 Hi everyone, it's MJ the fellow actuary and in this video, we're going to be talking about ruin theory. So to prevent ruin, people buy insurance. They pay the insurer premium and the insurer takes on the risk. Now if the risk is realized and the insurer cannot pay, then the insurer is ruined. Therefore, capital reserves need to be calculated to cover these risks. If the reserves are too low, the probability of ruin will be too high. And if the reserves are too high, then shareholders will get a poor return. They'll probably withdraw their capital and the insurer will close down. So it's very important that we can calculate an accurate amount of capital needed to both prevent ruin and give shareholders a very good return. And that's what the benefit is of the ruin theory model. It helps us calculate how much initial capital we need. What is the lowest amount of capital needed given a probability of ruin that is acceptable by the shareholders? And what we can also use ruin theory for is it can help us to see the impact various premium pricing strategies will have on ruin. As we know, higher premiums could reduce the probability of ruin, but we want to be careful because that might result in less policy holders. Now let's visually see what ruin theory looks like. So we've got this following graph where u is our initial capital and it's given in this graph by the blue. A very, very important parameter as well is duration and that is given on our x-axis. So on our x-axis we have time and on our y-axis we have our capital or our reserves and u is going to be the initial amount. We then have our premiums which we assume that we're receiving continuously, so some people are paying today, some people are paying tomorrow and we're getting this continuous stream of premiums and the premiums are going to increase our capital position. However, every now and again there is going to be a random loss amount and that is given by these red bars and they decrease our capital position. Now if our capital position reaches zero or some sort of solvency line set by the regulator, then we say that ruin has occurred. So in this graph we can see at time 3 which I've circled in pink, this is where ruin has occurred. Now let's look at the mathematics. So with ruin theory what we have is the following. We've got this function that has got two parameters, that u remember I said that is our initial capital and t is time and what this is essentially saying it's the probability that the capital goes below zero for some given time period and it is equal to alpha. So alpha will be our probability of ruin. So here we see u is the initial capital, t is time and alpha is the probability of ruin. We then have capital, remember that was the blue bar in our picture and it is given by the following. So our capital is fluctuating with time and we see that it is equal to u which is the initial capital and it's going to then increase with ct which is our premiums so c are going to be the premiums multiplied by time so that's where the ct is coming from and then the s of t is going to be the total claims at time t and total claims this is given by the compound distribution where we have the sum of all of these x's. So here we see xi is the size of the ith claim and how many of them well that's also given by a random number because we do not know how many claims are going to occur over a given time period but we can model it with another distribution so n of t is going to be the number of claims at time t. So again like I said this is the mathematics and then we also have the graphs. Now a lovely exam question is well okay how do we reduce ruin well not only just an exam question but I guess a discussion that is had at the various board meetings and in management's offices. So of course we can reduce ruin by increasing our initial capital and increasing the amount of premiums we receive. We can also reduce ruin by decreasing the duration because remember the probability of us being ruined in one day is going to be a lot lower than the probability of us getting ruined over a year because over a year we're exposed to a lot more risks and claims happening but we can also decrease the probability of ruin by decreasing the total claims and we can reduce the total claims by maybe tackling the severity so this is the size of claims and we can do that by maybe saying we're not going to pay anything more than a specific limit for our benefits or we might want to introduce a reinsurance scheme that then handles claims over a certain amount. We can also reduce total claims by tackling the other dimension of risk which is frequency or the number of claims and a way to do that very simply is to introduce an excess but in all the previous videos we've been talking about ways on how to manage all these risks. But what's lovely about ruin theory and specifically this model is that it will show an insurance company how effective the various risk management mechanisms are and we can actually quantify which one is the best which we can then use in our cost benefit analysis. Of course ruin theory does get a little bit more complicated like for instance let's talk about premium security loading so remember we had the whole idea of our capital where C is the premiums however one idea is that our premiums should be linked to the total claims and a loading factor so instead of C just being a random number that we came up with it should be given to the following formula where lambda is going to be the frequency of claims m1 is the severity of the claims and this theta is going to be the loading factor. Now what if this loading factor is too high well then people are going to rather buy insurance from our competitors. What if this loading factor is too low well then the risk of ruin is increased but there's this whole idea that the premium should be connected to the risk kind of like the risk premium plus an additional little bit so that you know this loading factor can allow for unexpected things so the company and the insurance firm is protected from ruin. But again it can get even more tricky with the mathematics especially when we're looking in continuous time with infinite duration like I said math gets a bit tricky and here the best that we can do is just provide an upper bound for the probability of ultimate ruin we use something called Lumberg's inequality as I know as the adjustment coefficient r and this is an inverse measure of risk and essentially the higher the value of r the lower the upper bound on the ultimate probability of ruin. Now for a compound Poisson process with parameter lambda then r is the unique positive root of the following equation we're here lambda is the Poisson parameter which we just mentioned c is the premium rate per unit of time of course you can even extend this by having that loading factor on top of it as well and mxr is the moment generating function of the individual claim amount at point r and then it'll be possible to drive the upper and lower bounds for r and of course this model can be extended even further to include the reinsurance impact on ruin. Now look this video has been focusing primarily on the theory I just wanted to give you a very broad overview what you do need to do is a lot of past papers now currently you will find ruin theory exam questions in the CM2 exam they used to be in the CT6 exam so if you want to find more you know past papers to do more practice questions go to CT6 in fact back in my day when I used to write the CT6 the old exams everything relating to lost distributions and insurance used to be in CT6 now they've put part of it in CS2 and part of it in CM2 there's a little bit of a mess they're probably gonna change that in the future but I will make a video soon on a typical past paper question and also for those of you who are looking at this and being like oh maybe I can apply this in practice just remember that this is also mainly a very theoretical model because in reality there is a claim delay that ruin theory models don't necessarily account for in fact to handle that delays you need something else known as run-off triangles which we will also discuss at a later stage but anyway thank you so much for watching this video and I'll see you soon for another one. Cheers