CM1 April 2022 Question 10) i) - dependent probability of surrender

[Danny]

I calculated the individual forces of decrement to get the dependent probability of surrender. This gives:

mu^death_63= -ln(1-0.004251)
mu^surrender_63= -ln(1-0.05)

aq^surr)_63=1-e^(-((-ln(1-0.004251)-ln(1-0.05)))*(-ln(1-0.05)/(-ln(1-0.05)-ln(1-0.004251)=0.498455804.

Note that this value of the dependent probability of decrement I calculated is different to the one provided in the solution (i.e they get 0.049787). Why is this? I am aware that the solution approaches the problem differently, but to my knowledge the relationship I expressed holds? I have calculated other dependent probabilities of decrement using formulae like this previously without issue.

Please could you explain?

Thanks,

Danny

 

[Joe Hook]

Hi Danny,

Surrenders are only allowed at the end of the year so it does not make sense to turn it into a force. The dependent probability of surrender is (prob the life survives mortality in the year) x surrender rate.

You can turn the independent rate into a force if you want but you've then calculated the dependent probability of surrender as though the decrements are competing, in other words that the life could die or surrender at any time. This will give a higher dependent probability of surrender than is really the case but some of those lives you are assuming surrender eg part-way through the year from 63 to 64 would actually have left via the death decrement instead since this is the only decrement operating over the year.

Since, this isn't the easiest to type out we could consider a multiple decrement table with lives.

Say there's 1,000 lives active at the start of the year and the independent prob of death is 0.004251 and independent probability of surrender is 0.05.

If surrender can only occur at the end of the year this means we expect 4.251 lives to die then 5% of the remaining 995.749 to surrender ie 49.79. This leaves 945.96 lives active at the end of the year.

If surrender can occur at any time and we use our typical multiple decrement formulae: force of mortality =-ln(1-0.004251) = 0.00426 and force of surrender = -ln(1-0.05) = 0.05129.

Now, the dependent prob of death is (0.00426/(0.00426+0.05129))*(1-exp(-0.00426-0.05129)) = 0.004144
Similarly, dependent prob of surrender is (0.05129/(0.00426+0.05129))*(1-exp(-0.00426-0.05129)) = 0.049895

So here of our 1,000 active lives we would have 4.144 lives dying and 49.89 lives surrendering again leaving us with 945.96 lives active at the end of the year. However, because we've allowed lives to surrender any time during the year some of these lives have surrendered before they otherwise would have died and so they have been removed via the surrender decrement when really, under the situation presented, they should have been removed via death.

Does this help?
Joe