Writers Den

Social Mobility Model

A study of social mobility across
generations was conducted and three social levels were identified: 1= upper
level (executive, managerial, high administrative, professional); 2= middle
level (high grade supervisor, non-manual, skilled manual); 3= lower level
(semi-skilled or unskilled). Transition probabilities from generation to
generation were estimated

to be
	:45	:48	:07
P=	0:05	:70	:251	:	(1)
	B:01 :50	:49C
	@		A

Suppose that Adam is in level 1 and
Cooper is in level 3, and that each person has one o?spring in each generation.
ConsiderT = 50 generations. Assume sample sizeN = 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
probabilityA(t)( resp. C(t)) that the 1st, 2nd, . . . , 10th generation

o?spring of Adam (resp. Cooper) is
in level 1, respectively. GraphA(t) andC(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 first level 3 o?spring? On average, how
many generations (mean and 95% CI) does it take for Cooper’s family to have the
first level 1 o?spring?

(d) What are the social and policy implications of these
results, in terms of eduction, taxa-tion, welfare programs, etc.?

1

Credit Risk Rating Model

Markov chains
are often used in nance 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
1

		0:81	0:18	0:01	C
P=		17	0:77	0:06	(1)
	B0:0	0	1
	@				A

In reality,
this transition matrix is updated every year. However if we assume no signi
cant 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 speci cation of a
Markov chain model.

Assume the
maximal lifetime of a rm 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 rm 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 re ection 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 rms each year.
Suppose that each year new rms of rating A and B are created with equal chance.
Suppose that the number of new rmed created in each year is the same as the
number of rms that default in that year. So we now assume that the number of
rms (non-defaulted bonds) is constant over time. For example, if in year t
= 10, there are 3 rms default, then 3 new rms 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 rms are in A in the long run? How would you
determine what is the expected fraction of rms that default each year in the
long run? How would you determine the expected number of time periods before a
‘A’ rated rm defaults?

2