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?

Login

+1 vote

Best answer

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 \(t+1\).

- All categories
- BUS 1003H - Introduction to Financial Risk (41)
- BUS 2016H - Financial Mathematics (50)
- BUS 3018F - Models (69)
- BUS 3024S - Contingencies (57)
- BUS 4028F - Financial Economics (20)
- BUS 4027W - Actuarial Risk Management (39)
- BUS 4029H - Research Project (5)
- Mphil (1)
- Calculus and Pure Mathematics (3)
- Statistics (16)

...