Bond Duration |
Let: B = maturity value of the bond c = coupon rate (per period) y = yield rate (per period) n = periods to maturity (... could be months, years, etc.)
Then: That means, a weighted average of the numbers 1, 2, 3, ...., n (these, after all, are the times we are averaging.) After these many periods all the n payments of $cB AND the maturity value of $B have been received.
Aah, but the weights are the present value of these payments, namely: A Definition: If we have weights called w_{1}, w_{2}, w_{3}, ... , w_{N} then the weighted average of the numbers 1, 2, 3, ..., N is (by definition) Numerator/Denominator where: Numerator = [1 w_{1} + 2 w_{2} + 3w_{3} + ... + N w_{N}] and Denominator = [w_{1} + w_{2} + w_{3} + ... + w_{N}] We start with the Denominator ('cause it's easier): Denominator = cB/(1+y)+cB/(1+y)^{2}+cB/(1+y)^{3}+ ... +cB/(1+y)^{n}+B/(1+y)^{n} which is so obscene that we let x = 1/(1+y) and get: Denominator = cB x {1 + x + x^{2} + x^{3} + ... + x^{n-1} } + B x^{n}
The sum is a geometric series so we use the magic formula: Denominator = cB {x (x^{n}-1)/(x-1)} + B x^{n}.
Now, on to the Numerator where, again for simplicity, we let x = 1/(1+y):
Surprise! This sum is the derivative of an earlier sum!
so 1 + 2x+ 3x^{2}+...+ nx^{n-1}
and DURATION = Numerator/Denominator becomes Now replace x by 1/(1+y) and simplify^{!?$#&*}: ^{!?$#&*}: You didn't expect me to simplify, right? But I can tell you that x/(x-1) = -1/y. See also Bonds 'n Stuff |