In the Q and A. chapter 21. Question21.9 asks us to calculate the reserve that covers the liability with probability of at least 99% i.e. P(L<=V) >=0.99. How are the inequalities p(k30<r) <0.01 and p(k30<=r+1) >= 0.01 obtained ?
This is Chapter 20, Q20.9 in the current (2021) version of the notes. In your expression P(L <= V) >= 0.99, L represents the liability for the policy (ie benefits less premiums net of expenses) and V is the reserve. We want it to be highly likely that the reserve exceeds the liability. The liability for the policy reduces over time. The liability is high at short durations as early death will mean that the sum assured will need to be paid even though only a few premiums will have been received. The liability is lower at longer durations as more premiums will have been received, and the death payment is further in the future. So, as the liability reduces over time, the reserve is more likely to cover the liability at later durations, and the concern for the insurance company is the early death of the policyholder. This means that, for it to be highly likely that the reserve exceeds the liability, we need there to be a small probability of early death, ie P(K < r) < 0.01. The condition P(K > r+1) >= 0.01 ensures that we find the first year where the probability switches from below 0.01 to above it.
In answering your question, from the reference you provided, I think I've identified that you are using an old version of the Course Notes (2019). The Course Notes are updated each year for changes to Core Reading, the syllabus, and to make corrections and improvements. Please see the Upgrade page on our website, which details the updates made since 2019: https://www.acted.co.uk/paper_cmp_upgrade.html You will need the CM1 2019/2020 and the 2020/2021 Upgrades, I think.
