Actuarial Outpost Common shock - expression for t_p_xy
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#1
04-16-2009, 07:15 PM
 PureKorean Member Join Date: Jun 2008 Posts: 104
Common shock - expression for t_p_xy

We know that the correct expression is

$_tp_{xy} = \exp\left[- \int_0^t \mu_x^*(s) \,ds\right] \exp\left[- \int_0^t \mu_y^*(s) \,ds\right] \exp\left[- \lambda t\right]$

but I am having a hard time convincing myself that this is true. Intuitively, I thought the expression would be

$_tp_{xy} = \exp\left[- \int_0^t \mu_x^*(s) \,ds\right] \exp\left[- \int_0^t \mu_y^*(s) \,ds\right] \exp\left[- \lambda t\right] \exp\left[- \lambda t\right]$

with the extra Pr(Z > t) added on, since we are dealing with 2 lives. Why isn't this so?

(Thank you to jraven -- digged up this TeX code from one of his earlier posts.)
#2
04-16-2009, 09:00 PM
 Jo.M. Member Join Date: Oct 2007 Posts: 288

The easiest way to understand this is to think about the common shock as a ''catastrophic event'' that would kill both lives instantaneously (such as an earthquake, if you like).

In this case, earthquakes happen at a rate lambda per unit of time. Having two lives, or even 100 lives will not increase the chances of an earthquake occurring. Therefore, you should only account for this hazard rates once.

I hope this helps.
#3
04-17-2009, 11:33 AM
 PureKorean Member Join Date: Jun 2008 Posts: 104

It does. Thanks a lot.

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