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-STAT 416 Stochastic Processes for Actuaries – Term 122
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
DEPARTMENT OF MATHEMATICS & STATISTICS
DHAHRAN, SAUDI ARABIA
STAT 416: Stochastic Processes for Actuaries
Semester 122
Major Exam Second
Monday, April 29, 2013
Allowed time 2 hours
:
Instructor
Adnan Jabbar
Name:
Student ID#:
Serial #:
Directions:
1) You must show all your work to obtain full credit for questions two and three.
2) You are allowed to use electronic calculators and other reasonable writing accessories that
help write the exam. Try to define events, formulate problem and solve.
3) Do not keep your mobile with you during the exam, turn off your mobile and leave it aside
Question No Full marks
1
6
2
5
3
3
4
3
5
6
6
8
7
9
Marks
obtained
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-STAT 416 Stochastic Processes for Actuaries – Term 122
Question one (6).
Claims occur on a portfolio of insurance policies according to a Poisson process at a rate. All claims are for
a fixed amount d, and premiums are received continuously. The insurer s initial surplus is U (< d) and the
annual premium income is 1.2 d. Show that the probability that ruin occurs at the first claim is:
1 – exp[ - 1/1.2(1 – U/d)]
Question two (5).
A stochastic interest rate model assumes that the annual interest rate during the next year will be 7.5% and
that the interest rate in subsequent years will be at a fixed but unknown level with probabilities in
accordance with the following probability distribution:
i
0.055 With probability 0.3
= 0.075 With probability 0.5
0.095 With probability 0.2
What is the expected accumulated amount by the end of the fifth year of an initial investment of $20,000?
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-STAT 416 Stochastic Processes for Actuaries – Term 122
Question three (3)
Consider an apple juice company controls 20% of the apple market. Suppose they hire a market research
company to predict the effect on an aggressive advertisement campaign. Suppose they conclude: someone
using brand A will stay with brand A with 90% probability. Someone not using brand A switch to brand A
with probability 70%. Find the probability of uses brand A after 2 weeks’ time?
3
-STAT 416 Stochastic Processes for Actuaries – Term 122
4
Questioіn four (3).
Consider an example where a company has a 10% market share. Using an advertisement campaign, the
transition matrix is given by
A
P = A [ .8
A' [ .6
Where
A'
.2]
.4]
A= uses brand A
A'= uses a different brand
If the probabilities in P remain valid over a long period of time, what will happen to the company’s market
share?
-STAT 416 Stochastic Processes for Actuaries – Term 122
5
Question five (2+4).
(a) The share price of a European company follows geometric Brownian motion with drift rate 0.2 per
annum and variance rate 0.15 per annum. A six month put option has exercise price of 120 Euros. If
the current value of the share price is 120 Euros, determine the probability that the put option will be
exercised.
(b) A discrete-time version of a model for the behavior of a stock price, S, is given by:
ΔS/S = μΔT + σε ΔT
Where T is time and ε is a random sample from a standard normal distribution.
A stock has expected return of 14% per annum and a volatility of return of 30% per annum. Using as
many required of the following random samples from a standard normal distribution, simulate the weekly
stock price movement over three weeks, given a starting price of $2 at the beginning of the first week.
Random Samples: 0.56, -0.34, 1.05, 0.07, -0.65
-STAT 416 Stochastic Processes for Actuaries – Term 122
Question six (2+3+2+1).
(a) Define the pure birth process. Give any example of pure birth process?
(b) Define the Galton-Watson process. Give 3 counter examples of the process?
(c) Suppose that possible number of offspring’s of an individual is
P0 = 0.2, P1 = 0.4, P2 = 0.4
The population starts with 1. Find the probability of extinction. What is the probability that the
Population will fie out if it initially consists of n individuals?
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-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question seven (4+5).
A Bonus-Malus system has four levels, 0%, 25%, 50% and 75%. The rules for moving between levels are as
follows:
If no claim are made in one year, the policyholder moves to the next higher level, or remain at the 75%
level. If one claim is made in one year, the policyholder moves down one level, or remains at the 0% level.
If two or more claims are made, the policyholder moves straight down to, or remains at, the0% level.
Policyholders at different levels are found to experience different rates of claiming. The number of claims
made per year follows a Poisson distribution with parameter λ as follows:
Level 0%
λ
0.29
25%
0.22
50%
0.18
75%
0.10
(a) Derive the transition matrix..
(b) Calculate the proportions at each different level when the system reaches a steady state.