Contingent Assurances

[r_v.s]

Would you plz explain the expression for a contingent assurance Ax1y (benefit pd on death of x, if x dies first) when payment is made at the end of year of death?
Isnt it the summation over t of the product of
1) v^t+1
2) tpxy
3) qx+t? (it is qx+t:y+t, with a one over the x+t in the CT notes)
What does the term in CT5, 2012 page 424 mean?

 

[John Potter]

Hi there,

You're nearly there with your product except that in 3), qx+t wouldn't quite cover it because it's still possible for y to die before x.

We can't write qx+t py+t either as we don't need y to live to the end of the year that x dies.

This is why we have the term that we do,

John

 

[r_v.s]

status x1y?

So, in effect the status x1y means that both x and y must die within the same year that x dies, and in that year x dies before y?

 

[John Potter]

No

"We can't write qx+t py+t either as we don't need y to live to the end of the year that x dies"

WE DON'T NEED Y TO LIVE - I'm not saying that we have to kill him or keep him alive after x dies. y just needs to be alive when x dies

John

 

[Delvesy888]

Is it possible to derive this expression from first principles? I.e. using double summation. I am having trouble doing this even though I understand the expression. My attempt:

A_x1y = Sum_{t=0 to inf}Sum_{s=t to inf} v^(t+1)*P(K_x = t, K_y = s)

Then P(K_x = t, K_y = s) = P(K_x = t)*P(K_y = s) by independence.

Then Sum{s=t to inf} P(K_y = s) = tpy

Putting that in the first formula will give:

A_x1y = Sum_{t=0 to inf} v^(t+1)*tpxy*qx+t

which I know isn't right.

Any help is greatly appreciated.

 

[Delvesy888]

Is anybody able to assist with this? It would help with deriving other annuities/assurances with payments made at the end of year of death - instead of being able to derive them intuitively.

Many thanks.

 

[John Potter]

Hi Delvesy,

Your derivation from first principles is a very good attempt but it doesn't quite work, again.

We need y to be alive at the time of x's death. You'd think that you could just stick a py+t in there but that doesn't do it either because we don't need y to be alive at the end of the year in which x dies.

John

 

[Johnson Adeleke]

Is there a way to do those without needing intuitive guesstimation?
I know that nqx1y = integrate{t=0 to inf} integrate{t=0 to inf} tpx(mu(x+t)) spy(mu(s+y)) ds dt
But now how do I translate this type of thinking into this question:
Two lives aged x and y take out a policy that will pay out £15,000 on the death of (x)
provided that ( y) has died at least 5 years earlier and no more than 15 years earlier.
Where the random variable is Z = 0 for Ty > Tx and v^Tx for Ty +15 >= Tx > Ty+5.