 Hi everyone, it's MJ, the Fellow Actory, and in this video we're going to go through a no-claim bonus question, which is part of Mark of Chains, which is part of CS2. Now what many of you don't know is that I offer tuition for the South African students, but because of COVID and the fact that we're all going virtual and I'm basically running all of this stuff from home and I'm not flying to Johannesburg and all over the place, I kind of feel like we can open this up to international students. So if you are interested in tuition for the actual exams, check out the description. There will be a link to a form and yeah, we might be welcoming international students for future tuition. But other than that, let's get started with our question. And as you can see, there's a lot in this question. So maybe pause the video and read through it on your own and then hit resume and we'll go through it together. Otherwise, it's going to take probably five minutes to read through all of that. And we are going to go through each section on its own. So let's maybe start up with the question preamble, which says, yeah, a simple no-claim discount system has four levels of discount. A new policy holder starts at 0% and at the end of each policy year, they can change their levels according to the following rules. Okay, they're either going to move up a level or remain at the maximum discount level or depending on how many claims they have, if there's been one claim, they're going to drop back one level or remain at the bottom. Or if there's been more than one claim, so two or more, then they're going to just go all the way back to zero. So what I'd like to do is draw out my state. So I should get a little exam pad, draw out my various states and let's see what are the key points in this question. So the idea here is that we have four levels, 0%, 20%, 40%, 60%. Great. We then are told that we're starting at the first level. And if there are no claims, we move up one level, or we stay at that maximum level of 60%. If there's one claim, we move down one level, or we stay at the bottom level. If there's two or more claims, we move straight to the bottom level, or we stay at the bottom level. So now let's visually represent what we've just said here. So if there's no claim, we're going to represent this by the green arrow. And we're going to show how you go from zero to 20, 20 to 40, 40 to 60, and 60 stays in 60 if there's no claim. If there has been one claim, we'll see that if you're in 60, you go down to 40. If you're in 40, you go down to 20. If you're in 20, you go down to zero. And if you're in zero, you stay in zero. And then if there's been two or more claims, essentially, all the states come back to zero, including zero stays at zero. And it's important to note that quite a few students forget to have the state that goes back to its own state. So they sometimes admit the fact that if there's no claim, the 60 doesn't come back to it. And they admit the idea that if you're at zero, you don't go back to zero. And then you'll see that you've done something wrong because then all your probabilities will not sum to one. Speaking of probabilities, this is now the next thing that we read in the question. It says for a particular policy holder in any year, the probability of a claim three year is seven divided by 10. So that means no claim. The probability of exactly one claim is one divided by five. And the probability of more than one claim is one divided by 10. So once again, we've got our little graph over here and how we basically put all these numbers together was we had our probabilities of no claim being the seven divided by 10. Well, I like to just put it into two decimal format, which is 0.7. So every time there's the green, I've got 0.7. One claim that is the yellow, that's one divided by five, or let's get it back into decimals, 0.2. And you can see all the yellows are now 0.2. And then more than or two claims or more, we have it with the red number one divided by 10, 0.1 in the decimal format. And like I say, I like to put it into decimals, because then it's easier to check that they all sum to one. I mean, in the exam, they could have actually made this a little bit more difficult, a little more tricky. And that is by admitting one of these probabilities. And we would have been able to figure it out, because all probabilities have to sum to one. So in the exam, that could be a little bit of a trick, knowing that they all sum to one, you might be like, I didn't get enough information. It's like, okay, you are, but think about it, all probabilities sum to one. Now what we need to do is the very first part of the question, which is write down the transition matrix for this time homogeneous Markov chain. Now, if you're going to jump straight into the step without drawing up that diagram, chances are you're going to make a couple of mistakes. But by drawing up the diagram, you're going to see how easy it is to make this transitional matrix. So we've got our diagram over here. And essentially, our matrix, we have zero 2040 60, zero 2040 60. And you can see from zero to zero, what we have to do is we have to add the yellow and the red, hence why I've made it orange, which is 0.3, because you're going to stay in zero, if you either have one claim, or you have two or more claims, then the only way to go from zero to 20, that's going to be if you don't claim, that's at 0.7, that's the green, but you cannot go from zero to 40, you cannot go from zero to 60. So that's where it's the 00. Then you can go through the same logic with the 20, the 40 and the 60. But this diagram should help you see the different colors relating to the type of claims. And like I say, you had to add the the one claim or the one or more claim in order to get that 0.3. And that happens for both the zero percent going to zero percent and the 20% going to that 0%. So you always want to, you know, combine your probabilities that start and end at the same place. That's why like I said, we're adding that 0.1 and that's 0.2 in order to get that 0.3. Again, that's a mistake that some students tend to make on this. And again, what you want to do is just check that all your rows add to one. So the first row adds to one, 0.3 plus 0.7 to one, one, one, one. I mean, if you're doing this in Excel, you can even run a little formula to make sure that that's all good. So let's look at the second part of the question, which is calculate the two step transition probabilities from state i to state j. So this is the probability ij over two steps. Okay, so our theory, and this is why it's always important to, you know, hopefully you guys have read up on Markov chains, you've gone through the act ed notes and not just jumping into this question and trying to follow along, because the theory will tell us that, you know, we have this two step transitional probabilities is going to be equal to the square of the transition matrix. So we either need to do the calculation by hand, which is very slow. It's what I had to do back in my day, or you can now use Excel because a lot of these exams are being written online because of of COVID. And you can see like back in my day, you had to do some matrix multiplication and then rely on your foundational maths. And it's very messy and all these kind of things where when it comes to Excel, you use a function called M multiple sense for matrix multiplication. And it spits this out for you instantly. So it's like three marks that you can get instantly. And we will go through how to do all the Excel things at the end of this video. Part three of the question says if the policy holder starts with no discount, calculate the probability that this policy holder is at the maximum discount level five years later. So again, if P is equal to our transitional matrix, essentially we need P to the power of five. Now then what we have that we can locate the top row and the final column. And that's going to give us our answer. This represents starting at 0% discount and ending in 60% discount after the five years. Now with Excel, this is very, very quick, nice and easy. Without Excel, like I say back in my day, we had to use the Chapman-Kolmogorov equation, which was given by this. And the whole idea is that remember, because it's over a five year period, you can actually have an instance where you actually have a claim and then no claims. But now when does that claim happen? Does it happen the first year, the second year, the third year or the fourth year? And you need to actually sum all of those possibilities together. Hence why you have the Chapman-Kolmogorov equation, lots of maths, lots of fun. But nowadays we just use Excel. And I'll show you how easy it is, essentially using that MALT. You do the range, you do it a few times and you can get to P5 very, very quick. Like I say, I'll show you how to do that on Excel towards the end of the video. But boom, it's a five more question. And you could split this out very, very quickly and get your 0.43 as your answer. Coming up to the final part of the question, it says if a large number of people having the same claim probability take up policies at the same time, calculate the proportion you would expect to be in each discount category in the long run. So once again, we need to rely on our theory. And the theory in this instance is that we need to find the stationary distribution. This is, you know, how does it stabilize in the long run? Again, we need a little bit of theory where we have this following equation where we've got pi times our transition matrix is going to be equal to pi, where pi is a vector that represents the stationary distribution of how many people, what percentage of people are in each of the various states. Now, what that basically means is we want, you know, P to the power of infinity. Because what's that's going to show is our distribution amongst the states is going to become stable. So over the long run, it's going to settle on this. So if we had to really just push it up, we will get that. Now I'll show you a hack on Excel and how to do that very, very quickly. But like I said, before my day, before you could use Excel on these online exams because of COVID, it was a lot harder back in my day, because we had to turn to the mathematics couldn't rely on the computer. But essentially, we needed to use simultaneous equations. And that trick here is also again to remember that all probabilities sum to one. And that's another equation that we can use. So what I'll do is I'll get our original transitional matrix remember this is P, and I've got our vector there at the top. And I just put it in these colors just to try and help you follow along, you know, what's going on. I kind of find with this subject, there's a lot of math, adding a little bit of color and explanation videos does go a long way to help some of the students see what we're talking about. So essentially, we have this vector times a matrix equaling a vector. And so we have our very first one, you know, at state the 0% discount. And we basically taking that 0.3, you know, the zero discount 0.3 in the 20% discount 0.1 in the 40% 0.1 in the 60%. And we essentially going to go through this on each of those columns. And you can see we now have our simultaneous equations plus the one where they all have to equal to one. Now what we do, this is mathematics, we rearrange the equations, we get a working variable, we do our substitutions, and then you can get your stationary distribution after doing a little bit of the hard work of getting 0.1639, 0.1556, 0.2042, and 0.4764. Now the reason I didn't show you all the math is because I'm going to show you a much quicker way on how to do this on Excel. So let's jump quickly into Excel. Okay, I'm using Google Sheets, but it uses the exact same formula as Excel. So essentially, what we want to do, we've got P over here, if we want to get, let's say, P squared. So over here, which is what we needed for one of the questions, essentially, what we're going to do is the following M mult, and then we just select our range, and then we just select the same one again. And it's going to push out the little probability for us. In fact, let me move that over there. And I mean, you could have your 20%, you could make it nice and pretty, but we don't have to do that too much. So if I wanted to get, say, P four, where's my little hat? P four. Essentially, what we're going to be doing here is M alt. And it's going to be number two, and number two again. And that's going to give us now our P four, which means if we want P five, which we need for the one question, essentially what we need to do again is M alt. We have four. And now we just need to add one. So we're going to go to this top one over here. And we're going to have, you know, P five. Now remember, we also wanted that P infinity thing. So we could easily get P 10, which would be M alt. And then essentially, we take this one here. And then we take this one here again. We get our answers. And then like I said, you want P infinity, but you'll see it'll start to stabilize very quickly. So again, we got M alt without P 10. And our P 10. And you can actually start seeing, you know, the answer is stabilizing. So we'll maybe do it one more time. Remember, you want to be very cautious of time in the exam. So don't go like all the way to like P 200. This should be fine for our last one. You can do this either to check your answer, or you can just tell the examiner this is what you what you do. There might even be a quicker formula on how to do it. And yet you can see we finally starting to stabilize to our answer. And then that over there is going to be your stable or your stationary distribution. But like I said, I hope you guys have enjoyed this video. If you want actual tuition and your international student, check out the form in the description below. And Joe, we can look to signing you guys up for the second semester. Or if you're watching this later on, you know, in 2023 2024, we're going to keep doing this. So if you're watching this video in the future, please feel free to use the form and we can get you guys into the course. Otherwise, thank you so much for watching. And I'll see you guys soon for another video. Cheers.