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Final Examination
Unit Code: ETC3430 / ETC5343
Unit Title: Financial Mathematics under Uncertainty
Exam Duration: 2 hours 40 minutes (includes reading, downloading, and uploading time)
Format: This is an individual assessment. This is a closed book exam.
No. of Questions: Answer all SEVEN questions. The total mark is 60.
Working: Show all your steps clearly. Start each question on a new page. Label each piece of paper with your Student ID.
Formulae & Tables: Formulae sheet and statistical tables are provided.
Calculator: Any calculator is permitted.
Submission: Your submission must occur within 2 hours and 40 minutes of the official commencement of this assessment task (Australian Eastern Standard Time). Upload photographs of your answer sheets.
Assessment: This final exam contributes 60% to the total mark of the unit.
Question 1 [15 marks]
(a) State one key property of a valid generator matrix for a Markov jump process. [3 marks]
(b) An insurance company has two old products which are no longer sold and for which there are only two policies still in force for each product. The transition rate at which an individual policy terminates is 0.2 and 0.3 per annum for each product respectively, and the policies are independent of each other.
(i) Give the state space for the number of these two products still in force at a future point of time. [4 marks]
(ii) Draw a transition graph for the process for the number of policies in force. [4 marks]
(iii) Write down the generator matrix. [4 marks]
Question 2 [10 marks]
An insurance company offers annual home insurance policies in partnership with a bank. The distribution deal involves taking part in the bank’s loyalty scheme called ‘1234’. Under ‘1234’, a customer gets a discount when buying or renewing the policy according to how many bank accounts they hold, as follows:
|
Number of Bank Accounts |
Discount |
|
1 |
0% |
|
2 |
1% |
|
3 |
15% |
|
4 or more |
25% |
An analysis of the data suggests that the transition matrix for the number of bank accounts held at annual intervals is as follows:
Question 2 (continued)
(a) A customer takes out a policy on 1st January 2018 at the 15% discount level. Calculate the probability that the customer remains at the 15% discount level on 1st January 2020. [2 marks]
(b) A customer takes out a policy on 1st January 2018 at the 15% discount level. Calculate the probability that the customer remains at the 15% discount level up to and including 1st January 2020. [2 marks]
(c) Does this admit a limiting distribution? If yes, derive the answer. [2 marks]
(d) Does this admit a stationary distribution? If yes, derive the answer. [2 marks]
(e) What is the long-term average discount? [2 marks]
Question 3 [5 marks]
For the following graphs of stochastic processes:
• classify them into different categories (hint: time / state-space);
• discuss their long-term behaviour.
Question 4 [8 marks]
A mortality investigation is conducted over a period of two and a half years. The population data below (t measured in years) are collected for lives aged 41, 42, and 43 next birthday. The observed number of deaths is 27 at age 41 nearest birthday and is 31 at age 42 nearest birthday. Use Trapezium approximation to estimate q41. Give your answer to 6 decimal places. Note that 
|
Age |
t = 0 |
t = 1 |
t = 2 |
t = 2.5 |
|
41 |
3,400 |
3,500 |
3,300 |
3,400 |
|
42 |
3,100 |
3,200 |
3,200 |
3,100 |
|
43 |
3,000 |
2,900 |
3,000 |
2,900 |
Question 5 [10 marks]
Apply the chi square test, cumulative deviations test, and signs test to assess the adherence to data of the following information. A mathematical formula with two parameters is used in the graduation process. Give specific and overall comments on the adherence to data of the graduated rates. You may also supplement your analysis with a suitable mortality graph.
|
Age |
Initial Exposed to Risk |
Observed Number of Deaths |
Graduated Mortality Rate |
|
51 |
5,062 |
63 |
0.011 |
|
52 |
5,444 |
83 |
0.012 |
|
53 |
4,706 |
62 |
0.013 |
|
54 |
4,560 |
109 |
0.014 |
|
55 |
3,968 |
115 |
0.017 |
|
56 |
4,411 |
128 |
0.020 |
Question 6 [6 marks]
In a small medical study, the observed number of months that each patient with liver disease survives until death (no asterisk) or leaving the study for other reasons (with asterisk) is recorded below. Use Kaplan-Meier estimation to find the distribution function of the survival time of a patient. Then use Greenwood’s formula to compute the variance of the Kaplan-Meier estimator of the distribution function at 10 months, and deduce the corresponding 90% confidence interval.
6, 17, 6*, 20, 18*, 16*, 3, 10
Question 7 [6 marks]
You are an actuarial analyst working in a life insurance company selling life and term annuities.
In a small medical study, the observed number of months that each patient with liver disease survives until death (no asterisk) or leaving the study for other reasons (with asterisk) is recorded below. Use Kaplan-Meier estimation to find the distribution function of the survival time of a patient. Then use Greenwood’s formula to compute the variance of the Kaplan-Meier estimator of the distribution function at 10 months, and deduce the corresponding 90% confidence interval.