Social Mobility Model and Credit Risk Rating Model

Social Mobility ModelA study of social mobility across generations was conducted and three social levels wereidentified: 1= upper level (executive, managerial, high administrative, professional); 2=middle level (high grade supervisor, non-manual, skilled manual); 3= lower level (semiskilledor unskilled). Transition probabilities from generation to generation were estimatedto beP =???.45 .48 .07.05 .70 .25.01 .50 .49??? . (1)Suppose that Adam is in level 1 and Cooper is in level 3, and that each person has oneoffspring in each generation. Consider T = 50 generations. Assume sample size N = 104and initial distribution [0.5, 0.0, 0.5].(a) What is the long-run percentage of each social level, i.e., steady-state distribution? Whatif initial distributions is [1, 0, 0] or [0, 0, 1]? Does the initial distribution matter in the longrun?(b) Compute the probability A(t)( resp. C(t)) that the 1st, 2nd, . . . , 10th generationoffspring of Adam (resp. Cooper) is in level 1, respectively. Graph A(t) and C(t) againstt = 1, 2, . . . , 10. What is A(10) and C(10)?(c) On average, how many generations (mean and 95% CI) does it take for Adam’s familyto have the first level 3 offspring? On average, how many generations (mean and 95% CI)does it take for Cooper’s family to have the first level 1 offspring?(d) What are the social and policy implications of these results, in terms of eduction, taxation,welfare programs, etc.?Credit Risk Rating ModelMarkov chains are often used in finance to model the variation of corporations’ credit ratings overtime. Rating agencies like Standard & Poors and Moody’s publish transition probability matricesthat are based on how frequently a company that started with, say, a AA rating at some point intime, has dropped to a BBB rating after a year. Provided we have faith in their applicability tothe future, we can use these tables to forecast what the credit rating of a company, or a portfolioof companies, might look like at some future time using matrix algebra.Let’s imagine that there are just three ratings: A; B and default, with the following probabilitytransition matrix P for one year:P =???0.81 0.18 0.010.17 0.77 0.060 0 1??? (1)In reality, this transition matrix is updated every year. However if we assume no significant changein the transition matrix in the future, then we can use the transition matrix to predict what willhappen over several years in the future. In particular, we can regard the transition matrix as aspecification of a Markov chain model.Assume the maximal lifetime of a firm is 200 years. Sample size N = 1000.a) We interpret this table as saying that a random A-rated company has an 81% probability ofremaining A-rated, an 1% probability of dropping to a B-rating, and a 1% chance of defaulting ontheir loans. Each row must sum to 100%. Note the matrix assigns a 100% probability of remainingin default once one is there (called an absorption state). In reality, companies sometimes come outof default, but we keep this example simple to focus on a few features of Markov Chains.Now let’s imagine that a company starts with rating B. What is the probability that it has ofbeing in each of the three states in 2 and 5 years? What is the probability that a currently A firmbecomes default within 2 years? And 5 years?b) Now let us imagine that we have a portfolio of 300 companies with an A-rating and 700companies with a B-rating, and we would like to forecast what the portfolio might look like int = 2, 5, 10, 50, 200, years. Plot the evolution of the portfolio.c) We mentioned that in this model ’Default’ is assumed to be an absorption state. This meansthat if a path exists from any other state (A-rating, B-rating) to the Default state then eventuallyall individuals will end up in Default. The model below shows the transition matrix for t = 1, 10,50 and 200. If this Markov chain model is a reasonable reflection of reality one might wonder howit is that we have so many companies left. A crude but helpfully economic theory of business ratingdynamics assumes that if a company loses its rating position within a business sector, a competitorwill take its place (either a new company or an existing company changing its rating) so we havea stable population distribution of rated companies.1So now we consider what happens if we introduce new firms each year. Suppose that each year newfirms of rating A and B are created with equal chance. Suppose that the number of new firmedcreated in each year is the same as the number of firms that default in that year. So we nowassume that the number of firms (non-defaulted bonds) is constant over time. For example, if inyear t = 10, there are 3 firms default, then 3 new firms are created, each with probability 0.5 ofbeing A or B.Under the new modelling assumption, what should be the transition matrix? How would youdetermine what fraction of firms are in A in the long run? How would you determine what is theexpected fraction of firms that default each year in the long run? How would you determine theexpected number of time periods before a ’A’ rated firm defaults?