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The decomposition of the investment profit and loss (p&l) for each business
year into different risk factors (e.g., interest rates, credit spreads, foreign
exchange rate etc.) is a task that is regulatory required, e.g., by Solvency 2.
Three different decomposition principles are prevalent: one-at-a-time (OAT),
sequential updating (SU) and average sequential updating (ASU) decompositions.
The SU and the ASU decompositions explain the p&l fully. However, the OAT
decomposition generates some unexplained p&l. The SU decomposition depends on
the order or labeling of the risk factors. The three decompositions can be
defined on different sub-intervals using annually, quarterly, monthly or daily
data. In this research, we empirically quantify: the unexplained p\&l of the
OAT decomposition; the dependence of the SU decomposition on the update order;
and how much the three decomposition principles depend on the size of the
sub-intervals.
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