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

  1. Aug 14, 2008 #1
    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 :(
     
  2. jcsd
  3. Aug 18, 2008 #2
    Hi, anyone? :(
     
  4. Aug 18, 2008 #3
    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?
     
  5. Aug 18, 2008 #4
    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
     
    Last edited: Aug 18, 2008
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