Actuarial Models: example sheet 3
*=easy, *=intermediate, *=dicult
* Exercise 1
Consider the no claims discount model as in Example 3.11 in the notes. The insurance
company is now considering to change the highest discount level from 40% to 50%. Research
Actuarial Models: solutions example sheet 5
Answer of 1
Since the only exit from the ill state is to the dead state and the dead state is absorbing, it
follows that Xt = d given X0 = i if and only if t is bigger than or equal to the rst jump
time (this fa
Actuarial Models: solutions example sheet 4
Answer of exercise 1
The Q-matrix for this model is given by
( + )
0
,
Q=
0
0 0
whereby h is the rst state, i is the second and d the third.
(a) The 3-dimensional system of Kolmogorov forward equations corresp
Actuarial Models: solutions example sheet 8
Answer to 1
(a) The table indicating the values of ti , di and ri are given in Table 1. We can then
compute the Kaplan-Meier estimator at the times ti using S(t0 ) = S(0) = 1 and the
recursion,
di
S(ti ) = S(ti1
Actuarial Models: solutions example sheet 7
Answer to 1
(a) Let we and wu be the observed total waiting time in state e respectively u and let e
and u be the observed total transitions out of state e respectively u. Then we know
from the lecture notes tha
Actuarial Models: solutions example sheet 10
Answer to 1
(a) Let x = (0, 100, 60, 0, 0, 40). (Note that the notation of the vector x here is used in
a dierent way than in the notes; whereas in the notes it corresponds to a vector
of covariates correspondi
Actuarial Models: solutions example sheet 9
Answer to
Using formula (9.1) in the lecture notes, we have
1
n
(, ) = C +
i log (ti ) A(ti ) ,
i=1
t
with A(t) = 0 (s)ds = (t) the cumulative hazard function and C R an unimportant
constant. Hence
n
i log + log
Linear models
Simon Wood
Mathematical Sciences, University of Bath, U.K.
Linear models
I
We have data on a response variable, y, the variability in
which is believed to be partly predicted by data on some
predictor variables, x1 , x2 . . .
I
We model this
Dr. Neal, WKU
MATH 429
The Gamma Distribution
The gamma distribution X ~ [ , ] is a special continuous probability distribution that
is a generalization of some other advanced distributions. We initially shall study the
gamma distribution without any cont
Lecture Notes
FINANCIAL MATHEMATICS I:
Stochastic Calculus, Option Pricing,
Portfolio Optimization
Holger Kraft
University of Kaiserslautern
Department of Mathematics
Mathematical Finance Group
Summer Term 2005
This Version: August 4, 2005
CONTENTS
1
Cont
TWO NOTES ON FINANCIAL MATHEMATICS
Daniel Dufresne
Centre for Actuarial Studies
University of Melbourne
The two notes which follow are aimed at students of nancial mathematics or actuarial science.
The rst one, A note on correlation in mean variance portf
Actuarial Models: solutions example sheet 6
Answer to 1
(a) To test for overall goodness of t, we use the chi-squared overall goodness of t test
with null hypothesis H0 : the graduates rates are the true rates. As the Makeham
2
model has three free parame
Actuarial Models: solutions example sheet 1
Answer to 1
The cumulative hazard function A(t) for t 0 is given by
t
A(t) =
t
(s)ds =
0
0
s1 ds = s |t = t ,
s=0
where the last equality follows because > 0 (and so 0 = 0).
The survival function is then given
Actuarial Models: solutions example sheet 2
Answer to 1
(i) In this case X1 , X2 , . . . are independent and thus X is trivially a Markov chain. (ii) In this
case X is a Markov chain as Xn = Xn1 + Zn and so given that we know the value of Xn1 ,
then Xn de
Actuarial Models: example sheet 6
*=easy, *=intermediate, *=dicult
* Exercise 1
A graduation is performed on a data set containing 10 age groups. The graduation was
done by using the Makeham model, cf. Section 1.2 in the notes. The vector of standardised
Actuarial Models: example sheet 1
*=easy, *=intermediate, *=dicult
* Exercise 1
Recall that for a Weibull distributed lifetime with parameters , > 0, the hazard function
is given by (t) = t1 , t 0. Find the cumulative hazard function and the survival
func
Actuarial Models: example sheet 2
*=easy, *=intermediate, *=dicult
* Exercise 1
Suppose you roll independently a die multiple times. Let Zn be the outcome of the nth roll
and let Sn = n Zi be the sum of the outcomes in the rst n rolls. In each of the foll
Actuarial Models: example sheet 4
*=easy, *=intermediate, *=dicult
* Exercise 1
Consider the following three state time homogeneous model for a terminal illness:
h:healthy
E
d
i:ill
d
d
d
d
d
d:dead
Let P(t) =
phh (t) phi (t) phd (t)
pih (t) pii (t) pid (
Actuarial Models: example sheet 5
*=easy, *=intermediate, *=dicult
Exercises 3 and 4 are part of the coursework (for MATH39511 and
MATH69511). Submit your answers to the reception on the ground
oor before noon on Wednesday 5 November. Do not forget to ll
Actuarial Models: example sheet 8
*=easy, *=intermediate, *=dicult
* Exercise 1
The following data give the burn time of a number of candles of the same type. During the
experiment some candles went out due to other reasons than the wax being fully usurpe
Actuarial Models: example sheet 7
*=easy, *=intermediate, *=dicult
* Exercise 1
Consider the employment-unemployment MJP model with two states e: employed and u:
unemployed and constant transition rates , where is the transition rate from employment
to un
Actuarial Models: example sheet 9
*=easy, *=intermediate, *=dicult
Exercise 4 is part of the coursework (for MATH39511 only). Submit your answers to the reception on the ground oor before noon
on Wednesday 3 December. Do not forget to ll in a cover sheet.
Actuarial Models: example sheet 10
*=easy, *=intermediate, *=dicult
* Exercise 1
The recorded survival times, in years, of six patients following a heart bypass operation are
given below.
10.4, 6.6, 4.2, 6.2, 15.9, 6.3,
where values followed by a indicate
Actuarial Models: solutions example sheet 3
Answer of 1
(a) Because of the higher discount level, the policyholders are more inclined to want this
discount and therefore will less likely report a claim to the insurance company in
order to stay or get to t
MATH39511
Two and a half hours
Statistical Tables provided
THE UNIVERSITY OF MANCHESTER
ACTUARIAL MODELS
22 January 2016
14:00 - 16:30
Answer ALL FIVE questions.
The total number of marks in the paper is 90.
Electronic calculators are permitted, provided