# Financial and Insurance Mathematics

Coordinated by: Department of Probability and Mathematical Statistics
Study branch coordinator: doc. RNDr. Martin Branda, Ph.D.

The aim of the study program Financial and Insurance Mathematics is professional training in the profession of mathematician in financial institutions or for independent scientific activity in the fields of finance and insurance, including a possible academic career. Graduates of this study program will gain in-depth knowledge of basic mathematical disciplines and special knowledge of probability theory and mathematical statistics, econometrics and stochastic processes, mathematical methods in finance, life and non-life insurance, risk management, accounting (including insurance companies accounting) and machine learning. The teaching is largely based on mathematical modeling and advanced data analysis using modern software tools. All this allows graduates to perform highly effective modeling of financial and insurance products, their analysis in terms of risk, profitability and other characteristics necessary for effective financial management.

Graduates of the Financial and Insurance Mathematics study program will be employed in banks, insurance companies, pension and investment funds and other financial institutions, as well as in state administration (Ministry of Finance, Czech National Bank and others). The professions they then hold include actuaries, risk managers and data scientists. The current demand of the mentioned institutions for these professions, especially from the commercial sphere, is high and this trend will probably not change in the future. Last but not least, the graduates will receive the education that is the basis for the award of a certificate of competence to perform actuarial activities issued by the Czech Society of Actuaries to its members meeting the criteria of internationally recognized actuarial education (the certificate is valid for the entire EU).

#### Assumed Knowledge

It is assumed that an incoming student of this program has sufficient knowledge of the following topics and fields:

Differential and integral calculus of one variable and several variables. Measure theory. Lebesgue integral. Vector spaces, matrix algebra.
Foundations of probability theory.
Foundations of mathematical statistics and data analysis, markov chain theory, linear regression.
Foundations of financial and actuarial mathematics.

Should an incoming student not meet these entry requirements, the coordinator of the study program may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.

#### Study Plans

The study program has two study plans. The study plan N is intended for students who start their studies in the academic year 2022/2023 (or later). The study plan P is intended for students who started their studies in the academic year 2020/2021 or 2021/2022.

#### Financial and Insurance Mathematics, plan N

Study Plan N is intended for students who start their studies in the academic year 2022/2023 (or later).

#### Branch selection

Program Financial and Insurance Mathematics is splitted into two branches:

1. Branch Insurance mathmematics is designed for students who want to become actuaries or risk managers in insurance companies.
2. Branch Data science and finance is designed to prepare for careers in data science, advanced data analytics or risk management primarily in financial institutions.

Choosing a branch involves three sequential steps:

Selection of compulsory elective courses according to the recommended course of study, typically from the summer semester of the first year.
Selection of one of the two variants of the subject "Diploma Thesis III". This course is usually enrolled at the beginning of the last semester of study.
Selection of an optional topic of the oral part of the final exam, when registering for the final exam.

#### Choice of elective courses

The choice of compulsory elective courses is guided by the prerequisites of individual variants of the "Diploma Thesis III". Each variant requires the fulfillment of certain requirements for passing the courses of the chosen specialization. These prerequisites are not verified when enrolling in the course "Diploma Thesis III", so this course can be enrolled without meeting all the prerequisites. However, they are verified during the control of the fulfillment of study obligations, so that a student who does not meet the prerequisites of the subject "Diploma Thesis III" at this stage cannot complete the study.

#### 1st year

In the summer semester of the 1st year of study, it is necessary to choose one of two specializations and enroll in the corresponding elective courses according to it. The specific composition of the study plan continues in the second year.

 Code Subject Credits Winter Summer NMFP401 Financial econometrics 4 2/1 C+Ex — NMFP403 Stochastic Processes 2 8 4/2 C+Ex — NMFP405 Probability for Finance and Insurance 4 2/1 C+Ex — NMFP407 Mathematics of Life Insurance 1 5 2/2 C+Ex — NMFP409 Mathematics of Non-Life Insurance 1 5 2/2 C+Ex — NMFP402 Generalized Linear Models 5 — 2/2 C+Ex NMFP404 Time Series 8 — 4/2 C+Ex NMFP406 Data Science 1 3 — 2/0 Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NMFP432 Mathematics of Life Insurance 2 5 — 2/2 C+Ex NMFP434 Mathematics of Non-Life Insurance 2 5 — 2/2 C+Ex Optional and Elective courses 2

#### 2nd year

 Code Subject Credits Winter Summer NMFP501 Seminar on Actuarial Science 1 2 0/2 C — NMFP503 Risk Theory 1 4 2/1 C+Ex — NMFP505 Stochastic Models in Finance 1 5 2/2 C+Ex — NMFP531 Risk Theory 2 4 2/1 C+Ex — NMFP537 Accounting and Solvency of Insurance Companies 5 2/2 C+Ex — NSZZ024 Diploma Thesis II 9 — 0/6 C NMFP532 Seminar on Actuarial Science 2 2 — 0/2 C NMFP558 Diploma Thesis III in Actuarial Mathematics 15 — 0/10 C Optional and Elective courses 14

#### 1st year

In the summer semester of the 1st year of study, it is necessary to choose one of two specializations and enroll in the corresponding elective courses according to it. The specific composition of the study plan continues in the second year.

 Code Subject Credits Winter Summer NMFP401 Financial econometrics 4 2/1 C+Ex — NMFP403 Stochastic Processes 2 8 4/2 C+Ex — NMFP405 Probability for Finance and Insurance 4 2/1 C+Ex — NMFP407 Mathematics of Life Insurance 1 5 2/2 C+Ex — NMFP409 Mathematics of Non-Life Insurance 1 5 2/2 C+Ex — NMFP402 Generalized Linear Models 5 — 2/2 C+Ex NMFP404 Time Series 8 — 4/2 C+Ex NMFP406 Data Science 1 3 — 2/0 Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NMEK412 Optimisation with Applications to Finance 5 — 2/2 C+Ex NMFP436 Data Science 2 5 — 2/2 C+Ex Optional and Elective courses 2

#### 2nd year

 Code Subject Credits Winter Summer NMFP501 Seminar on Actuarial Science 1 2 0/2 C — NMFP503 Risk Theory 1 4 2/1 C+Ex — NMFP505 Stochastic Models in Finance 1 5 2/2 C+Ex — NMFP533 Investment Analysis 5 2/2 C+Ex — NMFP535 Data Science 3 5 2/2 C+Ex — NSZZ024 Diploma Thesis II 9 — 0/6 C NMFP534 Stochastic Models in Finance 2 3 — 2/0 Ex NMFP556 Diploma Thesis III in finance 15 — 0/10 C Optional and Elective courses 12

#### Obligatory Courses

All courses in this group must be successfully completed.

 Code Subject Credits Winter Summer NMFP401 Financial econometrics 4 2/1 C+Ex — NMFP403 Stochastic Processes 2 8 4/2 C+Ex — NMFP405 Probability for Finance and Insurance 4 2/1 C+Ex — NMFP407 Mathematics of Life Insurance 1 5 2/2 C+Ex — NMFP409 Mathematics of Non-Life Insurance 1 5 2/2 C+Ex — NMFP402 Generalized Linear Models 5 — 2/2 C+Ex NMFP404 Time Series 8 — 4/2 C+Ex NMFP406 Data Science 1 3 — 2/0 Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NMFP501 Seminar on Actuarial Science 1 2 0/2 C — NMFP503 Risk Theory 1 4 2/1 C+Ex — NMFP505 Stochastic Models in Finance 1 5 2/2 C+Ex — NSZZ024 Diploma Thesis II 9 — 0/6 C

#### Elective courses

At least 21 credits must be obtained by passing the elective courses.

#### Branch Insurance Mathematics

The choice of compulsory elective courses in Insurance Mathematics is determined by the prerequisites of the course "Diploma Thesis III in Insurance Mathematics", which requires the completion of all of these courses:

 Code Subject Credits Winter Summer NMFP432 Mathematics of Life Insurance 2 5 — 2/2 C+Ex NMFP434 Mathematics of Non-Life Insurance 2 5 — 2/2 C+Ex NMFP531 Risk Theory 2 4 2/1 C+Ex — NMFP537 Accounting and Solvency of Insurance Companies 5 2/2 C+Ex — NMFP532 Seminar on Actuarial Science 2 2 — 0/2 C

#### Branch Data Science and Finance

The choice of compulsory elective courses in Data Science and Finance is determined by the prerequisites of the course "Diploma Thesis III in Finance", which requires the completion of all of these courses:

 Code Subject Credits Winter Summer NMEK412 Optimisation with Applications to Finance 5 — 2/2 C+Ex NMFP436 Data Science 2 5 — 2/2 C+Ex NMFP533 Investment Analysis 5 2/2 C+Ex — NMFP535 Data Science 3 5 2/2 C+Ex — NMFP534 Stochastic Models in Finance 2 3 — 2/0 Ex

#### Recommended Optional Courses

Not only this list serves as recommended optional courses, but students of the given branch can use elective courses of the second branch as optional courses.

 Code Subject Credits Winter Summer NMFP461 Credit Risk in Banking 3 2/0 Ex — NMFP465 Financial Derivatives 1 3 2/0 Ex — NMSA413 Optimisation Theory 8 4/2 C+Ex — NPFL129 Introduction to Machine Learning with Python 5 2/2 C+Ex — NMEK436 Computational Aspects of Optimisation 5 — 2/2 C+Ex NMFP462 Demography 3 — 2/0 Ex NMEK531 Mathematical Economics 5 2/2 C+Ex — NMFP463 Practical Aspects of Financial Risk Measuring and Management 3 2/0 Ex — NMFP466 Financial Derivatives 2 3 2/0 Ex —

#### Requirements for taking the last part of the final exam

Earning at least 120 credits during the course of the study.
Completion of all obligatory courses prescribed by the study plan.
Completion of all elective courses prescribed by the study branch.
Submission of a completed Master's Thesis by the submission deadline.

#### Requirements for taking other than the last part of the final exam

General rules are set in the internal regulation Rules Of Study at the Faculty. In the program Mathematical structures, the following are required:

Earning at least 105 credits during the course of the study.
If the other than the last part of the final exam is the oral part, completion of all obligatory courses prescribed by the study plan and all elective courses prescribed by the study branch except for Diploma Thesis III (NMFP556 or NMFP558 according to the study branch) is required.
If the other than the last part of the final exam is the defence, completion of all obligatory courses prescribed by the study plan and all elective courses prescribed by the study branch and submission of a completed Master's Thesis by the submission deadline are required.

#### Oral part of the final exam

The oral part of the final exam consists of three subject areas, from each the student gets one question. Two areas (Fundamentals of Probability, Mathematical Statistics and Econometrics; Fundamentals of Financial and Actuarial Mathematics) are compulsory, the third area is optional and corresponds to the chosen branch of study. The student can choose the third area:

Advanced Topics of Actuarial Mathematics and Accounting of Insurance Companies (corresponds to the branch Insurance Mathematics)
Advanced Topics of Finance and Data Analysis (corresponds to the branch Data Science and Finance)

A more detailed explanation of the requirements for the oral part of the final exam can be found on the website of the program guarantor: https://www2.karlin.mff.cuni.cz/~branda/garantfpm.html.

#### 1. Fundamentals of Probability, Mathematical Statistics and Econometrics

Probability for finance and insurance, econometric models, stochastic processes and time series, generalized linear models.

#### 2. Fundamentals of Financial and Actuarial Mathematics

Fundamentals of life and non-life actuarial mathematics, risk theory, basic stochastic models in finance.

#### 3. Optional areas

3A. Advanced Topics of Actuarial Mathematics and Accounting of Insurance Companies
Advanced topics of life and non-life actuarial mathematics, risk theory, accounting and solvency of insurance companies.

3B. Advanced Topics of Finance and Data Analysis
Optimization models with application in finance, investment analysis and models, risk measures, advanced stochastic models in finance, multivariate statistical models.

#### Financial and Insurance Mathematics, plan P

Study Plan P is intended for students who start their studies in the academic year 2020/2021 or 2021/2022. In the recommended course of study, it is necessary to pay attention to the following marked changes:

Most courses are no longer taught. For these courses, it is stated which new courses replace these non-taught ones.
The semester of some courses has changed.

#### 1st year

 Code Subject Credits Winter Summer NMFM401 Mathematics of Non-Life Insurance 1 1 5 2/2 C+Ex — NMFM405 Life Insurance 1 2 5 2/2 C+Ex — NMSA407 Linear Regression 8 4/2 C+Ex — NMSA409 Stochastic Processes 2 8 4/2 C+Ex — NMFM402 Mathematics of Non-Life Insurance 2 3 5 — 2/2 C+Ex NMFM404 Selected Software Tools for Finance and Insurance 4 3 — 2/0 Ex NMFM406 Life Insurance 2 5 3 — 2/0 Ex NMFM416 Life Insurance 2, exercises 5 2 — 0/2 C NMFM408 Probability for Finance and Insurance 6 3 — 2/0 Ex NMFM410 Insurance Companies Accounting 7 5 — 2/2 C+Ex NSZZ023 Diploma Thesis I 6 — 0/4 C Optional and Elective courses 7

1This course is no longer taught. It is replaced by a course NMFP409 Mathematics of Non-life Insurance 1.

2 This course is no longer taught. It is replaced by a course NMFP407 Mathematics of Life Insurance 1.

3 This course is no longer taught. It is replaced by a course NMFP434 Mathematics of Non-life Insurance 2.

4 This course is no longer taught. It is replaced by a course NMFP406 Data Science 1.

5 This courses are no longer taught. They are replaced by a course NMFP432 Mathematics of Life Insurance 2 (2/2 Z+Zk).

6 This course is no longer taught. It is replaced by a course taught in the winter semester NMFP405 Probability for Finance and Insurance (2/1 C+Ex).

7 This course is no longer taught. It is replaced by a course taught in the winter semester NMFP537 Accounting and Solvency of Insurance Companies.

#### 2nd year

 Code Subject Credits Winter Summer NMFM501 Seminar on Actuarial Sciences 1 8 2 0/2 C — NMFM503 Risk Theory 9 8 4/2 C+Ex — NMFM507 Advanced Topics of Financial Management 10 2 2/0 Ex — NMST537 Time Series 11 8 4/2 C+Ex — NSZZ024 Diploma Thesis II 9 0/6 C — NMFM502 Seminar on Actuarial Sciences 2 12 1 — 0/2 C NMFM505 Stochastic Models for Finance and Insurance 13 5 — 2/2 C+Ex NSZZ025 Diploma Thesis III 15 — 0/10 C Optional and Elective courses 10

8This course is no longer taught. It is replaced by a course NMFP501 Seminar on Actuarial Sciences 1.

9This course is no longer taught. It is replaced by courses NMFP503 Risk Theory 1 (2/1 Z+Zk) a NMFP531 Risk Theory 2 (2/1 Z+Zk).

10This course is no longer taught. It is replaced by a course NMFP533 Investment Analysis.

11 This course is no longer taught. It is replaced by a course taught in the summer semester NMFP404 Time Series.

12This course is no longer taught. It is replaced by a course NMFP532 Seminar on Actuarial Sciences 2.

13This course is no longer taught. It is replaced by a course NMFP505 Stochastic Models in Finance 1.

#### Obligatory Courses

 Code Subject Credits Winter Summer NMFM401 Mathematics of Non-Life Insurance 1 1 5 2/2 C+Ex — NMSA407 Linear Regression 8 4/2 C+Ex — NMFM405 Life Insurance 1 2 5 2/2 C+Ex — NMSA409 Stochastic Processes 2 8 4/2 C+Ex — NMFM402 Mathematics of Non-Life Insurance 2 3 5 — 2/2 C+Ex NMFM404 Selected Software Tools for Finance and Insurance 4 3 — 2/0 Ex NMFM406 Life Insurance 2 5 3 — 2/0 Ex NMFM416 Life Insurance 2, exercises 5 2 — 0/2 C NMFM408 Probability for Finance and Insurance 6 3 — 2/0 Ex NMFM410 Insurance Companies Accounting 7 5 — 2/2 C+Ex NMFM501 Seminar on Actuarial Sciences 1 8 2 0/2 C — NMFM503 Risk Theory 9 8 4/2 C+Ex — NMFM507 Advanced Topics of Financial Management 10 2 2/0 Ex — NMST537 Time Series 11 8 4/2 C+Ex — NMFM502 Seminar on Actuarial Sciences 2 12 1 — 0/2 C NMFM505 Stochastic Models for Finance and Insurance 13 5 — 2/2 C+Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NSZZ024 Diploma Thesis II 9 0/6 C — NSZZ025 Diploma Thesis III 15 — 0/10 C

1This course is no longer taught. It is replaced by a course NMFP409 Mathematics of Non-life Insurance 1.

2 This course is no longer taught. It is replaced by a course NMFP407 Mathematics of Life Insurance 1.

3 This course is no longer taught. It is replaced by a course NMFP434 Mathematics of Non-life Insurance 2.

4 This course is no longer taught. It is replaced by a course NMFP406 Data Science 1.

5 This courses are no longer taught. They are replaced by a course NMFP432 Mathematics of Life Insurance 2 (2/2 Z+Zk).

6 This course is no longer taught. It is replaced by a course taught in the winter semester NMFP405 Probability for Finance and Insurance (2/1 C+Ex).

7 This course is no longer taught. It is replaced by a course taught in the winter semester NMFP537 Accounting and Solvency of Insurance Companies.

8This course is no longer taught. It is replaced by a course NMFP501 Seminar on Actuarial Sciences 1.

9This course is no longer taught. It is replaced by courses NMFP503 Risk Theory 1 (2/1 Z+Zk) a NMFP531 Risk Theory 2 (2/1 Z+Zk).

10This course is no longer taught. It is replaced by a course NMFP533 Investment Analysis.

11This course is no longer taught. It is replaced by a course taught in the summer semester NMFP404 Time Series.

12This course is no longer taught. It is replaced by a course NMFP532 Seminar on Actuarial Sciences 2.

13This course is no longer taught. It is replaced by a course NMFP505 Stochastic Models in Finance 1.

#### Elective Courses

It is necessary to obtain at least 5 credits from elective courses.

 Code Subject Credits Winter Summer NMFM431 Investment Analysis 5 2/2 C+Ex — NMFM531 Financial Derivatives 1 1 3 2/0 Ex — NMFM532 Financial Derivatives 2 2 3 2/0 Ex — NMSA403 Optimisation Theory 3 5 2/2 C+Ex — NMST531 Censored Data Analysis 5 2/2 C+Ex — NMST539 Multivariate Analysis 4 5 2/2 C+Ex —

1 This course is no longer taught. It is replaced by a course NMFP465 Financial Derivatives 1.

2 This course is no longer taught. It is replaced by a course NMFP466 Financial Derivatives 2.

3 This course is no longer taught. It is replaced by a course NMSA413 Optimization Theory (4/2 C+Ex).

4 This course is no longer taught. It is replaced by a course NMFP535 Data Science 3.

#### Recommended Optional Courses

 Code Subject Credits Winter Summer NMEK432 Econometrics 1 8 — 4/2 C+Ex NMEK532 Optimisation with Applications to Finance 2 8 — 4/2 C+Ex NMFM461 Demography 3 3 — 2/0 Ex NMFM462 Practical Aspects of Financial Risk Measuring and Management 4 3 2/0 Ex — NMFM537 Credit Risk in Banking 5 3 2/0 Ex — NMFP436 Data Science 2 5 — 2/2 C+Ex NMFP534 Stochastic Models in Finance 2 3 — 2/0 Ex

1 This course is no longer taught. It can replaced by courses taught in the winter semester NMFP401 Financial Econometrics (2/1 C+Ex) or NMEK511 Econometrics (4/2 C+Ex).

2 This course is no longer taught. It is replaced by a course NMEK412 Optimisation with Applications to Finance (2/2 C+Ex).

3 This course is no longer taught. It is replaced by a course NMFP462 Demography.

4 This course is no longer taught. It is replaced by a course NMFP463 Practical Aspects of Financial Risk Measuring and Management.

5 This course is no longer taught. It is replaced by a course NMFP461 Credit Risk in Banking.

#### Requirements for taking the last part of the final exam

Earning at least 120 credits during the course of the study.
Completion of all obligatory courses prescribed by the study plan.
Earning at least 5 credits by completion of elective courses.
Submission of a completed Master's Thesis by the submission deadline.

#### Requirements for taking other than the last part of the final exam

General rules are set in the internal regulation Rules Of Study at the Faculty. In the program Mathematical structures, the following are required:

Earning at least 105 credits during the course of the study.
If the other than the last part of the final exam is the oral part, completion of all obligatory courses prescribed by the study plan except for NSZZ025 Diploma Thesis III is required.
If the other than the last part of the final exam is the defence, completion of all obligatory courses prescribed by the study plan and submission of a completed Master's Thesis by the submission deadline are required.
Earning at least 5 credits by completion of elective courses.

#### Oral part of the state final exam

The oral part of the final exam consists of three subject areas: Applied Probability, Insurance, and Finance and Accounting. One question is asked from each subject area.

#### 1. Applied Probability

Random variables, characteristics of their distributions. Random vectors, joint distribution, covariance, correlation, dependence modelling and measurement. Conditional distribution. Probability distributions in financial mathematics. Parameter estimators and their properties. Confidence intervals. Hypothesis testing principles. Maximum likelihood principle and the method of moments. One-sample, two-sample, paired tests. Analysis of variance. Linear regression model. The Bayes principle. Laws of large numbers and central limit theorems. Markov chains. Stationary processes. Time Series. Credibility theory. Collective risk models. Essentials of stochastic analysis.

#### 2. Insurance

Demographic model of life insurance. Capital and life policies. Reserves of life insurance. Multiple decrement models. Multiple life insurance. Solvency and reinsurance. Technical reserves of general insurance. Tariff rates.

#### 3. Finance and Accounting

Foundations of finance. Securities and their pricing. Stochastic models in finance. Financial risk and its measuring. Stock market analysis. Accounting.

Charles University, Faculty of Mathematics and Physics
Ke Karlovu 3, 121 16 Praha 2, Czech Republic
VAT ID: CZ00216208