STAT 4004 Actuarial Science
Tutorial 5
ZHENG,Xunze
Department of Statictics
The Chinese University of Hong Kong
Oct. 20, 2014
*Supplementary material:
Life insurance (or commonly life assurance, especially in the Commonwealth) is a contract between an in
3.7. Life Table
Given a survival model, we can construct a life table
From some initial age x0 to the limiting age
Define a function cfw_lx for x0 x as follows.
lx0 arbitrary positive number (radix of the table)
for 0 t x0,
lx0+t = lx0 tpx0
For x0
STAT 4004 Tutorial 5
Fall 2014
Life Insurance Continuous Cases
A life insurance contract offers benefit payment when a specified event occurs.
The contract value is represented by a present value random variable Z , which
depends on the time of occurrence
2.1. Valuation of insurance benefits
(a) Whole Life Insurance: continuous case
provides for a payment following the death of the insured at any time in the
future
bt
6

x+t
death
x
policy issued
If the payment is to be a unit amount at the moment of de
STAT 4004 Suggested Solution to Assignment 1
Fall 2012
1
(a)
i 1 i12
(12)
12
1
1 0.07
1
12
12
7.23%
(b)
(c)
d (4) 4 1 (1 i ) 4
6.92%
1
ln 1 i
6.98%
2
(a)
1
d ( 2)
2
2
e0.06
d (2) 2 1 e
5.91%
(b)
3
0.06
2
i (4) 4 e 4 1
6.05%
0.06
For Fund A,
a
STAT 4004 Suggested Solution to Assignment 2
Fall 2012
1
f ( x) s '( x)
(a)
x
d
exp
dx
1
x
x
exp
1
(b)
(c)
2
f ( x) x
( x)
s ( x)
2 5
s (5)
exp
s (2)
(a)
FK x y Pr K x y
y 1
qx
s x s x y 1
s x
y 1
100 x , for y 0,1, ,99 x
1
STAT 4004 Suggested Solution to Assignment 1
Fall 2015
1
(a)
i 1 i12
(12)
12
1
1 0.07 1
12
12
7.23%
(b)
(c)
d (4) 4 1 (1 i ) 4
6.92%
1
ln 1 i
6.98%
2
(a)
1
d ( 2)
2
2
e0.06
d (2) 2 1 e
5.91%
(b)
3
0.06
2
i (4) 4 e 4 1
6.05%
0.06
For Fund A,
a
STAT 4004 Assignment 2
Due: October 16, 2015
1. Suppose the survivor function of the newborn lifetime X is given by
s(x) = exp
x
,x > 0
Find
(a) the density function of X
(b) the hazard function of X
(c) the probability that X is alive at 5 given X > 2
2
STAT 4004 Suggested Solution to Assignment 2
Fall 2015
1
f ( x) s '( x)
(a)
x
d
exp
dx
1
x
(b)
( x)
x
exp
f ( x)
s( x)
1
x
(c)
2
2 5
s (5)
exp
s (2)
(a)
FK x y Pr K x y
y 1
qx
s x s x y 1
s x
y 1 , for y 0,1, ,99 x
100 x
STAT 4004 Assignment 1
Due: October 2, 2015
1. Suppose the nominal rate of interest convertible monthly is 7%. Find the equivalent
(a) eective rate of interest
(b) nominal rate of discount convertible quarterly
(c) force of interest, assuming it is consta
STAT 4004 Assignment 3
Due: November 18, 2015
1. For a 10year term insurance of 1000 on (x)with benet payable at the moment
of death:
(a)
t =
0.04 0 < t < 5
0.06 t > 5
t =
0.06 0 < t < 5
0.08 t > 5
(b)
Calculate the single benet premium for this insuranc
5. Life Annuities
5.1. Introduction
A life annuity is a series of payments made continuously or at
equal intervals while a given life survives
Annuitydue: payments made at the beginning of the payment
intervals
annuityimmediate: payments made at the
4. Life Insurance
4.1. Insurances Payable at the Moment of Death
bt
6
x+t
death
x
policy issued
bt : benet function
We are not interested in bt itself but the present value when the
policy issued
vt : discount function
Dene the present value function
2. Theory of Interest
2.1. The eective rate of interest
Consider a person invest an amount of money in a saving account at
a bank
the initial amount of money invested is called the principal
the total amount received after a period of time is called the
3. Survival Distributions and Life Tables
3.1. Probability for the Ageatdeath
Dene X the ageatdeath of a new born baby
X is a continuous random variable
To describe X, we can use
(a) the cumulative distribution of X
F (x) = Pr(X x)
(b) the density
STAT 4004 Mid—Term Test October 15. 2014
Time allowed: 2 hours
Marks distribution: 10 marks for each question
1. An investment fund accumulates with force of interest
K
6‘=1+(1—t)K>
0.
for 0 S t S 1. At time zero there is 100,000 in the fund. At time t =
3. Survival Distributions and Life Tables
3.1. Future Lifetime Tx of aged (x)
denote the lifeagedx by (x)
denote the future lifetime of (x) by Tx
x + Tx is the ageatdeath random variable for (x)
Let Fx be the distribution function of Tx
Fx(t) = Pr(
Tables for
Exam C
The reading material for Exam C includes a variety of textbooks. Each
text has a set of probability distributions that are used in its readings. For those
distributions used in more than one text, the choices of parameterization may not
SOCIETY OF ACTUARIES
EXAM C CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS
EXAM C SAMPLE QUESTIONS
The sample questions and solutions have been modified over time. This page indicates
changes made since January 1, 2014.
June 2016
Question 266 was moved t
CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS STUDY NOTE
REPLACEMENT PAGES FOR SECTION 16.5.3 FROM
LOSS MODELS: FROM DATA TO DECISIONS, FOURTH EDITION
by
Stuart A. Klugman
Copyright 2016 Society of Actuaries
The Education and Examination Committee provi
SOCIETY OF ACTUARIES
EXAM C CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS
EXAM C SAMPLE SOLUTIONS
The sample questions and solutions have been modified over time. This page indicates
changes made since January 1, 2014.
June 2016
Question 266 was moved t
Construction and Evaluation of Actuarial Models ExamOctober 2016
The Construction and Evaluation of Actuarial Models exam is a threeandahalf hour exam that
consists of 35 multiplechoice questions and is administered as a computerbased test.
For addit
STAT 4004 Tutorial 2
Fall 2014
Annuities
An annuity is a series of payments made at equal interval of time. There are
two types of annuities
Annuitycertain: the time horizon and amount of payments are deterministic;
Contingent annuity: payments are ran
STAT 4004 Tutorial 3
Fall 2015
Survival distribution
Survival function
Let X be a continuous random variable of a newborns ageatdeath, the survival
function is defined as the probability that the newborn will survive up to age x, i.e.
s ( x) Pr( X x) 1
STAT 4004 Tutorial 1
Fall 2014
Measurement of Interest
The accumulation function a (t ) is the amount of money accumulated at time
t 0 with an initial investment of $1.
e.g. Simple interest: a (t ) 1 it.
Compound interest: a(t ) 1 i .
t
Amount function is
2. Theory of Interest
2.1. The effective rate of interest
Consider a person invest an amount of money in a saving account at
a bank
the initial amount of money invested is called the principal
the total amount received after a period of time is called t
3. Survival Distributions and Life Tables
3.1. Future Lifetime Tx of aged (x)
denote the lifeagedx by (x)
denote the future lifetime of (x) by Tx
x + Tx is the ageatdeath random variable for (x)
Let Fx be the distribution function of Tx
Fx(t) = Pr(
27.
An actuary is modeling the mortality of a group of 1000 people, each age 95, for the next
three years.
The actuary starts by calculating the expected number of survivors at each integral age by
l95 +k = 1000 k p95 ,
k = 1, 2, 3
The actuary subsequentl
Appendix
STAT 4004 Fact sheet
Theory of Interest
Measurement
i
a(t)
(1 + i)t
d
(1 d)t
i(m)
(1 + i(m) /m)mt
d(m)
(1 d(m) /m)mt
Survival Probability
The distribution of T (x):
cdf: t qx
Survivor function: t px
Density function: t px (x + t)
The probabi