I don't have the latest study manual that you have to see how the answer on
Q19 is formulated but this is my answer:
"x" as you refer to it is not the number of exposures but represents the
reinsurer's QS percentage on the one risk with an ER of 50%. Try not to
think of the figures provided as expected losses and ER "per exposure" but
the E(L) & ER figures for the entire portfolio. The portfolio could have
any number of risks for which the expected loss is 1000 and an ER of 10%.
However, we are not told this and it really doesn't matter.
All that needs to be done is to determine what (x%) of the risk with
E(L)=100 and ER=50% would maintain the portfolio ER of 10%. First, the std.
dev. of this is calculated to be 50. Next remember that, with vertical
sharing, any portion of the risk selected will have the same ER as the whole
risk. Therefore, by taking x% of the risk, the ER of that portion will be
50(x%) [std. dev.] divided by 100(x%) [mean]. Therefore, the new portfolio
mean will be 1000 + 100(x%) and the standard deviation will be (100^2 +
(50(x%))^2)^(0.5). From here, just solve (x%) so that ER = 10% (=> x = 83%).
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>Cara and Andrea
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Wayne