(adsbygoogle = window.adsbygoogle || []).push({}); 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 :(

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# Homework Help: Markov chain calibration to a set of cumulated frequencies.

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