THE UNIVERSITY OF MELBOURNE
Centre for Actuarial Studies. Department of Economics
ACTL90008 Statistical Techniques in Insurance
Assignment 1, COVER SHEET
Due by 11:59 PM on Sunday 29 August 2021. Submission via LMS
This assignment contributes 10% of the total university assessment of this subject.
Please attach this cover sheet on top of your answers to your submission. Include your
name and student ID number in the table provided below
Student Number Name in full Signature
Declaration
I declare that this assignment is our own work and does not involve plagiarism or collu-
sion. I understand that penalties will be imposed if the instructions accompanying the
assignment are not followed.
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colluding.
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ACTL90008 Statistical Techniques in Insurance
Assignment 1, 2021
Instructions:
1. Complete the ACT90008 assignment cover sheet and include it in your
submission.
2. Write your answers to the questions below. You must show full working
in each question.
3. The total number of marks is 100. Up to 20% marks can be deducted if
your solutions are poorly presented.
4. You may include part of your code in your submission. You should
submit sufficient working so that the process you have implemented in
each question can be followed.
5. Your submission should be no longer than 10 pages excluding cover sheet.
You are reminded that heavy penalties apply to students who plagiarise or collude. These
terms are defined on the assignment cover sheet.
Part I
The the danish dataset in the SMPracticals R package contains losses on major
insurance claims due to fires in Denmark, 1980-1990 adjusted to reflect 1985 values. It
consists of 2492 measurements in millions DKK. The values of the claims have been
rescaled for commercial reasons. For this question we will use the Gamma and lognormal
distributions whose probability density functions are given in the Appendix.
(a) Use R to fit these two distributions by the method of moments to this set of data.
Comment your results.
[10 marks ]
(b) Use the estimates derived in part (a) as initial values to compute the maximum
likelihood estimates for each model. Comment your results.
[10 marks ]
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(c) Plot a histogram of the data and superimpose the graphs of the probability density
functions of the three distributions by using the estimates computed in Part (b).
[5 marks ]
(d) Let us suppose that the losses given in this dataset are drawn from:
(i) a gamma distribution, Ga(αˆ, βˆ), where αˆ, βˆ are the estimates obtained in Part
(b).
(ii) a lognormal distribution, LN(µˆ, σˆ), where µˆ, σˆ are the estimates obtained in
Part (b).
Using parametric bootstrap with a bootstrap sample of size B = 5, 000, find the
mean square error (MSE) of the maximum likelihood estimates for each model.
Discuss the results.
[20 marks ]
(e) The percentile premium principle for a risk X is defined as P(X) = {z|FX(z) ≥ 1−α}
where FX(·) is the cumulative distribution function of X, where the probability of
a loss X is at most 0 ≤ α ≤ 1.
By assuming the same conditions as in (d) (i) and (d)(ii) , use parametric bootstrap
with a bootstrap sample of size B = 5, 000 to calculate the percentile premiums at
security levels 0.1, 0.05 and 0.01. Comments the results. Discuss the advantages
and limitations of the premium principle.
[20 marks ]
Part II
You wish to examine the relationship between the number of vehicle accidents and the
unemployment rate in a particular region. For this purpose, you use the dataset traffic2
in the wooldridge R package and take as response and explanatory variables totacc
and unem respectively. Determine the simple linear regression model and fit the model.
Discuss the results. Is the model correctly specified? Justify your answer. Quantify the
degree of association between the two variables by using different correlation coefficients
discussed in lectures. Comment your results.
[35 marks ]
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Appendix:
• Probability density function of the gamma distribution:
f(x|α, β) = β
α
Γ(α)
xα−1 exp{−βx} with α > 0, σ > 0 and x > 0.
• Probability density function of the lognormal distribution:
f(x|µ, σ) = 1
x σ
√
2pi
exp
{
−1
2
(
log x− µ
σ
)2}
with µ ∈ R, σ > 0 and x > 0.
END OF ASSIGNMENT
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