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Actuarial Risk Theory Homework Assignment: Markov Processes and Transition Probabilities, Assignments of Risk Analysis

University of Illinois - Urbana-ChampaignRisk Analysis

A homework assignment from the university of illinois at urbana-champaign's actuarial science program, specifically for math 476 / 567 - actuarial risk theory, taught by prof. Rick gorvett. The assignment covers various problems related to markov processes, including calculating limiting probabilities for two-state markov processes, determining one-step transition probability matrices for markov chains, and analyzing the movement of policyholders between types. Students are encouraged to work in groups and to show their work for full credit.

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Uploaded on 03/10/2009

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Math 476 / 567 Prof. Rick Gorvett
Actuarial Risk Theory Fall, 2008
Homework Assignment # 2 (max. points = 10)
Due at the beginning of class on Thursday, September 11, 2008
You are encouraged to work on these problems in groups of no more than 3 or 4. However, each
student must hand in her/his own answer sheet. Please show your work – enough to show that
you understand how to do the problem – and circle your final answer. Full credit can only be
given if the answer and approach are appropriate. Please provide answers to two decimal places.
(1) Consider a two-state Markov process. If it rains today, the probability that it rains
tomorrow is 0.70. If it does not rain today, the probability that it rains tomorrow is 0.20.
Calculate the limiting probability that it rains on three consecutive days. (Hint: to
determine a multi-period limiting probability in this example, first calculate the limiting
probability of rain on the first day, and then you can easily determine the probability that
it also rains on each of the next two days.)
(2) Suppose that the economy, in any month, can be in one of three states: poor, fair, or
good. If the economy is currently poor, it will be poor next month with probability 80%,
and fair with probability 20%. If the economy is currently fair, it will be fair next month
with probability 50%, and either poor or good with probability 25% each. If the
economy is currently good, it will be good next month with probability 60%, fair with
probability 35%, and poor with probability 5%. What is the probability that the
economy, if it is currently good, will be poor two months from now?
(3) Using the same one-month transition probabilities for the economy as in problem (2)
above… What is the limiting probability of having a good economy for two consecutive
months?
(4) Suppose that the price of a share of stock can either move up (U) or down (D) each day
(it cannot stay the same). Whether the price moves up or down tomorrow depends upon
whether it moved up or down today and yesterday. The probabilities of the stock moving
up tomorrow are:
State Day t-1 Day t Probability of up movement on Day t+1
1 U U 0.70
2 U D 0.50
3 D U 0.60
pf3

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

Actuarial Science Program

DEPARTMENT OF MATHEMATICS

Math 476 / 567 Prof. Rick Gorvett Actuarial Risk Theory Fall, 2008

Homework Assignment # 2 (max. points = 10) Due at the beginning of class on Thursday, September 11, 2008

You are encouraged to work on these problems in groups of no more than 3 or 4. However, each student must hand in her/his own answer sheet. Please show your work – enough to show that you understand how to do the problem – and circle your final answer. Full credit can only be given if the answer and approach are appropriate. Please provide answers to two decimal places.

(1) Consider a two-state Markov process. If it rains today, the probability that it rains tomorrow is 0.70. If it does not rain today, the probability that it rains tomorrow is 0.20. Calculate the limiting probability that it rains on three consecutive days. (Hint: to determine a multi-period limiting probability in this example, first calculate the limiting probability of rain on the first day, and then you can easily determine the probability that it also rains on each of the next two days.)

(2) Suppose that the economy, in any month, can be in one of three states: poor, fair, or good. If the economy is currently poor, it will be poor next month with probability 80%, and fair with probability 20%. If the economy is currently fair, it will be fair next month with probability 50%, and either poor or good with probability 25% each. If the economy is currently good, it will be good next month with probability 60%, fair with probability 35%, and poor with probability 5%. What is the probability that the economy, if it is currently good, will be poor two months from now?

(3) Using the same one-month transition probabilities for the economy as in problem (2) above… What is the limiting probability of having a good economy for two consecutive months?

(4) Suppose that the price of a share of stock can either move up (U) or down (D) each day (it cannot stay the same). Whether the price moves up or down tomorrow depends upon whether it moved up or down today and yesterday. The probabilities of the stock moving up tomorrow are:

State Day t-1 Day t Probability of up movement on Day t+ 1 U U 0. 2 U D 0. 3 D U 0.

4 D D 0.

Determine the one-step transition probability matrix for this four-state Markov chain. (Note: this is not a violation of the Markov property that only the current state of a process matters for the future. Here, the current state is merely defined according to the movement of the market today and yesterday. However, the probabilities of the future movements and states do not depend upon the state of the process yesterday (which would be a function of the movement of the market both yesterday and the day before).)

(5) Using the Markov process and probabilities in problem (4) above… With respect to a particular week, you observe that the stock market moved up on both Monday and Tuesday. What is the probability that the stock market will move down on Thursday of that week?

(6) For the one-step transition probability matrix below, properly identify each of the three states (numbered 1 through 3) of the Markov chain as being either “recurrent” or “transient.”

P

(7) An insurance company insures three types of policyholders: Types A, B, and C. Type A policyholders have expected annual losses of $100; Type B policyholders have expected annual losses of $500; Type C policyholders have expected annual losses of $1,000. (Assume no loss inflation over time.) You are given the following annual transition probabilities regarding the movement of individual policyholders between types:

PAA = 0.50 PAB = 0.40 PAC = 0. PBA = 0.20 PBB = 0.60 PBC = 0. PCA = 0.10 PCB = 0.20 PCC = 0.

A policyholder is currently Type A. Find the expected annual loss of that policyholder two years from now.

(8) A global insurer is exposed to losses resulting from catastrophes, which occur according to a Poisson process at the rate of 0.50 per month. Find the probability that there will be at least six months between catastrophes.

(9) Suppose that you are the actuary for a large homeowners insurer. Your company wants to begin writing coverage for policyholders’ earthquake and windstorm exposures, and you have been assigned the task of analyzing the frequency of these catastrophes. You estimate that earthquakes occur according to a Poisson process at the rate of 0.05 per month, and windstorms occur according to a Poisson process at the rate of 0.15 per month. What is the probability that there will be three or more catastrophes (defined as earthquakes and windstorms) in any given calendar year?