Exam-Style Questions on Markov Chains and Percentile Calculations

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3L, authored by Mr. Stolyarov. This is Section 63 of the Study Guide. See an index of all sections by following the link in this paragraph.

Here is the general form of the kind of problem you need to solve in order to perform a percentile calculation. For 0 ≤ n ≤ 100, when you are looking for the nth percentile of the distribution of any random variable X, you are trying to find the value of c such that

Pr(X ≤ c) = n. The rest is algebra and is context-specific.

Some of the problems in this section were designed to be similar to problems from past versions of the Casualty Actuarial Society's Exam 3L and the Society of Actuaries' Exam MLC. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Sources: Broverman, Sam. Actuarial Exam Solutions - CAS Exam 3 - Spring 2007.

Broverman, Sam. Actuarial Exam Solutions - CAS Exam 3 - Fall 2007.

Broverman, Sam. Actuarial Exam Solutions - CAS Exam 3L - Spring 2008.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L63-1. Similar to Question 20 from the Casualty Actuarial Society's Spring 2008 Exam 3L. This question is also an excellent review of the concepts in Section 49.

You are given the following homogeneous Markov Chain with states X, Y, and Z and probabilities of transition among the states in each time period given by the following transition matrix.

...X......Y.....Z
X(0.1...0.5...0.4)
Y(0.4...0.3...0.3)

Z(0.1...0.7...0.2)

Find the probability that an entity in state Z at time 0 will be in state Z at time 2.