Hi,
So when we set up these EPV expressions we are looking for payment x discount factor x probability.
Payment = 20
Discount factor = exp(-0.05t)
The probability we need here is that life x dies at exact time t and life y has already died which is tpx * mu_x+t * tqy.
The probability that y has already died by time t (tqy) is 1-tpy = 1 - exp(-0.005t)
So here we need an integral from 0 to 20 [20,000 * exp(-0.05t) * exp(-0.005t) * (1 - exp(-0.005t)* 0.005]. If you solve that you'll get the answer given in the solutions.
On your point about E[Kx], I can appreciate why you think you've made a mistake but what you've calculated is actually fine. The reason it's coming out so high is because we're using a constant force of mortality that's very low (and not consistent with human mortality across the full range of ages, maybe more like a bowhead whale...). You could contextualise this by looking in the AM92 Tables and seeing that this force of mortality corresponds with someone who is 56/57 years old and eg 100px = exp(-0.5)=0.6065 which is very high!
Joe
Last edited: Feb 17, 2022