Duration: Formulas and Calculations - New York University
[Pages:2]Duration: Formulas and Calculations W.L. Silber
1. Definition
? D =
n Ct t=1 (1+ r )t
n Ct
(t )
?t=1 (1+ r )t
2. Explicit Sample Calculations
(a) For an 8% coupon (annual pay) four-year bond with a yield to maturity of 10%,
we have:
80 (1) + 80 (2) + 80 (3) + 1080 (4)
D = 1.10
(1.10)2
(1.10)3
(1.10)4
80 1.10
+
80 (1.10)2
+
80 (1.10)3
+
1080 (1.10)4
D = 3.56
(b) If the coupon were 4% rather than 8%, the formula would be:
D
=
40 1.10
(1)
+
40 (1.10)2
(2)
+
40 (1.10)3
(3)
+
1040 (1.10)4
(4)
40 + 40 + 40 + 1040
1.10 (1.10)2 (1.10)3 (1.10)4
D = 3.75
(c) Finally, for a zero coupon bond with four years to maturity we have:
1080 (4) D = (1.10)4 = 4
1080 (1.10)4
3. Duration Table for an 11.75% Coupon Bond
(1)
(2)
Coupon MAT
11.75 3YR
11.75 7
11.75 10
11.75 20
11.75 30
(3) YTM 11.75 11.75 11.75 11.75 11.75
(3a) DUR 2.70 5.14 6.38 8.48 9.17
(4) YTM 6.75 6.75 6.75 6.75 6.75
(4a) DUR 2.71 5.36 6.90 10.43 12.54
Notes: (1) Column 3a shows duration increasing with maturity, but less than proportionately
(2) Column 4a compared with 3a shows that a decline in yield to maturity (from 11.75% to 6.75%) increases duration, especially for the longer maturities.
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