A credit card company classifies its consumers in a given year in three groups:

Question

A credit card company classifies its consumers in a given year in three groups: nonusers (N), light users (L), and heavy users (H). The transition matrix for these states is P = N L H N L H [0.8 0.25 0.04 0.15 0.55 0.21 0.05 0.2 0.75] Find the probability that a customer goes from a light user to heavy user in two years. Suppose that there are 50,000 new customers, all of whom are light users. Find the number of customers that will be in each category after 2 years. Suppose the initial distribution for the three states is X_0 = [0.4 0.4 0.2]. Find the distribution after 3 years. What is the long-range prediction for the distribution of users?

Explanation / Answer

Formula:

we used solver to find the long run optimal solution.

Transition matrix N L H N 0.8 0.15 0.05 L 0.25 0.55 0.2 H 0.04 0.21 0.75 a. Conidering this transition matrix same across years so P(L->H)=0.2 b. initial distribution: N L H 0 50000 0 after 1 year N L H 12500 27500 10000 after 2 year N L H 17275 19100 13625 c. initial distribution: N L H 0.4 0.4 0.2 after 1 year N L H 0.428 0.322 0.25 after 2 year N L H 0.4329 0.2938 0.2733 after 3 year N L H 0.430702 0.283918 0.28538 d. Saturation stage prediction: initial distribution: N L H 0.394268 0.268438 0.293604 final optimised solution = = = N L H 0.394268 0.268438 0.293604

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