hello David Harper aonic turtle with a brief review of the Merton model for probability of default for frm candidates I continue to follow Michael on and credit risk models yesterday we looked at this key formula for credit risk where we saw that the expected loss of a credit or loan portfolio is equal to the product of three components adjusted exposure we looked at that yesterday lost given default and expected default frequency which for our purpose is the same as the probability of default or PD so let's look at that now first just to take one step back briefly let me just remind that we have two broad approaches to the estimation of the probability of default A reduced form which considers default as an exogenous process and we're not looking at that here instead we're looking at the structural approach to calculating the probability of default that treats default as an ad dogenous process meaning that we explain the default as a function of the firm's fundamentals specifically its balance sheet and so by far the most popular approach here is the meron model that we're looking at and probably the most popular commercial application is Moody's kmv which is based in large part on the Merton model but does have some nuanced differences in the application so similar definitely the same concept but different in application in order to compute the probably the default first I'll show you graphically and then we'll look at an expel Excel example we need some assumptions and here I'm going to follow Michael Long's example and assume the firm's assets are worth $11,000 so this is firm assets not Equity not debt but Equity plus debt all of the firm's assets at a th we also need an assumption about the default point or default threshold this is the point at which if the firm's assets drop below this we predict a default and so A good rule here a common rule here would be all of the short-term liabilities plus 1/ half the long-term liabilities now the structural model is really very simple in its idea and that is that we predict the firm will default if the firm's assets that's debt plus Equity drop below the default point but this is today the complexity comes up because we're trying to predict into the future so at the end of the period let's assume one period and we' like to predict the probability of default at the end of this one-year period already today we can say that there's a 51% difference between the assets and the default point in other words natural log of 1,000 divid 600 is 51% this is a continuously compounded growth rate really 600 growing continuously at 51% gets you a th000 and what we really mean is if a th000 drops by 51% on a continuous basis we would be at the default point but that's today so we need to know two other things we need the volatility of the firm's assets that's Sigma the standard deviation of the firm's assets all assume 25% again it's volatility of the assets not of equity or debt we also need an expected return of these firm's assets will assume 20% so this is really a growth on the firms assets and so now we can look to the end of the period if these firm assets grow by 20% Then on a continuous basis we expect the firm's assets to be here at 1,184 and we also know that this is the dispersion and so we can apply that same structural idea here's the average or expected value of the firm's assets here's the dispersion we simply ask what's the probability we could end up here in this tail way down here it's not likely but there is some probability if this is a random variable that probability under the structural approach becomes our probability of default or expected default frequency now in the future if the average or expected value of the firm is 1184 our default point is still 600 that means we expect a 68% difference here on a continuous basis between the expected value of the firm and here the default that 68% by itself doesn't really mean much to us we need to standardize it by dividing it by the dispersion here the 25% and when we do that we've now really converted this into standard normal units and what we found is that based on these assumptions this expected value of the firm's assets is 2.72 standard deviations away from this point here in the tail which represents our default and now it's just statistics if this is a normal distribution the cumulative distribution function tells us the probability that we could end up here in this tail 2.72 standard deviations away from the mean and so now we can see here is the formula signified by D2 or denoted by D2 this is the formula used in the meron model and this formula implements exactly what we just walked through here's the natural log of the firm value divided by default point that gives us this 51% that is already implicit on day zero that's the implicit cushion here plus what any kind of extra cushion we're going to get out of the positive growth rate in the firm's assets and then we're going to divide by the volatility or standard deviation in order to convert this into standard normal units that's going to give us D2 which is called the distance to default and then we only need to apply the inverse normal standard cumulative distribution in order to calculate our probability of default so now I'll show you the spreadsheet example but first let me note in none of this did we really use option pricing Theory option pricing theory in the Merton model is used at the very start to get the value of the firm's assets and the volatility of the firm's assets after we use option pricing Theory to get those this was all mechanical and is really not option pricing Theory so now let's look at the Excel spreadsheet so here we are at the top with the same assumptions that we just looked at at the bottom here I have the formula and then I'm just going to calculate this in the spreadsheet to show you the formula so first I'll just do the numerator here that's going to be the natural log of the firm's value today of a th000 divided by the default point of 600 notice if I just stop there I'm going to get the 51% that represents the cushion that's in the uh firm's structure on Day Zero but now I'm going to give credit for the expected growth in the assets so I'm going to add the expected return of 20% and then I'm going to subtract I'm going to depress that by the volatility squared which is the variance divided by two so what I've really done there is calculated the expected return on a geometric basis that's where volatility erodes the returns and I could multiply that by T the time which is one year and won't really change the equation in the numerator and that gets me the 6 68 or 68% so that's the cushion that we expect the difference between the firm's value and the expected value in the future and the default point and then the denominator is going to be the volatility multiplied by the number of periods so that's going to be 25% and then I'm going to divide that numerator by the denominator in order really to convert the 68% to standard normal units and that tells me that D2 or really my distance to default is 2.72 standard deviations now if that distribution is normal then I can use the standard normal cumulative distribution function norms disc to tell me if I put a negative in front of it to tell me the probability that I will end up in that tail and that's 33% or about a third of 1% is my probability of default or my expected default frequency as predicted by the meron model so I hope this was helpful this is David Harper the bonic Turtle thanks for your time [Music]