vilnius university faculty of mathematics and...

Projektas „Ekonometrijos bei finansų ir draudimo matematikos studijų programų atnaujinimas pritaikant tarptautinės rinkos poreikiams (EFDRA)“ Nr. VP1-‐2.2-‐ŠMM-‐07-‐K-‐02-‐008/ParS-‐13700-‐624

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VILNIUS UNIVERSITY FACULTY OF MATHEMATICS AND INFORMATICS

FINANCIAL AND INSURANCE MATHEMATICS

Master program Course descriptions

Vilnius, 2012


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CONTENT

1. Description of study program Financial and insurance mathematics ............................ 3 2. Selected chapters of analysis ......................................................................................... 8 3. Probability theory and mathematical statistics ............................................................... 11 4. Non-life insurance ......................................................................................................... 15 5. Time series analysis ....................................................................................................... 18 6. Stochastic analysis ......................................................................................................... 21 7. Life-insurance. Health insurance ................................................................................... 23 8. Financial mathematics ................................................................................................... 27 9. Risk theory ..................................................................................................................... 30 10. Dynamic aspects of survival theory ............................................................................. 34 11. Financial Derivatives ................................................................................................... 37 12. Stochastic models of financial mathematics ................................................................ 40

13. Risk management ......................................................................................................... 42 14. Pension Funds .............................................................................................................. 48


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Description of study program

Study program title National code Financial and insurance mathematics

Oficial name of awarding institution Language of instruction

Vilnius University, Faculty of Mathematics and Informatics English

Kind of study Cycle Level of qualification (according to LKS)

University studies second seventh

Mode of study and length of the program in years

Total ECTS credits

Total student‘s work hours

Contact hours Individual work hours

Full-time (1.5) 90 2400 712 1688

Study area Field of program (major) Field of program (minor) Physical sciences Mathematics

Degree awarded

Master of mathematics

Program director Contact information Prof., Habil. Dr. Vygantas Paulauskas [email protected]

Accreditating organisation Period of reference

Studijų kokybės vertinimo centras

Purpose of degree program

Training of qualified specialists in the area of financial and insurance mathematics, that are able of applying skills and mathematical knowledge both in academic and profesional fields as well as competing in the international labor market.

Profile of the program

Study content: subject areas Orientation of program Distinctive features • General mathematics.

(Probability theory and mathematical statistics, Selected chapters of analysis)

• Stochastic analysis (Stochastic analysis, Stochastic models of financial mathematics)

• Insurance mathematics (Non-life insurance, Life insurance. Health insurance, Risk theory, Pension

Master‘s level program geared towards academic as well as practical work

• The focus is on theoretical results, including current research, that allow understanding processes observed in practice. Students are taught to explain the possibilities and restrictions of applying theoretical models in practice

• Nurtured competences are in ageement with the “Groupe Consultatif” (EU group of


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funds, Dynamic aspects of survival theory)

• Financial mathematics (Financial mathematics, Time series analysis, Financial derivatives, Risk management)

actuaries) recommendations • Subjects are taught in English • Accquired qualification agrees

with the membership requirements of Lithuanian society of actuaries

• The competencies in Financial and Insurance Mathematics Master program match the requirements of international labour market.

Entrance requirements Recognition of earlier qualification

Bachelor degree in mathematics, statistics, or equivalent qualification.

Formally or informally aquired competences are recognised as long as they agree with the program.

Access to further studies

Graduates of the Master‘s program in financial and insurance mathematics can pursue further studies at the doctoral level in the areas of mathematics and/or statistics either in Lithuania or abroad.

Employability Program graduates can work as actuaries, financial analysts, risk assessors, consultants both for Lithuanian and foreign institutions supervising financial and insurance markets (e.g., in insurance companies, banks, pension and investment funds, consulting firms, government agencies, etc.)

Learning methods Assessment methods

Traditional lectures, recitation classes, seminars are complemented by case studies, problem-centered analysis, modeling, discussions, presentations, individual or group projects, portfolio creation.

Oral and written testing, presentation, term papers, portfolio, master‘s thesis

Generic competences.

Program graduate should be able to Learning objectives.

Program graduate will 1. Think abstractly and critically 1.1 Think abstractly and analytically

1.2 Be able to use mathematical language 2. Work in a team as well as individually

2.1 Show independence as a worker (He/She will be

able to set achievable goals and manage time.)

2.2 Be able to work in a team both as a leader and as a specialist

3. Responsibly complete assigned tasks and honor commitments

3.1 Be able to complete assigned tasks on time and as required

3.2 Be able to explain further action needed for further problem analysis

4. Carry on scientific research

4.1 Be able to find needed scientific literature and study it

4.2 Be able to select and apply appropriate methods to analyze a problem or situation


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Subject specific competences. Program graduate should be able to

Learning objectives. Program graduate will

Ugdyti bendravimo ir diskutavimo anglų kalba gebėjimus.

5. Apply mathematical knowledge and skills to solve actuarial problems

5.1 Be able to apply the methods of insurance mathematics

5.2 Be able to use acquired knowledge and skils in practical situations and know possible limitations

5.3 Be able to select appropriate solution method based on resources at hand

6. Apply mathematical knowledge and skills to investigate financial instruments and markets

6.1 Be able to model financial instruments and financial markets

6.2 Be able to use stochastic analysis theory to analyze financial markets

6.3 Be able to analyze the consequences of decisions taken

7. Disseminate aquired knowledge and work results both to specialists and nonspecialists

7.1 Be able to select appropriate presentation methods, form and content to convey information

7.2 Be able to present work results 8. Analyze models of financial and insurance

mathematics

8.1 Be able to modify the models of financial and insurance mathematics based on changing environment

8.2 Be able to estimate model parameters and properties

8.3 Be able to interpret obtained results and formulate conclusions


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PLAN OF PROGRAM (full time studies)

(STUDY SUBJECT CONNECTIONS WITH COMPETENCES AND OBJECTIVES)

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Study program objectives Generic competences Subject specific competences

1. 2. 3. 4. 5. 6. 7. 8. Main study objectives

1.1 1.2 2.1 2.2 3.1 3.2 4.1 4.2 5.1 5.2 5.3 6.1 6.2 6.3 7.1 7.2 8.1 8.2 8.3

1st YEAR 1600/60

1st SEMESTER 800/30

Compulsory subjects 800/30 324 476

Selected chapters of analysis

160/6 66 94 x x x

Probability theory and mathematical statistics

190/7 80 110 x x x x

Non-life insurance 138/5 48 90 x x x x x x x x

Time series analysis 158/6 64 94 x x x x x

Stochastic analysis 154/6 66 88 x x x

2nd SEMESTER 800/30


Life-insurance. Health insurance

226/9 96 130 x x x x x x x x x

Financial mathematics 174/6 66 108 x x x x

Risk theory 174/6 64 110 x x x x x x x x


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Study program objectives Generic competences Subject specific competences

1. 2. 3. 4. 5. 6. 7. 8. Main study objectives

1.1 1.2 2.1 2.2 3.1 3.2 4.1 4.2 5.1 5.2 5.3 6.1 6.2 6.3 7.1 7.2 8.1 8.2 8.3

Elective subjects 226/9

Dynamic aspects of survival theory

100/4 50 50 x x x x x

Financial Derivatives 126/5 50 76 x x x

Stochastic models of financial mathematics

126/5 50 76 x x x x x

2nd YEAR 800/30

3rd SEMESTRER 800/30


Master‘s thesis 674/25 16 658 x x x x x x x

Elective subjects 126/5

Risk management 126/5 50 76 x x x x

Pension funds 126/5 48 78 x x x x x x


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Course description

Course title Course code Selected chapters of analysis

Lecturer Department where the course is delivered

Prof. Vygantas Paulauskas Department of Mathematical Analysis Faculty of Mathematics and Informatics Naugarduko St. 24, LT-03225 Vilnius, Lithuania

Cycle Level of course Type of course

Second Advanced Compulsory

Mode of delivery Semester or period when the course is delivered

Language of instruction

Face-to-face 1st semester (Fall) English

Prerequisites and corequisites

Prerequisites: Calculus and basic knowledge of functional analysis and differential equations

Corequisites (if any):

Number of ECTS credits Student‘s workload Contact hours Individual work hours

6 160 66 94

Course objectives: programme competences to be developed In the course new (comparing with bachelor course) abstract spaces will be introduced, deeper insight will be given to linear functionals and operators, and methods of investigation of non-linear functios will be introduced. Students will be acquainted with solution of integral equations and differential heat equation.

Learning objectives. At the end of the course a student should: Learning methods Assesment methods

- Know main facts about topological spaces; - Know differentiation in normed spaces;

Traditional lectures on functional analysis. Practical training: solving problems that help to understand theory. Individual work: solving complimentary problems and studying the literature

Testing, written exam

- Be able to solve simple integral equations; - Understand the heat equation; - Be able to study literature on functional analysis and continue progress in this area.


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Course content: breakdown of the course

Contact hours Individual work hours and assignments

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Linear, metric and normed spaces 6 3 9 10

Homework: solving problems for each topic

Topological spaces 5 2 7 12 Linear functionals and operators 8 2 10 14 Generalized functions 5 2 7 10 Differentiation in normed spaces 6 2 8 12 Fredholm-Riesz-Schauder teorija 8 3 11 14

Newton method forsolving non-linear equations

4 1 5 8

Heat equation 4 1 5 8

Test and exam 4 6 Preparation for test and exam

Total 46 16 66 94 Assesment strategy Weight Time of

assesment Criteria

Test (written)

25 8-9th week Test consists of 2 theoretical questions and 2 problems from first two topics. Each question and problem is evaluated by points, the total sum of points is equal from 0 to 25.

Exam (written)

75 January Exam consists of 4 theory questions and 6 exercises (of diverse difficulty). Two questions require complete proof of some theorems or propositions. Exam is evaluated from 0 to 75 points. The points obtained from the test and exam are added, and the maximal possible sum is 100 points. The final mark is given according to the following principle: 10 – not less than 90 points 9 – not less than 82 points 8 – not less than 75 points 7 – not less than 65 points 6 – not less than 55 points 5 – not less than 45 points Students who collected fewer than 45 points get unsatisfactory mark (1 -4)

Author Publi

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Title Volume and/or number of publication

Publication place and publisher

Required reading Vygantas Paulauskas 2012 Selected chapters from

mathematical analysis (lecture notes)

In preparation

A. Račkauskas, A. Skūpas, A. Zabulionis

1989 Funkcinės analizės pratybų užduotys

I Vilnius University publishing house

A. Račkauskas, A. Skūpas,

1992 Funkcinės analizės pratybų užduotys

II Vilnius University publishing house


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A. Zabulionis Additional reading V. Paulauskas ir A. Račkauskas

2007 Funkcinė analizė I Vilnius, publishing house „Vaistų žinios“

V. Paulauskas ir A. Račkauskas

2007 Funkcinė analizė II Vilnius, publishing house „Vaistų žinios“


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Course description

Course title Course code Probability theory and mathematical statistics


Prof. Jonas Šiaulys Department of Mathematical Analysis Faculty of Mathematics and Informatics Naugarduko St. 24, LT-03225 Vilnius, Lithuania



Mode of delivery Semester or period when the course

is delivered Language of instruction

Face-to-face 1st semester (Fall) English.

Prerequisites and corequisites Prerequisites: Basic knowledge of mathematical analysis and probability theory.



7 190 80 110

Course objectives: program competences to be developed

Acquaint with the basic concepts and problems of probability theory and mathematical statistics. Get acquainted with the basic methods of probability and mathematical statistics usable for theoretical and practical problems.

Learning objectives. At the end of the course a student should: Learning methods Assessment methods

- Know the main objects of probability theory under consideration;

- Be able to employ the probability concepts such as of a random variable, random element, expectation, conditional expectation, characteristic function;

- Be able to employ the main concepts of mathematical statistics such as of a sample, parameter estimation, confidence intervals, linear regression;

- Know basic properties of the probability theory and mathematical statistics objects.

Problematic lecture, case analysis Written exam

- Be able to reasonably formulate the main statements Discussion lecture, Written exam


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of probability theory and mathematical statistics; - Be able to understand the proofs of various assertions from probability theory and mathematical statistics.

concept maps, demonstration, case by case analysis.

- Be able to prove the particular statements of the probability theory;

- Be able to choose suitable methods to solve various problems;

- Be able to choose appropriate methods for particular statistical tasks.

Debate, demonstration, preparation of readiness Presentation



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Elementary probability theory (repetition). 1 2 1 4 6 Study the first chapter of textbook [1], solve homework problems

Kolmogorov’s axioms, σ-algebra constructions in complex spaces.

1 2 3 5 Study §1 and §2 of the second chapter of textbook [1], solve homework problems

Introduction of probability measures on various measurable spaces.

1 2 3 5 Study §3 of the second chapter of textbook [1], solve homework problems

Random variables and random elements. 1 2 3 6 Study §4 and §5 of the second chapter of textbook [1], solve homework problems.

The expectation of a random variable and its properties (repetition)

1 3 1 5 6 Solve the assigned exercises, repeat the expectation properties according to §6 of the second chapter of textbook [1].

Conditional expectations and their properties 3 7 10 10 Study §7 of the second chapter of textbook [1], solve homework problems

Transformations of random variables and random elements

2 2 4 6 Study §8 of the second chapter of textbook [1], solve homework problems.

Preparation and presentation of readiness

2 2 8

The first midterm exam 2 2 8 Review the first part of the course.


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Various kinds of convergence of sequences of random variables

1 2 3 4 Study §10 of the second chapter of textbook [1], solve homework problems.

Characteristic functions, the method of characteristic functions.

2 6 8 8 Study §12 of the second chapter and §3 of the third chapter of textbook [1], solve homework problems

Discrete time martingales, the method of martingales.

2 8 10 10 Study §1, §3 and §4 of the 8th chapter of textbook [1], solve homework problems.

Population and sample, random sampling characteristics

1 2 3 2 Study §2.1-§2.3 of textbook [2], solve homework problems.

Order statistics of random sampling 1 2 3 2 Study §2.1 and §5.3 of textbook [2], solve homework problems.

Confidence intervals 1 2 3 2 Study §2.4 and §7.1-§7.5 of textbook [2], solve homework problems.

Testing hypotheses 1 2 3 2 Study §2.4 and §6.1-§6.4 of textbook [2], solve homework problems

Linear regression and correlation 1 2 3 2 Study §5.4 and §5.5 of textbook [2], solve homework problems.

Nonparametric tests 1 2 3 2 Study §6.5 of textbook [2], solve homework problems

Preparation and presentation of readiness

2 2 8

The second midterm exam

2 1 3 8 Review the second part of the course.

Total

21 48 8 3 80 110

Assessment strategy Weight Time of assessment

Criteria

General assessment strategy. A 10 point rating system is applied. It is possible to get 40 points on the first midterm exam. The same is possible on the second midterm exam. Additional 20 points can be collected for an individual or group self-study presentation. All collected points are added and divided by 10.

The first mid-term exam

40% During the semester

In this exam, students are tested on the material from the first half of the semester. Typically, the exam consists of one easy theoretical question (5 points), one hard theoretical question (10 points), and a long multi-stage exercise (25 points). To answer an easy theoretical question, a student should formulate some definition, theorem, or explain some concept. The answer to this question is assessed strictly: the student knows an appropriate definition or concept (5 points); the student does not know the appropriate definition or concept (0 points).


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A hard theoretical question is the proof of some assertion known from the syllabus. Given proof is assessed in a standard way: the student has not started proving the statement (0 points); the statement remains unproven, but the student made a few required correct steps of the proof (1-4 points); the assertion has been proved with large defects (5-6 points); the proof of the statement was presented with minor deficiencies (7-8 points); the proof of the statement was presented without any defects, all important places of the proof are fully explained (9-10 points). A long multi-stage exercise usually consists of five parts. In each of these parts, a student needs to find some characteristic of the same discrete time risk model. Each part of the exercise is assessed in points from 0 to 5 in a standard way: the student has not tried to find the desired model characteristic (0 points); the student in search of the required characteristic has made several essential errors (1-2 points); while finding the desired characteristic, the student made a few minor, e.g., arithmetic, errors (3-4 points); the student found correctly the desired characteristic of the model, all calculations and derivations are correct and accurate (5 points).


40% At the end of the semester

In this exam, students are tested on the material from the second half of the semester. The second midterm exam’s composition and assessment are similar to the composition and the assessment of the first midterm exam.

Presentation 20% During the semester

At the beginning of the semester, all students individually receive a task for readiness. The task consists of a theoretical problem, a complicated exercise, or of a practical problem. Topics are coordinated with students. Most of the topics require reading supplementary material. When the agreed time comes, each student presents a task done in electronic form.

Successfully completed tasks are presented during the seminars. One presentation takes approximately 15 minutes.

Author Publication year

Title Volume and/or publication number


Required reading

Shiryaev A.N. 1996 Probability Springer

Shao Jun 2003 Mathematical statistics Springer


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Course description

Course title Course code Non-life insurance




Second Intermediate Compulsory







5 138 48 90

Course objectives: programme competences to be developed Acquaint with the simplest theoretical models which are used to describe the non-life insurance business, develop the ability to apply these models for practical purposes, develop an abstract thinking, develop the ability for individual work.


- Know the basics of the discrete time risk model and the classical risk model and its main elements; - Know the main critical characteristics of these risk models; - Know the main formulas and procedures for calculation of these critical models characteristics.

Discussion lecture, case analysis Written exam

- Be able to apply the basic formulas and procedures for the estimate and finding all critical characteristic of discrete time and clasical risk models; - Know and identify the main probability distribution classes describing claims and losses.

Discussion lecture, analysis of model examples, presentation of individual and collective readiness.

Written exam

- Be able to choose the appropriate model describing non-life insurance business; - Be able to choose the appropriate model describing non-life insurance business; - Be able to apply the theoretical model to data held.

Demonstration, presentation of individual and collective readiness.

Presentation


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Discrete time risk model (model components, calculation of the ruin probability, finite time ruin probability, the Lundberg inequality, solution of recursive formulas)

6 10 16 25 Learn Chapter 1 of syllabus „Discrete time risk model“, consider the particular cases of the discrete time risk models described in the last subsection of the first chapter, consider the particular discrete time risk models provided in the homework exercises, prepare a presentation topic according to the designated subject

Presentation of readiness 2 2 The first midterm exam 2 2 10 Review the first Chapter of

syllabus, analyze a few standard cases of the discrete time risk model.

The classical risk model (model components, Poisson process, the compound Poisson process model, the ruin probability, net profit condition, the equilibrium coefficient, Lundberg‘s inequality, defective renewal equation, solution of the renewal equation, subeksponential distributions, asymptotic formulas for the ruin probability )

10 14 24 45 Learn Chapter 2 of syllabus „Clasical risk model“, consider the particular cases of the classical risk models described in the last subsection of Chapter 2, consider the particular classical risk models provided in the homework exercises, prepare a presentation topic.

Presentation of readiness 2 2 The second midterm exam 2 2 10 Repeat the second Chapter

of syllabus, analyze a few standard cases of the classical risk model.

Total 16 8 24 48 90 Assessment strategy Weight Time of

assessment Criteria

General assessment strategy. A 10 point rating system is applied. It is possible to get 40 points for the first midterm exam. The same holds for the second midterm exam. Additional 20 points can be collected for an individual or group self-study presentation. All collected points are added and divided by 10. The first midterm exam


In this exam, students are tested on the material from the first half of the semester. Typically, the exam consists of one easy theoretical question (5 points), one hard theoretical question (10 points), and a long multi-stage exercise in which a particular discrete time risk model must be considered (25 points). To answer an easy theoretical question, a student should formulate a definition, theorem, or explain a certain concept. The answer to this question is assessed strictly: the student knows the appropriate definition or concept (5 points); the student does not know the appropriate definition


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or concept (0 points). A hard theoretical question involves the proof of some assertion known from the syllabus. Given proof is assessed in a standard way: the student has not started proving the statement (0 points); the statement remains unproven, but the student has made a few correct steps of the proof (1-4 points); the assertion has been proved with large defects (5-6 points); the proof of the statement was presented with minor deficiencies (7-8 points); the proof of the statement was presented without any defects, all important places of the proof are fully explained (9-10 points). A long multi-stage exercise usually consists of five parts. In each of these parts, a student needs to find some characteristic of the same discrete time risk model. Each part of the exercise is assessed in points from 0 to 5 in a standard way: the student has not tried to find the desired model characteristic (0 points); while searching for the required characteristic the student has made several essential errors (1-2 points); while finding the desired characteristic, the student has made a few minor, e.g., arithmetic, errors (3-4 points); the student correctly found the desired characteristic of the model, all calculations and derivations are correct and accurate (5 points).



In this exam, students are tested on the material from the second half of the semester. The second midterm exam‘s composition and assessment are similar to the composition and assessment of the first midterm exam.


At the beginning of the semester, all students in groups or individually receive a task for readiness. When the agreed time comes, students present the work done. One presentation takes approximately 15 minutes. The topics are coordinated with the students. Most of the topics relate to the practical application of theoretical risk models.

Author Publi

cation year



Required reading J. Šiaulys 2012 Non-life insurance (syllabus

of lectures)

Additional reading D.C.M. Dickson 2005 Insurance risk and ruin Cambridge university

press T. Mikosch 2006 Non-life insurance

mathematics Springer


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Course description

Course title Course code Time series analysis

Lecturer Department where the course is delivered Lect. Danas Zuokas

Department of Econometrical Analysis Faculty of Mathematics and Informatics Naugarduko St. 24, LT-03225 Vilnius, Lithuania






Prerequisites and corequisites Prerequisites: Mathematical statistics, Probability theory



6 158 64 94

Course objectives: programme competences to be developed The course gives the basics of time series methods and models, their applications, modeling, forecasting of a real world (financial) data.


- Know and understand basic objects and their properties of the time series theory. - Know main time series models.

Lectures covering time series theory. Seminars for time series problem solving, analysis of specific questions or used case analysis. Individual work for additional problem solving and confirmation of theoretical knowledge.

Tests, mid term and final exam.

- Be able to identify, state and solve applied problems in economics, finance and other fields using time series methods. - Be able to select appropriate time series model.

Lectures covering applications of time series, employing used case analysis. Practical training for encouraging students state problems and find strategies for solutions.

Tests, practical tasks mid term and final exam.

- Be able to use time series literature and deepen theoretical knowledge.

Individual work for recommended reading.

Mid term and final exam.


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1. Estimation and elimination of time series trend and seasonal component.

4 2 2 8 6 Individual study of recommended readings and solving appointed exercises

2. Stochastic processes. Stationary time series. Autocovariance function.


3. Autoregressive moving average time series models (ARMA). Nonstationary ARIMA, SARIMA models.


4. Stylized facts of financial time series.


5. Models of conditional heteroskedasticity. ARCH and GARCH models. Volatility.


6. Parameter estimation of ARMA and GARCH models.


7. Tests 2 8 Prepare for the tests 8. Solution to individually assigned tasks presentation

8 8 Present solutions to the individually assigned tasks

9. Mid term exam 2 2 4 25 Recall theory and problem solving 10. Final exam 2 2 25 Recall theory and problem solving

Total

26 4 14 20 64 94

Assessment strategy Weight Time of

assessment Criteria

2 tests 1 hour written tests. First includes problems from topics 1-3, second – topics 4-6. Solutions are given points.

25% During seminars, when corresponding theoretical and practical part is finished

Mark 10 – student got no less than 90 % points






Mark 1-4 – student got less than 40 % points Solution to individually assigned tasks presentation Every student is assigned 2 practical tasks. First includes problems from topics 1-3, second – topics 4-6. Solutions are given points.

25% During practical training, when corresponding theoretical and practical part is finished







Mark 1-4 – student got less than 40 % points


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Mid term exam 2 hour open-book written exam, consisting of theoretical problems and exercises from topics 1-3. Solutions are given points.

25% During lectures, when corresponding theoretical and practical part is finished

Mark 10 – student has perfectly mastered the material, is able to analyze and generalize it. Understands and suitably uses concepts, knows main results of time series. Has gathered not less than 90% of points.

Mark 8-9 – student has mastered the material very well, is able to systemize and generalize it. Understands and suitably uses concepts, knows majority of the main results of time series. Has gathered not less than 80% (mark 9); 70% (mark 8) of points.

Mark 6-7 – student understands major time series concepts, knows main results of time series. Has gathered not less than 60% (mark 7); 50% (mark 6) of points.

Mark 5 – students understanding of time series concepts is superficial, she knows some results of time series. Has gathered not less than 40% of points.

Mark 1-4 – student does not know the material. The usage of terms and concepts is unsuitable. Has gathered less than 40% of points.

Final exam 2 hour open-book written exam, consisting of theoretical problems and exercises from topics 4-6. Solutions are given points.

25% During exam session Mark 10 – student has perfectly mastered the material, is

able to analyze and generalize it. Understands and suitably uses concepts, knows main results of time series. Has gathered not less than 90% of points.

Mark 8-9 – student has mastered the material very well, is able to systemize and generalize it. Understands and suitably uses concepts, knows majority of the main results of time series. Has gathered not less than 80% (mark 9); 70% (mark 8) of points.

Mark 6-7 – student understands major time series concepts, knows main results of time series. Has gathered not less than 60% (mark 7); 50% (mark 6) of points.

Mark 5 – students understanding of time series concepts is superficial, she knows some results of time series. Has gathered not less than 40% of points. Mark 1-4 – student does not know the material. The usage of terms and concepts is unsuitable. Has gathered less than 40% of points.

Author Publication

year Title Volume

and/or publication number


Required reading P.J. Brockwell, R. A. Davis

2002 Introduction to time series and forecasting.

2nd ed. New York : Springer.

R. S. Tsay 2002 Analysis of financial time series : financial econometrics

New York : Wiley.

Additional reading Hamilton J.D. 1994 Time Series Analysis Princeton, N.J. Princeton

University Press. Chan N.H. 2002 Time Series: Applications to

Finance. N.Y. Wiley.


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Course description

Course title Course code Stochastic analysis

Lecturer Department where the course is delivered Prof. Vigirdas Mackevičius

Department of Mathematical Analysis Faculty of Mathematics and Informatics Naugarduko St. 24, LT-03225 Vilnius, Lithuania





Face-to-face or distance learning 1st semester (Fall) English

Prerequisites and corequisites Prerequisites: Probability theory course (min 6 ECTS credits), calculus (one and several variables, min 16 ECTS credits)

Corequisites (if any): Minimal knowledge of elements of functional analysis is preferred


6 154 64 90

Course objectives: programme competences to be developed The course gives the basics of stochastic differential equations with driving Brownian motion, their applications, and modeling.

Learning objectives Learning methods Assesment methods

Basic understanding of the theory of stochastic integration and stochastic differential equations.

Traditional lectures of Stochastic analysis. Practical training: Solving problems that help understanding stochastic analysis. Individual work: Solving complemetary problems and studying the literature.

Written exam.

Basic skills in understanding, defining, analyzing, and simulating stochastic models.

A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis


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1. 1. Brownian motion (BM). Quadratic variation of a BM. Descrete- and continuous-time models and stochastic differential equations (SDEs). [1], Chs. 2-3.

4 2 6 8

Solving the problems from [1];

individual study of recommended readings [3],

Chapters 2-3; [4], Chapter 3.

2. Stochastic integral (SI) with respect to a BM. [1], Ch. 4. 6 2 8 8

3. Itô‘s formula for a BM. [1], Ch. 5. 5 2 7 8 4. SDEs. The existence and uniqueness of a solution. [1], Ch. 6. 4 4 8

5. Itô processes and SIs with respect to them. Itô‘s formula for an Itô‘ process. [1], Ch. 7. 5 2 7 8

6. Stratonovich integral and equations. [1], Ch. 8. 6 2 8 8 7. Linear SDEs. The expectation and variance of a solution of a linear SDE. Stochastic exponent. [1], Ch. 9.

6 2 8 8

8. Solutions of SDEs as Markov processes. Kolmogorov equations. Stacionary density. Application examples. [1], Chs. 10-11.

6 2 8 8

9. Simulation of SDEs. Strong and weak approximations of solutions of SDEs. Euler, Milstein, Itô–Teilor, Runge–Kutta approximations. [1], Chs. 13.

6 2 8 8

Preparation for exam and examination. 2 2 16

Total 48 16 66 88

Assesment strategy Weight Time of

assesment Criteria

Common evaluation scheme. The final mark (not exceeding 10) equals the sum of points (rounded to the nearest integer) obtained in written exam and practical training plus one. Written exam 80-100% 2.5 h The final examination includes 1-2 theoretical questions (6

points) and 4 problems (4x1 points). Problem solving 0-20% Practical

training Additional points for test results and activity at the lectures and practical training (up to 2 points).

Author Publ. year Title Volume Publisher Required reading 1. V. Mackevičius 2011 Introduction to Stochastic

Analysis: Integrals and Differential Equations

London, ISTE/Wiley

Recommended reading 2. T. Mikosch 1998 An Elementary Introduction to

Stochastic Calculus with a View Toward Finance

World Scientific, Singapore

3. D. Lamberton, B.Lapeyer

2000 Introduction to Stochastic Analysis applied to Finance

Chapman & Hall, London.


23

Course description

Course title Course code Life and Health insurance

Lecturers Department where the course is delivered Assoc. prof. Gintaras Bakštys, lect. Aldona Skučaitė Department of Mathematical Analysis

Faculty of Mathematics and Informatics Naugarduko St. 24, LT-03225 Vilnius, Lithuania





Face-to-face 2nd semester (Spring) English

Prerequisites and corequisites Prerequisites: Probability theory (First level); Actuarial mathematics (First level)



9 226 96 130

Course objectives: programme competences to be developed The aim of this course – to acquaint students with behaviour of life / health insurance markets and actuarial models used in life and health insurance. General competences developed: a) to be able to work independently and as team member; b) time management and accurate performance of tasks; c) ability to describe what further steps are needed for deeper analysis of problem. Professional competences developed: a) to be able to apply mathematical methods for solution of actuarial problems; b) to be able to present results for specialists and wide audience.

Learning objectives

At the end of the course a student should: Learning methods Assessment methods

- Understand and be able to describe possible failures of health insurance markets; be able to explain what methods may be used to deal with possible market failures as well as advantages and disadvantages of these methods

Problem based teaching Case study Reading of articles Modelling

Summative: Individual or group assignment Exam

- Be able to build actuarial model of life / health insurance: to select appropriate theoretical model and adapt it according to concrete situation; to choose suitable calculation method; to verify results

Traditional and / or interactive lectures Case study Reading of articles Modelling

- Be able to present and explain results; to describe main parameters and explain their impact for final result; describe what further steps are needed for deeper analysis of problem

Case study Individual or group assignment Reading of articles


24

- Be able to manage his / her time and accurately perform assigned tasks (when working individually and / or as team member)

Independent work (self assessment) Individual or group assignment



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Life insurance 1. Types of life insurance contracts. Setting of premium rates, reserves valuation. Profit recognition.

4 2 6 12 11 To read and realize assigned material. To be prepared for discussions

2. Possibilities of amortisation of acquisition expenses. Options of life insurance contract.

4 3 7 8 To read and realize assigned material. To be prepared for discussions

3. Life office risks management. Reinsurance. Legal supervision: solvency assessment. Solvency 2 regime.

10 6 16 22 To read and realize assigned material. To be prepared for discussions

4. Office model. Embedded value of life office. Valuation of options and guarantees.

6 2 3 11 12 To read and realize assigned material. To be prepared for discussions

5. Final test 2 2 12 To prepare for the test: refresh theory knowledge and practical skills

Total (Life insurance): 24 2 4 18 48 65 Health insurance

1. Differences between Health insurance market and „ideal“ market. Preferences and choice of individuals in insurance markets: utility theory; prospect theory.

8 2 2 4 16 18 To read assigned material (Utility theory; Papers by K. Arrow and J. Stiglitz, M. Rotschild). To prepare for discussions

2. Systems for financing of health care: public – mandatory; private – voluntary and their mix. Health care systems in Europe (European Union) and USA.

4 2 6 12 To read assigned material (funding of health care systems). To prepare for discussions

3. Actuarial models for health insurance: medical expenses insurance; critical illness insurance; disability insurance; long term care insurance

10 2 12 24 23 To read assigned material (actuarial models for health insurance). To prepare for discussions

4. Exam 2 2 12 To prepare for exam. To revise theory and its applications

Total (Health insurance):

22 6 4 16 48 65

Total: 46 8 8 34 96 130


25

Assessment strategy Weight Time of

assessment Criteria

Individual or group assignments (3-6 during semester)

40 During semester

Marks – 0; 0,25; 0,5; 0,75; 1 are given for each assignment. Summative assessment – marks for all assignments are added, the sum is then divided by number of assignments; obtained result is multiplied by 40. Quality of fulfilment of assignment, interpretation of results and answers to questions is assessed. Marks are given according to the following scheme: 1: assignment was carried out without mistakes; interpretation of results was correct; all questions answered exhaustively and correctly 0,75: some non essential mistakes when performing assignment and / or when interpreting the results; or no less than 75% of questions answered correctly 0,5: mistakes when performing assignment and / or when interpreting the results; less than 75% and no less than 50% questions answered correctly 0,25: serious (essential) mistakes when performing assignment and / or when interpreting the results; less than 50% and no less than 25% questions answered correctly 0: assignment was not carried out or performed but with many very serious essential mistakes; less than 25% of questions answered correctly

Exam (2 parts): Midterm and Final

60 (30+30)

Midterm – 7-8 week Final – June

During each part of exam student must fulfil 1-2 tasks. Each task may contain sub-tasks. Accuracy and exhaustiveness of fulfilment as well as application of acquired skills to solution of actuarial problems in life / health insurance are assessed. Marks 0-6 are given for every part of exam: 6: All tasks are fulfilled exhaustively and precisely, solutions are correct (correct model and assumptions, no logical mistakes in calculations), accurate written presentation. Proper and logical conclusions are drawn. 5: 1st case) No less than 75% of tasks fulfilled without essential imperfections. 2nd case) All tasks are fulfilled but with small deficiencies, e.g. 1-2 logical mistakes during calculations, conclusions are not logically grounded. 4: No less than 50% but less than 75% of tasks fulfilled without essential imperfections. Other tasks not carried out or carried out with serious mistakes (improper model or assumptions; logically unacceptable calculations, unreasonable method of solutions; no conclusions or conclusions contradict to result of calculation, etc.). 3: No less than 25% but less than 50% of tasks fulfilled without essential imperfections. Other tasks not carried out or serious mistakes made. 1-2: Some operations are correct (e.g. proper model selected, some calculations are correct, etc.), but no solution given, no conclusions, logical contradictions between parts of solutions, etc. 0: All solutions are logically unacceptable.. Remarks: Course coordinator or other lecturer may decide to give not


26

whole mark, e.g. 3,5. Summative assessment: Average of marks for both exam parts is multiplied by 10.

Overall assessment – sum of marks for 3 summative assessments

100 - 10: – no less than 90 points 9: – no less than 80 but less than 90 points 8: – no less than 70 but less than 80 points 7: – no less than 60 but less than 70 points 6: – no less than 50 but less than 60 points 5: – no less than 40 but less than 50 points 1-4: – less than 40 points

Author Publicat

ion year Title Volume and/or

publication number Publication place and publisher

Required reading G. Bakštys, A. Skučaitė

2012 „Life and Health insurance“. Lecture notes

Arrow, K. J. 1963 Uncertainty and the Welfare Economics of Medical Care

The American Economics Review, Vol. 53, Nr. 5

Bolnick, H. J. 2003 Designing a World Class Health Care System

North American Actuarial Journal, Vol. 7, No. 2

Booth P., Chadburn R., Cooper D., Haberman S., James D.

1998 Modern Actuarial Theory and Practice

Chapman \& Hall/CRC

Gerber H. 1979 Introduction to Mathematical Risk Theory

Richard D Irwin

Haberman, S.; Pitacco, E. 1999 Actuarial Models for Disability Insurance

- Chapman \& Hall/CRC

Mossialos, E; Thomson, S.

2004 Private health insurance and access to health care in the European Union

Euro Observer, Newsletter of the European Observatory on Health Systems and Policies. Vol. 6, no. 1.

Rothschild, M.; Stiglitz, J.

1976 Equilibrium in Competitive Insurance Markets: as Essay on the Economics of Imperfect Information

Quarterly Journal of Economics, Vol. 90, Nr. 4

Recommended reading Bowers N., Gerber H., Hickman J., Jones D., Nesbitt C.

1986 Actuarial Mathematics Society of Actuaries

Friedman, M.; Savage, L.J.

1948 The Utility Analysis of Choices Involving Risk

The Journal of Political Economy, Vol. 56, Nr. 4

Pauly, M.V. 1968 The Economics of Moral Hazard: Comment

The American Economics Review, Vol. 58, Nr. 3

Rotar, V. I. 2006 Actuarial Models: the Mathematics of Insurance

Chapman \& Hall / CRC

Sandstrom A. 2005 Solvency: Models, Assessment and Regulation

Chapman \& Hall / CRC


27

Course description

Course title Course code Financial mathematics

Lecturer Department where the course is delivered Assoc. prof. Martynas Manstavičius







Prerequisites and corequisites Prerequisites: basic linear algebra and measure-theoretic probability theory

Corequisites (if any) :


6 174 66 108

Course objectives: programme competences to be developed This course is designed to be an introduction to the main mathematical ideas needed to model financial markets and value contingent claims (put/call options, etc.) in discrete time setting.


- Have knowledge, understanding and formulation of the principles of risk-neutral valuation including some versions of the No-arbitrage theorem

Traditional lectures to explain the theory of discrete time financial mathematics models. Recitation classes to solve problems that help understand the concepts and methods presented.

Individual work: Solving complemetary problems and studying the literature; participating in seminars, discussions

Testing (open/closed book) - Have knowledge of replication and pricing of contingent claims in certain simple discrete time models - Have knowledge of the passage to the limit in CRR model leading to the Black-Scholes formula - Be able to demonstrate knowledge of the subject matter, terminology, methods and conventions covered in this course Testing, written exam - Be able to demonstrate the ability to solve problems involving understanding of the concepts


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1. Understanding financial markets and traded assets

3 1 4 4 Read through, e.g., [4, Ch. I-II]

2. Single period model of a financial market

6 2 8 8 Read through [1, Ch. I.1-3], solve end-of-section problems, study recommended literature

3. Valuation of contingent claims. Complete and incomplete markets

3 1 4 4 Read through [1, Ch. I.4-5], solve end-of-section problems, study recommended literature

4. Risk and return 3 1 4 4 Read through [1, Ch. I.6], solve end-of-section problems, study recommended literature

5. Test 1 (preparation and writing )

1 3 4 8 Review theory and problem solutions [1, Ch. I.1–6]

6. Multi-period model of a financial market

3 1 4 4 Read through [1, Ch. III.1-2], solve end-of-section problems, study recommended literature

7. Martingales and arbitrage-free market

6 2 8 8 Read through [1, Ch. III.3-4], solve end-of-section problems, study recommended literature

8. Binomial (CRR) model

6 2 8 8 Read through [1, Ch. III.5, IV.1-2], solve end-of-section problems, study recommended literature

9. Test 2 (preparation and writing) 1 3 4 8 Review theory and problem solutions [1, Ch. III.1-5, IV.1-2]

10. American options 6 2 8 8 Read through [1, Ch. IV.3-4], solve end-of-section problems, study recommended literature

11*. Consumption and investment problems

6 6 20 Read [1, Ch. II&V]; be ready for a discusion during the seminar, and/or prepare a presentation

12. Final exam (preparation and writing) 2 4 24 Review theory and problem solutions [1, Ch I, III-IV]

Total

36 4 6 18 66 108

* Free topic (time permitting); a presentation during a seminar could earn one point towards the final grade for a student

Assesment strategy Weight Time of assesment

Criteria

Tests 1&2 Each 3 hr test contains theoretical (closed-book) and problem solving (open book) parts. Points are awarded for each successfully answered question/problem. Test 1 contains material from topics I through V; Test 2 contains material

40% (20% each)

During recitation classes (approx. during 6th and 12th week)

Mark 10 – between 90% and 100% of available points on a test.

Mark 9 – between 80% and 89.99% of available points on a test.



Mark 6 – between 50% and 59.99% of available points on a


29

from topics VI through VIII.

test.


Mark 1-4 – less than 40% of available points on a test. Final exam The final 2 hr long written exam covers material from topics I through IV, VI through VIII, and X. It contains theoretical closed-book and practical open-book parts. Points are awarded for each successfully answered question/problem.

60% During exam session Mark 10 – A student shows excellent knowledge of the

course material, is able to analyze and generalize it, understands and correctly uses concepts, knows the main results of discrete time mathematical finance. He/she has collected between 90% and 100% of the available points.

Marks 8-9 – A student shows good/very good knowledge of the course material, is able to systematize and generalize it, understands used concepts, knows the majority of results of discrete time mathematical finance. 9 points are awarded for collecting between 80% and 89.99% of the available points; 8 points are awarded for collecting between 70% and 79.99% of the available points.

Marks 6-7 – A student understands the main concepts of the course and knows most of the main results of discrete time financial mathematics. 7 points are awarded for collecting between 60% and 69.99% of the available points; 6 points are awarded for collecting between 50% and 59.99% of the available points.

Mark 5 – A student shows skin-deep understanding of the concepts of discrete time financial mathematics. He/she has collected between 40% and 49.99% of available points

Marks 1-4 – A student does not know the studied material and inappropriately uses the terms and concepts of the course. Has collected less than 40% of the available points.

Presentation at the seminar*

10%* Approx. 15th

and 16th week An additional point towards the final grade can be collected for a well prepared presentation at the seminar. The presentation should cover part of the material of the topic No. XI.

The final grade is the sum of all collected grades multiplied by the corresponding weights and rounded to the nearest integer, e.g. , 8,5 is rounded upward to 9 and 8,49 is rounded downward to 8. Author Publication

year Title Volume



Required reading 1. S.R. Pliska 1997 Introduction to Mathematical

Finance: Discrete Time Models

Oxford, Blackwell Publishers Inc.

Additional reading 2. H. Föllmer, A. Schied

2002 Stochastic finance. An introduction in discrete time

Berlin-New York, Walter de Gruyter

3. J. Hull 2005 Options, futures and other derivative securities

6th ed. Pearson Prentice Hall

4. R.E. Bailey

2005 The Economics of Financial Markets

Cambridge University Press, http://library.northsouth.edu/Upload/Economics%20Finance.pdf (last checked 2012-03-01)


30

Course description

Course title Course code Risk theory





Mode of delivery Semester or period when the course

is delivered Language of instruction


Prerequisites and corequisites

Prerequisites: Basic knowledge of mathematical analysis and probability theory, initial knowledge on the risk models.


Number of ECTS credits

allocated Student‘s workload Contact hours Individual work hours

6 174 64 110

Course objectives: program competences to be developed

This course is a continuation of non-life insurance course. The goal of this course is to acquaint the students with the so-called risk renewal model. This model is often considered the last year in the scientific literature because of its broad application possibilities. Study focuses on the possibility of application of this model in the insurance practice. In order to develop a deep understanding of this theoretic model abstract thinking is developed as well as the ability to estimate suitable the model parameters, the ability to find and examine the known results individually and the ability to discuss with teachers and colleagues, using mathematical and insurance concepts.


- Know all components of the renewal risk model;

- Know the basic dynamical characteristics of such model;

- Know the main formulas and procedures for calculation of these characteristics.

Discussion lecture, case analysis Written exam


31

- Be able to apply the basic formulas and procedures for the estimate and finding the model characteristics;

- Know and identify the basic probability distributions classes describing the random claims and the inter-occurrence times;

Problematic lecture, demonstration, analysis of the model examples.

Written exam

- Be able to find individually the missing material to solving the problem;

- Be able to investigate complicated mathematical texts;

- Be able to choose the suitable version of renewal risk model for existing data;

- Be able to choose the optimal strategy to find different model characteristic.

Debates, demonstrations, preparation of individual and group readiness.

Presentation



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Components of the risk renewal model

1 1 2 2 Learn the introductory subsection of syllabus, solve the homework.

Counting renewal process, the law of large numbers for counting renewal process

1 4 5 6 Learn the subsection “The strong law of large numbers for renewal process” of syllabus, solve the homework.

Expectation of the counting renewal process, elementary renewal theorem

2 4 6 6 Learn the subsection

“Expectation of renewal counting process” of syllabus, solve the homework.

Renewal equations for quantities related to the renewal process. Residual life process, age process.


“Renewal equations” of syllabus, solve the homework.

Smith’s and Blackwell’s theorems for solutions of the renewal equation.

1 3 2 6 14 Solve the required exercises, study the recommended reading to prepare a presentation.

Higher order moments and the central limit theorem for renewal counting process.

1 3 1 5 12 Solve the required exercises, study the recommended reading to prepare a presentation.

Presentation of readiness

3 3


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The first interim exam

2 2 4 10 Review the first part of the course, analyze a few standard cases of the renewal process

The total claim amount process and its characteristics (mean, variance, the Laplace-Stieltjes transform, the central limit theorem)


“Total claim amount process” of syllabus, solve the homework.

Classical premium calculation principles 1 3 4 6 Learn the subsection

“Premium calculation principles” of syllabus, solve the homework.

Ruin probability and net profit condition for the risk renewal process

2 2 4 12 Study a supplementary reading, prepare a presentation.

Lundberg’s inequality for the risk renewal model

2 3 5 6 Learn the suitable subsection of syllabus, solve the homework.

Asymptotic formulas for the ruin probability in the risk renewal model

1 2 3 14 Study a supplementary reading, prepare a presentation.

Presentation of readiness

3 3

The second interim exam

2 2 4 10 Repeat the second part of the course, analyze a few standard cases of the risk renewal models

Total 15 28 10 11 64 110

Assessment strategy Weight Time of assessment

Criteria

General assessment strategy. A 10 point rating system is applied. It is possible to get 40 points for the first midterm exam. The same holds for the second midterm exam. Additional 20 points can be collected for an individual or group self-study presentation. All collected points are added and divided by 10.

The first midterm exam 40% During the semester

In this exam, students are tested on the material from the first half of the semester. Typically, the exam consists of one easy theoretical question (5 points), one hard theoretical question (10 points), and a long multi-stage exercise in which a particular case of the renewal process must be considered (25 points). To answer an easy theoretical question, a student should formulate a definition, theorem, or explain some concept. The answer to this question is assessed strictly: the student knows the appropriate definition or concept (5 points); the student does not know the appropriate definition or concept (0 points). The hard theoretical question involves the proof of some assertion known from the syllabus. Given proof is assessed in a standard way: the student has not started proving the statement (0 points); the statement remains unproven, but the student has made a few correct steps of the proof (1-4 points); the assertion has been proved with large defects (5-6 points); the proof of the statement was presented with minor deficiencies (7-8 points); the proof of the statement was presented without any defects, all important places of the proof are fully


33

explained (9-10 points). A long multi-stage exercise usually consists of five parts. In each of these parts, a student needs to find some characteristic of the same discrete time risk model. Each part of the exercise is assessed in points from 0 to 5 in a standard way: the student has not tried to find the desired model characteristic (0 points); while searching for the required the student has made several essential errors (1-2 points); while finding the desired characteristic, the student made a few minor, e.g., arithmetic, errors (3-4 points); the student correctly found the desired characteristic of the model, all calculations and derivations are correct and accurate (5 points).



In this exam, students are tested on the material from the second half of the semester. The second midterm exam’s composition and assessment are similar to the composition and the assessment of the first midterm exam.


At the beginning of the semester, all students in groups or individually receive a task for readiness. When the agreed time comes, students present the work done. One presentation takes 15-30 minutes. The topics are coordinated with the students. Most of the topics relate to the additional reading materials.

Author Publication year

Title Volume and/or number of publication


Required reading

J. Šiaulys 2012 Risk theory in insurance (syllabus of lectures)

Recommended reading

S.I. Resnick 1992 Adventures in stochastic processes

Boston, Birkhauser

F. Spitzer 1986 Principles of random walk Berlin, Springer

A. Gut 1988 Stopped random walks Berlin, Springer

P. Embrechts,

C. Klüppelberg,

T. Mikosch

1997 Modeling extremal events Springer


34

Course description

Course title Course code Dynamic aspects of survival theory

Lecturer Department where the course is delivered Prof. Jonas Šiaulys Department of Mathematical Analysis



Second Advanced Optional







4 100 50 50

Course objectives: programme competences to be developed Develop competences needed to mathematically model demographic processes; aquire the necessary mathematical aparatus


- Know the essential population characteristics and necessary concepts

Discussion lecture, map of concepts

Written midterm exam - Be able to describe the age structure of a population - Be able to characterize the density of a population

Discussion lecture, case study

- Be able to perform statistical analysis of a population density - Be able to estimate uknown model parameters

Case study Seminar presentation


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1. Evolution of a population. Linear, exponential and logistic evolution models. Trend of population evolution. Linear and nonlinear trends and their detection methods.

4 1 2 7 4 Study [2, Ch 1], solve homework problems, selectively read additional literature.

2. Age structure of a population. Survival function of a person, mortality rate and mortality curve. Life expectancy of a person and its characteristics. Examples of analytical survival functions. Population life table and its structure. Clasical rules for estimating survival function.

6 2 3 11 6 Study [1, Ch.2.], solve homework problems, selectively read additional literature

3. Population density. Lexis diagram. Birth density and population density. Mortality rate of a cohort. Interpretations of the product of the population density and death rate. Renewal equation for a population. Fertility rate of a woman. Stable and stationary population. Dependence of population stability on the fertility rate of a woman.

6 2 3 11 6 Study [2, Ch. 2-7] solve homework problems, selectively read additional literature

4. Statistical analysis of a population density. Estimators of population mortality rate and projections. Lee-Carter‘s method.

8 3 4 15 8 Study the paper [7]

5. The first midterm exam 2 3 13 Review theory and problem solutions. 6. The second midterm 2 3 13

Totals 24 8 16 50 50 Assessment strategy Weight Time of

assessment Criteria

General assessment strategy. A 10 point rating system is applied. It is possible to get 40 points for the first midterm exam. The same holds for the second midterm exam. Additional 20 points can be collected for an individual or group self-study presentation. All collected points are added and divided by 10. The first midterm exam


In this exam, students are tested on the material from the first half of the semester. Typically, the exam consists of one easy theoretical question (5 points), one hard theoretical question (10 points), and a long multi-stage exercise in which a particular discrete time risk model must be considered (25 points). To answer an easy theoretical question, a student should formulate a definition, theorem, or explain a certain concept. The answer to this question is assessed strictly: the student knows the appropriate definition or concept (5 points); the student does not know the appropriate definition or concept (0 points). A hard theoretical question involves the proof of some assertion known from the syllabus. Given proof is assessed in a standard way: the student has not started proving the statement (0 points); the statement remains unproven, but the student has made a few correct steps of the proof (1-4 points); the assertion has been proved with large defects (5-6 points); the proof of the statement was presented with minor deficiencies (7-8 points); the proof of the statement was presented without any defects, all important places of the proof are fully explained (9-10 points). A long multi-stage exercise usually consists of five parts. In each of these


36

parts, a student needs to find some characteristic of the same discrete time risk model. Each part of the exercise is assessed in points from 0 to 5 in a standard way: the student has not tried to find the desired model characteristic (0 points); while searching for the required characteristic the student has made several essential errors (1-2 points); while finding the desired characteristic, the student has made a few minor, e.g., arithmetic, errors (3-4 points); the student correctly found the desired characteristic of the model, all calculations and derivations are correct and accurate (5 points).



In this exam, students are tested on the material from the second half of the semester. The second midterm exam‘s composition and assessment are similar to the composition and assessment of the first midterm exam.


At the beginning of the semester, all students in groups or individually receive a task for readiness. When the agreed time comes, students present the work done. One presentation takes approximately 15 minutes. The topics are coordinated with the students. Most of the topics relate to the practical application of theoretical risk models.

Author Publi

cation year



Required reading 1. N.L. Bowers et al. 1980 Actuarial Mathematics Itasca 2. S. Preston, P. Heuveline, M. Guillot

2002 Demography: Measuring and Modeling Population Processes

Wiley-Blackwell

Additional reading 3. N. Keyfitz, D. Smyth

1977 Mathematical demography Springer-Verlag

4. J. Impagliazzo 1984 Deterministic aspects of mathematical demography

Springer-Verlag

5. S. Haberman, E. Pitacco

1999 Actuarial models for disability insurance

Chapman & Hall / CRC

6. M. Iannelli, M. Martcheva, F.A. Milner

2005 Gender-structured population modeling: mathematical methods, numerics, and simulations

Philadelphia: Society for Industrial and Applied Mathematics

7. L. Carter, R.D. Lee

1992 Modeling and Forecasting U.S. Mortality: Differentials in Life Expectancy by Sex

International Journal of Forecasting 8, no. 3 393–412

http://pagesperso.univ-brest.fr/~ailliot/doc_cours/M1EURIA/regression/leecarter.pdf (last checked: 2012-03-29)


37

Course description

Course title Course code Financial Derivatives

Lecturer Department where the course is delivered Lect. K. Liubinskas






Face-to-face 2nd semester (spring) English

Prerequisites and corequisites Prerequisites: Calculus, Financial Mathematics



5 126 50 76

Course objectives: programme competences to be developed Be able to analyse the effect of taken decisions; be able to interpret the obtained results and to formulate conclusions; be able to apply in practice the theoretical knowledge of simulating of financial derivatives.


- Understand the main financial derivatives (options, forward contracts, futures contracts) and their combinations. - Understand the main valuation methods of financial derivatives.

Traditional lectures provide theory on financial derivatives Practical trainings are designed to analyse the problem-oriented questions, focus on problem solving and on discussion of self-study problems Seft- study is designed for individual homework to firm up the theoretical knowledge of financial derivatives

Evaluation of self-study problem solutions, presentations, test, final exam

- Be able to model financial derivatives with suitible software, formulate and solve problems on financial derivatives based on theoretical

Traditional lectures provide theory on financial derivatives valuation and modelling

Evaluation of self-study problem solutions, presentations, test, final exam


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knowledge. - Be able to analyse the results on financial derivatives valuation. - Be able to interpret results and formulate conclusions on finanicial derivatives.

Practical training is designed to discuss the follow-up questions, focus on problem solving, analysis and results interpretation Seft- study is designed for individual homework to firm up the theoretical knowledge of financial derivatives

- Be able to use the theoretical literature on financial derivatives, deepen theoretical knowledge and financial derivatives valuation and modelling skills.

Seft- study is designed for reading extra material, to be discused during practical trainings

written exam



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1. Options 6 3 9 9 Solve assigned problems and study additional literature

2. Forward contracts and futures contracts 4 2 6 4

3. Valuing forward contracts and futures contracts 4 2 6 8

4. Hedging using futures contracts 2 1 3 4

5. Properties of options 4 2 6 8

6. Binomial trees 7 3 10 12

7. Processes for stock prices and derivatives prices 4 2 6 6

8. Test 1 1 2 10 Review problem solutions and prepare for the test

9. Final examination 2 15 Review the theory and problem solutions

Total 32 16 50 76


assesment Criteria

Test 2 hour long written test. Test contains problems of topics 1-3. Solutions give marks by points.

30% During practical training, soon after first three topics are covered

Mark 10 – student got at least 90 % of possible points






Mark 4-1 – student got less than 40 % of possible points

Written examination

70% During exams session Mark 10 – student soaks up taught material perfectly, is


39

2 hour long written exam. Exam contains theoretical assignments and practical problems. Task give marks by points.

able to analyse and to generalise it, understand and properly uses concepts, knows the essential results of financial derivatives. Student got at least 90% of possible points.

Mark 9-8 – student soaks up taught material very good/good, is able to systematise and to generalise it, understands concepts, knows most essential results of financial derivatives. Student got at least 80% (9 points); 70% (8 points) of possible points.

Mark 7-6 – student understands basic concepts of taught material and knows fundamental results of financial derivatives. Student got at least 60 % (7 points); 50% (6 points) of possible points.

Mark 5 – student understands concepts poorly, knows some results of financial derivatives only. Student got at least 40% (5 points) of possible points.

Mark 4-1 – student does not know teaching material, uses concepts in a wrong way. Student got less than 40% of possible points.

The overall mark will be calculated as the weighted average of the test mark and the exam mark and will be rounded according to the arithmetical rules. Author Publi

cation year



Required reading K.Liubinskas 2012 Lecture Notes Vilnius J.C. Hull 2005 Options Futures and other

Derivatives Pearson Education, Inc.,

Upper Saddle River, New Jersey

S.E. Shreve 2004 Stochastic Calculus for Finance I: The Binomial Asset Pricing Model

Springer

Additional reading P. Wilmot, S. Howison, J. Dewynne

1995 The Mathematics of Financial Derivatives

Cambridge University Press


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Course description

Course title Course code Stochastic models of financial mathematics

Lecturer Department where the course unit is delivered Prof. Vigirdas Mackevičius



Second Intermediate Optional



Face-to-face or distance learning 2nd semester (Spring) English

Prerequisites and corequisites Prerequisites: Probability theory course (min 6 ECTS credits), Stochastic analysis (6 ECTS credits)

Corequisites: Minimal knowledge of elements of functionsl analysis is preferred


5 126 50 76

Course objectives: programme competences to be developed The course gives the basics of continuous-time stochastic models of financial mathematics, their applications, and modeling.


- Have basic understanding of continuous-time models of financial mathematics and pricing of financial derivatives.

Traditional Lectures of Stochastic analysis. Seminars: Seminar talks on additional topics of financial mathematics. Individual work: studying the literature and preparation of seminar talks.

Written exam.

- Have basic skills in understanding, defining, analyzing, and simulating stochastic financial models.

- Have a critical awareness of current problems and research issues in the fields of stochastic analysis and financial mathematics.

- Have the ability to use appropriate mathematical tools and techniques in the context of a particular financial model.


41



Lect

ures

Con

sulta

tions

Sem

inar

s

Tot

al c

onta

ct h

ours

Indi

vidu

al w

ork

hour

s

Assignments

1. Common distribution of Brownian motion and its maximum process. [1], Ch. 1. 3 3 4

Individual study of recommended readings [4], 4-6 sk.

Preparation for seminar talks: [4], Ch. 7.; [3], Sec. 3.4; [5], Sec. 5.4, p. 224-234; [5], Sec. 7.4, p. 308-

320; [5], Sec. 7.5, p. 320-331; [5], Sec. 10.2.1, p. 406-420.

2. Change of measure. Girsanov theorem. [1], Ch. 5

3 3 4

3. European options. Self-financing strategies. [1], Ch. 6-7.

3 3 5

4. Black–Scholes model. Option pricing problem. Black–Scholes equation and formula. [1], Ch. 8-9.

6 6 5

5. Interpretation of Black–Scholes formula in terms of change of measure. Martingale probability. [1], Ch. 10.

3 3 4

6. Exotic options. Their pricing. [1], §12. 5 3 8 10 7. Clasdical interest rate models. Partial differential equation for classical models. [1], Ch. 13.

3 3 4

8. Vasicek model. CIR (Cox–Ingersoll–Ross) modeli. HJM (Heath–Jarrow–Morton) model. [1], Chs. 14-16.

6 3 7 10

9. Other interest rate models. 10 10 24 Preparation for exam and examination. 2 2 12

Total 32 2 16 50 76


assesment Criteria

Common evaluation scheme. The final mark (not exceeding 10) equals the sum of points (rounded to the nearest integer) obtained in written exam and practical training plus one. Written exam 80% 2 h The final examination includes 1 theoretical question (6 points)

and 1 problem (2 points). Seminar talks 20% Up to 2 points for a seminar talk. Author Publi

cation year



Required reading 1. V. Mackevičius 2012 Lecture notes on Stochastic

models of Financial Mathematics. www.mif.vu.lt\~vigirdas

Additional reading 2. T. Mikosch

1998

An Elementary Introduction to Stochastic Calculus with a View Toward Finance

World Scientific, Singapore

3. R.-A. Dana and M. Jeanblanc

2007 Financial Markets in Continuous Time

Springer

4. D. Lamberton and B. Lapeyere

1996

Introduction to Stochastic Calculus Applied to Finance

Chapman & Hall, London

5. S.E. Shreve 2003 Stochastic Calculus for Finance II. Springer


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Course description

Course title Course code Risk management

Lecturer Department where the course is delivered Assoc. prof. Martynas Manstavičius



Second Intermediate Optional



Face-to-face 3rd semester (Fall) English

Prerequisites and corequisites Prerequisites: measure-theoretic probability theory Corequisites (if any) : functional analysis would be a plus Number of ECTS credits Student‘s workload Contact hours Individual work hours

5 126 50 76

Course objectives: programme competences to be developed This course is designed to provide an axiomatic foundation to risk measures used in applications. Concepts, ideas and many theoretical results that are scattered throughout many scientific papers will be studied. Ability to search for and critically analyze relevant material will be fostered.


- Have knowledge, understanding and formulation of the axioms of coherent risk measures, their generalizations, and sets of acceptable risks

Lectures, recitation classes, seminars, discussions Testing (open/closed book) - Have knowledge of at least two characterizations

of coherent risk measures - Have knowledge of the usage examples of coherent risk measures - Be able to demonstrate knowledge of the subject matter, terminology, methods and conventions covered in this course Recitation classes, seminars,

consultations Testing, written exam - Be able to d emonstrate the ability to solve problems involving understanding of the concepts


43



Lect

ures

Con

sulta

tions

Sem

inar

s

Rec

itatio

n ho

urs

Tot

al c

onta

ct h

ours

Indi

vidu

al w

ork

hour

s

Assignments

1. Axioms for sets of acceptable risks and risk measures; their correspondence

9 9 9 Read through [1, Sect. 1-2], study recommended literature

2. Characterizations of coherent risk measures and their examples

9 9 9 Read through [1, Sect. 3-5], study recommended literature

3. Generalizations of coherent risk measures (spectral, convex and other types of measures)

9 9 9 Read through [2],[3], study recommended literature

4. Midterm (preparation and writing)

1 5 6 10 Review theory and problem solutions

5. Risk measures in more general spaces 12 12 17 Read [4]; be ready for a discusion during the seminar, and/or prepare a presentation.

6. Final exam (preparation and writing) 3 5 22 Review theory and problem solutions

Total 27 4 12 5 50 76

Assesment strategy Weight Time of assesment

Criteria

Midterm This 3 hr midterm exam contains theoretical (closed-book) and problem solving (open book) parts. Points are awarded for each successfully answered question/problem. The midterm contains material from topics I through III.

35% During recitation classes (approx. 11th week)

Mark 10 – between 90% and 100% of available points on a test.






Mark 1-4 – less than 40% of available points on a test. Presentation at the seminar

15% Weeks 12

through 16 Mark 10 – a student gave a well-prepared and thorough presentation on a selected topic; he/she was able to answer questions from fellow students, made handouts (up to 4 pages long).

Mark 5 – a student only partially understood the topic, presentation was skin-deep; only part of the questions were answered; handouts were not prepared. Mark 0 – a student did not understand the selected topic, presentaton was poorly prepared; not one question was answered; handouts were not prepared.

Final exam The final 2 hr long written exam covers material from topics I

50% During exam session Mark 10 – A student shows excellent knowledge of the

course material, is able to analyze and generalize it, understands and correctly uses concepts, knows the main results of discrete time mathematical finance. He/she has


44

through III. It contains theoretical closed-book and practical open-book parts. Points are awarded for each successfully answered question/problem.

collected between 90% and 100% of the available points.

Mark 8-9 – A student shows good/very good knowledge of the course material, is able to systematize and generalize it, understands used concepts, knows the majority of results of discrete time mathematical finance. 9 points are awarded for collecting between 80% and 89.99% of the available points; 8 points are awarded for collecting between 70% and 79.99% of the available points.

Mark 6-7 – A student understands the main concepts of the course and knows most of the main results of discrete time financial mathematics. 7 points are awarded for collecting between 60% and 69.99% of the available points; 6 points are awarded for collecting between 50% and 59.99% of the available points.

Mark 5 – A student shows skin-deep understanding of the concepts of discrete time financial mathematics. He/she has collected between 40% and 49.99% of available points

Mark 1-4 – A student does not know the studied material and inappropriately uses the terms and concepts of the course. Has collected less than 40% of the available points.

Author Publication

year Title Volume



Required reading 1. P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath

1999 Coherent measures of risk

Math. Finance 9(3), pp 203-228

Willey Periodicals, Inc. Available for free at http://onlinelibrary.wiley.com/doi/10.1111/1467-9965.00068/pdf (last checked 2012-03-05)

2. C. Acerbi

2002 Spectral measures of risk: A coherent representation of subjective risk aversion

Journal of Banking & Finance 26, pp. 1505-1518

Elsevier

3. H. Föllmer and A. Schied

2002 Convex measures of risk and trading constraints

Finance and Stochastics 6, pp. 429-447

Springer

4. F. Delbaen

2000 Coherent risk measures on general probability spaces

Preprint available at http://www.math.ethz.ch/~delbaen/ (last checked on 2012-03-05)

Additional reading 5. E. Jouini, W. Schachermayer, and N. Touzi

2006 Law invariant risk measures have the Fatou property. In: Advances in Mathematical Economics (S. Kusuoka and A. Yamazaki ed.)

9, pp. 49-71 Springer

6. A.S. Cherny, P.G. Grigoriev

2007 Dilation monotone risk measures are law invariant

Finance and Stochastics, 11, pp. 291-298

Springer

7. C. Acerbi, D. Tasche

2002 On the coherence of expected shortfall

Journal of Banking & Finance, 26(7), pp. 1487-1503

Elsevier


45

Course description

Course title Course code Pension Funds

Lecturers Department where the course is delivered Lect. Aldona Skučaitė Department of Mathematical Analysis






Face-to-face 3rd semester (Fall) English

Prerequisites and corequisites Prerequisites: Probability theory (First level); Actuarial mathematics (First level)



5 126 48 78

Course objectives: programme competences to be developed The aim of this course – to acquaint students with actuarial models used for assessment of pension systems. General competences developed: a) time management and accurate performance of tasks; b) ability to describe what further steps are needed for deeper analysis of problem. Professional competences developed: a) to be able to apply mathematical methods for solution of actuarial problems; b) to be able to present results for specialists and wide audience.

Learning objectives

At the end of the course a student should: Learning methods Assessment methods

- Know 3 pension pillars and be able to classify pension systems. Understand the concept of „solidarity between generations“ and its importance for every pension system. Be able to critically assess pension systems and analyse its advantages and disadvantages. Be able to explain differences of pension systems (demographic, financial etc.) for wide audience.

Problem based teaching „Idea‘s shower“ Reading of articles Search and edition of information Preparation for oral presentation

Summative: Short oral presentation about chosen country‘s pension system. Advantages and disadvantages of the system must be critically assessed.)

- Understand and be able to explain demographic and financial model of pension system: understand differences between pure and relative probabilities; be able to forecast demographic parameters using Lexis diagram; - Understand and be able to explain financial peculiarities of pension systems and mechanism of

Traditional and / or interactive lectures Case study Reading of articles Modelling

Summative: Individual or group assignment

Exam


46

funding; be able to construct actuarial financial model and find unknown parameter; be able to explain mechanism of funding, its advantages and disadvantages. - Understand and be able to explain differences between deterministic and stochastic approach to pension annuities; be able to calculate risk measures (of pension annuities). - Understand what is longevity risk, explain differences between cohort and period mortality tables; be able to apply main methods of construction of projected mortality tables and explain advantages and disadvantages of different methods. - Be able to assess impact of longevity risk: calculate risk measures; to distinguish between pooling and non pooling risk; explain advantages and disadvantages of main methods of risk management (hedging, reinsurance etc.) - Be able to describe what further steps are needed for deeper analysis of problem - Be able to manage his / her time and accurately perform assigned tasks (to get ready for exam; plan his / her time when carrying out individual assignment etc.) .

Independent work (self assessment)

Summative: Oral presentation Individual or group assignment Exam



Lect

ures

Con

sulta

tions

Sem

inar

s

Labs

Tot

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hour

s

Indi

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hour

s

Assignments

1. Main features of pension systems. Three pension pillars. Classification of pensions. Calculation of pension size. Necessity of solidarity between generations. Pension systems in European Union and USA.

2 4 6 10 To read assigned paper. To prepare and present short presentation (5-10 min.) about specific countries‘ pension system

2. Demographical models of pension systems: pure and relative probabilities; Lexis diagram. Funding models of pension systems: terminal funding; funding methods; individual / aggregate funding methods.

5 1 8 14 15 To read assigned material (demographical – financial models). To prepare for discussions.

3. Pension annuities: deterministic vs. stochastic approach. Longevity risk: cohort and period mortality tables. Projected mortality tables and methods of projection: extrapolation; single entry mortality table (age adjustment method); parametric methods; Lee Carter method.

5 1 8 14 15 To read assigned material (pension annuities; projected mortality tables; Lee Carter method). To prepare for discussions.

4. Longevity risk: coefficient of variation and other risk measures; risk and mortality scenario; pooling and non pooling parts of risk. Methods for management of Longevity risk: hedging, reinsurance, longevity bonds and others.

5 1 4 10 16 To read assigned material (longevity risk; methods for longevity risk management). To prepare for discussions.


47

5. Exam 4 22 To prepare for exam. To revise theory and its applications

Total: 17 3 4 20 48 78 Assessment strategy Weight Time of

assessment Criteria

Individual or group assignments (2-4 during semester)

40 During semester

Marks – 0; 0,25; 0,5; 0,75; 1 are given for each assignment. Summative assessment – marks for all assignments are added, the sum is then divided by number of assignments; obtained result is multiplied by 40. Quality of fulfilment of assignment, interpretation of results and answers to questions is assessed. Marks are given according to the following scheme: 1: assignment was carried out without mistakes; interpretation of results was correct; all questions answered exhaustively and correctly 0,75: some non essential mistakes when performing assignment and / or when interpreting the results; or no less than 75% of questions answered correctly 0,5: mistakes when performing assignment and / or when interpreting the results; less than 75% and no less than 50% questions answered correctly 0,25: serious (essential) mistakes when performing assignment and / or when interpreting the results; less than 50% and no less than 25% questions answered correctly 0: assignment was not carried out or performed but with many very serious essential mistakes; less than 25% of questions answered correctly

Oral presentation

10 February or September (3-4 week)

Summative assessment – given marks (0; 0,25; 0,5; 0,75 or 1) are multiplied by 10. Structure, form and duration of presentation as well as reliability of references are assessed: 1: essence of presentation is clearly stated; fluent presentation; all parts are logically connected; time limit is not exceeded; duration of presentation was equal to stated limit or shorter by no more than 25%; presentation based on at least 2-3 reliable references; no less than 75% of questions were answered. 0,75: some non essential imperfections, e.g. time limit is a bit exceeded, no less than 50%, but less than 75% of questions were answered etc. 0,5: 1-2 essential imperfections, e.g. essence of presentation was not stated clearly; no logical connection between parts of presentation; duration is significantly shorter or longer than limit, but presentation is based on at least 1-2 reliable references; no less than 50% of questions were answered. 0,25: more than 2 essential imperfections and less than 50% but no less than 25% of questions were answered. 0: presentation is not presented at fixed time (without valid excuse) or presented but with many essential deficiencies and less than 25% of questions was answered.

Exam (2 parts): Midterm and Final

50 (25+25)

Midterm – 7-8 week Final – June (January)

During each part of exam student must fulfil 1-2 tasks. Each task may contain sub-tasks. Accuracy and exhaustiveness of fulfilment as well as application of acquired skills to solution of actuarial


48

problems in pensions systems are assessed. Marks 0-5 are given for every part of exam: 5: All tasks are fulfilled exhaustively and precisely, solutions are correct (correct model and assumptions, no logical mistakes in calculations), accurate written presentation. Proper and logical conclusions are drawn. 4: 1st case) No less than 75% of tasks fulfilled without essential imperfections. 2nd case) All tasks are fulfilled but with small deficiencies, e.g. 1-2 logical mistakes during calculations, conclusions are not logically grounded. 3: No less than 50% but less than 75% of tasks fulfilled without essential imperfections. Other tasks not carried out or carried out with serious mistakes (improper model or assumptions; logically unacceptable calculations, unreasonable method of solutions; no conclusions or conclusions contradict to result of calculation, etc.). 2: No less than 25% but less than 50% of tasks fulfilled without essential imperfections. Other tasks not carried out or serious mistakes made. 1: Some operations are correct (e.g. proper model selected, some calculations are correct, etc.), but no solution given, no conclusions, logical contradictions between parts of solutions etc. 0:. All solutions are logically unacceptable. Remarks: Course coordinator or other lecturer may decide to give not whole mark, e.g. 3,5. Summative assessment: Average of marks for both exam parts is multiplied by 10.

Overall assessment – sum of marks for 3 summative assessments

100 - 10: – no less than 90 points 9: – no less than 80 but less than 90 points 8: – no less than 70 but less than 80 points 7: – no less than 60 but less than 70 points 6: – no less than 50 but less than 60 points 5: – no less than 40 but less than 50 points 1-4: – less than 40 points

Author Publicat

ion year Title Volume and/or

publication number Publication place and publisher

Required reading Bowers, N.L.; Gerber, H.U.; Hickman, J.C.

1986 Actuarial Mathematics - The Society of Actuaries

Pitacco, E., et. al.

2009 Modelling Longevity Dynamics for Pensions and Annuity Business

- Oxford University Press

A. Skučaitė

2012 Pension systems. Lecture notes

Recommended reading Barr, N.

2002 Reforming Pensions: Myths, Truths and Policy Choices

International Social Security Review, Vol. 55 2 /2002

Blackwell Publishers

Werding, M.

2003 After Another Decade of Reform: Do Pension Systems in Europe Converge?

CESifo DICE Report 1/2003

http://www.ifo.de/portal/pls/portal/docs/1/1193630.PDF

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