The curtate future lifetimes of a beneficiary aged x and her spouse
aged y are subject to the following independent probabilities of
death.
k q(x+k) q(y+k)
0 .25 .50
1 .75 .50
2 1.00 1.00
You may assume i=10%. Determine a(xy). The actuarial present value
of an annuity payable at the beginning of each year as long as either
You may assume i=10%. Determine a(xy). The actuarial present value
of an annuity payable at the beginning of each year as long as either
beneficiary is alive.(note: a bar should be place over xy and dots
over a).
The thing here is that they are giving the information to find the
probabilities that we really need.
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First, recall that a(xy) = ax + ay - a(xy)
So we need Px, 2Px, 3Px,..., Py, 2Py, 3Py, and P(x,y), 2P(x,y),
2P(x,y), 3P(x,y),...
To find those Ps:
Px = 1 - q(x+0) = 0.75
2Px = Px * P(x+1) = Px * [1-q(x+1)] = 0.75 * 0.25 = 0.1875
3Px = Px * P(x+1) * P(x+2) = 0.75 * 0.25 * 0 = 0
Hence, kPx = 0 , for k = 3, 4, 5,...
The same happens with Ps for (y):
Py = 0.5
2Py = 0.5 * 0.5 = 0.25
kPy = 0 , for k = 3, 4, 5,...
Finally, the Ps for (x,y):
P(xy) = Px * Py = 0.75 * 0.5 = 0.375
2P(xy) = 2Px * 2Py = 0.1875*0.25 = 0.046875
kP(xy) = 0. for k = 3, 4, 5,...
The rest is easy. Just use the regular formulas. Hope was clear :-)