Does it make sense to combine different modified durations?

Question

Does it make sense to aggregate different modified durations into one overall duration measure?

In the context of insurance liabilities:

Total Value Discounted Outstanding Claims: $1500
Duration of Discounted Outstanding Claims: 0.73    
Total Value Discounted Recoveries: -$650
Duration of Discounted Recoveries: 0.43
Total Value of Discounted Net Outstanding Claims: $850
Duration of Discounted Net Outstanding Claims: ????

So I know that this is a simple example, and we could have a number of other components for which we have calculated the duration.

If I take a sum product / weighted average, I get a "Total" duration of 0.959 which I'm finding hard to make sense of since, both the duration components are less than 0.959.

Should the underlying cashflows be brought together upon which a duration calculation can be made, rather than some sort of weighted average portfolio approach?

Answer 1

It does make sense, both are a stream of projected and discounted cashflows, combined they are also a portfolio of cashflows.

Duration as a weighted average (macaulay duration) is additive. As modified duration, it is also additive if the discount rates are the same or similar.

Your weighted average approach is correct. Your duration of 0.959 years is scaled to $850 (which is nearly half the size of the biggest contributor to your duration) hence it's larger than the individual duration. Maybe for a more intuitive number consider Dollar Duration which is equal to Value * Duration * 1bp

DD of claims = 1500 * 0.73 * 10^-4 = 0.1095 DD of recoveries = -650 * 0.43 * 10^-4 = -0.02795

Sum of these is a dollar duration of 0.08155 (or also equal to 850 * 0.959 * 10^-4)