STAT3801_3909 AS2_revised

pdf

School

The University of Hong Kong *

*We aren’t endorsed by this school

Course

3801

Subject

Statistics

Date

Nov 24, 2024

Type

pdf

Pages

Uploaded by KidNeutron12719

14 / 15 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801/STAT3909 Advanced Life Contingencies Assignment 2 Due: March 27, 2015 (Friday) 1. Assume a constant force of decrement for each decrement over each year of age for a triple decrement table. You are given: (i) p 01 50 = p 03 50 (ii) p 02 50 = 2 p 01 50 (iii) μ 01 50+ t = ln 2 for 0 < t < 1. (a) Calculate 0 . 5 p 02 50 and 0 . 5 p ′ 02 50 . (b) Calculate 0 . 5 p 02 50 . 5 and 0 . 5 p ′ 02 50 . 5 . 2. Suppose that p 01 x = 0 . 15 and p 02 x = 0 . 21. Find 0 . 5 p 01 x +0 . 1 and 0 . 5 p ′ 01 x +0 . 1 under the following assumptions: (a) each of the decrements is uniformly distributed over each year of age in its associated single decrement table; (b) each of the decrements is uniformly distributed over each year of age in the double decrement table; (c) each of the forces of decrement is constant over each year of age. 3. For a double-decrement table where cause 1 is death and cause 2 is withdrawal, you are given: (i) Deaths are uniformly distributed over each year of age in the single-decrement table. (ii) Withdrawals occur only at the end of each year of age. (iii) l 00 x = 1000 (iv) p 02 x = 0 . 40 (v) d 01 x = 0 . 45 d 02 x Calculate p ′ 02 x . 4. A special whole life insurance on ( x ) pays 10 times salary if the cause of death is an accident and 500,000 for all other causes of death. You are given: (i) μ 00 x + t = 0 . 01, t ≥ 0. 1

(ii) μ (accident) x + t = 0 . 001, t ≥ 0. (iii) Benefits are payable at the moment of death. (iv) δ = 0 . 05. (v) Salary of ( x ) at time t is 50 , 000 e 0 . 04 t , t ≥ 0. Calculate the actuarial present value of the benefits at issue. 5. For a special fully continuous whole life insurance on ( x ), you are given: (i) Mortality follows a double decrement model. (ii) The death benefit for death due to cause 1 is 3. (iii) The death benefit for death due to cause 2 is 1. (iv) μ 01 x + t = 0 . 02, t ≥ 0. (v) μ 02 x + t = 0 . 04, t ≥ 0. (vi) The force of interest, δ , is a positive constant. Calculate the net premium for this insurance. 6. For a fully discrete 4-year term insurance on (40), who is subject to a double- decrement model: (i) The benefit is 2000 for decrement 1 and 1000 for decrement 2. (ii) The following is an extract from the double-decrement table for the last 3 years of this insurance: x l 00 x d 01 x d 02 x 41 800 8 16 42 – 8 16 43 – 8 16 (iii) v = 0 . 95. (iv) The net premium, based on the equivalence principle, is 34. Calculate the net premium reserve at the end of year 2. 7. For a multi-state model of a special 3-year term insurance on ( x ): (i) Insureds may be in one of three states at the beginning of each year: ac- tive (State 0), disabled (State 1), or dead (State 2). The annual transition probabilities are as follows for k = 0 , 1 , 2: i p i 0 x + k p i 1 x + k p i 2 x + k Active (0) 0.8 0.1 0.1 Disabled (1) 0.1 0.7 0.2 Dead (2) 0 0 1 2

(ii) A 100,000 benefit is payable at the end of the year of death whether the insured was active or disable. (iii) Premiums are paid at the beginning of each year when active. Insureds do not pay any annual premiums when they are disabled at the start of the year. (iv) v = 0 . 90. Calculate the level annual benefit premium for this insurance. 8. You are given the following sickness-death model. 0. Healthy 1. Disabled 2. Dead μ 01 x + t μ 10 x + t μ 02 x + t μ 12 x + t You are given the following assumptions for the model (these are the standard assumption): – Assumption 1: The state process is a Markov process – Assumption 2: The probability of ≥ 2 transitions in any period of length h years is o ( h ). – Assumption 3: For all states i and j , and for all ages x ≥ 0, t p ij x is a differen- tiable function of t , and for i negationslash = j , μ ij x = lim h → 0 + h p ij x h . (a) (i) Under Assumptions 1 and 2 above, the probability t + h p 00 x can be expressed as t + h p 00 x = t p 00 x h p 00 x + t + t p 01 x h p 10 x + t + o ( h ) . Interpret in words each of the three terms on the right side of this equation. (ii) Use this equation along with the assumptions above to derive the Kol- mogorov forward differential equation for d dt t p 00 x . (b) Write down the Kolmogorov forward differential equation for t p 0 j x , j = 1 , 2 for this model. 3

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Related Documents

35.docx

APS Ch 11 Quiz 2019 Answer Key.pdf

Reporte teleclase Finanzas I.docx

Screenshot_56.jpg

Self-Quiz Unit 3_ Attempt review _ Homejh.pdf

GUIA EJERCICIOS III UNIDAD(A.C.I) MEDIDAS DISPERSION.pdf

Interest Rate Determination and Yield Curves-98b765.xlsx

actividad4_uII.docx

vertopal.com_RNN_C&M&E.pdf

34.pdf

jasp.docx

5.png

Related Questions

Recommended textbooks for you

MATLAB: An Introduction with Applications
Statistics
ISBN:9781119256830
Author:Amos Gilat
Publisher:John Wiley & Sons Inc
Probability and Statistics for Engineering and th...
Statistics
ISBN:9781305251809
Author:Jay L. Devore
Publisher:Cengage Learning
Statistics for The Behavioral Sciences (MindTap C...
Statistics
ISBN:9781305504912
Author:Frederick J Gravetter, Larry B. Wallnau
Publisher:Cengage Learning
Elementary Statistics: Picturing the World (7th E...
Statistics
ISBN:9780134683416
Author:Ron Larson, Betsy Farber
Publisher:PEARSON
The Basic Practice of Statistics
Statistics
ISBN:9781319042578
Author:David S. Moore, William I. Notz, Michael A. Fligner
Publisher:W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:9781319013387
Author:David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:W. H. Freeman