R代写 - ACTL5106 Insurance Risk Models

INSTRUCTIONS:
• Time Allowed: 2 hours
• 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
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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.
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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.
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(b) [3 marks] Give two reasons why the condition θ > 0 is important from the in-
surer’s point of view.
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(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.
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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, . . ..
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(b) [6 marks] Define µ(θ) = E[Xi|Θ = θ]. Show that
E[µ(Θ)] =

β − 1 .
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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.
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(d) [5 marks] Prove that the Bayes premium is
PBayes =
2T
β + 2T − 1
∑T
t=1 xt
T
+
β − 1
β + 2T − 1

β − 1 .
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(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.
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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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(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.
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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.
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(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.
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(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.
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(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.
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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.
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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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