Bond Duration
...years, $100 in 3 years, $100 in 4 years, $100 in 5 years and the $1,000 principle will be returned in 5 years. To compute the weighted average of a set of numbers, first, we need multiply the numbers by the weights and add those up, then add all the weights up and divide the former by the latter. In this case the weights are $100/1.07, $100/1.07^2, $100/1.07^3, $100/1.07^4, $100/1.07^5, and $1,000/1.07^5, or $93.46, $87.34, $81.63, $76.29, $71.30, and $713.00. The numbers being average are the times the payments are received, or 1 year, 2 years, 3 years, 4 years, 5 years, and 5 years. So the duration is: 1*$93.46 + 2*$87.34 + 3*$81.63 + 4*$76.29 + 5*$71.30 + 5*$713.00 $4739.69 D = ---------------------------------------------------------------------------------------- = ----------------- $93.46 + $87.34 + $81.63 + $76.29 + $71.30 + $713.00 $1123.02 D = 4.22 years Bond has a face value of $P, coupon of c, YTM of y, maturity of M years, M c*[Sum i/(1+y)^i] + M/(1+y)^M i=1 D = --------------------------------------- M c*[Sum 1/(1+y)^i] + 1/(1+y)^M i=1 For bond GOC 4% with face value of 1,000 due January 1, 2008, YTM is 7%, 0.04*[1/ (1+0.07)^1+ 2/(1+0.07)^2+3/(1+0.07)^3+4/(1+0.07)^4+5/(1+0.07)^5]+5/(1+0.07)^5 D = --------------------------------------------------------------------------------------------------------------- 0.04*[1/(1+0.07)^1+1/(1+0.07)^2+1/(1+0.07)^3+1/(1+0.07)^4+1/(1+0.07)^5]+1/(1+0.07)^5 0.04*[0.9346+ 1.7469+2.4489+3.0516+3.5649]+3.5649 4.0349 = ---------------------------------------------------------------------------- = ------------- = 4.60 years ...