DHW Exercise 6.1 How to find the probability of making a profit

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We are asked to find the probability of making a profit. The problem is that in the expression for $$L_0$$ there are two random variables $$K_{[30]+1}$$ and $$min(K_{[30]+1},n)$$ so we can not simply use $$P[L_0<0]$$. I understand that much but have no idea what to do. PS: I recall this being done in a Lecture and I still didn't understand it then. Please help.

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That was beautiful . Thanks Njabs ❤

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The question refers to a whole life assurance issued to select lives aged 30 with premiums payable annually in advanced for at most 20 years. The first part asks to find the net premium, which is \$1179,73.

The future loss, in this case, is a piecewise function of the curtate future lifetime, i.e:

$$L = 200000v^{1+K_{[30]}} - 1179.73\ddot{a}_{\overline{1+K_{[30]}}|}$$ when $$K_{[30]} \lt 20$$

$$L = 200000v^{1+K_{[30]}} - 1179.73\ddot{a}_{\overline{20}|}$$ when $$K_{[30]} \ge 20$$

Note that the future loss is a decreasing function of $$K_{[30]}$$ which makes sense given that in order for the insurer to make a profit on a whole life assurance policy they want the policyholder to survive for as long as possible.

When  $$K_{[30]} \lt 20$$ the future loss is positive hence the insurer cannot make a profit when the policyholder dies within 20 years of the policy inception. We only need concern ourselves to the condition when the policyholder survives for more than 20 years.

Therefore $$P[L < 0] = P[K_{[30]} \ge n ]$$. This equality holds because the future loss is a decreasing function of the curtate future lifetime. So what this is saying is that the the probability of not making a loss equals the probability of surviving for at least n years.

So all you have to do is find the minimum age that the policyholder needs to survive to that makes the future loss negative.