1. 1
Week 1 1
Principles of Actuarial Science;
Games of Chance (1)
Outline:
1. Traditional Actuarial Application: Insurance
2. Principles of Actuarial Science
3. Probability Basics (Revision?)
4. Poisson and Binomial Distributions
Reading
(Req) Sherris, 1.3-1.4, 2.1-2.3
(Recc) Sherris 1.1-1.2
1
Version: 15 July 2010. Copyright UNSW Actuarial Studies
2. 2
Traditional Application of Actuarial Science - Insurance
What is insurance?
Examples:
– Life (Death)
– Life (Survival)
– Health
– Disability
– Motor
– House
– Home Contents
– Workers Compensation
Fundamental Idea: Payment(s) that are incurred when a (random) event occurs
Other actuarial applications: Superannuation, Risk Management
(Will discuss in much more detail later in the course)
3. 3
Discussion
Consider a life insurance policy that:
– pays $100 at the end of the year if the policyholder ("Harry P") dies during the year
– assume that Harry has a 50% chance of dying
Q: Suppose that you are the actuary of the insurance company. Suggest how you
would determine the premium of this policy (making any additional assumptions if
required)
4. 4
Basic Principles of Actuarial Science
Probability and Statistics
– quantify future uncertainty
– probability models - Binomial, Poisson, Normal, Log-Normal
– Survival models
Risk Pooling
– Independent Risks
– Risk Factors (age, sex, smoker status etc)
Law of Large Numbers
– Expected values
– Variability of averages less for pools of independent risks
5. 5
Interest Rates
– Time Value of money
– economics of interest rates
– present values
Risk
– Utility Theory and Ranking of risky outcomes
– expected utility principle, certainty equivalents
6. 6
Discussion (ctd)
Suppose the premium for the policy has been set at $55, and that interest rates are
0.
What are some of the practical issues the company must now consider? Consider:
–(i) What to do with the $55 received as a premium?
(ii) How should this policy be reflected in the company’s financial reports?
7. 7
Basic Principles of Actuarial Science (ctd)
Financial Principles
– Accounting and Market values
– Fair Value of Liabilities
Actuarial Modelling
– Contingent Cash flows
– Death, survival, illness, financial asset returns
8. 8
Risk Management
– What is risk?
– Pooling and Diversification of Risk
– Experience Rating
– Profit Sharing
– Solvency
Risk Classification
– Moral Hazard
Careless or deliberate actions of insured affect payments
– Adverse Selection
Worse than average risks have incentives to purchase insurance
– Information Asymmetry
9. 9
Probability and Games of Chance
Probability
Permutation and Combinations
Useful Result: Binomial Theorem:
(x + y)n
=
n
X
r=0
n
r
xn r
yr
Simple method of determining probabilities (where/if possible!):
– Evaluate all possible equally likely cases
– Count the Frequency or proportion of times that the event occurs
12. 12
Random Variables
A random variable X is a function that assigns values to possible outcomes.
Discrete random variables - fixed number of finite values or countably infinite possible
values.
Continuous random variables - values on a continuous scale
Examples in Actuarial Science:
– Number of motor vehicle accidents occurring during a particular month
– Number of lives who die from cancer who are aged 40 last birthday during a par-
ticular time period.
– Time until the first accident for a particular car insurance policy
– Claim payment for a fire insurance policy
13. 13
Probability Density and Distribution Functions
Probability a discrete random variable X takes a particular value x is denoted
f(x) = P(X = x)
Probability density has the properties
f(x) 0 and
X
all x
f(x) = 1
Distribution function, (= "cumulative distribution function") of X; is the probability that
the random variable takes a value less than or equal to a specified value x
F(x) = Pr (X x)
14. 14
For continuous random variables the probability density function (p.d.f.) f(x) is de-
fined such that
Pr (a X b) =
Z b
a
f(x)dx
with
f(x) 0 for 1 < x < 1
Z 1
1
f(t)dt = 1
CDF:
F(x) = Pr (X x)
=
Z x
1
f(t)dt
15. 15
Expectation
Discrete random variable - expectation of a function of a random variable g(X) is
defined as
E [g (X)] =
X
all x
g (x) f(x)
Continuous random variable
E [g (X)] =
Z 1
1
g (x) f(x)dx
16. 16
Moments
Moments (about the origin) of a random variable are defined as
E [Xr
] r = 1; 2; : : :
First moment (r = 1) is the mean of the distribution of the random variable - usually
denoted by .
Discrete random variable
E [X] = =
X
all x
xf(x)
Continuous random variable
E [X] = =
Z 1
1
xf(x)dx
17. 17
Moments - Variance
The mean is a measure of central tendency for a distribution.
The second moment about the mean is referred to as the variance of the distribution.
Variance usually denoted by 2
V ar [X] = 2
= E
h
(X )2
i
The standard deviation is the square root of the variance, .
The variance is a measure of dispersion or spread of the distribution.
18. 18
Example
Consider a random variable X which takes the value 1 if Harry P dies and 0 if he
survives the year.. Find the mean, variance and standard deviation of X:
19. 19
Common Distributions in Actuarial Science
Poisson
Binomial
Geometric
Exponential
Normal
Log-Normal
Weibull
20. 20
Poisson Distribution
The Poisson distribution
- often used to model the probability of occurrence of rare events such as insurance
claims
- also as an approximation for the chance of dying for a small enough parameter .
Poisson ( )
Pr (X = x) =
e x
x!
x = 0; 1; 2 : : :
E [X] =
V ar [X] =
21. 21
Exercise
Assume that the probability of an insurance claim on a particular insurance policy
during a year has a Poisson distribution with = 1
10:
Calculate the probability that there will be
(a) no claims during the year on the policy?
(b) exactly 2 claims?
(c) at least one claim?
22. 22
Binomial Distribution
Number of successes (x) in a series of n independent trials where each success has
probability p.
In actuarial science
– probability that a number of lives in a group of lives with the same chance of dying
will die assuming independence between the lives.
Binomial (n; p)
Pr (X = x) =
n
x
px
(1 p)n x
x = 0; 1; 2 : : :
E [X] = np
V ar [X] = np (1 p)
23. 23
Example
Suppose we have 100 policyholders, each with a probability of 0.2 of dying within the
next year. What is the probability that we observe a total of 10 or less deaths by the
end of the year?
