- #1
ibimbo
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Homework Statement
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
Homework 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.
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 :(