s price to changes in interest rates. The three common measures of duration are:

Macaulay duration: The weighted average of the time to receipt of coupon interest and principal payments.

Modified duration: A linear estimate of the percentage price change in a bond for a 100 basis points change in its yield-to-maturity.

     $$Modified\ duration=Macaulay\ duration\ /\ (1\ +\ r)$$

 

     $$Approximate\ Modified\ Duration={{\left({PV\ }_-\right)\ -\ \left(P{V\ }_+\right)}\over {2\ *\ \Delta \ yield\ *\ PV_0}}$$

A 12% annual-pay bond has 10 years to maturity. The bond is currently trading at par. Assuming a 10 basis-points change in yield-to-maturity, calculate the bond’s approximate modified duration.

Solution:

The bond is priced at par which means that the initial YTM = coupon rate = 12% and V0 = 100.

ΔYTM = 0.001

V = 100.57

N = 10, PMT = 12, FV = 100, I/Y = 11.9; CPT à PV = 100.57

V+ = 99.44

I/Y = 12.1; CPTà PV = 99.44

     $$Approximate\ modified\ duration={{V_--V_+}\over {2{*V}_0*\Delta YTM}}={{100.57-99.44}\over {2\times 100\times 0.001}}=5.65$$

Effective duration: The linear estimate of the percentage change in a bond’s price that would result from a 100 basis points change in the benchmark yield curve.

     $$Effective\ Duration={{\left({PV}_-\right)-\left(PV_+\right)}\over {2*\Delta curve*PV_0}}$$