Actuarial Statistics Module 7: Exposed to risk
Actuarial Statistics
Benjamin Avanzi
c University of New South Wales (2013)
School of Risk and Actuarial Studies
[email protected]
Module 7:
Exposed to risk
1/35
Actuarial Statistics Module 7: Exposed to r
Actuarial StatisticsWeeks 8&9
Outline:
Exposed to risk
Exact calculation
Census approximations
Age Denitions
Rate Intervals: Life Year, Calender Year, Policy Year
Reading
CT4Chapter 11
1/59
Exposed to risk
Central exposed to risk versus initial exposed
Actuarial Statistics
Exercises 4
1. It is known that, if the model is correct, then the CoxSnell residuals should
be exponentially distributed with parameter 1. We wish to check this graphically.
(a) Suppose there are no censored observations. Suggest on
Actuarial Statistics
Exercises 3
1. For proportional hazards models the explanatory variables for subject i,
T
i , act multiplicatively on the hazard function. If i = exi then the hazard
function for subject i is
T
(t; xi ) = 0 (t)exi .
(a) For a binary e
Actuarial Statistics
Exercise 1
(Questions have been adapted from IAA Course Notes and the book by London.)
1. Let X, Y or T be the survival time (or failure time) random variable.
(a) Suppose that f is the probability density function of X. Prove that
f
Actuarial Statistics
Exercises 5
1. A population of insects has a total number of n, experiences a constant force
of mortality n . The population is maintained at level of n insects by replacing
the deaths that occur. The probability of exactly one death
Actuarial StatisticsWeeks 10  12 Part II
Graduation, standardization and new developmentsPart II
Outline:
Graduation
Methods of Graduation: Parametric Formula, Reference to a
Standard Table, Graphical Graduation, Splines, adjusted
average formula
Tests
Actuarial StatisticsWeeks 1012: Part I
Graduation, standardization and new developmentsPart I
Outline:
Graduation
Introduction
Testing smoothness
Statistical tests of the adherence of a graduation to data
1.
2.
3.
4.
5.
6.
ChiSquare Test of t
Standard
Actuarial StatisticsWeek 7
Outline:
General Markov modeldata issue
Binomial Model
Poisson Model
Model Choice: General Markov, Binomial, or Poisson Models
Reading
CT4Ch10
1/38
Data issue
The calculation of the estimates need to compute the total
waiting
Example (i) Comparing a mortality experience of a certain
population with the standard table:
s q (from
Age Exposed
Observed
x
x
to Risk
70
600
71
72
Ex
x
Ex
sq
zx =
x
I
NIEN qN
I
tables)
Number
23
0.03776
22.656
EN qN (
0.073676
750
31
0.04170
31.275
0.
Actuarial StatisticsWeek 6
Outline:
General Markov models and Applications
Reading
(Req) CT4 Chapters 6 and 10
(Rec) CT4, Chapter 3 (covered in Stochastic Models)
1/37
General Markov Model (multiple state model)
J = cfw_1, 2, . . . , n nite set of states
Actuarial StatisticsWeeks 1012: Part I
Graduation, standardization and new developmentsPart I
Outline:
Graduation
Introduction
Testing smoothness
Statistical tests of the adherence of a graduation to data
1.
2.
3.
4.
5.
6.
ChiSquare Test of t
Standard
Actuarial StatisticsWeek 1
Outline:
Future lifetime, probabilities and mortality
Complete and curtate expectation of life
Actuarial Notation
Life tables
Simple laws of mortality
Reading: ACTED CT4 Chapter 7, London 3rd Edition Chapter 3
1/29
The main obj
Actuarial StatisticsWeek 6
Outline:
General Markov models and Applications
Reading
(Req) CT4 Chapters 6 and 10
(Rec) CT4, Chapter 3 (covered in Stochastic Models)
1/37
General Markov Model (multiple state model)
J = cfw_1, 2, . . . , n nite set of states
Actuarial Statistics
Exercises 2
Some questions have been adapted from IAA Course Notes and the London
textbook.
Note: In the following you should derive the following by hand unless stated
otherwise. (You may also wish to solve them using R after becomin
Actuarial Statistics
Exercises 6
1. (IAA Course notes) Show that
gg
p = t pgg
x
t t x
gj
x+t
j=g
2. Show that in a model with states 0 active, 1 dead, 2 retired and transition
intensities 01 , 02 where both the dead and retired are absorbing states
that
Actuarial Statistics
Exercise 2 Solutions
1. Given the following failure times (a + indicates a censored observation)
17, 13, 15+ , 7+ , 21, 18+ , 5, 18, 6+ , 22, 19+ , 15, 4, 11, 14+ , 18, 10, 10, 8+ ,
17 we have
(a) N is the number of lives under invest
Actuarial Statistics
Sample solutions to Exercises 11&12
1. It would be appropriate to graduate the results of a mortality investigation
using a mathematical function if
a suitable mathematical formula can be found that describe mortality
rates adequatel
Actuarial Statistics
Solutions to Exercise 10
1. Due to natural forces one would expect mortality rates to progress
smoothly from age to age, i.e. there should be no sudden change in the curvature
of the mortality curve. Death rates are initially produced
Actuarial Statistics
Solutions to Exercises 3
1. (a) The hazard ratio for presence vs. absence of exposure is
(t; 1)
= e .
(t; 0)
b) If is given by 0.2, then the hazard ratio is
e = e0.2 = 1.2214.
If is given by 0.2, then the hazard ratio is
e = e0.2 = 0.
Actuarial Statistics
Sample Solutions to Exercise 8
1. (a) We can consider the central exposed to risk year by year for each
person.
Pele:
c
E30
c
E31
c
E32
c
E33
c
E34
1.1.99 to 10.11.99
11.11.99 to 10.11.00
11.11.00 to 10.11.01
11.11.01 to 10.11.02
11.1
Actuarial Statistics
Sample Solutions to Exercise 7
1. For the binomial model where the number of deaths in a year, D, for a
population of N individuals is such that
Binomial (N, q)
N d
N d
P r [D = d] =
q (1 q)
d = 0, 1, 2, . . . N
d
D
We have
D
N
var (
Actuarial Statistics
Sample Solutions to Exercise 6
1. We have
t pgg
x
dt
gg
t+dt px
gg
= lim+
tp
t x
dt0
Now
=
gg
t px
dt pgg
x+t
=
gg
t+dt px
gg
t px
1
gg
t px
+ o(dt)
j=g
=
gj
dt px+t
1
gj dt + o (dt)
x+t
j=g
Therefore
gg
tp
t x
=
lim+
gg
t px
1
Actuarial Statistics
Solutions to Exercises 4
1. (a) One method is just to compare the observed cumulative distribution of
the CoxSnell residuals with the theoretical CDF of an exponential(1) distribution.
(b) It should be a straight line with a slope of
Actuarial Statistics
Exercises 5: Sample Solutions
1. Probability of no deaths in time (0, t + h) is probability no deaths in (0, t)
times probability of no deaths in (t, t + h) so that
D0 (t + h) = D0 (t) (1 n h o (h)
so that
o (h)
D0 (t + h) D0 (t)
= n
Actuarial Statistics
Solutions to Exercise 1
Sample Solutions to Exercises 1 (note: most students will nd quicker and better ways of deriving the
solutions  these solutions give full and lengthy answers.)
1. (a) For y > x,

f (y
=
=
=
d
d
F (y  X > x)