Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
HW3 Solution
1
1. X U [0, 1] and thus, E(X) = 0.5 and V ar(X) = 12 . E(S) = 100(0.5) = 50 and V ar(S) = 100V ar(X) = 25 .
3
P r[S 0.95] = 0.95 implies that 0.95 = 50 + (1.6449) 25/3 54.75 using Normal approximation.
1
For X, we have E(X) = 1 and V ar(X)
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Homework Assignment 3
1. X1 , , X100 are independent random variables each uniformly distributed on the interval (0, 1). Use the
100
S
normal approximation to nd (a) the 95th percent
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Homework Assignment 1
1. The probability density function (pdf) of a random variable X with the uniform distribution on the interval
[0; 2] is given by
f(x) =
1
,
2
0,
if 0 x 2;
other
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2012
Final Exam Review Problems
Basic Conceptual Problems
B1.
What is the mathematical modeling in actuarial science?
What are two basic criteria for model selection?
What are the basic st
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY
EXAM C CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS
EXAM C SAMPLE QUESTIONS
The sample questions and solutions have been modified. This page indicates
changes made to Study Note C0908.
January 14, 2014:
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
1. (a) E(X d) =
d
0
xf (x)dx +
d
Midterm Test Solution
d
d
d
df (x)dx = 0 xd[s(x)] + ds(d) = [xs(x)]d + 0 s(x)dx + ds(d) = 0 s(x)dx.
0
2. Y P = (Y 150Y > 150) is a conditional random variable. Its variance is
V ar(Y P ) = E[(Y 150)2 Y > 150] [E(Y 150Y
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 5 Outline
1. Methods for creating new distributions (Cont.)
Mixture model: For (X) fX (x) and f (), we have that fX (x) =
Example. If (X) has exponential distribution for given =
1
fX (x)f ()d.
and has a distribution (, 1 ).
Determine the
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 3 Outline
1. Properties of E(X k )
Denition. The kth moment of a random variable X exists usually means that xk f (x)dx <
. We have the inequality:  xk f (x)dx xk f (x)dx. Here are some basic properties of
E(X k ):
E(X 2 ) (E[X])2 since V a
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
CITY UNIVERSITY OF HONG KONG
Department of Mathematics
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Lecture2 Outline on Basic Quantities of Distributions
(1) Basic quantities: Moments, percentiles, and mgf & pgf.
(2) Moments: m
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
CITY UNIVERSITY OF HONG KONG
Department of Mathematics
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Lecture1 Outline
(1) Mathematical modeling in actuarial science is to build a model to forecast
or predict insurance costs in t
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Ma6622: Midterm Test, Summer 2014
There are 10 questions in total, please sovle problems in order. Also, No group work, no discussions. Do all problems
independently!
1. Prove that E(X d) =
d
0
s(x)dx for a random variable X 0 with a continuous pdf. s(x)
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY
EXAM C CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS
EXAM C SAMPLE SOLUTIONS
The sample questions and solutions have been modified. This page indicates
changes made to Study Note C0908.
January 14, 2014:
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2012
Final Exam
Introduction. There are twelve questions in this nal exam. Please show ALL your work. You are allowed
to use a calculator and one page formula sheet prepared by yourself. G
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 6 Outline
1. Risk Measure: V aR and T V aR
A risk measure is a mapping from a set of random variables to the real numbers (X) : X X R+ .
Examples are the moments, variance, percentiles of random variables, etc. A risk measure of X is
said to be co
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 7 Outline
1. Topics in Severity Coverage and Modications
There are two distributions that are important in estimating potential operational risk losses for a
certain risk type/business line combination. One is the loss frequency distribution and t
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 8 Outline
Aggregate Loss Models
There are advantages in modeling the claim frequency and claim severity separately, and then combine
them to obtain the aggregate loss distribution. For example, expansion of insurance business may have
impacts on
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Ma6622 Hw2
Solve problems in order.
1. You are given the pdf of a random variable X
f (x) =
2 2x
e
+ 2e3x , x > 0.
3
Find out
(a) the survival function S(x);
(b) the hazard rate function h(x), and
(c) compute the mean of X.
2. (a) Suppose X B(m, p) and Y
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
CITY UNIVERSITY OF HONG KONG
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Homework Assignment 4
1. Suppose fX (1) = 0.60 and fX (2) = 0.40 gives the distribution of the claim amount variable X, and P r[N =
0] = 0.70 and P r[N =
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
HW4 Solution
1. (i) From the distribution of X: fX (1) = .6, fX (2) = .4, we have E(X) = 1.4, E(X 2 ) = 2.2, and V ar(X) =
2.2 1.42 = .24. Similarly, from P r(N = 0) = .7, P r(N = 1) = .2, P r(N = 2) = .1, we have E(N ) =
.4, E(N 2) = .6 and thus, V ar(N
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
CITY UNIVERSITY OF HONG KONG
Department of Mathematics
MA6622: Model Calibration in Finance and Actuarial Sciences
Summer 2014
Lecture: MWF 7:00pm9:50pm at AC1 LT7 (MWF)
Instructor: Dr. Don Hong Oce: Y6624
Phone number: 34428352 Email: [email protected]
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 13 Outline
Credibility Theory (cont.)
Bhlmann Credibility
u
For a risk group or block of insurance policies with loss measure denoted by X, which may be claim
frequency, claim severity, aggregate loss or pure premium. If the risk proles of the g
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 12 Outline
Credibility Theory (cont.)
Full Credibility for Claim Severity
We now consider the standard for full credibility when the loss measure of interest is the claim severity.
Suppose there is a sample of N claims of amounts X1 , X2 , , XN
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 11 Outline
Some basic facts on statistical estimator
For a parameter with a distribution pdf f(), let denote its estimator. We consider the estimation
error by introducing the following error functions (also called loss functions) of :
(1) L(, )
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 10 Outline
Tail Value at Risk and Linear Exponential Family
Tail Conditional Expectation (TCE=TVaR), Tail Value at Risk
Insurance companies often set aside amounts of capital from which they can draw in the event
that premium revenues become insu
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
Lecture 9 Outline
Aggregate Loss Models (cont.)
(2) Collective risk model:
Let N be the number of losses in the block of policies, and Xi be the amount of the ith loss, for
i = 1, , N . The aggregate loss S is given by
S = X1 + + XN ,
where N is a random
Model Calibration in Finance and Actuarial Science
MA 6622

Summer 2014
HW1 Solution
1. (a)
x
F (x) =
1,
if x > 2;
1
x, if 0 x 2;
2
0,
if x < 0.
0,
if x > 2;
1 1 x, if 0 x 2,
2
1,
if x < 0.
f(x)dx =
S(x) = 1 F (x) =
1
2x , if 0 x < 2;
f(x)
h(x) =
= 0,
if x < 0.
S(x)
U D, if x 2
2
2
1
1
(b) E[X] = xf(x)dx = 0 2 xdx = 1. E[X