# What is the relationship between a term contingent insurance and a joint life term assurance.

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With regards to Tutorial 4 question 5 2018, how can we equate a term contingent insurance with x=30, y=30 , n=30 to half of a joint life term assurance with x,y,n as above ?

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As the lives are identical $$x=30$$ and $$y=30$$, the term contingent assurance $$A_{\overset1{30}:30:30}$$ is equivalent to $$\frac12a_{\overbrace{30:30}^1:30}$$.

This is because the probability of either one dying first is equal (i.e. 1/2 each) as they are identical lives, and the joint life term assurance is the equivalent of a term contingent assurance on both lives, but with identical lives this leads to the equation as in question 5 of the tutorial.

The maths is as follows:

$$A_{\overbrace{30:30}^1:30}=A_{\overset1{30}:30:30}+A_{30:\overset1{30}:30}$$

But

$$A_{\overset1{30}:30:30}=A_{30:\overset1{30}:30}$$

as the lives are identical.

Hence

$$A_{\overbrace{30:30}^1:30}=2A_{\overset1{30}:30:30}$$

Hence

$$A_{\overset1{30}:30:30}=\frac12A_{\overbrace{30:30}^1:30}$$