 Hi everyone, it's MJ the fellow actuary and in this video we're looking at a question on extreme value theory which is part of loss distributions which is part of the CS2 actuarial exam and like I've been saying in the previous videos if you don't know I do tuition to the South African students but because of COVID and we're doing everything virtual if you are an international student check the description there will be a link to a form if you want to join in on the tuition for the following semester but with that said let's jump straight into the question maybe pause it and give it a read through but we are going to go through it and we're going to start off with the first question which is explain why extreme value theory models can be useful. So this is for two marks it's a nice theory question so hopefully everyone's getting 100 percent for the theory questions in the actual exams and essentially it's a very nice easy answer extreme value theory models they allow us to more accurately estimate the likelihood of extreme events that could cause ruin since traditional models tend to place more emphasis on more frequent events to improve overall model fit. So you're going to get one mark for saying that the model helps us understand extreme events and you're going to get one mark for saying why traditional models aren't that good at helping us understand extreme events for example if we had our data given by the black bars and the pink is the fitted distribution you will see that in the tails we're kind of underestimating these extreme events because our model is trying to fit all the medium claims and the claims closer to zero and it tends to ignore the extreme ones. Of course extreme value theory doesn't only have to imply to to insurance as we're going to see in the following question we can use it in sports. So the question reads as follows a sports scientist is interested in analyzing the probability that the javelin world record may be broken next year and is intending to use EVT to do this. The sports scientist has obtained data for the distances of all javelins thrown from all javelin competitions in 2019. The total number of throws recorded was three thousand. The sports scientist has carried out an EVT analysis using the generalized Pareto distribution by selecting only those throws that exceeded 50 meters. This resulted in the longest 150 throws being selected for the analysis. The following parameters were obtained from the EVT analysis. So essentially they've already done all the hard work they've already done everything we now just need to apply it. So the key points that we're getting from this question the total number of throws was three thousand. The extreme throws were 150. We're using the generalized Pareto distribution otherwise known as the peak over threshold approach and that has got the cumulative distribution while our limiting function is going to be as follows and we've been told what the various parameters are. Our threshold is 50 our gamma is 3 our beta is 15. Now when the question says determine the percentage of javelins thrown that would be expected to exceed 70 meters next year again we now just need to apply our theory. So step one we're going to say what is the probability that x is greater than 70 given that x is greater than 50 and the reason why we want this because now we can use our generalized Pareto distribution and we have the thing that we're trying to exceed minus the threshold. We plug that into our formula substituting those parameters for the values given and we get our nice little answer of 0.331816 which means there is a 33.1816 chance that a javelin exceeds 70 given that it first exceeded 50. Now interestingly enough a lot of students stop here they're like okay my answer is done and I'm like no that's just step one because step two we need to see well of the 3000 javelins thrown only 150 exceeded 50 because think about it your answer if you're keeping it at 33 percent you're saying that 33 percent of the javelins are going to exceed 70 but that that should bring a little bit of a long bell we're looking at extreme values so your value or your answer is always going to be a very very small percentage so whenever you come up with 33 percent as an answer to an extreme value question that should be yeah checking yourself and saying hold on something doesn't sound right and of course because you've only done step one step two is to say well of those 3000 javelins thrown only 150 exceeded 50 and that is 5 percent now we can combine that 5 percent with that 33 percent to get our final answer that is step three so the percentage of javelins that are expected to exceed 70 meters next year is going to be the probability that they exceed 50 which is your 5 percent multiplied by the probability that they exceed 70 given that they had exceeded 50 which is what we had calculated earlier that's our 33 percent so we're taking our 5 percent multiplying by the 33 percent and we're getting our final answer of 1.6591 percent and that answer sounds a lot better remember it's an extreme event so we're expecting the probability to be quite low now look the theory of extreme value theory is is crazy it's actually very very beautiful i mean especially this approach there's multiple approaches of doing it but the beautiful thing about this approach the peak over threshold is what it's saying is that the extreme values regardless of what the distribution actually is the limiting distribution is going to be this generalized Pareto distribution given down there below and no matter where we take those extreme values we're going to get this type of distribution of course that is like a 10 second explanation and you can see i've created an entire Udemy course and before you can even get to the extreme value theory video which is 18 minutes long you do want to have a little bit of a background in loss distributions and see why we need loss distributions by understanding you know an introduction to insurance and how insurance is designed of course once you have extreme value theory you can then move into reinsurance and collective risk models and like i said it's a great thing but i'll put a maybe a link to the Udemy videos if you haven't already watched them yet because we are going to finish up on the question which is just by saying comment on the limitations of this analysis and of course extreme value theory it's a very very powerful mathematics but it can only really be used in some very select cases in the physical world you know sometimes those criteria are met for instance it does require that you know these observations are independent and identically distributed is that going to be the case with these javelins well not really because we're looking at all the competitions which means it's likely that you know the same person was in a couple of these competitions also you sometimes throw three times so we're not going to have independence as the same person is going to be throwing a few also they're not going to be identically distributed as throwing javelins might depend on factors like weather condition location you know maybe it's easier to throw when you're at the higher altitude than at a competition that's at lower altitude with wind blowing and all these kind of things then do we expect javelin throws to be constant year on year well not necessarily maybe there's been upgrades in javelins to lighter carbon fibers maybe a new throwing technique has been discovered maybe there's been a pandemic that is limited the training time for athletes maybe new rules and regulations are introduced and this is kind of the thing about as in actuaries we're not just focused merely on the statistics and the mathematics we also want to have a little bit of a holistic understanding of what we're applying this thing to to see if it actually is the perfect fit of course we can step back into our statistics and say yeah maybe that sample size of 150 is probably too small you can also go statistically well would we get the same answer with a different hurdle rate because remember the actual distribution of these extreme values will be a little bit different the general perito is just a limiting distribution but like I said if you want to know more about the theory of extreme value theory check out the udemy course where like I say there's quite a lot of prior um or yeah pre-information that you're going to need to know before we can tackle this but it's absolutely beautiful I love how the other type of extreme value theory ties in so beautifully with the central limit theory which is such a foundational concept of statistics like I say this whole theory is is beautiful but it is complicated it does require quite a lot of of prior knowledge so that's why if you need it go check out the udemy course as always thanks so much for watching and I'll see you guys all for another video in the future cheers