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Difference between \( \mu_{x+t;y+t}^{01}\) and \(\mu_{x;y}^{01}\)

+2 votes
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asked Oct 9 in BUS 3024S - Contingencies by anonymous

When we are calculating continuous joint life assurances, they have \( \mu_{x+t;y+t}^{01}\)  in the formula, is that the same as \(\mu_{x;y}^{01}\) which is what I see given in a question an if not does the bounds of the integral change?

1 Answer

+1 vote
answered Oct 9 by asilmotala (2,610 points)

Hi there. Firstly, thank you for using the code to correctly code the notation, it makes it much easier to answer.

Those two symbols are not the same.

\( \mu_{x+t;y+t}^{01}\)  represents the force of transitioning at a certain time t from state 0 to state 1, for a male aged x and female aged y at time 0. In practical terms, I will consider myself as an example. It is the force of transitioning from state 0, where both my wife and myself are alive, to state 1 (where I assume one of us is dead, not sure on the exact model you are referring to). This force is the force at time t=10 i.e. 10 years in the future, where I am currently aged x=21 and she is aged y=22 at time 0.

Now \(\mu_{x;y}^{01}\) represents the force of transitioning at time t=0 from state 0 to state 1, for a male aged x and female aged y at time 0. I hope you can see the difference, and using myself as an example again, it is the force of transitioning from state 0, where both my wife and myself are alive, to state 1 (where I assume one of us is dead, not sure on the exact model you are referring to) immediately, where we are currently aged x=21 and y=22.

I hope this clears up your question. I cannot comment on how the question in reference changes, because it depends on the question. However I would assumed the integral bounds would change, and you may not actually be able to use \(\mu_{x;y}^{01}\) if a non-constant transition force is given.

commented Oct 9 by 1107 (150 points)

Thanks Asil, i do understand the concept now. The bounds of the integral are for the following case.

Lets say \(u_{xy}^{01}= 0.01 +0.05*1.05^x\) and \(u_{xy}^{02}= 0.01 +0.05*1.05^y\). Where x is the male and y the female. And state 0 is both alive, state 1 is male dead, female alive and state 2 is female dead, male alive.

I would like to calculate 

$$\overline{A}_{xy}^{01} = \int_0^\infty e^{-\delta t}  {}_{t}p^{00}_{xy} \mu_{x+t;y+t}^{01} dt$$

Where 

$${}_{t}p^{00}_{xy} = e^{-\int_0^t (\mu_{x+s;y+s}^{01} + \mu_{x+s;y+s}^{02}) ds}$$


(Please correct me if any of the above formulae are wrong)

My issue comes in when we are given \(u_{xy}^{01}\) instead of \(\mu_{x+t;y+t}^{01} \) and the same goes for \(u_{xy}^{02} \), how do I use the above equations and those integrals.

commented Oct 12 by asilmotala (2,610 points)

Glad my explanation helped!

Now that you understand it, applying it will be easy. You basically substitute (x+t) wherever x is, and (y+t) wherever y is.

So for the above example, the exponents of 1.05 would be x+t/y+t, and then you integrate as normal. I hope you find that helpful.

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