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R代写 - ACTL5106 Insurance Risk Models

时间：2020-11-28

INSTRUCTIONS:

• Time Allowed: 2 hours

• Reading time: 10 minutes

• This examination paper has 24 pages

• Total number of questions: 8

• Total Marks available: 100 points

• Marks allocated for each part of the questions are indicated in the examination

paper. All questions are not of equal value.

• This is a closed-book test and no formula sheets are allowed except for the For-

mulae and Tables for Actuarial Exams (any edition). IT MUST BE WHOLLY

UNANNOTATED.

• Use your own calculator for this exam. All calculators must be UNSW ap-

proved.

• Answer all questions in the space allocated to them. If more space is required,

use the additional pages at the end.

• Show all necessary steps in your solutions. If there is no written solution,

then no marks will be awarded.

• All answers must be written in ink. Except where they are expressly required,

pencils may be used only for drawing, sketching or graphical work.

• THE PAPER MAY NOT BE RETAINED BY THE CANDIDATE.

Page 1 of 24

Question Marks

1

2

3a)

3b)

3c)

4a)

4b)

4c)

4d)

4e)

5a)

5b)

6a)

6b)

6c)

6d)

7

8

Total

Page 2 of 24

Question 1. (2 marks)

Let X be a loss random variable of a risk. Write down a formula for the premium

of this risk using the expected value principle.

Question 2. (2 marks)

Which of the following statements are true?

(A) de Pril’s algorithm is for calculating convolutions of discrete non-negative

integer valued random variables with positive probability mass at 0.

(B) de Pril’s algorithm is for calculating the distribution of non-negative integer

valued compound random variables with positive probability mass at 0.

PLEASE TURN OVER

Page 3 of 24

Question 3. (15 marks)

Consider the Crame´r-Lundberg surplus process

C(t) = c0 + pit−

N(t)∑

i=1

Yi, t ≥ 0,

where

• C(t) is the insurer’s surplus level at time t;

• c0 is the initial surplus;

• pi is the constant premium rate;

• N(t) is a Poisson process with rate λ; and

• Yi’s are claim amounts that are independent and identically distributed and

are independent of the above Poisson process.

(a) [4 marks] Assume that each claim amount follows the probability density function

fY (y) =

2

5

e−2y(3 + 4y), y > 0.

Suppose λ = 5, pi = 6 and c0 = 1.5. Calculate the relative security loading θ.

PLEASE TURN OVER

Page 4 of 24

(b) [3 marks] Give two reasons why the condition θ > 0 is important from the in-

surer’s point of view.

PLEASE TURN OVER

Page 5 of 24

(c) [8 marks] Suppose

• Yi ≡ 1000 for i = 1, 2, 3, . . ., and pi = 1300λ;

• The values of λ and c0 are unknown;

• The probability that ruin occurs at the first claim is 1%.

Determine the numerical value of the initial surplus, c0.

PLEASE TURN OVER

Page 6 of 24

Question 4. (20 marks)

The annual claims X1, X2, . . . for a given policyholder in an insurance portfolio are

known to be (conditional on the policyholder’s risk parameter Θ = θ) independent

and identically distributed with probability mass function

fX|Θ(x|θ) = (x+ 1)(1− θ)2θx, x = 0, 1, 2, . . . ,

where 0 < θ < 1. The (unobservable) risk parameter Θ is assumed to follow a Beta

distribution with parameters α, β, where α > 0 and β > 2.

(a) [2 marks] State the name of the distribution that has probability mass function

fX|Θ(x|θ) and identify its parameter(s). Hence, deduce that

E[Xi|Θ = θ] = 2θ

1− θ

for i = 1, 2, 3, . . ..

PLEASE TURN OVER

Page 7 of 24

(b) [6 marks] Define µ(θ) = E[Xi|Θ = θ]. Show that

E[µ(Θ)] =

2α

β − 1 .

PLEASE TURN OVER

Page 8 of 24

In the parts (c)-(e) below, suppose that we have observed T years of claim amounts

X = (X1, X2, . . . , XT ) to be x = (x1, x2, . . . , xT ).

(c) [4 marks] Show that the posterior distribution of Θ|X = x is a Beta distribution

with parameters

α˜ = α +

T∑

t=1

xt and β˜ = β + 2T.

PLEASE TURN OVER

Page 9 of 24

(d) [5 marks] Prove that the Bayes premium is

PBayes =

2T

β + 2T − 1

∑T

t=1 xt

T

+

β − 1

β + 2T − 1

2α

β − 1 .

PLEASE TURN OVER

Page 10 of 24

(e) [3 marks] Without performing any calculation, determine whether the Buhlmann’s

credibility premium is greater than, smaller than, or equal to the Bayes premium

with justification.

PLEASE TURN OVER

Page 11 of 24

Question 5. (12 marks)

Consider two random variables X and Y , where both follow exponential distribution

but with parameters α > 0 and β > 0 respectively. They are linked through the

Farlie-Gumbel-Morgenstern copula defined by

C(u, v) = uv + θuv(1− u)(1− v), u, v ∈ [0, 1],

where θ ∈ [0, 1] is the parameter of the copula.

(a) [4 marks] Explain whether the copula allows for possibility of independence

between X and Y .

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Page 12 of 24

(b) [8 marks] Show that the joint density of X and Y can be represented as

fX,Y (x, y) = A(αe

−αx)(βe−βy) +B(2αe−2αx)(βe−βy)

+ C(αe−αx)(2βe−2βy) +D(2αe−2αx)(2βe−2βy), x, y > 0,

and determine the constants A, B, C and D.

PLEASE TURN OVER

Page 13 of 24

Question 6. (23 marks)

Recall that a distribution is from an exponential dispersion family if its density has

the form

fY (y) = exp

[

yθ − b (θ)

ψ

+ c (y;ψ)

]

, θ ∈ Θ, ψ ∈ Π.

(a) [4 marks] Describe the two main components of a generalized linear model and

explain how the two components are linked.

PLEASE TURN OVER

Page 14 of 24

(b) [7 marks] Show that the distribution corresponding to the following probability

density function belongs to the exponential family of distributions:

g(y) =

yα−1e−y/β

βαΓ(α)

, y > 0.

PLEASE TURN OVER

Page 15 of 24

(c) [6 marks] Consider a distribution from the exponential dispersion family with

b(θ) = 10 log(1 + eθ).

Derive the expressions for the natural link function and the variance function.

PLEASE TURN OVER

Page 16 of 24

(d) [6 marks] Assume that you know that the following three Poisson general linear

models (GLM) with the same link function, g(·), all fit the data well:

Model 1: g(µi) = β1xi1

Model 2: g(µi) = β1xi1 + β2xi2

Model 3: g(µi) = β1xi1 + β2xi2 + β3xi3 + β4xi4

The scaled deviances are given as below

Model Deviance

Model 1 72.23

Model 2 70.64

Model 3 67.13

Which model is the best based on the available information and the likelihood ratio

test at 5% significance level? Explain why.

PLEASE TURN OVER

Page 17 of 24

Question 7. (13 marks)

The cumulative paid claims on a portfolios of insurance policies are given in the

following table:

Accident year Development year

1 2 3

2014 5,496 x 7,982

2015 5,162 8,028

2016 6,434

where x is a positive number.

Suppose the claims will completely run off in 3 years and the development factor from

development year 2 to development 3 is 1.04504. By assuming that the ultimate loss

ratio is 0.85, you have found that the Bornhuetter-Ferguson estimate of outstanding

claims at the end of year 2016 for accident year 2016 is 6,464. Determine the

numerical value of the earned premium for year 2016.

PLEASE TURN OVER

Page 18 of 24

(This page can only be used to answer Question 7.)

PLEASE TURN OVER

Page 19 of 24

Question 8. (13 marks)

Consider the following payoff matrix of a zero-sum game with two players, A and

B. The payoff matrix lists the gains for A and losses for the player B.

B

A

Strategy 1 2 3

a 10 34 7

b 22 14 8

c X 30 26

where X is an exponential random variable with mean 1/λ. Determine the numerical

value of λ so that the probability that there is an optimal solution is 10%.

END OF PAPER

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