Lake Michigan Credit Union finances auto loans for its members. Each loan can be classified into one of four categories: paid in full, good standing, poor standing (with one or more missing payments), and delinquent (where the debt is sold to a collection agency).
Past records indicate that each month 10% of the accounts in good standing are paid in full, 70% stay in good standing, and 20% become poor standing.
Also, 50% of the accounts in poor standing move to good standing, 40% remain in poor standing, and 10% are delinquent.
Using excel, construct the transition matrix T for this situation. Be sure to label the states.
Suppose Lake Michigan Credit Union currently has 600 outstanding auto loans, 400 of which are in good standing, and 200 of which are in poor standing. In the long run, approximate how many of these auto loans will be paid in full, and how many will end up as delinquent.
***When I found the solution after putting the transition matrix into my calculator and then selecting rref the last two rows had zeros all the way across and I am stuck as to what it means when the fourth row has zeros all the way across. Please help!
To construct the transition matrix, we need to define the states and their corresponding probabilities of transitioning from one state to another. In this case, the states are:
1. Paid in full
2. Good standing
3. Poor standing
4. Delinquent
Based on the given information, we can determine the transition probabilities between these states. Let's denote the probabilities as follows:
PFF: Probability of transitioning from paid in full to paid in full
PGS: Probability of transitioning from paid in full to good standing
PPS: Probability of transitioning from paid in full to poor standing
PDL: Probability of transitioning from paid in full to delinquent
GSG: Probability of transitioning from good standing to good standing
GPF: Probability of transitioning from good standing to paid in full
GPS: Probability of transitioning from good standing to poor standing
GDL: Probability of transitioning from good standing to delinquent
PSG: Probability of transitioning from poor standing to good standing
PPF: Probability of transitioning from poor standing to paid in full
PPS: Probability of transitioning from poor standing to poor standing
PDL: Probability of transitioning from poor standing to delinquent
DPS: Probability of transitioning from delinquent to poor standing
DDL: Probability of transitioning from delinquent to delinquent
Now, let's populate the transition matrix T using the given probabilities:
PFF | PGS | PPS | PDL
T = [
GPF | GSG | GPS | GDL
PPF | PSG | PPS | PDL
0 | 0 | PPS | PDL
]
To find the approximate number of loans that will be paid in full and end up as delinquent in the long run, we can use matrix multiplication.
Let's assume the initial distribution of loans is [0, 400, 200, 0], representing the number of loans in each state: [Paid in full, Good standing, Poor standing, Delinquent].
1. Multiply the initial distribution by the transition matrix T: [0, 400, 200, 0] * T = [x1, x2, x3, x4]
2. The resulting vector [x1, x2, x3, x4] represents the approximate number of loans in each state after one transition.
Repeat this process iteratively to see how the loans distribute over time.
However, it seems like you encountered an issue with your calculator when using the rref function. Having zeros across a row may indicate that the system of equations is either inconsistent or has infinitely many solutions. Please double-check your input matrix and be sure the calculations are correct.
To construct the transition matrix T, we can use the given probabilities:
Let the states be:
State 1: Paid in full
State 2: Good standing
State 3: Poor standing
State 4: Delinquent
We use the probabilities provided to fill in the matrix:
```
...............................................
| Paid in full | Good standing | Poor standing | Delinquent |
...............................................
Paid in full | 0.1 | 0.9 | 0.0 | 0.0 |
...............................................
Good standing | 0.0 | 0.7 | 0.2 | 0.1 |
...............................................
Poor standing | 0.0 | 0.5 | 0.4 | 0.1 |
...............................................
Delinquent | 0.0 | 0.0 | 0.0 | 1.0 |
...............................................
```
This is the transition matrix T.
Now, to find the number of auto loans that will be paid in full and end up as delinquent in the long run, we need to find the steady-state solution of the transition matrix.
The steady-state solution can be found by solving the equation T * X = X, where X is a column vector representing the steady-state probabilities.
To solve this equation, we construct the augmented matrix [T - I | 0], where I is the identity matrix.
```
.........................................................
| 0.1, 0.9, 0.0, 0.0, | 1.0, 0.0, 0.0, 0.0, 0.0 |
| 0.0, 0.7, 0.2, 0.1, | 0.0, 1.0, 0.0, 0.0, 0.0 |
| 0.0, 0.5, 0.4, 0.1, | 0.0, 0.0, 1.0, 0.0, 0.0 |
| 0.0, 0.0, 0.0, 1.0, | 0.0, 0.0, 0.0, 1.0, 0.0 |
.........................................................
```
Applying row reduction techniques, we get:
```
.........................................................
| 1.0, 0.0, 0.0, 0.0, | 0.000 |
| 0.0, 1.0, 0.0, 0.0, | 0.000 |
| 0.0, 0.0, 1.0, 0.0, | -1.786 |
| 0.0, 0.0, 0.0, 1.0, | 0.000 |
.........................................................
```
The fourth row of zeros means that there is no equation governing the probabilities for state 4 (Delinquent) since it is an absorbing state where the loans cannot move out of.
Now, let's approximate the long-run probabilities:
From the solution matrix, we see that the probability of being in state 4 (Delinquent) in the long run is 0.
To find the probabilities for states 1, 2, and 3, we normalize the corresponding row of the solution matrix:
State 1 (Paid in full) probability ≈ 0.000 / (-1.786) ≈ 0
State 2 (Good standing) probability ≈ 0.000 / (-1.786) ≈ 0
State 3 (Poor standing) probability ≈ -1.786 / (-1.786) ≈ 1
Therefore, in the long run, approximately all loans in poor standing (state 3) will remain in poor standing, while no loans will be paid in full or end up as delinquent.
So, out of the 600 outstanding auto loans, approximately 600 will end up in poor standing, and none will be paid in full or become delinquent in the long run.
Please note that this is an approximation based on the given transition probabilities, and actual results may vary.