1 Markov Analysis In an industry with 3 firms we could look at the market share of each firm at any time and the shares have to add up to 100%. If we had information about how customers might change from one firm to the next then we could predict future market shares. This is just one example of Markov Analysis. In general we use current probabilities and transitional information to figure future probabilities. In the notes here we study an accounts receivable example.

2 Say in the accounts receivable department, accounts are in one of 4 states, or categories: state 1 - s1, paid, state 2 – s2, bad debt, here defined as overdue more than three months and company writes off the debt, state 3 – s3, overdue less than one month, state 4 – s4, overdue between one and three months. Note the states are mutually exclusive and collectively exhaustive. At any given time there will be a certain fraction of accounts in each state. Say in the current period we have the % of accounts receivable in each state. In general we have a row vector of probabilities (s1, s2, s3, s4).

3 Say now there are 25% of the accounts in each state. We would have (.25,.25,.25,.25). This set of numbers is called the vector of state probabilities. Next I show the matrix of transition probabilities: The first row is being in the first state in the current period, the second row is being in the second state in the current period, and so on down the rows.

4 Now, in the matrix of transition probabilities let’s think about each column. The first column says an account is in state 1 in the next period. The second column says an account is in state 2 in the next period, and so on. Note the first row has values 1, 0, 0, 0. The values add to one. If an account is all paid this period then it must be all paid next period. So the 1 means there is a 100% chance of being all paid next period and 0 % chance in being in any other category. In the second row we have 0, 1, 0, 0. If an account starts as bad it will always be bad. So it has a zero chance of being paid, less than one period overdue or be between 1 and 3 periods overdue.

5 In row three we have.6, 0,.2,.2. If an account is less than 1 month overdue now, next period there is a 60% chance it will be all paid, 0% chance it will be bad because it can not be over 3 months bad, 20% chance it will be less than a month - wait, wait wait. How can an account be bad less than one month now and less than one month next period? Any account can have more than one unpaid bill and we keep track of the oldest unpaid bill for the category. Note that each row has to add up to 1. Now we are ready to ask a question. If each state has 25% of the accounts this period, what percent will be in each state next period? We take the row vector and multiply by the matrix of transition probabilities, as seen on the next screen.

6 (t, u, v, w) defghijklmnopqrsdefghijklmnopqrs We will end up with (a1, a2, a3, a4), where a1 = t(d) + u(h) + v(l) + w(p) a2 = t(e) + u(i) + v(m) + w(q) a3 = t(f) + u(j) + v(n) + w(r) a4 = t(g) + u(k) + v(o) + w(s) Matrix multiplication

7 (.25,.25,.25,.25) We will end up with (a1, a2, a3, a4), where a1 =.25(1) +.25(0) +.25(.6) +.25(.4) =.5 a2 =.25(0) +.25(1) +.25(0) +.25(.1) =.275 a3 =.25(0) +.25(0) +.25(.2) +.25(.3) =.125 a4 =.25(0) +.25(0) +.25(.2) +.25(.2) =.1

8 So, it we start with 25% of accounts in state 1, then next period we have 50 % of accounts in state 1, and so on. If you wanted to see what the %’s in each state would be two periods from the start we would do the same calculation, but use the row vector that we ended with in the first period (.5,.275,.125,.1) Now we can do this problem in QM for Windows. In the module Markov Analysis make the new data set have the initial probabilities and the matrix of transition probabilities. Specify the number of transitions. In my initial problem I would have put one and in the Markov Analysis Results section would have seen the answer in the ending probability section. If I wanted to see the probabilities of being in each state at the end of two months I would put 2 for number of transitions and would get (.615,.285,.055,.045).

9 Now, in this particular problem we have what are called absorbing states. Not all problems have absorbing states and if not just do what we have done up to now. An absorbing state is one such that once in it one stays in that state. For instance, once debt is bad it is always bad. Now, in the long run all debt will either be bad or paid. In QM for Windows when we do a Markov Analysis problem that has absorbing states, no matter how many transitions you put there is always an output section called matrices and it includes the FA matrix. In our problem we have The rows represent the non-absorbing states and the columns represent the absorbing states.

10 The first row is state 3, debt of less than one month, and row 2 is state 4, debt of 1 to 3 months. Column 1 is paid debt and column 2 is bad debt. So, the first row says 96.55% of less than one month debt will be paid over the long term and only 3.45% of this debt will not be paid. The second row means that 86.21% of 1 to 3 month debt will be paid over the long terms and 13.79% of this debt will go bad. Say that there is $2000 in the less than one month overdue category and $5000 in the 1 to 3 month overdue category. How much can the company expect to collect of this $7000 and how much will it not collect? See the next screen (and QM for windows can not do this for us)

11 We have to do matrix multiplication, here (2000, 5000) ([{2000*.9655} + {5000*.8621}], [{2000*.0345} + {5000*.1379}]) or ( , 758.5). So of the $7000 in states 3 and 4, $ can be expected to be collected and $758.5 would not be collected.