Markov chain calibration to a set of cumulated frequencies.

[ibimbo]

1. The problem statement, all variables and given/known data

Hi!
I have been given such a task:
A population of firms can assume three states: good-bad-bankrupt (default)
The cumulated frequencies of default (DP) from year 1 to 10 are given.
Find an appropriate transition matrix (TM)

I'm given a matrix of historical cumulated frequencies of default like this:

DP =

firm type/year
1 2 3 and so on
good 0.7 0.5 0.3
bad 0.8 0.6 0.4

and i have to find a transition matrix which looks like the following

TM=
good bad default
good ? ? ?
bad ? ? ?
default 0 0 1

2. Relevant equations
TM^n
gives the transition matrix from year 1 to n, and specifically the column "default" will show the cumulative frequencies of defaults in year n.

3. The attempt at a solution

Basically i have to minimize the difference between the defaults column of the TM and the cumulated frequencies (DP) i am given for TM^n, with n from 1 to 10 years, therefore i have 10 equations like

Min --> TM^n(last column)-DP(n)

Constraints:
- 1st and 2nd row have to sum to 1
- last row has to be 0,0,1

I would appreciate if someone could help me to frame this problem ;)

Hint: i read on a paper that was doing that exercise they used "least squares", but in my studies i have never gone beyond fitting a time series, while here i have a matrix annd i am completely lost :(

[ibimbo]

Hi, anyone? :(

[cellotim]

If you want help, you'll have to frame it in mathematical terms. Let me try and understand. You have a list of probabilities for a firm to default, one vector for each year, D(n), given a firm's state in year 1. You want to find the 3x3 transition matrix, T, such that a firm in default stays in default with probability 1 and that after n years the probability that a firm will go into default is as close as possible to the given probabilities for that year, i.e. T^n - D(n) is minimal.

Is the problem asking for an exact, symbolic minimization or some sort of regression, best fit algorithmic approach?

[ibimbo]

hi thanks for helping out!

The Default state is absorbing, meaning the prob for a defaulted firm to become good or bad is 0, hence the last row of my transition matrix is [0,0,1].

I was told to find an approximation method, suggesting least squares.

However i would not know how to set up the problem, as in my studies i have just come across rather simple OLS or linear programming problems, while this is a bit more complicated, because the objective function doesn't look any linear.


Unknown: Transition Matrix (T)

Problem: Min(T^n-D(n))

-by T^n i mean the last column which contains the probabilities to migrate to Default State.
- D(n) data is available for n=1 to 40

sub
- T(Good,Good)+T(Good,Bad)+T(Good,Default)=1
- T(Bad,Good)+T(Bad,Bad)+T(Bad,Default)=1

- T(Default,Default)=1 T(Default,Good)=0 T(Default,Bad)=0