 Hi everyone, it's MJ the fellow actuary and in this video. I want to talk about the Merton model So let's maybe start off with the history. It was created by Robert Cox Merton And he is famous for actually being part of the black skulls model Of which he won the Nobel Prize and he was part of the long-term capital management fund that Used the black skulls model was very very effective for a couple of years getting plus 40% returns before getting absolutely Destroyed when Russia defaulted and the Thai currency fell apart Ending to lots and lots of money getting lost But because the model was you know, it's often referred to just as the black skulls, but its full name is the black skulls Merton model Very interestingly is he has a student called Robert Jarrow Who we will be talking about a little bit late in another video who came up with the Yarrow Turnbull model which uses mock of jump processes to determine The probability of default anyway coming back to the Merton model We will see that's three people with the acronym of KMV They would take this model and they would extend it and Moody which is a credit rating agency would then buy out that KMV model and use it in their proprietary models known as lost Colk and Risk Colk so this model that we're using is very much a foundation to something that is used today And it's built very much on this idea that a company is financed by both equity and Debt and the Merton model It's a structured model in the sense that it's going to be trying to use the share price or the equity of a firm in order to Calculate the probability of default And essentially less this maybe explore this idea a little bit more so a company is financed by equity and debt Now we know that equity this is when you buy a share You get a form of ownership which allows you to vote on the strategy of that firm And it also gives you unlimited potential return You can think if you buy a stock today at ten dollars The price in the future could go up indefinitely and therefore there's this unlimited potential return Whereas if we look at debt debt in a way gets a fixed return So normally you say I'm gonna lend you money at a rate of 10% and then that return is essentially what you get and In in in the fact that you're getting less potential return comes with the idea that you're getting more Security so what we see with debt and equity is that if a company has to fall apart debt gets repaid first and this is because they have this higher Security or lower risk. They are gonna get lower return Whereas equity that doesn't have as much security gets paid out lost because they're taking higher risk They're gonna get higher return. Now part of this whole idea With the merchant model is we are going to be assuming that our equity isn't paying any dividends And we're also going to assume that debt is simply a zero coupon bond Which means all interest repayments are combined in the final redemption date So those are just two assumptions to to keep in the back of your mind Now let's look at the formula So essentially what we have is we're saying the Value of the company at time zero or if you want to think of it as the assets of the company at time zero But we're gonna go with value so the value at the company at time zero is Equal to the equity at time zero plus the debt at time zero and this is not time zero of the company This is just we'd say time zero for the present value right now, and then we have the future Date which is given by You know the letter t and we can see that in the future So debt t the future value of the debt It's either going to be equal to the smaller of of two values either It's going to be equal to The debt the original debt times the interest rate remember because this is a zero coupon bond Or it's going to be equal to the value of the company at that future time It's going to be equal to the smaller of those two values Whereas equity is going to be equal to the maximum of either the future value minus the future debt or zero So the whole idea with equity is remember to remember its limited liability if a company fails The shareholders don't have to put in more money. So this actually maybe just explain this a little bit more with an example So we're gonna look at two situations when the company does well and when the company does badly so when the company does well it's still in Business and what this means is that equity is therefore going to be equal to the value of the company less the debt and The debt is going to be repaid and it's going to be equal to the original amount Times the interest that was that was paid and this is kind of what we're seeing here This is that debt zero times the interest rate or here value Of the future minus debt of the future. So that's what happens if the company does well Now, let's look at what happens when the company does badly. So the idea here is that the company gets Liquidated it's it's basically it's game over So in the situation the debt was not able to be paid or the debt is more than the value So what happens is equity then receives zero now? If the future value is going to be less than the debt times the interest rate Then that is what the debtors are going to get they do get that lost from a remaining amount over there So when the company does well, these are the payoffs when the company does badly These are the payoffs over here Now if we had to draw the diagrams of these payoffs, we will see that equity is Basically acting like a long call on the company. You can think of this as the value of the company You can see as soon as it's above debt the equity Holders are going to be as we say in the money. Although when they're below You know when the value of the company is below the debt and the company is being liquidated We can see that the equity holders are going to get nothing and they're gonna lose that original investment that they put in so you Can think of that original investment as the premium for a long call and if we had to look at say The debt payoff remember the debt payoff we spoke about over here It's the minimum of the debts at the current date times the interest Or it is the future value of the company. It's gonna be the minimum of the two We're gonna see that this is the payoff diagram And we're gonna see that debt is behaving very much like a short put on a company So this was the big idea of the Merton's model was that okay? Hold on hold on what we're essentially dealing with is we can use the black skulls model that is used to value Options specifically call and put options and the premiums that one must pay in order to to get the right That these these contracts grant we see that we can use that when it comes to Equity and debt because here we can see that okay equity Like I say is is like the the premium for the long call and we're gonna see that the interest Repayment so the interest times the debt is effectively going to be the premium for the put option That the debt the person who's lending the money is going to be receiving so What we can do is if we had it like I say look at that black skulls model We can equate a whole bunch of these formulas or these these values that relate to call options We can relate it equal to our company So for instance equity a time zero is going to be the premium of a call option The debt at times t is the same as the strike price the interest times the debt at time Zero is equal to the premium for the put option the value of the company is going to be like the share price Of a company at a future time t and the value of the company time zero is going to be equal to the current share price of the company So like I said The big thing to to realize with the Merchant model And sometimes people do get a little bit tricked up with this is when we're referring to value It's the company's assets and this is not necessarily the share price What we're just seeing is that in the Merchant model assets take the place of share price in the black skulls model So also be aware that it is difficult in reality to know what this asset value is at all times Plus we're also assuming like what we'll see a little bit later the black skulls is that there's this volatility You know figuring out volatility of a share price is quite difficult You know, you've got to use the gawk model or a whole bunch of other other techniques It's it's also a little bit of the Achilles heel for the Merchant model Is that not only assuming that we know what the asset value is at all times? That's an observable security, but we're also assuming that we know the volatility of this asset value at all times And and like I said that is probably something that trips up this model quite a bit but bearing that in mind We can come back to our black skulls model Using these different values for it We can see that we can get the following formulas and how I'm getting these two formulas So I'm essentially taking the black skulls model and changing each of these values for what the Merchant model suggested And I've done that by linking up the payoff diagrams with those of an option diagram And this is where the fun begins because we know with the black skulls model if you're familiar with that We know that the probability that the future share price is less than the strike price is equal to the normal cumulative value of negative d2 Which the d2 is given by that equation over there this is basically the probability that the put will be in the money and What that means is we can use this with the Merchant model to say well Okay, if that is the probability that the puts gonna be in the money that is the same as saying the probability of Default at times t is going to be equal to or essentially this normal cumulative of negative d2. So This is where it gets powerful It's because if we know when the probability that the put will be in the money And we know the put will be in the money when it comes down here This means that the long person in the put isn't is enjoying the benefit But the person who's on the short side of it is going to experience a downfall We can use that logic to link it up and say, okay So the probability that the put will be in the money is the same as the probability of default of this company So this is one of the powerful things is that we can figure out this formula here Plug it in and it is going to tell us the probability of default, but that's not all Also, we're knowing from our formula that the interest times the the original debt is going to be equal to The debt at a future time Discounted by the risk-free rate Subtracted by the current debt and it's going to give us this current formula over here Which means we can do a little bit of mathematics and what we find is we can put the two together Because what we also know is that current debt is going to be equal to the future debt Discounted at the actual interest rate that is being applied to that debt instrument, which we can denote as B So deep sorry B is the implied interest rate or is the risk-free rate We can subtract the two together and we can find out what is the implied credit spread of this bond So we can know okay and remember we do expect corporate bonds to pay out an interest rate higher than the risk-free rate That should get on a government bond because of this higher likelihood of default and the Merton model is going to help us Calculate what that credit spread is so I can say this is the powerful thing of The Merton model is it tells us the probability of default Okay, it can also be used to calculate the company's debt at various times and We can use it to calculate the credit spread now look This was very much a high Overview of the Merton model it is something that is tricky to get your head over and like I said the big thing that I would Recommend is that you go back and you look at your the whole material around calls and option prices so for instance Go and watch videos on call and put options go and study the payoff diagrams of those instruments And then even go tackle the black skulls model and once you've done that come and rewatch this video again And that should give you just a clearer indication of what's happening with this Merton model But very very powerful We can use the share price of a company to determine probability of default on its corporate loans What the company's debt is and the credit spread so very very powerful? Things that we can bring from the black skulls. Sorry from the Merton model Anyway, thank you so much for for watching