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ETC3430 Financial Mathematics under Uncertainty
创建日期：2024-08-20 17:16:56
标签： ETC3430
Financial Mathematics under Uncertainty

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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.

Your manager has heard about the Lee-Carter (LC) model and the Cairns-Blake-Dowd (CBD) model. She knows that you took some excellent actuarial courses back in Monash and now asks you to explore the use of these state-of-the-art mortality projection methods in pricing and reserving practices. Write a memo to your manager - explain the key rationales of the LC and CBD models, compare their major strengths and weaknesses, and suggest how they can be applied in the valuation of the products offered by your company.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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