About Atkinson's P.272 Excercise 4.9

[Sherwin]

It is about joint last-to-die. You are given qx(t) and qy(t), and are required to calculate LTDqxy(t).

LTDqxy(t)=qx(t)*qy(t), isn't it? But the book maybe means not.

While the book said that
1-SurvFactorxy(t)=[1-SurvFactorx(t)]*[1-SurvFactory(t)]

In my familiar form, 1-tpxy=[1-tpx]*[1-tpy].

Which one is correct then?

[wat?]

It is about joint last-to-die. You are given qx(t) and qy(t), and are required to calculate LTDqxy(t).

LTDqxy(t)=qx(t)*qy(t) (1), isn't it? But the book maybe means not.

While the book said that
1-SurvFactorxy(t)=[1-SurvFactorx(t)]*[1-SurvFactory(t)]

In my familiar form, 1-tpxy=[1-tpx]*[1-tpy]. (2)

Which one is correct then?

Both are correct. Formulas (1) and (2) as noted in your quote are identical - just substitute. Recall that qx(t) = 1 - px(t).

[inexactuary]

The problem is that qx(t) and qy(t) are the mortality rates for that year, so multiplying them together gives you the probability that they both die in that year, given they both survived to the beginning of the year. However, LTDqxy(t) is the probability that the LAST death occurs during that year, given that AT LEAST ONE survives to the beginning of the year. If they had given you tqx and tqy, then you could just multiply them together, but in this case the easiest way is to calculate the survival factors for each, then use your equation (2) above. I'm guessing that the confusion was probably the meaning of qx(t) and qy(t).

[wat?]

The problem is that qx(t) and qy(t) are the mortality rates for that year, so multiplying them together gives you the probability that they both die in that year, given they both survived to the beginning of the year. However, LTDqxy(t) is the probability that the LAST death occurs during that year, given that AT LEAST ONE survives to the beginning of the year. If they had given you tqx and tqy, then you could just multiply them together, but in this case the easiest way is to calculate the survival factors for each, then use your equation (2) above. I'm guessing that the confusion was probably the meaning of qx(t) and qy(t).

Sorry for the confusion - I should actually refer to the text when answering a question.

inexactuary's interpretation is spot-on: simply multiplying the qx's together implies both lives survived to x and both die by x+1. This does not take into account the probability that only one of the lives is still living.