Social Mobility Model & Credit Risk Rating Model

Social Mobility Model
A study of social mobility across generations was conducted and three social levels were
identi?ed: 1= upper level (executive, managerial, high administrative, professional); 2=
middle level (high grade supervisor, non-manual, skilled manual); 3= lower level (semiskilled or unskilled). Transition probabilities from generation to generation were estimated
to be .45 .48 .07 (1)
P = .05 .70 .25 .
.01 .50 .49
Suppose that Adam is in level 1 and Cooper is in level 3, and that each person has one
o?spring in each generation. Consider T = 50 generations. Assume sample size N = 104
and initial distribution [0.5, 0.0, 0.5].
(a) What is the long-run percentage of each social level, i.e., steady-state distribution? What
if initial distributions is [1, 0, 0] or [0, 0, 1]? Does the initial distribution matter in the long
run?
(b) Compute the probability A(t)( resp. C(t)) that the 1st, 2nd, . . . , 10th generation
o?spring of Adam (resp. Cooper) is in level 1, respectively. Graph A(t) and C(t) against
t = 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 family
to have the ?rst level 3 o?spring? On average, how many generations (mean and 95% CI)
does it take for Cooper’s family to have the ?rst level 1 o?spring?
(d) What are the social and policy implications of these results, in terms of eduction, taxation, welfare programs, etc.?

Credit Risk Rating Model
Markov chains are often used in finance to model the variation of corporations’ credit ratings over
time. Rating agencies like Standard & Poors and Moody’s publish transition probability matrices
that are based on how frequently a company that started with, say, a AA rating at some point in
time, has dropped to a BBB rating after a year. Provided we have faith in their applicability to
the future, we can use these tables to forecast what the credit rating of a company, or a portfolio
of 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 probability
transition matrix P for one year: 0.81 0.18 0.01 (1)
P = 0.17 0.77 0.06
0
0
1
In reality, this transition matrix is updated every year. However if we assume no significant change
in the transition matrix in the future, then we can use the transition matrix to predict what will
happen over several years in the future. In particular, we can regard the transition matrix as a
specification 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 of
remaining A-rated, an 1% probability of dropping to a B-rating, and a 1% chance of defaulting on
their loans. Each row must sum to 100%. Note the matrix assigns a 100% probability of remaining
in default once one is there (called an absorption state). In reality, companies sometimes come out
of 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 of
being in each of the three states in 2 and 5 years? What is the probability that a currently A firm
becomes 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 700
companies with a B-rating, and we would like to forecast what the portfolio might look like in
t = 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 means
that if a path exists from any other state (A-rating, B-rating) to the Default state then eventually
all 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 how
it is that we have so many companies left. A crude but helpfully economic theory of business rating
dynamics assumes that if a company loses its rating position within a business sector, a competitor
will take its place (either a new company or an existing company changing its rating) so we have
a stable population distribution of rated companies. 1 So now we consider what happens if we introduce new firms each year. Suppose that each year new
firms of rating A and B are created with equal chance. Suppose that the number of new firmed
created in each year is the same as the number of firms that default in that year. So we now
assume that the number of firms (non-defaulted bonds) is constant over time. For example, if in
year t = 10, there are 3 firms default, then 3 new firms are created, each with probability 0.5 of
being A or B.
Under the new modelling assumption, what should be the transition matrix? How would you
determine what fraction of firms are in A in the long run? How would you determine what is the
expected fraction of firms that default each year in the long run? How would you determine the
expected number of time periods before a ’A’ rated firm defaults?