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STAT 3058 - Risk Modelling

创建日期：2024-06-01 17:26:56

所属分类：统计学statistics

标签： STAT 3058

Risk Modelling

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STAT 3058 - Risk Modelling 2

Assignment
Total Marks: 60 marks.
No late assignments will be accepted.
Please show all workings for full marks.
Where appropriate, round to 5 decimal places.
No separate code file should be submitted. All code should be part of your answer
booklet.
Notes:
1. No late assignments will be accepted. If extension is required, you need to contact the
lecturer MORE THAN 24 HOURS before submission time.
2. Assignments that are not accompanied by a COMPLETED Assignment cover sheet will
get a grade of 0.
3. Please name your assignments answer book with your uni id as ‘uxxxxxxx’ and submit
it under Part 1 of the Assignment 1 Turnitin submission in Wattle.
1
1. Only use R in part (a) and part (f) of this question. The rest are to be done by
hand/typeset.
You work in an insurance company, where you are tasked with estimating aggregate claim amount
arising from a portfolio of insurance policies.
Suppose that S represents the aggregate claim amount from the portfolio which makes a Binomial
number of claims with parametersm = 20 and q = 0.25 and that these claims are Gamma distributed
with shape parameter α = 15 and scale parameter θ = 0.4.
You are given that the third raw moment for a Gamma distributed random variable is: αθ3(α +
1)(α+ 2)
(a) (3)Make appropriate adjustment to the R-code provided on Wattle to simulate G(s), and generate
the probabilities for G(15), G(30) and G(45).
In your answer script, copy and paste the code used and use the R function, data.frame(), to
display the required s values and their corresponding CDF (G(s)) values.
(b) (3)The cumulant generating function (CS(t)) of S ∼ CompBin(m, q,Gamma(α, θ)) is given by
CS(t) = m ln
[
1− q + q(1− θt)−α] , t < 1/θ
Find expressions for C(1)(t), C(2)(t) and C(3)(t), where the superscript (i) denotes the ith
derivative.
Hint: Define g(t) = 1− q + q(1− θt)−α, and use it in your final expressions.
(c) (4)Hence or otherwise, find expressions for the mean (µS = C(1)(0)), variance (σ2S = C
(2)(0)) and
the standardised coefficient of skewness (γS =
C(3)(0)
(C(2)(0))3/2
) of S.
Compare these expressions to those generated from using the formula in slide 19 of the lecture
notes.
(d) (2)Hence or otherwise, calculate the mean (µS), variance (σ2S) and standardised coefficient of
skewness (γS) of S for the given portfolio.
(e) For s = {15, 30, 45}, estimate G(s) using:
i. (2)the Normal approximation
ii. (3)the Lognormal approximation
iii. (3)the translated Gamma approximation
(f) (4)Generate 10, 000 simulated aggregate claim values for this portfolio, and then generate 10, 000
values each from the Normal distribution and the Lognormal distribution in part (e). For each
of these three simulations, set the random generator seed to 123.
Plot the empirical density functions of these three sets of simulated values in the same plot,
and use different colors for each curve. Make sure your plot has proper title, axis names and
legends.
2
In your answer script, copy and paste the code used and use the R function, head(), to display
the first 7 values for each of the three simulations. Also, attach the plot required in your answer
script.
(g) (5)Comment on the main features of the plot generated in part (f). Hence or otherwise, compare
the suitability of the Normal and the Lognormal approximation methods to this portfolio.
(h) (4)A colleague has claimed that a Normal Power (NP2) approximation is better than the translated
Gamma approximation.
Let Φ denote the CDF of a standard normal distribution, and Z = (s − µS)/σS . Under NP2
approximation we get:
G(s) ≈ Φ
((√
1 +
9
γ2S
+
6Z
γS
)
− 3
γS
)
Verify the validity of the claim.
This question is worth 33 marks.
2. Only use R in part (c) of this question. The rest are to be done by hand/typeset.
Let X be a random claim amount. Let us assume that insurer charges premium loading of 80%, and
that the claim size density follows
f(x) = (1 + 6x) e−3x, x ≥ 0
(a) (4)Find the mgf of X, mX(r).
(b) (5)Solve for the value of the adjustment coefficient R.
(c) (9)Employ Monte Carlo Simulation in R to generate 2000 surplus processes using time step of
0.1 and time period [0, 100]. For the ith simulation, set the random number generator seed to
100+i. To simulate the claim size, apply the Newton-Raphson method via using the uniroot()
function in R. You can assume that the claim frequency λ = 1 per time step.
Using the simulation results, estimate the probability of ultimate ruin for U0 = 2, 3, 4, 5. Ignore
the probability that ruin happens outside of simulated time period.
In your answer script, copy and paste the code used and use the R function, data.frame(), to
display the probabilites along with their coresponding U0 values.
Hint: The “Monte Carlo Simulation example - R” file on Wattle is a good starting point!
Cramer’s asymptotic ruin formula states that the ruin probability satisfies
ψ(U) ∼ Ce−RU , as U →∞
where, 0 < C ≤ 1. An estimate of C is C = µ1θm′X(R)−µ1(1+θ) . m
′
X(R) denotes the first derivative of
mX(R).
3
(d) (5)While Cramer’s asymptotic ruin formula is most appropriate for high values of U , it can provide
a good approximation for smaller values of U . Check the validity of this claim by comparing
probability of ultimate ruin calculated using Cramer’s asymptotic ruin formula to the simulated
probability of ultimate ruin for U0 = 2, 3, 4, 5.
(e) (4)The simulated probability of ultimate ruin can be seen as the actual probability of ultimate
ruin. Show the actual probability of ultimate ruin, the Cramer’s asymptotic probability of
ultimate ruin, and the Lundberg’s upper bound for U0 = 2, 3, 4, 5 in a table, and comment on
their relationship.
This question is worth 27 marks.

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