Aggregating Modified Duration for Claims and Recoveries
Summary
The document explains how to combine the interest-rate sensitivity of discounted insurance claims and recoveries. It treats the cashflows as a portfolio and uses value-weighted durations, with recoveries represented by a negative value. The resulting net duration can exceed the durations of the individual components because the net value is smaller than the largest component, while its dollar sensitivity reflects the difference between their sensitivities.
The example gives claims valued at $1,500 with duration 0.73 and recoveries valued at -$650 with duration 0.43. It calculates dollar duration for each and sums them, yielding the same net sensitivity as applying the combined duration of 0.959 to the net value of $850. The discussion says modified duration aggregation assumes the discount rates are the same or similar; differing rates or cashflow assumptions may limit the simple calculation.
Key ideas
- Discounted claims and recoveries can be treated as a combined portfolio of cashflows.
- A value-weighted duration can describe the net position, including negative-valued recoveries.
- Net duration may exceed each component duration when offsetting values shrink the combined position.
- Dollar duration provides a way to compare the components’ absolute rate sensitivities.
- Modified duration aggregation assumes component discount rates are the same or similar.
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Full text
# Does it make sense to combine different modified durations? # Does it make sense to combine different modified durations? 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 by BG25 (score 0, accepted) https://quant.stackexchange.com/a/45032 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)
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