 So question 5 is a ruin theory exam question And what we're gonna do is read through the preamble and then answer the four parts And it starts by saying an insurer has issued two Five-year term assurance policies to two individuals involved in a dangerous sport Premiums are payable annually in advance and claims are paid at the end of the year of death We're then given a table that includes who the individual is their annual premium their summer shirt and their annual probability of death Now we're told to assume that the probability of death is constant over each of the five years of the policy and Suppose that the insurer has an initial surplus of you Now the very first part of the question asks us to define what is meant by phi u and phi u t Now this is pure theory So you want to aim for a hundred percent with a question like this But you want to be careful and make sure you include all the relevant details So phi u is going to be the probability of ultimate ruin This means ruin at any time in the future where phi u t is the probability that ruin occurs Before time t and of course ruin is defined as when our surplus at any time is less than zero Now note There are some students who are not getting full marks here because they're writing out a mathematical Expression, but then they fail to tell me what that t meant So you have to make sure that you include the word time in order to get full marks And you don't just say you know for any t it's like okay But what do you mean by t make sure you tell me that t means time? We're then gonna go on to the the second third and fourth part of the question But let's first come back to this preamble and kind of just see what it's talking about So first of all we're dealing with just two policies and this is a very Interesting or unusual case normally an insurer is dealing with thousands of policies We're then told that the duration is five years and we're also told that the premiums are received in the beginning of the year and Again, this is a little bit of a unique situation because it's not continuous like standard ruin theory assumption We're also told that the claims are paid at the end of the year and again Not immediately like the standard ruin theory assumption So what this means is we need to approach this question from first principles and not just turn to the notes try find a Formula slap in some numbers and hope for a result We're going to have to think and remember ruin occurs if you know The claims are greater than the capital plus the premiums That's going to be the condition that we need to look at and like I said We don't have a fancy formula to just plug these values into we need to go back to first principles So if we look at part two of the question and I've also just included the table there because we're going to be Needing to rely on these these values It says assuming that our initial capital is a thousand determine the distribution of you know the capital after one year and Calculate what is going to be the probability of ruin after one year? So what we want to do is we want to start out with our surplus at the stock member surplus capital reserves These are three words that are kind of interchangeable when it comes to ruin theory But essentially what we're going to have at the start is our initial surplus Plus the premium from policy a plus the premium from policy B So our initial surplus that's that one thousand that we see in the question the premium from policy a we're getting this From the table that's a hundred plus the premium from policy B. That's 50. We're also getting that from the table We're adding that all together and we're getting one thousand one hundred and fifty So this is going to be our capital at the start and like I said There is a little bit of a confusion because we've got this initial capital of a thousand But now our capital at time zero is one thousand one hundred and fifty and the reason for that is because the Premiums are being received at the beginning of the unlike traditional ruin theory where we assume that they are received continuously So instead of it being CT, you know in your formula that we've seen in the notes We now have to make sure that we're getting those premiums at the start Now what we want to do is kind of determine the distribution of where this capital is going to be in one year's time But before we can get there what we first need to do is just look at all the various death Possibilities so this is the step that we needed to take before we can look at the distribution of the capital at the end of The first year so death possibilities There is the chance and this is the best case scenario. No one dies So what we're essentially doing is we're taking the complement of the annual probability of death of both a and b and Because we're again, we're assuming that they're independent that they're not related to each other They're not like in a bobsled team where if one dies, there's an accident. There's a correlation between the two we're making this assumption and What that means is the complement of you know, basically their survival probabilities We multiplying that together and we're getting zero point eight five five Then there's the probability that only a dies So this is the probability that a dies times the probability that b survives and we get zero point zero four five Only be dies this means the probability that a survives and be dies And that's going to give us zero point zero nine five and then the worst case scenario is when both of them die Which is zero point zero five times zero point one, which is going to be equal to zero point zero zero five Now what we want to do is we can now look at the distribution of our capital at the end of the first year and If no one dies it means there's no claims to pay out Which means our capital is essentially going to be the same and that means we have 1150 with the probability of zero point eight five five now if only a dies this means that We need to pay their summer shirt, which is one thousand seven hundred and we can see that that is greater than our initial Surplus, which means at the end of the year. We're gonna have a shortfall of 550 with the probability of zero point zero four five This is a bad situation to be in because this is ruined This meets the criteria that our claim is greater than our capital plus the premiums Now if only be dies It's actually not that bad from a company's point of view because their summer shirt of four hundred is less than the initial Capital of one thousand one hundred and fifty which means we will still have a surplus of seven hundred and fifty if only be dies And this will happen with the probability of zero point zero nine five However, if both of them die then the summer shirts sum together to get 2100 which gives us with a shortfall of nine hundred and fifty with the probability of zero point zero five again This is ruin So we now have our distribution of the capsule at the end of the first year Which is part one of the question now the question wants us to calculate You know what is the probability of ruin the probability of ruin? We now need to add those two conditions So when only a dies or when both of them die because those are the criteria that lead to the shortfall that lead to ruin So we add those probabilities together and we get zero point zero five Now most students get this part of the question and they're very very happy It's this next part of the question that Can get a little bit confusing quite a few students struggle with this instead of just going straight into the answer because in a way You can calculate this in two lines I'm actually going to go through the whole thinking process because you need to think for this question Again, we're not just applying blind formulas. That's not what actuaries do We are required to think and actually understand the material that we're studying to be conscious of what we're learning and like I say not just taking a formula and plugging in some values So we need to think in this question Now we know from the previous question that the probability of ruin of the one year is going to be equal to zero point zero five Okay, now this occurs if only a dies or if both of them die What we can now see is that the probability of ruin of the time to Well, you know probability of ruin of the two years is going to be equal to the probability of ruin of the one year Plus the probability of ruin occurring in year two given that it didn't occur in year one So what we see is that ruin doesn't occur in year one if a is a lie Remember if B dies, that's problems if a dies that was problems because the summer shirt for a was 1,700 which was greater than our initial surplus plus the premiums of 1,150 so it's a big problem if a dies However, if B died the summer shirt was lower than our initial surplus So we didn't really have to worry and that means we only have to focus on A a is the life that matters when it comes to whether this insurer is ruined or not They have enough capital enough surplus to handle the summer shirt of life B But they don't have enough for the summer shirt of life a Now there's no coincidence that if we look at it the probability of ruin of the one year Was actually equal to the annual probability of death of life a we kind of calculated a little different way But if you look they're both zero point zero five Now the probability of ruin occurring in year two Given it didn't occur in year one is the same as saying the probability that a dies in year two Given that a didn't die in year one and what that's gonna give us is That survival probability so one minus the probability one minus the probability that they died Multiplied by the probability that they died, which is gonna be one minus zero point zero five times zero point zero five Thus the probability of ruin off the time two is equal to the probability of ruin off the time one Plus this probability of ruin occurring in year two given that it didn't occur in year one Which like we've just said is the probability that a dies in year two given that a didn't die in year one And what we can see is we simply going to add these two calculations together at zero point zero five Plus one minus zero point zero five times zero point zero five and we get our answer of zero point zero nine seven five So the whole thing with this question was that you needed to think you need to solve this question from first principles and not try and Apply some formula and of course in this situation We saw that a lot of the assumptions of standard ruin theory were broken with premiums being received in the beginning claims being made at The end and we had to do it by basically brute force Which wasn't a problem because we only had two policies So it wasn't too difficult Mathematically and then we kind of end off the exam question by asking You know asking us to explain how runoff triangles work and how they relate to the classic ruin theory model Now whenever you've been given two different things and you need to explain them You also want to focus, you know, not only how they relate to each other But also focus on how they differ some students just explained ruin theory and then just explained runoff triangles And they didn't kind of look at well Where do they differ and of course the big thing is that ruin doesn't account for delay runoff triangles do and it's a five-point Question so at this level five points will give you five marks So you want to have this your first point ruin theory and runoff triangles both Determine how much capital is needed in reserves to meet upcoming claims But ruin theory assumes claims are settled as soon as they occur in reality This isn't the case and there can be a significant delay due to the claim process cycle Runoff triangles account for the delays and investigate claims in the year They occurred and not when they are settled They do this by tabulating claims and projecting the past patterns onto current claims to estimate their future development Now that's five points. That's all you need to get your five marks Of course, you could have written this question for for 20 marks and given an essay and gone into more detail All of that and of course That's another exam technique is to know when you've written enough so that you don't run out of time for the rest of the exam Anyway, let's maybe move on to the next question, which will be coming up soon