# Why is there a discontinuity in the calculation of central exposure to risk when the rate interval is a Calendar year ?

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edited Jun 17, 2017

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?

+1 vote
answered Jun 17, 2017 by (1,490 points)
selected Jul 9, 2017

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.
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 $$t+1$$).
$$\frac{1}{2}(P_{x,1 Jan} + P_{x,31 Dec}) = \frac{1}{2}(P_{x,1 Jan} + P_{x+1,1\ Jan\ in\ next\ year})$$
So there is a discontinuity at time $$t+1$$.