Dear Sir/Madam/Tutors, I am able to derive the gross future loss random variable (GFLRV) for the other except renewal expenses. In the solutions, it shows that the PVRV for expenses (I am going to omit the claim expenses from it since I understand how to obtain that) is: 300 + 0.25P + 0.05P*(annuity_advance(min{K[55]+1, 10}) - 1) + 50min{K[55], 9} The bit that I do not understand is the last bit where 50min{K[55], 9} as I thought it would have been at least 50(1.04)^min{K[55], 9}. Could you explain to me how is that worked out? I have attached my working along with this thread. It'll probably show how my current (probably flawed) thought process to working out the GFLRV. Additionally, any tips on how to tackle complex GFLRV? I typically stumble on them if they are non-standard. I try to work from first principles using GFLRV of existing assurances/annuities and adjust my way to what I think it should be. Thanks.
The renewal expenses on each policy anniversary are \( 50(1.04)^{t} \) so, assuming that the policyholder doesn't die during the policy term, the present value of these will be \( 50(1.04)v + 50(1.04)^{2}v^{2} + 50(1.04)^{3}v^{3} \: + \: ... \: + \: 50(1.04)^{9}v^{9}\). Since the interest rate we have is \( 4 \% \: pa \), then the \( v \) terms are going to be \( (1.04)^{-t} \), so the \( (1.04)^{t} \) and \(v\) terms in the expression above cancel each other out and we just have \( 50 * 9 \). If the policyholder does die during the term, then there will be no renewal expenses for \( t > K_{[55]} \), so we would have \( 50 * K_{[55]} \). Combining the two scenarios, we get the final answer of \( 50 \, \text {min} ( K_{[55]} , 9 ) \). In terms of how to tackle complex questions like this in general, I agree that writing things out from first principles is a good idea. It can be useful to consider what's being paid in each individual policy year under all possible scenarios, as I've done above.
