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: DURATION = weighted average of the times when all the interest AND the value-at-maturity value are received. 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: cB/(1+y), cB/(1+y)2, cB/(1+y)3, ..., cB/(1+y)n (them's the coupons) and, finally, B/(1+y)n (the present value of the maturity value of the bond) A Definition: If we have weights called w1, w2, w3, ... , wN then the weighted average of the numbers 1, 2, 3, ..., N is (by definition) Numerator/Denominator where: Numerator = [1 w1 + 2 w2 + 3w3 + ... + N wN] and Denominator = [w1 + w2 + w3 + ... + wN] 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 + x2 + x3 + ... + xn-1 } + B xn The sum is a geometric series so we use the magic formula: 1 + x + x2 + x3 + ... + xn-1 = (xn - 1)/(x - 1) and get: Denominator = cB {x (xn-1)/(x-1)} + B xn. Now, on to the Numerator where, again for simplicity, we let x = 1/(1+y): Numerator = cB x {1 + 2x+ 3x2+... + nxn-1} + B n xn Surprise! This sum is the derivative of an earlier sum! (Uh ... that's Calculus ya know.) so 1 + 2x+ 3x2+...+ nxn-1 = d/dx {x + x2 + x3 + ... +xn } = d/dx {x (xn - 1)/(x - 1)} = d/dx { xn - 1 + (xn - 1)/(x - 1) } = nxn-1 + nxn-1/(x - 1)- (xn-1)/(x-1)2 = n xn/(x-1) - (xn-1)/(x-1)2 and DURATION = Numerator/Denominator becomes (after cancelling the B's) {cn xn+1/(x-1) - c x (xn-1)/(x-1)2 + n xn}/ {c x (xn-1)/(x-1) + xn} Now replace x by 1/(1+y) and simplify!?\$#&*: DURATION = (1+y)/y - {1+y + n(c-y)}/ {c[(1+y)n - 1] + y} !?\$#&*: You didn't expect me to simplify, right? But I can tell you that x/(x-1) = -1/y. See also Bonds 'n Stuff