# “Babcock duration”: a weighted average of two factors

Lately, I have had to explain what Babcock duration is. Further to saying it’s an alternative gauge of duration to Macaulay it simply is a weighted average of two factors [1]:

• $n$ (maturity) and
• $\left(\sum_{i=1}^n \frac{1}{\left(1 + YTM\right)^i}\right) \left(1 + YTM \right)$

The outright formula [2], the weighted average, is:

$D = n \left(1 - \frac{Y}{YTM} \right) + \frac{Y}{YTM} \left(\sum_{i=1}^n \frac{1}{\left(1 + YTM\right)^i}\right) \left(1 + YTM \right)$.

There’s a zero coupon with n years of maturity and a par-bond duration.

[1] Babcock, G., (1985), Duration as a Weighted Average of Two Factors, (March-April 1985) Financial Analyst Journal.

[2] Benninga, Simon, (2001), Financial Modeling, MIT Press.

Posted in Economics