 Hi everyone, it's MJ the fellow act tree and in this video I want to go through a very simple capital calculation example So let's pretend that we are playing a game against some thugs And they have agreed to pay a $60 and in return we need to flip a coin And if the coin lands on heads, I have to pay them $100 Now if we had to look at the expected payoff We're getting $60 and then there's a 50% chance that we have to pay out $100 So we can see that I'm actually expected to make $10 per game However, these are thugs and if I can't pay they will break my legs Also a whole bunch of thugs want to play the game at the same time Now let's say I have a risk appetite of 5% That means I am prepared to risk my legs getting broken with a 5% chance of that happening The question now is how much capital do I need to hold for an X amount of thugs So that my return on capital is maximized given this 5% risk limit Now what we're going to do is we're going to let X be anywhere between 1 and 7 thugs And we're going to see what is the probability of ruin and how much capital I need to hold And what is going to be the expected return And you can see that suddenly it's gone from being a very simple game of flipping a coin And when it comes to capital modeling, there's ruin, there's how much capital to hold There's expected return, you can see how can get quite complicated So that's how we're going to be doing this very simple example Just to go through the various steps and things that we have to see So what I've done is I've drawn out this table and in the first column We've just got the bunch of thugs that we're playing against So we either play against one thug or we can play all the way up to seven thugs And we can see that this is the the income that we're making So the 60, 120, 180, all the way up to 420 That is the fact that each thug is playing $60 in order to play the game And because we're expected to make $10 per game, we can see that if we play against a seven thugs We're going to make $70 as expected profit and you can see 60, 50, 40, 30, 20, 10 So because this is a profitable game, we actually want to increase it as much as possible However, we need to be aware of situations where we can get ruined or in our situation our legs get broken And we see that in the very first instance, so let's say we agree to play with just one thug And we flip a coin This means that if the coin lands on heads, we have to pay $100 And we can see that we're going to be ruined. It's actually show the probability of ruin as well So we can see that there's a 50% chance that I have to pay out $100 and that I am ruined When we come to the second game, we can see that there's another chance where I get Ruined and that is when two heads come up with the two coins that we're flipping And this is a 25% chance because it's 50% heads on the first coin 50% heads on the second coin you join the two together and get 25% so Here I now have a 25% chance of being ruined because it is when both land on heads If just one lands on heads, and this is the interesting part Then my income is enough to cover the loss. So if there's just one head The income of 120 will be able to cover the loss of that one game So it's only if both lands on heads and that's we've got minus 200 If we come to three thugs, we see that once again I am ruined if If the the payout that I have to make is more than 200 and this happens if there's two heads Or if there's three heads, so what I'm looking at actually now in these situations Is a bit of a cumulative probability of the binomial Distribution So with three hugs three three thugs If two or more heads are played then I am ruined my income is not allowed to cover it And this has a chance of 50 and we can keep going four thugs It's if there's 300 and this is if I get three or more heads out of the four coins With five thugs if I get four or more heads, then I'm ruined um This one's actually quite quite interesting in the sense that it has to be more than 400 because Of the 60 60 60 that actually comes up to a multiple of Well, it's it's 300 and that's you know, that means we can actually pay up three three games um So it's only if it's minus 400 that we are ruined and this happens with an 18.75 percent Like I said, I'm using the cumulative probability of the binomial distribution in order to calculate my probabilities of ruin We can see it for six and we can see it for seven Now the interesting part comes in with okay Once we know the probability of ruin and the amount that's going to cause us to be ruined We can calculate the capital that we required and what the percentage of ruin will then be So we can see that when we're playing against one thug in order to make up this hundred dollars over here We've got 60 we need to pay another 40 So we need to hold 40 In capital and if we hold 40 in capital We're going to see that our chance of ruin is zero percent Why because the 60 from the income and the 40 Where we're holding in capital is going to be able to handle the worst-case scenario of us having to pay a hundred dollars when it's heads um, we see with the when there's two thugs We now need to hold 80 dollars with regards to to capital because we've got 120 which is enough to cover the first loss But we need another 80 in order to cover the second loss So essentially we're taking these two amounts subtracting them together to calculate how much capital we need Things do start getting a little bit interesting the further down the thugs we go So if we look at just three thugs, we have the situation where We're getting 180 dollars in and we're going to be ruined if we have to pay 200 or more dollars So what we can do is we can hold just say $20 and then we'll have 200 and that will be able to cover the situation where we get two heads out of three However, we are still exposed to the situation that three of the coins lands on heads And that has a 12.5 chance of happening So we can see that if we hold 20 in capital We can reduce the ruin to 12.5 percent, which is much better from 50 percent And if we hold 120 we can reduce the ruin all the way down to zero percent But let's maybe actually look at expected returns while we do this as well because what we can see specifically for one and two thugs Our expected return is 25 percent. We're putting up 40 dollars. We're making 10 Profit so 10 divided by 40 is 25. Here. We're making 20 dollars. We're putting up 80 once again Our expected return is 25 percent But look what happens when it's three thugs and we put just 20 dollars Our expected return goes up to 150 percent. However, it's 12.5 percent risk that our legs get broken And remember our appetite is we want it to be around 5 percent So 12.5 percent is a little bit too high So the next step that we can go to is having to put in 120 dollars Because any other amount that we put between 20 and 120 is still going to have a 12.5 percent chance of ruin 120 is the lowest amount of capital that we can put in in order to reduce our ruin a little bit further And this is just the way the game has been designed And we can see once we put up 120 and we're expected to get 30 I expect a return drops to 25 percent to gain But let's now come up to to four thugs. So once we have four thugs There is a 31.25 percent chance of us being ruined because if three or four heads Land up of our on our coins then we are ruined So if we're flipping four four coins if three or more lands on heads Then we ruin because if just one or two we can use our income and that is going to cover those losses But where things get interesting is if we add 60 as capital Then we move our total cash holdings up to 300 Which means that we only get ruined now if there are four heads So we flip four coins and all four of them have to land on heads And that is you know 0.5 to the power of four, which is 6.25 percent And you can see I expected return is then going to be 40 divided by 60 Which is 66.67 percent and that's quite cool because what we've done is I mean we've almost doubled our expected return And that risk of 6.25 percent is quite close to our 5 percent limit You can see the other thing that we would have to do is add in 160 And that would reduce our risk down to zero percent But then our expected return will be back to 25 percent if we have say five thugs Well, we know that we are getting in 300 dollars And we're going to be ruined if only of the five coins that we're flipping four or more have to land on heads This has got a probability of ruin of 18.75 percent So this was actually the lowest probability of ruin Before we started holding any capital and the reason for this is that because you are 60 60 60 Makes 300 which is a lowest common denominator or a highest common factor something to do with math But you can see how the 300 links up with the 100 100 100 and the 60 60 60 60 Anyway, coming back we can see that If we hold a hundred dollars as capital Then what we're doing is we can reduce the probability of ruin of 18.75 percent Down to 3.125 percent and that's quite good. That's we see it's it's 50 percent So it's double the expected return of all the other ones and it is below our risk appetite of 5 percent The next thing that we would have to do if we would want to reduce ruin even further would be to hold 200 dollars Which would bring it down to zero percent. However, we then have our expected return coming back down to 25 percent And we can see for six and seven thugs What tends to happen is We can hold 40 percent Sorry 40 dollars in capital and that is going to reduce the probability of ruin of 34.375 percent to 10.93 percent And that gives a 150 percent return But 10.93 is quite higher than our risk appetite Or we could hold 140 And this is where things also get quite interesting. We can reduce the risk down to 1.56 percent Because now it's it's almost you need a roll Or you need to flip six coins and all six need to land on heads 0.5 to the power of six and you can see that that Has a quite a good expected return of 42.85 percent. So quite a high return for quite a low risk And then with seven we see that okay What we're doing here is if we hold 80 in capital and we can see 70 divided by 80 is equal to 87.5 percent So for 6.25 percent We're now getting an expected return of 87.5 percent Which is the same risk as if we're played with four thugs But now we're getting a higher expected return The reason why we're getting high expected returns as we start increasing the amount of thugs that we start playing with Is because we can see that the variance is kind of linked to the number of thugs that we're playing against So that's one of the things to to keep in in consideration And then what we can see here is we can also hold 180 dollars And that will reduce our capital of return to 0.78 percent And we would have an expected return of 38.8 percent And where this starts getting interesting is you can start seeing that okay the more Or the higher expected return I want to take on the more risk. I need to expose myself onto and Not only that but you can also reduce your your your probability of ruin by increasing the number of thugs because of the impact That's going to have on the variance So we can see that if we want to absolutely maximize our expected return We would have to go with 150 percent and have a Probability of ruin of 10.93 percent. Of course if you're not holding any capital Then your return is is infinite because you're not risking any capital And you you know, whatever you earn is is, you know, you can't calculate your return there So it might even be worth considering playing five thugs And not holding any capital and getting 18.75 percent, you know taking that chance Or you can say I want 150 percent return. I'm going to put 40 dollars I'm going to have a 10.93 percent chance of getting ruined Or I'm going to have put 80 in capital It's going to give me a ruin of 6.25 percent And then I'm going to have an expected return of 87.5 percent or you could be saying Hold on. I only want to put up a hundred in capital And what I'll do then is play against five thugs and this will give me a 3.125 percent chance Um of ruin which is below my risk appetite. It's going to give me a 50 return So at the end of the day because we couldn't find one that was exactly at 5 percent because this is also one of the things is because the more return The more risk we take on the more return we can get We don't want to come in too low. We don't want to go I think there was like a 1.56 percent or a 70 0.78 percent We don't want to take those ones on because we've made this this Or we've come to this conclusion that we are prepared to take on five percent So it is a bit of a problem if you're under your risk appetite because it means that You've got room to to take on more risk which will allow you to take on more expected return So that's why we won't necessarily go for the lowest probability of ruin We need to coincide it with our expected return. Remember, we're trying to minimize ruin and maximize return So these are the two things that you want to consider which means the final answer would either be Five thugs 100 capital get a return of 50 and a risk of 3.125 percent in this case We're a little bit under our expected risk Or we can go a little bit over our risk and 6.25 percent is actually it's it's less of a deviation than 3.125 percent And here what we can do is say that now there's seven thugs We have 80 capital this will give us a return of 87.5 percent with a risk of 6.25 percent And what I wanted to basically just show with this video is when we have a very very simple situation Which is literally we've got some thugs and we're flipping coins. They're paying us and we're then paying them if it's a heads We can see how complicated This whole capital calculation can become. I mean it's once you've done all the maths is still going on straight forward On what exactly we should do like I said, do we play against five thugs or seven thugs? You know, do we hold 100 capital? Do we hold 80 capital? You know, it's these are things that the the board of directors will then sit around and have to have a vote Or have to have a discussion to determine the optimal strategy of the business But you can now imagine that when we stop going from just flipping coins To investing in sophisticated structured products on this on you know, public financial markets And you start going into crazy derivatives and arbitrage and all these other weird and wonderful things You can see how capital modeling Can become very very confusing and very very tricky But at the end of the day, you need to come back to the first principles and understand the relationship between capital ruin and uh return And hopefully y'all that will prepare you because in these exams It's unlikely that they'll give you a very very difficult mathematical question around capital modeling But they will maybe throw in something where they ask you or they'll test you on some of the fundamental first principles So make sure you have a good grasp of those. Anyway, thank you so much for watching and I'll see you guys soon. Cheers