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in BUS 3018F - Models by (430 points)
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The central exposed to risk is calculated using :  \(\int_{0}^{T}P_x (t) dt\)

But when the rate interval is a calendar year, there is said to be a discontinuity. What is meant by this?  

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Let us suppose that we have \(P_{x,t} =\) number of lives under observation, age \(x\), next birthday, at time \(t\), where \(t\) = 1 January in calendar year.

When finding the central exposed to risk, we need to be careful in ensuring that we measure the exposed to risk for the relevant group of lives at the start and end of the year.

The number of lives under observation aged \(x\) next birthday, at time \(t\) is given by \(P_{x,t}\). This same group of lives will change age classification to age \(x+1\) next birthday at the the same time one year later (at time \(t+1\)). The number of lives in this group is given by \(P_{x+1,t+1}\).

We are in effect, using the fact that \(P_{x+1,t+1}\) represents the number of lives under observation, age \(x\) next birthday, just before midnight on 31 December (ie. just before time \(t+1\)). 

Therefore our trapezium approximation for any one year, if age labels change on 1 Jan each year, is given by:

\(\frac{1}{2}(P_{x,1 Jan} + P_{x,31 Dec}) = \frac{1}{2}(P_{x,1 Jan} + P_{x+1,1\ Jan\ in\ next\ year})\)  

So there is a discontinuity at time \(t+1\).