Auto Insurance Company Classifies Its Customers In Three

An Auto Insurance Company Classifies Its Customers In Three Catagories

An auto insurance company classifies its customers into three categories: poor, satisfactory, and preferred. Each year, 35% of those in the poor category move to satisfactory, 20% of those in the satisfactory category move to preferred, 20% of those in the preferred category move to satisfactory, and 20% of those in the satisfactory category move to poor. Customers are never moved directly from poor to preferred or from preferred to poor in a single year. Assuming these percentages remain consistent over the long term, determine the expected long-term number of customers in each category.

Paper For Above instruction

Understanding customer classification dynamics within an auto insurance company requires analyzing the probabilistic transitions between categories over time. The problem involves a Markov process, where the states are the customer categories: poor, satisfactory, and preferred. The key goal is to determine the long-term, or steady-state, distribution of customers across these categories, assuming the transition probabilities remain constant over time.

To address this, we model the problem using a Markov chain with a transition matrix that encapsulates the probabilities of moving from one category to another in a single time period (one year). By computing the steady-state vector (or stationary distribution), we identify the proportion of customers expected in each category in the long run, regardless of the initial distribution.

Constructing the Transition Matrix

Let's define the states:

  • P: Poor
  • S: Satisfactory
  • F: Preferred

Based on the problem statement, the transition probabilities are as follows:

  • From Poor: 65% stay in Poor (since 35% move to Satisfactory), 35% move to Satisfactory, 0% move directly to Preferred.
  • From Satisfactory: 20% move to Preferred, 60% stay Satisfactory, 20% move to Poor.
  • From Preferred: 20% move to Satisfactory, 80% stay Preferred, 0% move to Poor.

Using these, the transition matrix \( P \) is:

| Poor | Satisfactory | Preferred |


Poor  | 0.65 |     0.35     |     0    |


S     | 0.20 |     0.60     |    0.20  |


F     | 0    |     0.20     |    0.80  |

In matrix form:

P = \begin{bmatrix}


0.65 & 0.35 & 0 \\


0.20 & 0.60 & 0.20 \\


0 & 0.20 & 0.80


\end{bmatrix}

Calculating the Steady-State Distribution

The steady-state vector, \(\pi = [\pi_P, \pi_S, \pi_F]\), satisfies:

\pi P = \pi

and the sum of the probabilities is 1:

\pi_P + \pi_S + \pi_F = 1

Expanding the stationarity equations:

\pi_P = 0.65 \pi_P + 0.20 \pi_S + 0 \\


\pi_S = 0.35 \pi_P + 0.60 \pi_S + 0.20 \pi_F \\


\pi_F = 0 \pi_P + 0.20 \pi_S + 0.80 \pi_F


\end{pre>


Rearranged as:


(1 - 0.65) \pi_P - 0.20 \pi_S = 0 \\


- 0.35 \pi_P + (1 - 0.60) \pi_S - 0.20 \pi_F = 0 \\


- 0.20 \pi_S + (1 - 0.80) \pi_F = 0


\end{pre>


0.35 \pi_P - 0.20 \pi_S = 0 \quad (1) \\


-0.35 \pi_P + 0.40 \pi_S - 0.20 \pi_F = 0 \quad (2) \\


-0.20 \pi_S + 0.20 \pi_F = 0 \quad (3)


\end{pre>


From equation (3):


0.20 \pi_F = 0.20 \pi_S \Rightarrow \pi_F = \pi_S




Substitute \(\pi_F = \pi_S\) into equation (2):


-0.35 \pi_P + 0.40 \pi_S - 0.20 \pi_S = 0 \Rightarrow -0.35 \pi_P + 0.20 \pi_S = 0


\end{pre>


-0.35 \pi_P + 0.20 \pi_S = 0 \Rightarrow 0.20 \pi_S = 0.35 \pi_P \Rightarrow \pi_S = \frac{0.35}{0.20} \pi_P = 1.75 \pi_P




Recall from earlier, from equation (1):


0.35 \pi_P = 0.20 \pi_S \Rightarrow 0.20 \pi_S = 0.35 \pi_P


\end{pre>


which is consistent with \(\pi_S = 1.75 \pi_P\). Now, apply the normalization condition:


\pi_P + \pi_S + \pi_F = 1 \Rightarrow \pi_P + 1.75 \pi_P + 1.75 \pi_P = 1


\Rightarrow (1 + 1.75 + 1.75) \pi_P = 1


\Rightarrow 4.5 \pi_P = 1


\Rightarrow \pi_P = \frac{1}{4.5} \approx 0.222


\end{pre>


Then:


\pi_S = 1.75 \times 0.222 \approx 0.389


\


\pi_F = \pi_S \approx 0.389


\end{pre>


Interpretation and Conclusion


The long-term steady-state distribution of customers is approximately:
  • Poor: 22.2%
  • Satisfactory: 38.9%
  • Preferred: 38.9%

This distribution indicates that, over time, the company's customer base will stabilize around these proportions. The majority of customers will be in either the satisfactory or preferred categories, with a smaller proportion in the poor category. These insights help the company in planning resources, marketing strategies, and customer service priorities to optimize their customer retention and classification policies.

References

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