# markov chain modelling

#### ace123

##### New Member
Hi guys,

Got a complex question regarding markov chain modelling.

I'll start off with an example. Say you own a credit card and you can only ever be in 3 states.
1)non-user - you're currently not using your credit card
2)full payer - you pay off your balance in full each month
3)partial payer - you only pay some of your balance off eg. 50%, 60% etc.

I'm not having any problems calculating the probabilities of moving from one state to the next. i just dont know how i would use these probabilities to calculate someone's expected outstanding balance on their credit card.

Any ideas or help?

Much appreciated

#### JohnM

##### TS Contributor
I don't know much at all about Markov chains, but wouldn't this model need to include some knowledge of a distribution of price(\$) of purchase(s) with the credit card?

#### ace123

##### New Member
Ok say for example a customer, at the end of month 0, has a balance of £50. His average spend per month is £5 and the monthly interest rate is 0.5%.

Also at the end of month 0 the customer is sitting as a partial payer - state 3.

The prob of going from :

State 3 to 3 is 60% (partial payer to partial payer)
State 3 to 2 is 30% (partial to full)
State 3 to 1 is 10% (partial to non-user)

Say we wanted to look at the projections for the next 12 months.

How do we use the above info and calculate an expected balance at the end of each month??

#### Dragan

##### Super Moderator
Ok say for example a customer, at the end of month 0, has a balance of £50. His average spend per month is £5 and the monthly interest rate is 0.5%.

Also at the end of month 0 the customer is sitting as a partial payer - state 3.

The prob of going from :

State 3 to 3 is 60% (partial payer to partial payer)
State 3 to 2 is 30% (partial to full)
State 3 to 1 is 10% (partial to non-user)

Say we wanted to look at the projections for the next 12 months.

How do we use the above info and calculate an expected balance at the end of each month??
I would say that what you need to do is define what your initial probability distribution is for the Marko Chain (V) and your transition matrix (P).

That said, you can then determine the projections for the next 12 months from:

V^(1,...,12) * P

For example, if the person is intially in state 3 we would have:

[0,0,1] ^(1,...12) * P.

Further, in terms of your transition matrix, I think you need to consider the probabilities of going from not only 3 to 1, 2, and 3 but all 9 cases i.e. a 3X3 matrix associated with P.

#### Dragan

##### Super Moderator
What do you mean by the above?

Sure, I mean:

V^1 * P = V^0 *P
V^2 * P = V^1* P
.
.
.
where P is the transition matrix.

I think this can be more appropiately written as:

V^k = V^0 * P^k

where k is the nuimber of stages where k=1,....,bla

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#### Martingale

##### TS Contributor
Sure, I mean:

V^1 * P = V^0 *P
V^2 * P = V^1* P
.
.
.
where P is the transition matrix.

I think this can be more appropiately written as:

V^k = V^0 * P^k

where k is the nuimber of stages where k=1,....,bla
??? I'm even more confused now.

...what to you mean by V^2. isn't V a vector.

#### Martingale

##### TS Contributor
Sure, I mean:

V^1 * P = V^0 *P
V^2 * P = V^1* P
.
.
.
where P is the transition matrix.

I think this can be more appropiately written as:

V^k = V^0 * P^k

where k is the nuimber of stages where k=1,....,bla
I see now...

I would use the subscript notation

V_1 * P = V_0 *P
V_2 * P = V_1* P=V_0*P*P=V_0*P^2