STAT3057/6057: Assignment 2 Questions

Semester 1, 2023

INSTRUCTIONS:

Total Marks: 80

The due date for this assignment is: 5PM, FRIDAY 26 MAY (Week 12).

A submission link for the assignment will be placed on Wattle prior to the due date. Please submit your assignment as a PDF document. Do not submit in other formats.

This assignment involves the use of R for Questions 2, 3, 4 and 5. For questions that require the use of R, you are required to submit the R code that you use and the corresponding R output. Do not submit the R code and R output as separate files. Include the relevant R code and R output with your answers in a single PDF document.

Make sure to include all working out steps in your answers.

DATA FOR ASSIGNMENT

You will need to use the data files Assignment2_Q3Data.csv and Assignment2_Q5Data.csv that have been placed on Wattle for questions 3 and 5.

Run the code below. You may have to specify the correct path to the csv file corresponding with where you have saved the file:

Q3data <- read.csv("Assignment2_Q3Data.csv", header = TRUE)

Q5data <- read.csv("Assignment2_Q5Data.csv", header = TRUE)

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QUESTION 1 (10 marks)

An insurer offers a pet insurance product with the following features:

• a policy excess (deductible) of $250, so that policyholders pay the first $250 of any loss amount.

• a coinsurance factor of c%. This means that the insurer pays c% of loss amounts and the policyholder pays the remaining (1-c)%.

• a policy limit of $4000. The insurer only compensates the policyholder up to this loss limit. If we denote the loss amount as X, and the loss amount for the insurer as Y , then:

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Y = c(X−250) c(4000 − 250)

if X ≤ 250 if250<X≤4000 if X > 4000

If X follows a log-normal distribution with parameters μ = 6.7 and σ2 = 1.6, find the coinsurance factor c that the insurer should apply, if:

a) we want to ensure that the expected amount that the insurer will pay, per payment, is $1200 (i.e., E(Y |X > 250) = 1200).

b) we want to ensure that the expected amount that the insurer will pay, per loss, is $1000 (i.e., E(Y ) = 1000).

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QUESTION 2 (15 marks)

Note: You will need to use R for parts (a) and (b) of this question.

An insurer offers a specialised insurance product that pays compensation when a person suffers a temporary injury due to medical negligence. Each compensation claim consists of two payments: A (medical expenses) and B (temporary loss of earnings).

The insurer has approached a large reinsurance company to purchase the excess-of-loss (XoL) contract. The insurer has provided analysis of their claims to the reinsurer and has based their analysis on the assumption that the dependency between payment amounts A and B follows a Gaussian copula with ρ = 0.3.

You are the head actuary for the reinsurer, and you have expressed concern that the insurer’s analysis is incorrect. You believe that there is strong dependency between payments A and B, and that a Gaussian copula is also not appropriate.

For the questions below, assume that the marginal distribution for payment amount A is Gamma distributed with parameters α = 0.63 (shape parameter) and θ = 4780 (scale parameter), and the marginal distribution for payment amount B is log-normally distributed with parameters μ = 6.2 and σ2 = 1.3.

a) Using R, simulate 10,000 claim amounts assuming that the dependency between A and B follows: i) a Gaussian copula with ρ = 0.3.

ii) a Gumbel copula with parameter α = 10.

Plot out scatterplots of the simulated values of A versus B for both cases (i) and (ii) and comment

on the results. (7 marks)

b) The insurer wishes to take out an excess-of-loss reinsurance contract with a retention level of M =

$15, 000 per claim. Use the simulations from (a) to estimate the following:

i) the probability that a compensation claim, X, will exceed $15,000. (2 marks)

ii) the conditional and unconditional expected claim amounts, in excess of the retention level. i.e., estimate E(X − M|X > M) and E(X − M). (3 marks)

c) Based on the results of your analysis, discuss the implications to the reinsurer if the dependency follows a Gumbel copula with α = 10, but if the reinsurer had instead charged premiums for the XoL contract based on the insurer’s assumption of dependency. (3 marks)

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QUESTION 3 (15 marks)

Note: You will need to use R for parts (c) and (e) of this question. You will need the data for Question 3

(Q3data).

Assume that we have a loss random variable X, and that XM is the block maximum for X, such that

XM =max(X1,X2,...,Xn).

Assume that XM follows the Generalised Extreme Value (GEV) distribution with parameters α,β and γ,

and that γ ̸= 0.

Recall from lectures that the CDF of the GEV distribution, when γ ̸= 0, is:

? ? γ(x−α)?−1/γ? H(x)=exp − 1+ β

a) If we define VaRp∗ such that Pr(XM ≤ VaRp∗) = p∗, write down an expression for VaRp∗ for XM in terms of γ, α, β and p∗. (3 marks)

b) If we define VaRp such that Pr(X ≤ VaRp) = p, where X is the original loss random variable, use your result from part (a) to write down an expression for VaRp for X in terms of γ, α, β, p and n. (5 marks)

You have invested $5 million in a fund that tracks the ASX200 index.

Q3data contains daily return data for the ASX200 for 567 trading days. You can assume that the return data is serially uncorrelated and there is no time series structure in the data.

Let X denote the daily losses (e.g., a daily return of 0.01 in Q3data corresponds with a 1% return and a -1% loss.)

c) You decide to model extreme losses in the ASX200 data by fitting a GEV distribution to X. Using R, fit the GEV and obtain estimates of α, γ and β using a block size of length n = 21. (2 marks)

d) Use the expression that you derive for VaRp from part (b) and your parameter estimates from part (c) to compute the 1-day VaR0.95 for the investment of $5m. (2 marks)

e) Compute the 1-day VaR0.95 and the TVaR0.95 of the $5m investment by using the empirical approach. (3 marks)

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QUESTION 4 (15 marks)

Note: You will need to use R for part (c) of this question.

Consider the time series process xt = φxt−3 + wt + θwt−2,

where wt is Gaussian white noise with variance σ2. Assume that this process is stationary and invertible.

a) Find the coefficients ψj if we write the process in the following form: wt = P∞j=1 ψj xt−j . (5 marks)

b) Use the form of the process from part (a) to forecast xt+1 using the following data and the estimates obtained from a realisation of the time series process:

(xt−6 , ..., xt ) = (−0.41, 2.39, 0.69, −0.09, 2.13, −0.65, −1.49) and φˆ = 0.4, θˆ = 0.3.

You can assume that all other past values of x are zero. i.e., xt−7 = xt−8 = ... = 0. (5 marks)

c) Assuming that φ = 0.4, θ = 0.3 and σ2 = 1, simulate the time series in R using the function ‘arima.sim’ with n = 100 (Use set.seed(123) when generating the simulated values). Use the function ‘arima’ to fit a ARMA(3,2) model to the simulated data (do not include a mean). Use the ‘forecast’ function to generate a forecast from the fitted model with a forecast period of h=50. Plot the simulated time series and forecast. Describe the key features in the simulated data and forecasts. (5 marks)

You do not need to include the simulated values or forecast numbers with your solution, but include the R code that you use and the plots that you have generated.

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QUESTION 5 (25 marks)

Note: You will need to use R for all parts of this question. You will need the data for Question 5 (Q5data).

Let xt denote the time series ‘Q5data’ (where t = 1, ..., 200). The data represents daily returns for a particular asset over 200 days.

a) Plot the series xt and the sample ACF and sample PACF functions. Describe the key features in the plots and the correlograms. (3 marks)

b) Use the R function ‘arima’ to fit AR(p) models for p=1,2,3,4,5 and 6 to xt. By comparing AIC values for the different models, which model would you prefer for xt? Write down the equation of this model for xt including all parameter estimates. (4 marks)

[NOTE: The ‘arima’ function in R outputs a parameter called the “intercept” in R. This is an estimate of the mean μ, and not an estimate of the constant φ0.]

c) Undertake model diagnostics on the residuals of the model you selected from part (b). What do you conclude as a result of these tests? (4 marks)

d) Plot out the squared residuals and the sample ACF and sample PACF of the squared residuals of the model you selected from part (b). Describe the key features in the plots and the correlograms. Based on these results, what do you conclude about the model? (3 marks)

e) Using the model you selected from part (b), re-estimate the model allowing for ARCH(1) errors. Write down the full model for xt including all parameter estimates. (2 marks)

f) Using the model you selected from part (b), re-estimate the model allowing for GARCH(1,1) errors. Write down the full model for xt including all parameter estimates. (2 marks)

g) You invest $100 in the asset and you expect that future returns will follow the model from part (f). Using the function ‘ugarchsim’, undertake 1000 simulations of returns over the next 100 days based on the fitted model from part (f), and then use these 1000 simulations to find the 10-day, 50-day and 100-day 99% Value at Risk (VaR) for your investment. (7 marks)

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