 Hi everyone, it's MJ the fellow Actree and in this video I want to talk about capital modeling and we're gonna see that we're gonna be taking a similar approach to that of ruined theory and in the previous video we spoke about ruined theory and we said that it Along with value at risk can be used as a capital modeling framework something that talks about capital Confidence and duration those are kind of the three things we want in a model so that we can use it to see how much capital we need However, the ruined theory is very much designed around the insurance industry. You've got the premiums coming in You've got the losses going out where what we want to maybe do in this video is make a more General capital model that can be used across the financial sector. So let's define a couple of things We're gonna say let kt equal the value of additional capital at time t now I'm referring to it as additional capital some textbooks will refer to it as economic capital or risk capital So just be aware of that But we're also gonna say let a t equal the value of assets at time t and let LT equal the value of liabilities at time t and then we want to kind of see how are they all related to each other and we say then Kt is equal to a t minus LT or essentially the additional capital at time t Is gonna be equal to the value of our assets at time t less the value of our liabilities and for those of you who have Done pensions you'll see that this links up very nicely with idea there known as surplus But let's have a very quick example if I owe ten million dollars in one year's time and I invest twelve million into stocks Then I currently have two million in additional capital We can kind of see we plug in the values into our formula and it looks very simple Now trust me the maths is gonna get a lot more difficult But in this case we see that we have a positive amount of additional capital and this is desired We would actually be a little bit worried if that capital is a negative amount because that means we wouldn't be able to meet Our obligations when they fall due later in the future But the fact that we have a positive amount of capital that is good But the question now is is this enough is just having an additional two million enough to cover our liabilities in the future and What we actually need to do is we actually need to think about capital and how it is relative to liabilities So we want to kind of look at the statement and what the statement is essentially saying is we're saying the probability That our additional capital is greater than some ratio of our liabilities We want it to be greater than one minus alpha where alpha is a representation of our risk tolerance so Essentially we can think of KLT Or this ratio of liabilities is some sort of threshold that we want our additional capital to cover and Alpha is our risk tolerance and it could be 0.5% it could be 5% it depends very much on the risk appetite of the institution Although if it's a financial institution the regulator might have a say in both what the KLT should be and What the risk tolerance should be so let's look at our example? let's say the regulator or Business or the board kind of says we need to have this comfort ratio of 0.2 and we want it to be our risk appetite or risk tolerance is 0.05 Then we can say well, okay Do we have more than 20% of the value of our liabilities stored up in additional capital? And we can see that in our example 2 million is greater than or equal to 20% of 10 million which represented our liabilities. So we see that that Inequality holds which means everything is good. In fact, this means that we have sufficient capital at time 0 But now we might want to ask ourselves well, okay, we've got enough capital at time 0 But will we have enough capital at time t? You know, we know that the additional capital time t is going to be equal to the assets at time t minus the liabilities at time t and Since the assets have been invested in shares We know that their value isn't going to stay static for anyone who's traded the stock market You know that these prices bounce up and down and they go all over the place Fortunately for us, we do have a model which allows us to predict possible futures of Our stocks and it's this following formula over here Now if you're seeing that for the first time I would want to recommend my market risk videos Where we go and look at okay, this is an ito's lemmas process something that allows us to integrate A squiggly line or saying that it's got Brownian motion So I want you to go maybe watch that video if you want to see how we actually got to this formula But at the end of the day, this is how I said how the maths can start getting a little bit more tricky is capital modeling we're essentially looking at a framework and we can start plugging in different Models for what we're dealing with so we could even have had another whole Complicated model for our liabilities. That's maybe based on the term structure of interest rates But what I want to do is just maybe home in on the one with regards to assets and we can see that assets the the log of assets is Distributed normally According to the following formula, of course having this formula is not enough We still need to estimate what certain parameters is specifically the mu and and the sigma squared which in as you know link up to the mean and the variance and There's various ways that you could do that you could take a forward-looking approach using the implied probability Gain from the black skulls model or you could do a backward-looking gawk model Which is a time series model that looks at Historical observations and fits a curve that also takes the previous volatility into consideration Which allows for volatility clustering and and all that kind of stuff But like I said, this is all discussed in the market risk video Essentially though at the end of the day why we are interested in these models is because they are gonna have an impact on the amount of capital we need to hold and essentially what we're gonna see is The higher the variance is of our assets The larger K needs to be in our KLT in order to meet the risk tolerance limits of alpha So what this means is that let's say I need to buy some stocks. I've got 10 million in in my liabilities Yes, I can put 12 million into into stocks But if I had to put say 12 million into a very very volatile stock Then that actually might not be enough because a very volatile shock could go down 20% It could go up 30% So the more safer or the lower the variance of the stock that I'm investing in the less of it I actually need to hold all the less additional capital I need to hold because it's gonna be able to be greater than our liabilities But the more riskier I start playing with my investment strategy the more support capital I actually need Now this starts getting a lot more complicated when we realize that our total assets can actually be split For instance, we could have 30% in bonds 20% in equity 10% in cash 40% in property and Our liabilities as well could be split We could have a 10-year fixed amount that we need to pay Maybe a 15-year inflation adjusted amount that we need to pay or for an insurance company Especially specifically a life insurance We might have to pay out of death benefit and we don't actually know exactly when that is going to be Now once we start splitting our assets and we start splitting our liabilities amongst all these different things We now need to know not only the variance of each of them But we also need to know the correlation of all the parameters Because all of these things are kind of be interacting on top of each other For example, if the interest rates had to fall It's gonna have an impact on both our assets and our liabilities And we need to see how sensitive the additional capital is needed in order to see if we need to make any adjustments So this is the whole idea and this is what makes capital modeling incredibly difficult Is because not only do we need to know all of these variances and these parameters for correlation We also need to understand that volatility starts changing in an unstable way over time And also the dependency structures of our correlations are also going to change in extreme times Now if we want to make it even more complicated this model here essentially is just looking at assets and liabilities What about income and expenses because this is also going to be depending on Expectations of future business, you know, are we gonna make more sales in the future? Are we gonna grow or we're gonna start losing market share and it also needs to start including unexpected events that might disrupt Businesses because you can think about it, you know There are some factors that affect liabilities Assets and income and expenses for example This whole coronavirus has impacted all four of these things in ways And it has changed the dependency structure of the correlation between the various assets and liabilities So we can see that we went from something very very simple. Oh, I owe 10 million I've got 12 million in stocks how much additional capital I've got 2 million great The math is nice and easy But we see if we start looking into the future if we start trying to model our assets and our liabilities things do start to get a lot more crazy and What essentially we start doing in capital modeling is we will maybe try and create separate models for say Market risk. This is the stock prices and those type of assets will make a separate one for credit risk This is if we're lending money out and we're a bank and something like that If we're an insurance company, then there's the actuarial risks of mortality Longivity and even morbidity that we need to maybe model and then there's of course operational risk Which we also need to factor in so what we will sometimes do when it comes to capital modeling is We can actually design separate models for each specific risk And then what we do when it comes to deciding how much capital we need We can then use something known as a copula that then combines them all together Now there's also a course on copulas that I recommend you go and see Where we'll see that there's different types of copulas that we can use and essentially what a copula is doing is It it's allowing the dependency structure of our assets and liabilities or our various risks to flex under certain conditions So if there is a pandemic like we experiencing now with the corona virus Then what can happen is we can use maybe say the Clayton copula Which as you can see has got a very fat tail in the lower ends and means when things are going down things are going to become a lot more correlated, but of course using copulas and Modeling these things is just one approach and it is quite a sophisticated approach So what instead kind of happens is there are also other things that we can do so One idea is to look at stress testing and scenario analysis Which we spoke about in the course on the principles of modeling We can also look at a modified version of ruin theory where instead of looking at say insurance losses We can change those losses to to something else that is more relevant to our business or we could even take a Very very simplistic and crude approach such as a generic model from a regulator And in fact, we actually gonna talk about that in the next video something known as the risk weighting approach So what I wanted to do in this video I didn't want to maybe scare you guys too much But I wanted to show that capital modeling can get incredibly complicated very very quickly And so because of that we do kind of fall back to more simpler methods That might be a little bit more crude But at least they're possible and they can give us some indication of how much capital we need to hold Because at the end of the day, we actually don't know exactly how much capital We need to hold and therefore you'll see some businesses try to be prudent by holding a little bit more than is Necessary from the regulator Of course depending on their shareholders They might have pressure on them not to hold too much because then it is gonna reduce the returns that the shareholders Can enjoy and at the end of the day the main source of capital is from the shareholders So is this balancing act that needs to be done? But anyway, let's go to the next video. We will talk more about the risk weighting approach